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vii HPE ESRN EWRSWT EAIELINKS NEGATIVE WITH NETWORKS MEASURING 3 CHAPTER PRELIMINARIES AND FOUNDATIONS 2 CHAPTER INTRODUCTION 1 CHAPTER ALGORITHMS OF LIST FIGURES OF LIST TABLES OF LIST . oeCnrlt esrmn nSge ewrs...... 38 ...... Networks Signed in Measurement Centrality Node 3.2 20 ...... 9 . Networks Signed . in Measurements Relevance . 5 Node . 6 . 3.1 4 . . . . 6 ...... 2 ...... Theories . . and . . . Properties, . . Datasets, . . . Network . . Signed ...... 2.3 ...... Theories . . and . . Properties . Network . . Unsigned . . . . Deﬁnitions . and . . Notations 2.2 . Basic ...... 2.1 ...... Organization . . . 1.3 Contributions . Challenges 1.2 Research 1.1 .. xeiet 33 . . . . . 41 ...... 25 ...... 39 ...... 22 ...... Function . . . Objective Measurement Measurement . . . Centrality (DeSCent) Signed Centrality . . . Signed Deep . . . of Overview . 3.2.2 . . An ...... 3.2.1 ...... Experiments . . . . . 3.1.3 . . . 15 ...... 10 . . Methods . Global 9 ...... 3.1.2 ...... 8 . 11 ...... Methods . . . . Local ...... 8 . . 3.1.1 ...... 7 ...... Theories . Network . . . . Signed Properties . Networks . . . . Signed . on . . . . Analysis 2.3.3 Data ...... Datasets Network . . . . 2.3.2 Signed ...... 2.3.1 ...... Homophily . . Network . . Coeﬃcient . Clustering . . and 2.2.4 Transitivity . . . Reciprocity 2.2.3 Network Density Network and Distribution 2.2.2 Degree 2.2.1 ... inPeito 33 41 35 ...... 25 ...... 29 ...... Theory . . . Status . on . . . 23 Based . Centrality . . . Signed ...... 24 . . . . . 3.2.2.1 ...... 23 ...... Prediction . . . Strength . . Tie . . . 28 ...... Prediction . . . . Sign 3.1.3.2 ...... 3.1.3.1 ...... Restart . . . with Walk . . . . Random Networks Weighted . . for . . Measure . . . . Similarity Asymmetric . 3.1.2.3 ...... Katz . 3.1.2.2 . . 17 . . . . 3.1.2.1 ...... 15 Attachment . . Preferential ...... Index . Jaccard 3.1.1.3 . . . . . Neighbors . Common 3.1.1.2 . . . . . 3.1.1.1 ...... Networks Signed . in Theory . Status Networks Signed in Theory Balance 2.3.3.2 2.3.3.1 ...... AL FCONTENTS OF TABLE ...... viii ...... xvi xiv xii 19 1 6

viii HPE IIGNTOK IHNGTV LINKS NEGATIVE WITH NETWORKS MINING 5 CHAPTER LINKS NEGATIVE WITH NETWORKS MODELING 4 CHAPTER . indGahCnouinlNtok 102 ...... Networks Convolutional Graph Signed 5.1 79 ...... Networks Bipartite Signed in Balance 4.2 58 ...... Networks Signed of Modeling Generative 4.1 .. xeiet 114 ...... 104 ...... 95 ...... 104 Experiments ...... 5.1.3 ...... Framework . Network . Convolutional . Graph . Signed . Proposed . The . . . Statement 5.1.2 Problem . . 5.1.1 . . 87 ...... 73 . . . Experiments . . . . . 4.2.3 . . . 80 ...... 72 . Networks . Bipartite . Signed . for . . Prediction . Sign ...... 4.2.2 ...... 65 ...... 59 . . . 62 . Networks . . Bipartite . . Signed . . . in . . Theory . . . . . Balance ...... 4.2.1 ...... Experiments 47 . . . BSCL . of . . Complexity . 4.1.6 . Time . . 59 ...... 4.1.5 ...... BSCL . . for . . Learning . . Parameter . . . BSCL . Model for (BSCL) Generation . . 4.1.4 Network Chung-Lu Signed . . Balanced of . . Overview 4.1.3 An . . 46 . Statement . 4.1.2 Problem . . . . . 4.1.1 ...... Experiments Framework Network Deep DeSCent 3.2.4 Overall 3.2.3 ... efrac oprsn...... 115 ...... 109 ...... 105 . . . . . 107 . . . 99 ...... 96 . Comparison . . . Performance ...... 5.1.3.1 . . . . . Network . Convolutional . . Graph . Links Signed Negative . . and . Positive . . with . Paths . Aggregation . 5.1.2.3 . . Networks . Convolutional . Graph . Unsigned 89 5.1.2.2 . . . . . 92 . 5.1.2.1 ...... 88 ...... 84 ...... Analysis . . . . Parameter 86 ...... Results Comparison 4.2.3.2 . . . . . 83 ...... 4.2.3.1 . Prediction . . . . Sign . 81 Based . . Walk . . . Random ...... Prediction . . . Sign . Low-Rank . 4.2.2.3 . 82 . . Classifier . . Based . . Caterpillars . . . Signed . 4.2.2.2 . . . . 77 ...... 4.2.2.1 . . . Networks . . Bipartite . . . in 73 . . Caterpillars . . . Signed ...... Analysis . . . Butterfly . Signed 4.2.1.5 . . . . Classes . Isomorphism . . Butterfly Networks . . Signed 4.2.1.4 Bipartite . Signed . . in . Butterflies . . Signed 4.2.1.3 . . Datasets Network . . Bipartite Signed 4.2.1.2 ...... 4.2.1.1 ...... Experiment Learning . Parameter . Experiment Generation Network 4.1.6.2 4.1.6.1 54 . . Learning . . Learning 4.1.4.3 . . 50 . 52 Learning 4.1.4.2 . . . . . 4.1.4.1 ...... 45 . . . Analysis . . Parameter . . . Datasets . Across . Generalization 3.2.4.3 . . . Comparison Performance 3.2.4.2 . 42 . . . 3.2.4.1 . . . . . Constraints . Measurement DeSCent Structures Additional Higher-order and Theory Balance Harnessing 3.2.2.3 3.2.2.2 U d V 70 . 66 . . . . 65 ...... ix ...... 101 56

ix HPE INDNTOKAPPLICATIONS NETWORK SIGNED 6 CHAPTER . ogesoa oePeito 155 ...... Prediction Vote Congressional 6.2 134 ...... Prediction Polarity Interaction and Link 6.1 119 ...... Embedding Network Signed Role-based 5.2 .. xeiet 164 ...... 163 ...... 160 ...... 158 . 156 ...... 145 ...... Experiments . . . . . MFCVP . of . . . Details . 6.2.6 Classiﬁcation ...... 6.2.5 ...... 157 ...... MFCVP . of . Factors . . Social . . . MFCVP . . of . Factors . (MFCVP) . 6.2.4 Ideology Prediction Vote . . Congressional Multi-factor . . of 6.2.3 Overview . . . Statement . 6.2.2 Problem . . 6.2.1 . . . . . 134 ...... Experiments . . 135 . 6.1.4 . 139 . . . 128 ...... Framework . . (LIP) . Prediction . . Polarity . Interaction . . and . Link . . Joint . The ...... 6.1.3 ...... Analysis . Data . Opinion . User . Network . Signed . . 121 . Statement 6.1.2 . Problem . . . . 6.1.1 . 123 ...... Experiments . . . . . 5.2.3 ...... Embedding Network . Signed Role-based . . . 5.2.2 . . . Statement Problem 5.2.1 ... oiia atrAayi 170 ...... 165 ...... 164 . . . 168 ...... 167 160 ...... 161 ...... Analysis . Factor . . . Political . . . Predictions . . . . . Outcome Vote . . Roll-call . . . Overall 6.2.6.5 . . . . Predictions . Vote . . . Representative Individual . 6.2.6.4 ...... Settings . . . Experimental 6.2.6.3 153 . . Collection Data . . . and Dataset . 6.2.6.2 ...... 6.2.6.1 152 ...... 149 . . . . . Features . Network . 147 . . Bipartite . Signed ...... Features . Party . . 6.2.4.2 ...... 6.2.4.1 ...... 143 ...... 142 ...... 141 . . . . Analysis . . Parameter ...... Discussions . . . Experiment . 6.1.4.4 ...... Experiments . . . Cold-Start 6.1.4.3 ...... 136 . Experiments Sparsity 6.1.4.2 ...... 139 . . . 6.1.4.1 . LIP . . for . . 137 Method . . Optimization . . . An . . . . Framework . Joint . . 136 . Proposed . The . 6.1.3.4 . . . Correlations . . Opinion Models . User . . Prediction Modeling 6.1.3.3 Polarity . . . Interaction and . . . Link Basic . 6.1.3.2 . . . . . Perspective . . 6.1.3.1 Local A 129 . . Opinions: . . User Perspective 127 Correlated . Global . A . . . Opinions: . User . . Correlated 6.1.2.3 . . . . Dataset . Epinions . Extended 6.1.2.2 . 131 ...... 6.1.2.1 . 124 . . . . 126 ...... 122 ...... Role-nodes: . . . . of . Encodings . . . . the . 121 of . . . . . Interpretation . 118 ...... Comparison Performance . 5.2.3.2 ...... 5.2.3.1 ...... Justiﬁcation . . . Model . . . . Network . Original . . . the . Embedding . 5.2.2.3 ...... Transformation . . Network 5.2.2.2 ...... 5.2.2.1 . . . . Embedding Network . Signed . . . . Embedding Network Unsigned . 5.2.1.2 . . 5.2.1.1 . . . . Analysis Parameter 5.1.3.2 x ...... 133

x BIBLIOGRAPHY DIRECTIONS FUTURE AND CONCLUSION 7 CHAPTER . umr 173 . 176 ...... Directions . Future . 7.2 Summary 7.1 ...... xi ...... 173 178

xi al .0 ttsiso indbprientok...... 81 . 85 ...... 80 ...... networks. . . bipartite signed . . on statistics . . butterﬂy Signed . . 4.11: Table . . . . networks. . bipartite 76 signed . on Statistics . . 4.10: Table . . networks. . bipartite signed regarding . Notations . 4.9: networks . Table 76 signed . real the . . to networks . . generated 75 the . . from . diﬀerence . Absolute 76 . . dataset. . Epinions 4.8: . the . Table . in . types . . triangle . . signed . 75 of . . Distribution . dataset. . . . Bitcoin-OTC 4.7: the Table . . . in types . . . triangle signed . . . of Distribution . . dataset. . Bitcoin-Alpha . 4.6: the Table . in . types 52 . triangle . signed . of . . Distribution . 73 . . networks. 4.5: . signed Table . . generated . 60 in . . balanced . triangles . . of . . Proportion ...... 4.4: . . Table ...... distribution. . sign . . link Positive/negative . . modeling. . 4.3: generative . Table for . networks . social . signed . three . of . Statistics . . . 4.2: Table . . modeling. generative . network signed . regarding Notations . 37 4.1: Table AUC. . with results . prediction link Signed setting. directed the 3.6: Table under 37 35 prediction strength tie of . . comparison Performance . setting. undirected 21 . the 3.5: Table under . prediction . strength . tie 34 . of . comparison Performance . . setting. . directed 3.4: . Table the . under . prediction . 10 link . of . comparison . Performance . . setting. . . 14 undirected 3.3: Table the . . under prediction . . . link 13 of . . . comparison Performance ...... 3.2: . Table networks. . . signed . in . . relevance . node . regarding . Notations networks. . social . signed . 3.1: in . Table links . negative . and . positive . of . strengths . Tie . networks. social 2.3: . Table signed in . reciprocal being . links of . Probability . 2.2: . Table networks. social signed four of Statistics 2.1: Table vrgdoe h he aaesfrec epciepoet...... 78 ...... property. respective each for datasets three the over averaged ITO TABLES OF LIST xii

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16 D tahrst ttster.Freape fw aetecci indtinl ossigo the of consisting triangle signed cyclic the have we if links then example, positive acyclic, three For is triangle theory. the status if 2) to and adheres direction; it their reverse and positive to links negative flip first acyclic. take are 8 can and triangles cyclic signed are directed 4 that where note forms, We different 12 relations. directed the consistency of of concept deductions the logical on the based is in theory status perspective, triangle a from Thus, network. the signed directed in perhaps perspective from that considers link separate positive a a assuming takes than theory Rather status networks. links), disliking negative or Unlike liking or users [2]. positive on based in network (i.e., developed signed later undirected was an in theory defined status is [3] which theory, in balance found observations the of some on Based Networks Signed in Theory Status 2.3.3.2 [28]. time over increases triangles unbalance to balance of ratio the and networks that signed shown in been balance also of has level it the measuring works prior with line in is This respectively. that find and combinations sign four the 94 of each count we network signed rected hm(i.e., them that that imply respectively h hr ikwith link third the osau hoyqiesmlrt aac hoy[] tcnb acltdta hnconverting when that between calculated agree be 6 only can that triangles It network signed undirected [2]. to triangles theory network balance signed directed to similar quite adhering triangles theory all) status almost (and to majority the have networks signed real-world many that shown 8 . to 5% eemnn fatinl deet ttster a edn ytefloigtreses 1) steps: three following the by done be can theory status to adhere triangle a if Determining D D 9 9 and a ihrsau hnte (i.e., them than status higher a has ih o ml that imply not might D 92 8 > . 4% D 9 .Hwvr ete rmtefis w ik ehv that have we links two first the from then we However, ). D ftid nBtonApa ici-T,Saho n pnosaebalanced, are Epinions and Slashdot Bitcoin-OTC, Bitcoin-Alpha, in triads of D 9 8 hnigthat thinking −→ + D D 8 8 nta hnsthat thinks instead D eivsthat believes : , D D 8 : dislikes −→ + D D 8 9 D > and , D D : 9 u perhaps but , D 9 a ihrsau hnte (i.e., them then status higher a has 9 rae otaito.W oeta ro ok have works prior that note We contradiction. a creates D D > 9 9 D −→ a ihrsau.Smlry eaieln from link negative a Similarly, status. higher a has 17 + : ,and ), D 8 hnbsdo ttster hs he links three these theory status on based then , D D 8 D 8 to utblee that believes just 9 eivsthat believes D 9 mle that implies D 8 D a ihrsau than status higher a has D D 9 8 D so oe ttsin status lower a of is 9 8 D < safin of friend a is < D : 92 : D < ), . 0% D 9 : u then but , , believes 91 D . 5% 8 it , ,

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18 ewr.Ms ftepeiu okhsbe o nindntok.Det h nlso of inclusion the to Due networks. signed to networks. applicable directly unsigned the not are for in measures are been centrality unsigned they has existing “important” links, work negative or previous “central” the how on of based Most nodes of network. general, ranking In a [49]. construct applications to of urban is plethora and task a [46], the has biology later the [45], which economics sociology, as to such 48] domains, [47, many infrastructure across 49] 48, 47, 46, [45, tions theory. balance to relationship their discuss and proposed networks our signed present we for 3.1, measures Section relevance in node Thus, [44]. Restart with Walk of Random information as such structural network whole the the utilize measurements global information while neighborhood neighbors; local common use as only such measurements local – used measurements information global the and local to into according divided positive be can only measurements these with that networks note We social 43]. relevance (or [11, links) node networks existing unsigned of for majority designed vast been have The measurements [42]. recommendations and search community [40], [41], classiﬁcation detection node 39], [38, prediction network link social as diverse such in applications usage and their tasks by analysis shown been has This analysis. network social of keystones to networks. entire used or be communities), (i.e., can subgroups that users, of network formula pairs deﬁne mathematical users, can rank we and/or and/or that compare algorithms metrics of these via use then the is It through measurements form. numerical in output be in to contained information network of a expression the allows that expression mathematical a is metric network chapter this In nieSge oilNtok. D 4hItrainlWrso nMnn n erigwith Learning and Mining 2018. on Workshop (MLG), International Graphs 14th KDD Networks.” Social Signed Online 1 oecnrlt safnaetlntokmaueetta a ies e fapplica- of set diverse a has that measurement network fundamental a is centrality Node the of one is network, social a in are nodes two relevant how measures which relevance, Node ye er hnigWn,Shn ag n iin ag RlvneMaueet in Measurements “Relevance Tang. Jiliang and Wang, Suhang Wang, Chenxing Derr, Tyler 1 eivsiaentokmaueet o ewrshvn eaielns A links. negative having networks for measurements network investigate we , ESRN EWRSWT EAIELINKS NEGATIVE WITH NETWORKS MEASURING HPE 3 CHAPTER 19

19 aac hoy[0,ta ol ehlfli ulignd eeac esrmnsi indsocial signed in measurements relevance networks. node building in helpful i.e., be networks, could that social [30], signed theory to balance related the theories stimulates social fundamental This most the [54]. of networks one for social investigation unsigned for measurements relevance node building in prediction. strength mining tie network signed signed and two prediction on sign measurements of these tasks of effects the are what relevance (b) signed to and in links measurements; aim negative and we positive both networks, of use unsigned make to in how (a) research in following: relevance relevance the investigate node node to measuring Analogous of problem networks. the social on signed study comprehensive and initial the perform Hence, we explored. relevance been not signed previously had on analysis network investigation signed systematic on effects and their and general measurements a However, 53]. [52, prediction similarities node link designing for in works recent relevance very few novel a develop are us There help networks. to signed for potential measurements the recommender offer improve could similarly links negative can Thereby, and 6]. 4], [5, [3, systems prediction link positive of performance significantly the can boost links negative of number small a even example, For tasks. analytical various in links Networks Signed in Measurements Relevance Node 3.1 balance and status the theories. namely links, negative and positive both having social networks triadic on prominent defined both theories upon built measurement centrality signed a develop we Thus, 3.2, Section networks. in signed in information network they higher-order exploiting 51], of 50, use into the [24, explored take measurements not have that centrality unsigned proposed existing measurements extending some a by links in been negative users have account “infamous” there and recently “famous” Though the between network. differentiate links negative signed to of need introduction now the we with because associated also complexities and added the to due partially is This aaAayi Discussions: Analysis Data positive over value added significant have links negative that research recent from evident is It oiltere uha oohl 8 lya motn role important an play [8] homophily as such theories Social 20

20 oteue nomto,w a ogl iiete olcladgoa esrmns Local measurements. global and local to them divide roughly can we information, used the to of neighbors of set where the 3.1 and Table degree in notations above the summarize We symmetrical. necessarily R unsigned to analogous section, this networks. In signed for measurements theory. relevance node balance develop we and networks, properties unique networks as signed such makes aspects links negative many of in availability the analysis, data preliminary our to novel building guide to this use can measurements. we relevance theory, signed balance satisfy networks social signed in of triads most the as Furthermore, measurements. relevance signed investigate dedi- systematically needs networks. to still efforts social it cated studied, unsigned extensively been from have distinct measurements relevance be node networks though Therefore, social signed makes which links, positive 9 8 aynd eeac esrmnshv enpooe o nindntok.According networks. unsigned for proposed been have measurements relevance node Many Definitions: and Notations According networks. unsigned in studied extensively been have measurements relevance Node from different are links negative of properties 2.3, Chapter in performed analysis the on Based ersnstend eeac rmuser from relevance node the represents al .:Nttosrgrignd eeac nsge networks. signed in relevance node regarding Notations 3.1: Table X R A A Notations 3 3 3 3 # # # | A 8 8 8 8 8= 8= 8= 9 8 8 8 8 + 8= + | − + ( ( ( ( A 3 # ( ( # 3 8 3 >DC − 8 − 8 8 >DC >DC 8 >DC ) ) ) ) + − ) ) h ij nr ftematrix the of entry (i,j) node the for neighbors (negative) positive of node Set for neighbors (outgoing) incoming of Set node for neighbors of node Set of links negative of (Outdegree) node Indegree of links positive of (Outdegree) Indegree node of (Outdegree) Indegree node of Degree matrix relevance Node matrix adjacency Absolute links positive(negative) only of matrix Adjacency matrix Adjacency Descriptions euse We D 8 na nindnetwork. unsigned an in R ∈ D 8 R ouser to 21 D # 8 × # odnt h eeac cr arx where matrix, score relevance the denote to D 9 oeta oerlvnevle r not are values relevance node that Note . D 8 X D 8 D 8 D D D 8 8 8 3 8 and # 8 denote

21 o odsg h indoeadfial ics t oncinwt indntokpoete and properties network signed with connection theory. its balance discuss finally detail and then one it, signed introduce the briefly design Preferential first to how will and we measurement, Index, unsigned each Jaccard For neighbors, 17]. local [56, common Attachment representative including on networks based measurements unsigned relevance for signed methods local build we subsection, this In Methods Local 3.1.1 measurements. relevance node unsigned representative to to strategy theory third the balance we apply subsections, and to following how links the detail will In negative ones. of unsigned advantage on take based measurements to relevance signed is develop strategy third Our networks signed applicable. for theory not balance makes also but links; positive and negative be between can differences into network a network Such signed matrix network. the the unsigned by in an represented links into network negative signed convert the to making be thus links, would positive strategy second positive of The impact the of [55]. over-estimation in links result could that links negative the ignores use completely only to is first The ones. signed global and local as measurements relevance signed group will methods. we work, this has In edge each where directions. an networks both for directed network simply directed are a networks requires undirected that since method network, any undirected use could we that undirected Note to networks. corresponding directed directed, and and undirected be Mean- can measurements Restart. relevance with node Walk Random while, as such information structural global whole while the neighbors; utilize common measurements as such information neighborhood local use only measurements ihnd eeac esrmnsfrusge ewrs hr r he taeist design to strategies three are there networks, unsigned for measurements relevance node With A ˜ where A A ˜ 9 8 + ntecluaino oerlvnesoe.Ti strategy This scores. relevance node of calculation the in = | A 9 8 22 | goigsgso ik o nyoelosthe overlooks only not links of signs Ignoring .

22 D ecnitrrtSNa ubro omnnihosof ( sign neighbors the common of of polarity number as SCN interpret can We follows: as SCN define we Therefore, links. where as: defined formally between is score which relevance neighbors, the common defines of UCN number intuition, this on Based relevant. be to (UCN) neighbors Common Unsigned Neighbors Common 3.1.1.1 egbr,i.e., neighbors, ( sign the on disagree they illa omr aacdtid.Ohrie hyhv oedsgemnso h in,i.e., signs, the on disagreements more have they Otherwise, triads. balanced more to lead will nindJcadIdx(UJI) Index Jaccard Unsigned Index Jaccard 3.1.1.2 with triads more force to aims SCN Therefore, triads. balanced more in eaie h C crsb h ubro nqenihostoueshv as: have users two neighbors UJI unique effect, of such number mitigate the To by scores scores. UCN relevance the high penalizes neighbors of numbers large with users give to (| 9 # u tinrstenme fuiu egbr hs w sr ae hrfr,UNi likely is UCN Therefore, have. users two these neighbors unique of number the ignores it but , 8 indCmo egbr (SCN) neighbors Common Signed C + neto oBlneTer:If Theory: Balance to onnection ∩ | G # | − 9 eoe h ieo h set the of size the denotes | + | # (| 8 − # R ∩ 8 + 9 8 # ∩ | # = + 9 # 8 + |) (| + 9 ∩ | > | + | # # # 8 + 8 (| + + 9 ∩ # # ∩ | + | C nycniestenme fcmo egbr of neighbors common of number the considers only UCN : 8 − 8 + # # ∩ ∩ + 9 − 9 # | + | G | + | # # 8 − . + − 9 9 ftondssaealto omnfins hyaelikely are they friends, common of lot a share nodes two If : ∩ # | + | |) # D R C antb ietyetne oicuenegative include to extended directly be cannot UCN : R 8 # − 8 8 − > 9 8 9 8 − and # 9 ∩ ∩ (| = | = 8 − n hnsbrcigtenme fnihosthat neighbors of number the subtracting then and ) # # # ∩ | | | 23 − 9 D # # # 8 + + 9 )−(| − |) # 9 | 8 8 8 ∩ ). ∪ ∩ ∩ − 9 ge ihtemjrt ftesgso their of signs the of majority the with agree # |) # # # − 9 then , # 9 9 9 | + | | | | 8 + ∩ # R # 8 − D 9 8 − 8 9 ∩ sngtv,wihwl loresult also will which negative, is | + | and # + 9 # |) D 8 − then , 9 D ∩ hr hyareo the on agree they where 8 and # + 9 R |) 9 8 D 9 spstv which positive is D ob balanced. be to 8 and D 9 D sthe as 8 (3.2) (3.1) and

23 *% ewrsflo oe-a itiuin.Terlvnesoefor score relevance The distributions. power-law follow both in networks degrees since separately, networks, negative and positive the from scores the relevance split calculate first We SPA. define to way a from us network paves which links, negative getting and rich positive “the both observe for we words, richer” other In distributions. power-law the follow links negative and [11]. users two the the Therefore, of degrees future. the multiply the to in is friends UPA new of create score relevance users to analysis, node likely network more social are of friends terms In many have richer. already gets that rich the that is realm, finance the from taken (UPA) Attachment Preferential Unsigned Attachment Preferential 3.1.1.3 as: neighbors unique of number total the eeac score relevance where indPeeeta tahet(SPA) Attachment Preferential Signed C hnw en P between SPA define we Then (SJI) Index Jaccard Signed 9 8 + neto oBlneTer:Smlrt C,SItrest oc oetid balanced. triads more force to targets SJI SCN, to Similar Theory: Balance to onnection B86= n iial ednt h eeac as relevance the denote we similarly and ( G ) A = *% oapstv network positive a to ,0 r- if -1 or 0, 1, 9 8 + slre hntengtv one negative the than larger is R 9 8 *% = B86= G iia ofo C oUI J sdfie sSNdvddby divided SCN as defined is SJI UJI, to UCN from to Similar : D R 9 8 + 8 slre,eulo mle than smaller or equal larger, is 9 8 ( and = *% D = 8 3 A 8 and + D | + # 9 8 9 + R × n omnyue nepeainbhn hsmethod, this behind interpretation used commonly One : n eaienetwork negative a and 8 + as: nteScin232 edmntaeta ohpositive both that demonstrate we 2.3.2, Section the In : 9 8 − 3 D ∪ + 9 9 = *% *% , # *% 24 have: 3 (# 8 − 8 × ∪ 9 8 − 9 8 − 3 ) # 9 8 5 *% 9 from + 9 9 8 − ( *% ∪ = # 9 8 − 3 A − 9 8 − h overall the , 9 8 + − | . × *% , *% 3 8 A − 9 0 and − nutvl,i h positive the if Intuitively, . 9 8 − 9 8 + hnw a s P to UPA use can we Then . ) and 9 R from 9 8 *% hudb positive; be should A 9 8 − + r computed are sdntdas denoted is (3.4) (3.3) (3.5)

24 *% xoeta ea ntewih soitdwt h on fptsa h eghicess[57]: increases length the as paths of count the with associated weight the on decay exponential (UK) Katz Unsigned Katz 3.1.2.1 theory. balance to them connect then and relevance ones signed unsigned global representative design on to based how measurements detail for we subsection, methods this In global relevance. the high of with users neighbors Most two that network. assume whole networks the unsigned through pass to information propagation the relevance for allow also of but neighborhoods, local the only not of use make methods global The Methods Global 3.1.2 higher have to degrees higher with users allow others. will with scores which relevance SPA, design we links, negative and ﬁnd empirically we Actually richer". getting rich that the “ with contradicts which other, each cancel and set to is way strength relevance The otherwise, ogrptswl easge eswih hnsotrpts hscnb omltdrcrieyas recursively formulated be can This paths. shorter than weight less assigned be will paths longer where C D 5 9 8 − neto otesge ewr rpry codn otepwrlwdsrbtoso positive of distributions power-law the to According property: network signed the to onnection 9 ( | *% | paths aebt agrpstv n eaiedges oiieadngtv eeac crswill scores relevance negative and positive degrees, negative and positive larger both have . R 9 8 + ,9 8, ; 9 8 5 *% , ( | hudb eaie hrfr h inof sign the Therefore negative. be should *% stecuto ah flength of paths of count the is 9 8 − 9 8 + ) *% , hsmto usoe h olcino l ah from paths all of collection the over sums method This : | R = 9 8 max | st aggregate to is 9 8 − ) ( *% R = 9 8 | *% = 9 8 + Õ ; *% , ∞ = 1 9 8 + V − ; *% D | · *% 9 8 − 8 ; paths ) from 25 and ok etrthan better works 9 8 + 9 8 − and ,9 8, ; | D tmyntwr el o xml,when example, For well. work not may It . 8 9 | to *% = hudhv ihrlvnei hyhave they if relevance high have should R 9 Õ ; oeta esol have should we that Note . 9 8 ∞ = 1 9 8 − sdcddby decided is V ; i function a via A ; 5 ( *% 9 8 + B86= *% , 5 straightforward A . ( *% 8 9 8 − to ) 9 8 + < V 9 = n a an has and − | *% *% 1 othat so (3.6) 9 8 + 9 8 − D − ) 8 .

25 motneo h ogrpts u omlto n t eurnerlto o h aclto of calculation the for relation recurrence length its of and paths formulation Our paths. longer the on importance and relevance of number odd an unbalanced users than involving k-cycles contains ones balanced it more expect if we SK, unbalanced from and scores relevance edges With negative edges. negative of number even an contains it if as deﬁned user is each and from term scores diagonal a is It “self-similarity”). (i.e., that Note length: varying of paths the of counting the handle to follows with ic aigapstv dei rval aiga vnnme fngtv ik napt flength of path initializing a for in reasoned links negative similarly set of and we number 1, even nodes), an two having the trivially is connecting edge edge positive direct a having a since (i.e., 1 length of paths counting When length of paths in links negative links. where indKt (SK): Katz Signed 9 ie,alteptsbetween paths the all (i.e., B 5 ( ; B and X R ; 9 8 , 9 8 U sue oesr hteeynd ntentokhsahg eeac othemselves to relevance high a has network the in node every that ensure to used is ob ihrpstv rngtv,sc hti piie vraltecce involving cycles the all over optimizes it that such negative, or positive either be to U ; ) ; ; safnto ocmietecut fptswt vnadodnme fnegative of number odd and even with paths of counts the combine to function a is aiga vno d ubro eaieegsi enda follows: as deﬁned is edges negative of number odd or even an having r h arcsta odtenme fptswt nee n d ubrof number odd and even an with paths of number the hold that matrices the are 8 and D 8 aac hoysae htakccei indsca ewr sbalanced is network social signed a in k-cycle a that states theory Balance ae ntedegree the on based 9 oaheeti,w ol hrfr edt hoetesg ftenode the of sign the choose to need therefore would we this, achieve To . 8 ; and epciey etw ildsusteinrwrigo SK. of working inner the discuss will we Next respectively. , R U B 9 8 9 ; ; R .A oei K eas a iial lo h ea of decay the allow similarly can also we UK, in done As ). B = = = = 1 A 3 B B V = − Õ G ; ; ; = W easm that assume We . − − A Õ : 3 1 1 1 # = 8 + A A V . 26 1 , ; − + A 5 + 8: + ( U B U R U ; 1 , ; 9 : ; − − U = 1 1 ; + A A A ) X − − + 9 8 B X ; = − 1 I tnraie h relevance the normalizes It . and U ; − 1 ersn h paths the represent B 1 as (3.7) (3.8) A + 8

26 ftersligvlei oiie h oerlvnebetween relevance node the positive, is value since resulting links, the between of if sign number of even choice an optimal having the paths us give of will number this the from links negative of number odd paths an of number the count we if of Therefore, number between relevance. odd negative an a have having to when want Similarly, we edges, them. negative between relevance node positive a have should between paths of in ( path link positive one Adding nodes. of length of aacdtid o -yls,wieS uhsmr o any for more pushes SK while 3-cycles), (or triads balanced U Let Step: Inductive for holds theorem the Suppose Hypothesis: Inductive Let Basis: as: induction adjacency Proof. network signed the on (3.6) Eq in as Katz defined unsigned matrix applying to equivalent is (3.8) Eq in of rule update the to leads oiieln to link positive U hoe 1. Theorem ; : A = C + neto oBlneTer:S sbitbsdo aac hoy C n J ocsmore forces SJI and SCN theory. balance on based built is SK Theory: Balance to onnection B ) ; opoeteterm eol edt hwthat: show to need only we theorem, the prove To U − 8 = 1 ; ; and − A ; ( − 1 − B = ilrsl napt flength of path a in result will hnw choose we When 1 : + 9 1 D aiga vnadodnme fngtv de,rsetvl,btenalpairs all between respectively, edges, negative of number odd and even an having − iha vno d ubro eaieegs hnw a utattenme with number the subtract can we then edges, negative of number odd or even an with ae nordfiiinof definition our on based , U 9 , 8 U A ; − − and . : ; 1 1 )( A = if + A : D 9 . + 8 + aea vnnme fngtv ik,acrigt aac hoy we theory, balance to according links, negative of number even an have rae eaieln to link negative a creates − 1 B hnorlf ieis size left our Then . A ; − = 5 ) ( B B = ; ; − A , 1 U : A A ; ( + ) A + oapt in path a to ) = ) + ; B = ( U iha vnnme fngtv ik.Ti intuition This links. negative of number even an with 1 B A ; and − ; : 27 1 − + A 1 U ( U − hc opee h proof. the completes which , D B 8 ; iial,w a banteudt ueof rule update the obtain can we Similarly, . 1 9 and ) : and , ehave we , ; + and B = 1 ; − 9 − : 0 smnindaoe oespecifically, More above. mentioned as 1 nohrwords, other In . A U ; B 8 crlst eblne.I h majority the If balanced. be to -circles when radn eaieln ( link negative a adding or : and ; ∈ + − 1 ( R B ) U 9 # 1 = D ; spstv,ohrienegative. otherwise positive, is × − 8 # = a oln to link no has U ( where , B A 1 : ) ; A euemathematical use We . = + ( ( + B A A U : + 9 8 : − − A D A U = − 9 − : indKatz signed , ( − ) 1 ) ) if = = A D B A A − 8 : : oa to ) a a has = A . − A ; + .

27 ihblneter.T aeoraayi ntefloigcs,let case, following the in analysis our ease To theory. balance with the as value undefined to an edges to incoming lead all would over this summation Therefore, negative. are half other the while positive, node a that Assume works. 0. to close them maps it small, are weights the when and large, are weights edge into weights incoming the into all coming weights of edge the of each normalize now they that is adjustment The of neighbors is ASCOS of formulation The networks. weighted following: handle the to [58] ASCOS the of enrichment an is Networks Weighted for Measure Similarity Asymmetric 3.1.2.2 de.Tefruaini hw below: shown is formulation The edges. ` nindAymti iiaiyMauefrWihe ewrs(UASCOS++) Networks Weighted for Measure Similarity Asymmetric Unsigned = nte su si edrcl pl SO+,tersligrlvnesoecudcontradict could score relevance resulting the ASCOS++, apply directly we if is issue Another (SASCOS++) ASCOS++ Signed incoming the from relevance normalized of summation the as relevance node the defines It Let ( 1 − P 9 8 4 − = 8: 8 A 3 ) to 8 8= 9 8 If . n ecnrwietefruainas: formulation the rewrite can we and 9 h oictosfrACS+wr efre ohnl egt nthe on weights handle to performed were ASCOS++ for modifications The . A 8: R = 9 8 1 and = 8              a nee ubro noigegs hr afteegsare edges the half where edges, incoming of number even an has 1 2 ^ 8 R h term The . : sngtv,hence negative, is 9 8 ∈ Í # = 8 8= SO+ a iclist ietyaatt indnet- signed to adapt directly to difficulties has ASCOS++ : R 8            @ @ = 1 ∈ ∈ | Í # Í # # 2 2 A 8 8= 8 8= 8 P 8= 8: | ( > A A 1 : R 28 8@ ∈ 8@ − Í # ( + ( 8 4 szero. is 8= 1 − _ 1 − R sngtv and negative is − 8: 9 : 4 − ) 2 ) astewihst ecoet when 1 to close be to weights the maps I 8: 8 8 ≠ = ) R 9 9 9 : 8 8 ^ ` ≠ = = spstv.Tu,if Thus, positive. is 9 9 @ ∈ Í # 8 8= 8 A ytesummation the by 8@ , hsmethod This : _ = A ^ 8: R (3.9) 9 : and is

28 hsw antdrcl pl RRt indntok.Teeoe esuysge admwalk random signed study we Therefore, networks. signed to URWR apply directly cannot we thus W matrix at transition a on based to move to of probability (URWR) Restart with Walk Random Unsigned Restart with Walk Random 3.1.2.3 triads. balanced more push will it words, other In theory. balance with aligning measurements C we below: networks, shown signed is SASCOS++ in links negative and positive of this numbers leave the negative of a imbalance seeing when the times) to three Due about (by link. similarity the in push stronger a providing is it Thus, signed with ASCOS++ using when that networks, note We SASCOS++. build to us motivates which theory, ( triad resulting when Similarity, theory. balance follow not terms does three these of product the then positive, also neto oBlneTer:I ses ovrf htSSO+ sal ohv h relevance the have to able is SASCOS++ that verify to easy is It Theory: Balance to onnection 8 9 8 u otefc sn SO+ ihsge ewrs ol neetydsge ihbalance with disagree inherently could networks, signed with ASCOS++ using fact the to Due indRno akwt etr (SRWR) Restart with Walk Random Signed ileda node at end will = 0 tews ie,n ikbetween link no (i.e., otherwise ` ` seult prxmtl .3ad-.2when -1.72 and 0.63 approximately to equal is ema s u aeacag otenraiain(i.e., normalization the to change a make but is, as term ( + 1 , − − , 2 + ) 9 sas o balanced. not also is ) .W en hstasto arxas matrix transition this deﬁne We ). ortr to return to R 9 8 R = =              2 1 2 WR : 8 ∈ Í # n ihprobability with and 8 8= ( + @ 1 W 8 ∈ Í # and − A 8 8= (where 2 8: ) | A 9 I .Wt h nuto,UW sfruae s[44]: as formulated is URWR intuition, the With ). 29 8@ = h rniinmatrix transition The : | admwle trigo node on starting walker random A : R ( ( R 1 1 W 9 : 9 8 − − 9 8 sngtv,tepouti oiieadthe and positive is product the negative, is sngtv n h eutn ra ( triad resulting the and negative is 2 4 stepoaiiyta h akrstarting walker the that probability the is )( − 2 A I hoe egbro h urn node current the of neighbor a chooses 8: W − 8: ) 9 8 2 R spstv rngtv,respectively. negative, or positive is W 9 : = > 3 1 ) 8 − 8 8 if 1 = ≠ 8 W ^ 9 9 and .Tefruainfor formulation The ). a ob non-negative, be to has 9 r once and connected are 8 hthsa has that + , (3.10) (3.11) + , − )

29 D D crsof scores Firstly, similarly. of score links. semantic signed the of capture can meanings thus and scores relevance negative with other each aﬀect can links negative of estimation the in where of neighbors of number the for account to is of score relevance of portion the Intuitively eas red’finsaefins ntecnrr,if contrary, the R On friends. are friends friends’ because R of score of relevance the theory, balance on Based restart. with R (e.g., (1) that indicates This friend". my is egto h ikfrom link the of weight way, Let SRWR. build to way 8 9 8: 8: 9 8 oec fisniho ssal Thus, small. is neighbor its of each to D n h eeac crsfrom scores relevance the and ihteaayi bv,w r ed odsustedtiso RR efcso h relevance the on focus We SRWR. of details the discuss to ready are we above, analysis the With codn oaoeetoe intuitions, aforementioned to According 9 < > > D ¯ D to 0 88 0 : B86= 0 eas red’eeisaeeeis hc sipidfo teeeyo yenemy my of enemy “the from implied is which enemies, are enemies friends’ because ) hthv ik to links have that ) D (or D steotdge of degree out the is D (or 8 9 fteesaln from link a there’s if 9 9 , ( D A to 8 D = 9 : and 8 D and 8 1 ) ,9 =, , . . . , sukon while unknown; is sue oecd h mato h ino h ik.Wt inintroduced sign With links. the of sign the of impact the encode to used is D R : 9 8 R D r ieyt efins,i a ugs that suggest may it friends), be to likely are : 9 , 9 8 D h eeac cr a ebt oiieadngtv.Toueswith users Two negative. and positive both be can score relevance the , r ieyt efins,i a niaethat indicate may it friends), be to likely are 8 D ¯ = to D eadaoa arxwt t ignlelement diagonal its with matrix diagonal a be ≠ 1 D 9 D ,9 =, , . . . , n 2 h siainas eed ntesgso ik from links of signs the on depends also estimation the (2) and ; 8 8 : osdrn ohpstv n eaielns hs h normalized the Thus, links. negative and positive both considering w.r.t sgvnas given is D D : R 8 to 9 8 to D R D 8 D ∝ 88 D 9 ic h eeac crswrtohrndscnb derived be can nodes other w.r.t scores relevance the since 9 srlvnesoeto score relevance ’s ≠ : o xml,if example, For . siiilzdto initialized is Õ R hs nutossgetdb aac hoypv sa us pave theory balance by suggested intuitions These . : 8 W 9 8 r ntaie o0 hc en htterelevance the that means which 0, to initialized are , ¯ D D B86= : 8: a eetmtdas: estimated be can : R otiue to contributes If . = 30 8: ( A | D A D a eue oestimate to used be can ¯ ¯ 9 : 88 8: 88 ) slre then large, is | W ¯ 1 A 9 : because A 9 : R 9 : D R D 8: < 8 9 8 : > a eidctdb hs fnodes of these by indicated be can 0 w.r.t D hudb egtdby weighted be should 0 8 (or D (or and W 8 ¯ hudb oiieyrelevant positively be should D D 9 8 D 8 D D : : a eueu oifrthat infer to useful be can 8 ssaladteeet of eﬀects the and small is 9 and and and r red (or friends are D ¯ 88 R D D D = 9 8 9 9 9 Í r nme)but enemies) are r red) and friends), are r nme (or enemies are with : | A 8: A W ¯ R | 9 : nthis In . 9 8 9 8 (3.12) This . D ≠ > : 0 to 0 ) .

30 D hoe 5] h eyDslnustermi ttda follows as stated is theorem Levy-Desplanques The [59]. theorem qaini ie as given is equation with form matrix in that noticing By from estimated score relevance the that sure is make Eq.(3.13) to is of term side second the right-hand and the neighborhood, of term ﬁrst the where at arrives node a at arrives walker the time Each links. outgoing negative and from positive starting through walker neighborhood random a that considering Now itself. to akrkesmvn until moving keeps walker at arrives walker between scalar where together, Eq.(3.13) and Eq.(3.12) Combining ( I 9 twl update will it , − C 2 retes eew hwta RRi orc,i.e., correct, is SRWR that show we Here orrectness: S I ( ) ,9 8, − D 1 8 ) then , a epofduigtefloiglma hc skona Levy-Desplanques as known is which lemma, following the using proofed be can sabnr niao ucinwith function indicator binary a is 0 D B86= and R R 9 9 8 9 , 88 R ( 1 sudtdas updated is yterlvnesoe fndsta aelnsto links have that nodes of scores relevance the by A ≠ = hc sue ocnrltecnrbto ftetoprs fterandom the If parts. two the of contribution the control to used is which , 9 : 8 R 2 R , RS ) 9 8 R R R 9 8 W ¯ 88 9 8 ← os’ hne hc gives which change, doesn’t = ( + 9 : ← sudtdas updated is 2 2 R 1 = Õ 2 Õ 9 8 : − : Õ D A ¯ : ← 2 B86= R :: 9 : B86= ) B86= I = edeﬁne we , 2 where ( Õ ( ( A : 1 A ( 9 : A − 9 : R B86= :8 ) 9 8 2 31 ) I W )( W ) I ¯ ¯ W ( sudtdas updated is steiett arx h ouint h above the to solution The matrix. identity the is ( ¯ ,9 8, I 9 : 9 : A − :8 S R 9 : R ) R 2 = 8: 8: = S ) 8: R W ) D ( + ¯ ¯ ( + 1 − 88 ( + − 9 : 1 if 1 1 > 1 A R 1 ( 8 − I − 0 8: D − = n hnE.31)cnb written be can Eq.(3.15) then and i.e., , − 2 8 2 tcnieaieytasi oits to transmit iteratively can It . 2 ) ) 9 I 2 ∗ ) I ( S and ( ,9 8, ,9 8, ) D 1 − 8 1 ) ) 0 srlvn oitself. to relevant is D xss h xsec of existence The exists. tews.Terandom The otherwise. 9 fterno walker random the If . (3.16) (3.15) (3.13) (3.14) 2 sa is

31 Proof. 2. Theorem have we lemma, above the on Based nonsingular. 1. Lemma crsta nraetesrcua aac fagvnsge network. relevance signed learn given to a of tends balance actually structural SRWR the Thus, increase that triads. scores other for observations similar give can we until reduced to added be will score relevance positive the increases which (3.14), Eq. during 3.1(a), Figure structures in unbalanced example, reducing For show while we process. Next structures updating the balanced unbalanced. the keep are to three likely remaining is the SRWR while that balanced are 3.1(e) and 3.1(d) 3.1(a), to leads which h paepoes h oi iewt / en oiiengtv ik.Tedse iewt +/- with line dashed The links. positive/negative means means +/- with line solid The process. update the Thus, C neto oblneter:Fgr . ie ersnaietilt htwl apnduring happen will that triplets representative gives 3.1 Figure theory: balance to onnection I R Let + − a +++ (a) 9 8

2 + S > P Let

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2 R − 9 9 8 eoe eaie(rtetidbcmsblne) olwn iia process, similar a Following balanced). becomes triad the (or negative becomes S ≠ iue31 rpesecutrddrn indrno walk. random signed during encountered Triplets 3.1: Figure ∈ 9 8 , 8 2 < S 0 | R P htrdcstepstv eeac score relevance positive the reduces that Since . = 0 + < 2 < 9 8 b + - + (b) ×

codn otesca aac hoy[9,tersligtid nFigures in triads resulting the [49], theory balance social the to According . | = Õ 9 = ≠ - easur matrix.If square a be

8 2 | 1 S S Í + snon-singular. is , 9 8

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2 | hnw have we Then . exists. - =

P Õ 32 88 9 | P = 88 | A D 1 R + ¯ | d - - + (d) Also, .

9 8 88 8: > R | S - 9 8 Í

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Í P 8 9 D | 88 ¯ 0 P | 9 R A 88 ≠ | 9 8 ilb de to added be will 9 8 8 > 9 8 | | S - | Í > o all for

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8 . | R P + 8 9 8 9 8

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32 h rdcino ik 6] hrfr,tesg rdcinpromnecnrflc h ult of quality the reflect can performance prediction sign measurements. for relevance the good Therefore, are [60]. generally links measurements of relevance prediction study node the previous is good A links that network. suggested unlabeled signed networks the an unsigned in whether signs in predict link other to of is knowledge given networks negative signed or positive in prediction sign of problem The Prediction Sign 3.1.3.1 directed and undirected both with experiments conduct settings. we Therefore, undirected links. one directed considering two before by as ones mentioned link undirected As to applied networks. naturally be directed can for measurements are directed RWR and for ASCOS designed while are networks; measurements (PA)based undirected Attachment Preferential and neighbor common (JI), section, Index last Jaccard the in (CN), discussed measurements Among tasks. the of each for tuning different in perform they how built – have is We tasks. question second measurements. The better measurements. to global leads and strategy local which numerous – is answer to want we The of question separately. signs, first strategies ignoring the and links denote negative removing “UCN-I” by networks and signed to “UCN-R" “UCN” adapting example, For respectively. signs, links ignoring negative removing and by networks signed to to “X-I" applicable and “X-R" measurements use corresponding we the “X”, denote measurement Note unsigned (3) an theory. given and balance subsections, following and the signs; properties in network ignoring that signed (2) on based links; unsigned versions negative adapt signed removing to advanced strategies building (1) three – have networks can following we signed the section, for answer last measurements to the aim in We mentioned As prediction. strength questions. tie two and prediction sign network signed i.e., two on tasks, measurements analysis relevance signed of impact the investigate we section, this In Experiments 3.1.3 o aho h aaeeie esrmns epromdcosvldto o h parameter the for validation cross performed we measurements, parameterized the of each For 33

33 nteln rdcineprmn.Udrteudrce etn,w goetedrcin flinks of directions the ignore used [4]. we in be setting, practice can undirected common all following the they Under thus experiment. available prediction is link information of the network performance in datasets, the four assess all to For metric real-world the as prediction. in (AUC) link imbalanced Curve the usually Under are Area links use link negative we negative networks, a and signed predict positive from we Since threshold, threshold than optimal less otherwise. an is positive search score “X-I", relevance and we and the data, “X-R" if training For then and the data, links. From training of the sign “[0,1]". the in indicate is to score score relevance relevance the from the score of relevance sign of a the pair obtain use each can for directly measurements scores speciﬁc relevance signed the get The to set users. training the on measurements relevance perform inPeito Performance: Prediction Sign choose randomly we dataset, each For al .:Promnecmaio fln rdcinudrteudrce setting. undirected the under prediction link of comparison Performance 3.2: Table UASCOS++-R UASCOS++-I SASCOS++ URWR-R URWR-I UCN-R Metrics UPA-R UCN-I SRWR UPA-I UJI-R UK-R UJI-I UK-I SCN SPA SK SJI h inpeito oprsnrslsaesoni al 3.2 Table in shown are results comparison prediction sign The Bitcoin- Alpha 0.765 0.496 0.530 0.751 0.500 0.531 0.730 0.488 0.517 0.559 0.481 0.497 0.669 0.497 0.499 0.671 0.501 0.500 80% 34 Bitcoin- 0.481 0.628 0.766 0.482 0.587 0.628 0.475 0.587 0.725 0.489 0.524 0.716 0.497 0.523 0.774 0.484 0.603 0.775 OTC stann,adtermiiga etn.We testing. as remaining the and training, as Slashdot 0.494 0.569 0.693 0.498 0.542 0.641 0.484 0.571 0.550 0.503 0.513 0.549 0.508 0.520 0.663 0.497 0.554 0.677 Epinions 0.530 0.566 0.702 0.538 0.560 0.634 0.498 0.634 0.630 0.512 0.522 0.629 0.508 0.520 0.705 0.537 0.573 0.703 [− 1 , 1 ] ec we hence ;

34 ag -,]t nuees apnsfo u rsne oerlvnemaueet otetie the to measurements the relevance in node strength presented their our have from to mappings datasets each easy two with ensure the associated to normalized strength [-1,1] have truth range we ground that Note a have network. that the datasets in four edge the of two only an the are is they algorithm prediction strength connection tie the network. a weighted indicate a of is to input output the link the and words, network a binary) other to (or unweighted In weight a 64]. assign 63, to [62, aims strengthen measurements which relevance prediction, of strength application tie possible is another Therefore, indicate can strengthen. also but connection links of the signs the indicate can only not networks signed for score relevance The Prediction Strength Tie 3.1.3.2 performance. best the obtains SRWR) (i.e., RWR Under signed the while links. signs; of ignore negative (2) remove signs (1) and the that links variants predict ASCOS++ helping the outperforms in also information SASCOS++ setting, rich directed contain the circles long – with consistent [61] is observation in This that triads. consider methods only methods global local that while note circles; We long consider measurements. signed local than obtain performance consistently relevance prediction measurements sign node better signed building global in Meanwhile, links negative networks. of signed (2) for importance and measurements the links suggest negative speciﬁc results remove signed These (1) that that signs. note these ignore We than better respectively. much settings, perform directed measurements relevance and undirected for 3.3 Table and ehv nyue h w ici aaes(ici-lh n ici-T)frti akas task this for Bitcoin-OTC) and (Bitcoin-Alpha datasets Bitcoin two the used only have We al .:Promnecmaio fln rdcinudrtedrce setting. directed the under prediction link of comparison Performance 3.3: Table UASCOS++-R UASCOS++-I SASCOS++ URWR-R URWR-I Metrics SRWR Bitcoin- Alpha 0.791 0.556 0.606 0.644 0.562 0.588 35 Bitcoin- 0.809 0.590 0.644 0.705 0.639 0.630 OTC Slashdot 0.627 0.500 0.541 0.578 0.519 0.524 Epinions 0.687 0.563 0.565 0.580 0.493 0.516

35 etn,w a e htaanSW stebs efrigmeasurement. performing best the is directed SRWR the again For that see 62]. can [63, we information setting, local use Thus, strength only tie networks existing correctly. unsigned most relevance for fact, algorithms the In prediction of strength. relevance strength predicting the at predict good be to could need information local also in we prediction, links, strength of tie for signs However, prediction to accurately. sign addition sign better the achieve To predict to prediction. need sign only of we predic- that performance, strength from different tie is in observation measurements This signed global tion. than performance obtain better measurements even signed or local of comparable Meanwhile, importance the measurements. supports relevance further signed This in links measurement. negative specific measure- signed best a overall was The dataset prediction. each strength in tie ment for signs ignore (2) that or these relevance links outperform the negative measurements remove specific of (1) signed results time, the the of discuss most that further note we We given performance, Now, measurements. baseline performance. random worst the the in of results context [-1,1] the range the in uniformly values picking of tion be can It respectively. settings, setting: directed undirected the and for 3.4 undirected Table for the from 3.5 observed Table and 3.4 Table in strated error root-mean-square prediction. use strength tie we of Therefore, performance the evaluate [-1,1]. to the metric to from the scores as threshold (RMSE) relevance optimal the an search map we to – data prediction training strength tie similar for the use prediction we sign “X-I", and as “X-R” strategy for signed While of scores strength. relevance tie the predicted use the directly as we measurements that specific Note network. the of edge each with associated edges. datasets these with associated strengths h rtosrainTbe34frteudrce etn sta h admtesrnt predic- strength tie random the that is setting undirected the for 3.4 Table observation first The Performance: Prediction Strength Tie strength tie the predict to attempt then and input as network binary entire the provide We h i teghpeito efrac sdemon- is performance prediction strength tie The 36

36 al .:Promnecmaio ftesrnt rdcinudrteudrce setting. undirected the under prediction strength tie of comparison Performance 3.4: Table al .:Promnecmaio ftesrnt rdcinudrtedrce setting. directed the under prediction strength tie of comparison Performance 3.5: Table UASCOS++-R UASCOS++-R UASCOS++-I UASCOS++-I SASCOS++ SASCOS++ URWR-R URWR-R URWR-F URWR-I Random UCN-R Metrics Metrics UPA-R UCN-I SRWR SRWR UPA-I UJI-R UK-R UJI-I UK-I SCN SPA SK SJI Bitcoin-Alpha Bitcoin-Alpha 0.302 0.298 0.298 0.277 0.291 0.286 0.277 0.291 0.286 0.301 0.319 0.318 0.320 0.319 0.321 0.648 0.299 0.302 0.292 0.291 0.296 0.294 0.284 0.295 0.290 37 Bitcoin-OTC Bitcoin-OTC 0.335 0.333 0.333 0.308 0.332 0.324 0.308 0.332 0.324 0.338 0.363 0.361 0.364 0.364 0.362 0.664 0.334 0.345 0.328 0.328 0.331 0.329 0.320 0.333 0.326

37 utpesca hoisadhge-re tutrlifrain hl lohvn h inductive the having also while information, structural including higher-order measurement and centrality theories signed allow social a would multiple defining network in neural perspectives deep multiple a of of incorporation use the the Furthermore, have them deep understand [86]. to seen improvements efforts have continued and we 85] seen 84, addition, [83, In applications other of generative [82]. plethora a classification 77], in node utility 76, learning’s and 75, 81] [74, 80, networks tasks 79, of mining [78, modeling Given representations and network analytical learning 73]. various as [72, advance such functions networks to approximate complex used to in been able has being learning also deep but advantages, 71] these 70, 69, [68, data in patterns handle desired. to still insufficient are also measures are centrality group signed new second Hence, the the networks. in not signed methods are so they and and [67] links links positive positive from of also negation properties is different It very them. have is between links links interactions negative negative the that capture and evident to positive fail the they as of information separation the vital the losing or Apparently, inherently links positive [66]. weak links either positive as of links negation negative treating by positive handling simultaneously 2) centrality links and negative unsigned [65]; results and existing isolated the applying combining finally then and network, networks, each to independent measures two positive the into separating links 1) negative categories: two following and the into grouped roughly be can they and links media. social online in especially have networks), can signed that (or networks links many negative are and the there positive of today most However, but networks, network?”. unsigned a on in focused answer users only central has to or literature seeking important efforts most these the are driven “Who primarily – has they question analysis important) the network (i.e., central Social network, how a on network. based of the nodes nodes to the are the of on ranking a function provide real-valued can a values define these where to seek measures These networks. for Networks Signed in Measurement Centrality Node 3.2 eplann a enpoe ontol epwru nlann n xrcigcomplex extracting and learning in powerful be only not to proven been has learning Deep negative of inclusion the considering when centrality define to attempts recent been have There measures centrality of development the on focused has research of volume large a years, the Over 38

38 ohpstv n eaielns )i loalw st osrc u ento fsge etaiyto centrality signed of definition our construct to us allows also it 3) links, of negative interactions and the positive from both coming capture input better feature the can in network found network deep signed the networks the 2) in larger) network, patterns complex other much the the for (perhaps model other new signed a in train one calculation to in needing centrality without learned signed parameters for network utilzed deep be the for to allows network that benefit significant the has 1) value centrality signed framework learning mapping deep centrality signed a a use other learn to to to propose relations therefore We form links. can negative users and positive that both fact with the users to due complex very inherently are networks signed negative or positive a either in users). important infamous or be famous can (e.g., user way a to note due we users links, “important” negative and of “normal” introduction between the differentiating to definitions. addition respective their in on networks, based signed centrality higher In a have to targeted being These are users which values. to lower interpretations as physical have intuitive have typically users also and “normal” structure network typical the on other based are while measures value, more a higher that such a network has a in user users of “central” status the measure to is networks unsigned in centrality Measurement Node (DeSCent) Centrality Signed Deep of Overview An 3.2.1 theories. score balance centrality and signed status by a guided learning user for each framework for deep a propose We networks. signed for specific networks. signed in centrality node advanced an for the have complexities even models the and deep capture benefits, to networks stated potential previously unsigned the in another with along inherent in links, negative nodes already by the complexities introduced more of the centralities to the due calculating Therefore, for it network). on utilizing model then deep the and training network (i.e., signed networks one across calculated be can centrality that such properties ec,w edaddctdwyt elz h indcnrlt au o ahue.However user. each for value centrality signed the realize to way dedicated a need we Hence, measurement centrality dedicated a developing of problem the investigate to aim we Therefore, c 8 h eet ftede erlntokfrsge etaiyaetrefold: three are centrality signed for network neural deep the of benefits The . 5 : D 8 → c 8 htpoet user a projects that 39 D 8 oterlearned/corresponding their to

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41 D ik to links with triad unbalanced siaigsge desrnt,adcnb ute nesodtruhtefloigeape fa if example: following the user through understood further for way be principled can and a their provides strength, therefore edge This signed and weight. estimating another higher a one have of should status link) the signed directed judging (i.e., at opinion sense better a have they triangles, more in from link opinion the the trust a to define likely accurately more more are to we attempt that an is in intuition triangles One with centrality. also signed but node’s (3.19)), Eq. in far so shown (as signed our to information higher-order) measurement. (i.e., not centrality clustering do that local we those this and that adding balanced) while such (i.e., unbalanced) triangles theory (i.e., of social the types to both adhering parameterize those likely and between less differentiate differences are can they these are reason utilize the triangles can is We unbalanced this and such exist. triads) any to social that those in implies frustration theory higher The to even (due an unstable having links. as negative defined of is number triangle odd) unbalanced) (or (or balanced A ones. unbalanced to compared signed of types the differentiate further triangles. to 29] we [30, networks theoretical theory signed balance in social structural Similarly with a triangles. triangles local from local use of (both use the links is of [62]) empirically strength and the [91] standpoint determine to networks, unsigned heuristic In studied connection. heavily fact every in with one and associated networks implicitly social strength in of strength) spectrum equal a of (i.e., is same there the are links all not that it However, shown links. been signed has directed single from information includes only formulation our now Until Structures Higher-order and Theory Balance Harnessing 3.2.2.2 8 fte aea“togr oncin(..hv oecmo egbr) oeta whether that Note neighbors). common more have (i.e. connection “stronger” a have they if eteeoepooet tlz o nyterltosisbtenueso h igeln level link single the on users between relationships the only not utilize to propose therefore We as networks social in form to likely more are triangles balanced that us tells theory Balance D 8 spstvl once otreusers three to connected positively is D 8 r o qa rmaltrenihos oeseicly ewn oprmtrz the parameterize to want we specifically, More neighbors. three all from equal not are D 9 to D 8 D spstv rngtv easm htwe hs w sr r involved are users two these when that assume we negative or positive is H n otinlswith triangles no and , D G , D 42 D I H emgtwn oifrtesrnt ftegiven the of strength the infer to want might we , and , D I u hrsablne ra with triad balanced a shares but , D 9 ie to gives D G an ,

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43 q 32)cnb ovre noteblwobjective: below the into converted be can (3.20) Eq. future one as it leave will we and information global work more work. our capture extend to can we circles Thus, longer length. consider any of 2-hop to circles include to to applied triangles be can on theory focus balance only information, we Although wanting not connections. but “weaker” network, existing the remove in edges to certain strengthen to we However, information triangle zero. the have include all would to matrices seek triangle four the in values corresponding the then edge, ac u eusv ento.Ti ed otefloigmnmzn problem: minimizing following the to leads This definition. recursive our match A be rankings. to shrink) measurement (or the boost allows to attempting also users and malicious users to susceptible other less centrality to absolute much high too very of influence user their a distributing prevent to from done is this specifically, More [92]. PageRank in vector normalizing the of utilization the that Note user hr h atterm last the where h bv nue u ewr orcl asteue etr etr ocnrlt crsthat scores centrality to vectors feature user the maps correctly network our ensures above The − hsi eas fteeeit otinl novn aro oe,btte aea existing an have they but nodes, of pair a involving triangle no exists there if because is This . ntrso u betv o eCn,i elet we if DeSCent, for objective our of terms In epitotta ecniu oicroaetesnl ietdeg nomto from information edge directed single the incorporate to continue we that out point We D I as: min p \ ∀ I '46 D = 8 L( − Õ U ∈ ( H Õ \ \ : ) ) = sarglrzto emo h epnua ewr parameters. network neural deep the on term regularization a is : A 5 \ 5 D + IH p \ ( Õ 8 x : ( ∈U x + : 8 ) ) A

A = − IH 5 \ − :8 − " ( + Õ x Õ + 9 8 : V − ) V + − T 5 T 5 \ IH \ p − :8 Õ ( p ( + 44 x 9 9 x : 9 + : + p ) ) 5 V \ [ sbsdo h aeie stenormalization the as idea same the on based is A p A − ( − x + 98 9 T − T :8 5 9 \ IH − :8 ) + + ( * − V A x V + + 8 + − 98 ) T ! [ T replace 2 + + 98 + − :8 + T V + + * '46 IH + T [ + + [ + 98 + − ( T c \ T [ + + 8 98 ) * − − :8 o ie user given a for [ * T + * IH T − + # 98 * D A 8 then , (3.21) + and

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46 2 a h s fde erigeal etaiycosntok?ad()Hwd h parameters the do How (3) performance? and its networks? affect cross DeSCent centrality of enable learning deep of measurements? Can use centrality the (1) signed Can existing (2) other questions: than three scores following centrality the signed better answer learn to DeSCent signed seek deep We proposed the (DeSCent). of effectiveness measurement the centrality evaluate to experiments conduct we section, this In Experiments 3.2.4 relation in derivative the using only and error the to calculating when constant a as (3.24)) Eq. (i.e., matrices adjacency urn rpsdsge etaiyvle.Nx nln ,t efr h paeto update the perform to 7, line on Next values. centrality signed proposed current vector constant a calculate construct to We how objective. discuss our and convergence of until gradient the loop 8 to 5 parameters Lines the initialized. 4, randomly line extracts are on 3 network Then, Line structure. network unbalanced). the or on balanced based features are node triangles the whether on based (separated matrices treating 5 \ 9 5 4 2 1 3 8 7 6 loih 3.1: Algorithm ( 5 x loih . eal h tp o piiigormdl nln ecntuttetwo the construct we 1 line On model. our optimizing for steps the details 3.1 Algorithm vector centrality signed Construct Input: while parameters network neural the initialize Randomly Construct create Respectively Output: Use \ 9 ( ) x 8 paeparameters Update of gradient Calculate vector constant Create and A ) and hsudt rcdr srpae ni ovrec sn ac rdetdescent. gradient batch using convergence until repeated is procedure update This . + o convergent Not Í 9 G and c 5 k 5 = T \ \ : p ( ( respectively , (U A x x 9 + : 9 − , ) ) piiainpoeuefrDeSCent. for procedure Optimization , T oetatnd features node extract to E A with A + − + 98 A , + , E + + do T − and k \ and * V ) 9 sn ac rdetdescent gradient batch using + + L( k , T and A A T + \ where 98 − * − ) − n hratro ie2cet h orsge ragemotif triangle signed four the create 2 line on thereafter and , using from k + : from [ c epciey n yhligtervle,w r effectively are we values, their holding by and respectively, , k + where T I E 5 \ ← + + 98 A * ( X + x and 8 5 c and ) \ − 8 ( fe replacing after ← E x Í : I − A ) 47 5 − 5 o user for \ \ p ( \ ( x x : 8 : ) ) A D − :8 I 5 \ k + ( x nln ,wihcnan DeSCent’s contains which 6, line on V 9 ) − and T − :8 5 \ + ( [ x − : ) T ihconstants with − :8 * \ ftede neural deep the of norobjective our in \ ereplace we k 9

47 hspoie nadtoa 6faue.Freape n fteefaue ol eteaverage the that be neighbors would of features set these the of for degree one negative example, incoming For features. 16 on additional based an averages provides This these of with sets linked that different neighbors four over averaging obtain we However, features. positive/negative and user given the for degrees positive/negative in/out the are features 4 extracted. feature Below signed each in. discuss neighbors involved and are their define they we distribution, triangles balanced/unbalanced degree of signed number node’s the and given distribution, the degree – groups node manual three each extract try to for first propose features We will of work. we future one Here as embedding automatic structure. leave network and extraction the from embedding via automatically learnt ( theory balance of contribution the observe [ we to question, third DeSCent the on address analysis To parameter datasets. a learned other perform the the utilizing in nodes then of and centrality dataset the single calculate a to corresponding network on their training to by features evaluated evaluate node is the to which mapping experiments scores, for centrality further are perform networks we deep in learned question, later the second experiments general the these how answer of to results Then, the will and We section. problem this problem. prediction prediction link link signed signed the the for discuss them further utilizing quality by the values evaluate centrality to signed approach the indirect an of perform we common [65], the following literature Therefore, the the in [95]. thus evaluation status centrality node [67]; to distributions related is law is It networks power signed the network). in follow formation signed links link a negative of and users positive the both of that ranking observed centrality signed a (i.e., values centrality signed + and , o h ru fsge eredsrbto of distribution degree signed of group the For user a for features distribution degree signed node the extract to how discuss we First Features: Node Extracting “ground-truth” have not do we because straight-forward not is questions two first the Addressing | # 8 >DC [ − − n the and ) | ). BD< and BC0CDB sntdbfr,tend etrscnb xrce aulyor manually extracted be can features node the before, noted As osrit cnrle by (controlled constraints D 8 sn n ftefu osbedrce indlinks. signed directed possible four the of one using 48 D 8 D snihos eetatteaeaein/out average the extract we neighbors, ’s 8 a ie oiieln o fw eeto were we If to. link positive a given has _ 1 and D 8 (i.e., _ 2 respectively). , | # 8 8= + | , | # 8 8= − | D , 8 | V These . # + 8 >DC , V + − | , ,

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60 loihs nln fAgrtm41 efis construct first we of 4.1, learning Algorithm parameter of the 1 and line process On generation network algorithms. the both discussing processes level later high before the BSCL through step of we Here 4.1. Algorithm in shown is (BSCL) model Chung-Lu rpriso h nu ewr eamt rsre hs nld h ecnaeo oiielinks positive of percentage the include These preserve. to aim we network input the of properties of use the through perspective. created sign be will that edges negative or positive of bias using of introduce (instead we probability Therefore, corrected network. a input the of distribution sign true the from deviating on based sign the choose simply we if network the and (i.e., perspective) perspective sign sign local global (i.e., the theory on balance not on based is closure wedge for sign edge the mining triangles of percentage networks varied [4]. signed a balanced have all being can not networks since real-world fact necessary in is and This balanced, able completely networks. is are signed model in our balance being parameter, of this triangles range of the a introduction capture of the With to majority balanced. the are ensure edge to new this sign by edge created parameter the a assigning introduce of we probability Thus, the denotes vertices. two be these could between there neighbors but walk), common two-hop other our (through constructed explicitly only we not wedge are triangles single we these procedure, the closure of closing wedge the most performing and when networks that, Note signed theory. in balance the to property adhere mentioned, key a previously is as triangles However, local formed of the process. distribution also construction and the distribution during degree coefficient the clustering maintaining for mechanism the allows automatically walk. two-hop the by created procedure) closing wedge (i.e., wedge the off closing E ihteitouto ftreprmtr (i.e., parameters three of introduction the With deter- of process above the However, distribution. sign maintain to want also we Meanwhile, which model, TCL the on based is (BSCL) model Chung-Lu Signed Balanced proposed The ecncntuttevre apigvector sampling vertex the construct can we , [ .Ti mle htwe admyisriga deinto edge an inserting randomly when that implies This ). [ o admyisre ikadi sdt orc the correct to used is and link inserted randomly a for ) 61 [ hnti ol edt u eeae networks generated our to lead could this then , π d ssono ie2 etw aclt the calculate we Next 2. line on shown as , U and E hn sn h eredistribution degree the using Then, . V ,tepooe aacdsigned balanced proposed the ), V hc sfo h local the from is which U hc is which , V which ,

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62 ewr ae ntettlblnei h nu indntok(i.e., network signed input probability the with in balance total the on based network parameter that neighbors common other any with triangles vertices of composed and h eeto of selection the procedures. insertion edge two these discuss further will we Next use. parameter our on use based we sign 5, its line On select and distribution. edge random a insert or balance, the E 18 17 11 15 13 12 16 14 10 8 1 3 2 5 4 9 7 6 loih 4.2: Algorithm 8 hog neighbor a through h eg lsn rcdr sslce ihprobability with selected is procedure closing wedge The using and triangle a into wedge a closing by edge an insert either will We return for E E 4 > > + 9 : , = oM to 1 eoeods defrom edge oldest Remove else if E E ocoeit rage ent htatog eaeepiil osrcigtetriangle the constructing explicitly are we although that note We triangle. a into close to 8 > ! − wedge_closing_procedure apefrom sample = V else if else if E E ← E netarno edge random a insert o eemnn fw huditouemr aacdo naacdtinlsit the into triangles unbalanced or balanced more introduce should we if determining for 9 9 create_positive_edge close_for_balance( > + apefrom sample = efr w-o akfrom walk two-hop Perform = do ( admypartition Randomly , Add Add E E M, E E egbr ohv oeblne triangles balanced more have to neighbors > > − + V 8 > − BSCL_Network_Generation( ecos oslc h desg of sign edge the select to choose we , nfrl trno from random at uniformly π E ← E ← 4 4 ) 9 8 9 8 E E 8 to to > + > − , : π E { ∪ { ∪ E E oln on land to : > + > + and 4 4 π or or 9 8 9 8 } } E E V E {E > ) − > − 9 ( E then U ae ntesg htcoe h eg n te common other and wedge the closes that sign the on based ( htedge that ohv oeublne triangles unbalanced more have to > + > E d ) E ∪ ) ae on based 9 then ehv utslce h eg ossigo edges of consisting wedge the selected just have We . then > − } E π respectively d 4 8 hn nln ,w efr w-o akfrom walk two-hop a perform we 6, line on Then, . 9 8 hog neighbor through ," ,d ,V U, d, c, ", [, 63 odtriewiheg neto ehdw will we method insertion edge which determine to [ E ol loipiil ecoigwde oform to wedges closing be implicitly also would 8 and E 4 9 9 8 ih ae ec,w s u learned our use we Hence, have. might uhta h aoiyo h triangles the of majority the that such d ). nln ,btsat nln with 4 line on starts but 5, line on E U : ocretymiti h sign the maintain correctly to Δ .Teeoe nln 7, line on Therefore, ). V ohl maintain help to 4 8:

63 probability network. generated the into triangles unbalanced (1- probability with Similarly, respectively. uu sepy hnw rce osml from sample to proceed we then empty, Then, is queue. queue the from to edge added an are selecting vertices the time collision, each a before such have network we the time in every exists For already that self-loop). edge a an or insert to selecting (i.e, collisions having when for queue a and positive resulting the return sets can edge generator negative network 17 the and to removed 3 been lines have from will FCL loop from this performing After from. selected was it (i.e., sets edge negative and positive maintains it that such select and 16 line to go we insert closing wedge the during made selections by sign controlled local procedure the from induced bias the parameter incorporates learned using it that than our rather use therefore we and selection, distribution sign sign the global the disrupt will procedure closing edge the distribution, for sign sign correct the the maintain to network generated from our desire sampled we since is However, network. vertex second a 12, line on balance whether on depending 8, line suggest on would mentioned theory As theory. balance to adhere common will other through neighbors) implicitly and walk two-hop the through explicitly those (both created being fntpromn h eg lsn rcdr,BC ilisedisr admeg with edge random a insert instead will BSCL procedure, closing wedge the performing not If n tpw i o eto nAgrtm42fres fdsrpini htw lomk s of use make also we that is description of ease for 4.2 Algorithm in mention not did we step One network generated the in edge oldest the remove to is step next the insertion, edge After 4 9 8 sapstv ikadadi oteset the to it add and link positive a as ( 1 − d ) E hspoessat iial sln yslcigteﬁs vertex ﬁrst the selecting by 4 line as similarly starts process This . > + " 4 4 and 9 8 9 8 V de.Ln 7sosta eslc h leteg rmteuino the of union the from edge oldest the select we that shows 17 Line edges. hrfr,wt probability with Therefore, . ed ocrflyb eemnd speiul icse,tewedge the discussed, previously As determined. be carefully to needs obigpstv rngtv,w iladteeg oteset the to edge the add will we negative, or positive being to E 4 > − 9 8 respectively. , ob eaieadteeoeadi otesto eaieedges negative of set the to it add therefore and negative be to E > + E ∪ π ie nlns4ad1) h uu scekd fthe If checked. is queue the 12), and 4 lines on (i.e. V > − ,tesg of sign the ), n hnrsetvl eoei rmteeg set edge the from it remove respectively then and ) π 64 E π uhta ecnte netedge insert then can we that such > + oee,i h uu snnepy hnwe then non-empty, is queue the if However, . U nteohrhn,wt probability with hand, other the On . eaannt that note again We . U nln 3 ecos og oln 4and 14 line to go to choose we 13, line on , 4 9 8 ilb eetdt nrdc more introduce to selected be will " ie,alteiiiledges initial the all times, U ilb ertsuch learnt be will 4 9 8 E E > + 8 ( nothe into Then, . 1 or − [ E E > − U G for > − . ) > , ,

64 au of value if 1 be to function indicator an edge the let edge speciﬁcally, each More to assigned triangle. variable latent a the the into be to closed added being was wedge edge a deﬁning the through after whether or learned randomly determine be network which can edge, it each that with is associated idea variable general hidden The a [102]. model TCL the in process similar a parameter the For Learning 4.1.4.1 these learning for algorithm proposed the discuss to iteratively. we and edge alternatively next inserted parameters Hence, perspective this via sign created unbalanced. global being or the are balanced on triangles be solely whether based of perspective is the local insertion because the edge is ignores random This and a network. of the sign in balanced the are for that decision triangles of percentage the when and on Similarly, triangles based of sign perspective. a sign with local edge the random on a based inserting only are when decisions example, these For since on distribution other. (based unbalanced each or to balanced related be are to triangles parameters constructing these that notice We network. signed based process parameters generation network the and on model BSCL the introduced have we subsections, last the In BSCL for Learning Parameter 4.1.4 two a perform to unable if queue vertex the from utilize walk we Similarly, hop queue. the of front the from take instead I 9 8 = Let 0 niae htteeg a rae i admsmln from sampling random via created was edge the that indicates π 4 d 8 9 8 ersn h rbblt fselecting of probability the represent titeration at a rae hog h w-o akwdecoigprocedure. closing wedge walk two-hop the through created was d emk s fteEpcainMxmzto E)lann ehdfollowing method learning (EM) Expectation-Maximization the of use make we , ,U, d, d E 8 C etw ildsushww a er h parameters the learn can we how discuss will we Next . uigteE rcs.Nx eaayetetopoeue fwedge of procedures two the analyze we Next process. EM the during and V eew ics o olanteeprmtr rmteinput the from parameters these learn to how discuss we here , E 9 si h egbrstof set neighbor the in is 4 9 8 hs aetvralscnb qa o1o ,where 0, or 1 to equal be can variables latent These . U 65 hshsteptnilt irp h distribution the disrupt to potential the has this , E 8 rmtesmln vector sampling the from E : n tews,and otherwise, 0 and V ,ti ildsuttegoa sign global the disrupt will this ), π and I 9 8 π ,U, d, 8 = , I 1 [ d E ugssthat suggests and C 9 eoethe denote ∈ V I # . 9 8 : ] ∈ as /

65 htwr deigt aac hoy ese oapoiaeteepce ubro triangles of number expected the approximate to seek We theory. balance to adhering were that calculated have we that Note Learning 4.1.4.2 in edge edges of set fact a sampling via calculated be can expectation of edge expectation the probability the over closure wedge by of probability the calculates which of probability with represented is (that procedure closing wedge or insertion random the either for method the to of walk neighbor two-hop mutual a perform to able were we probability the probability with insertions random the for selecting (1) following: the on node) selected based first probabilities (i.e., node starting a given insertion random or closing a omlt h odtoa rbblte o lcn h edge the placing for probabilities conditional the formulate can selecting I 9 8 r sflos respectively: follows, as are ) o h aclto fteepcainof expectation the of calculation the For I 9 8 S scniinlyidpnet ecnidvdal aclt h xetto of expectation the calculate individually can we independent, conditionally is E E n hntk h vrg costesto de ape as: sampled edges of set the across average the take then and 9 9 rmthe from stescn oewt probability with node second the as I % I 9 8 9 8 ( I a edfie sn h ae’Rl sfollows: as Rule Bayes’ the using defined be can 9 8 to V E = 8 E 3 and % % : 1 [ I | ( ( egbr of neighbors 4 9 8 4 4 9 8 9 8 9 8 E | d E , 9 | | C I I Δ (i.e., ] 9 8 9 8 8 d , = = = C rmteiptntokta eoe h ecnaeo triangles of percentage the denotes that network input the from ) % 1 0 E I = E , E , ( 9 8 : ? I C E 9 8 8 8 ∈ + % en ae ntepoaiiyo h debigcreated being edge the of probability the on based 1 being d , d , : 1 ( nearvn ttemta neighbor mutual the at arriving once ) # = 4 = C C 9 8 I ) ) 8 9 8 1 and S 1 = = | I | 4 given 9 8 4 ( d 9 8 Õ 9 8 66 1 C E = E , 4 π ∈ E − : 9 8 S : % Õ 8 1 8 2 h eg lsn ihprobability with closing wedge the (2) ; ∈ d d , E d ∈ E , ( sepce ogtcetd hslast the to leads This created. get to expected is # C 4 C [ )( , 8 # 9 8 8 C I d , 4 ) 9 8 π utemr,temxmzto o the for maximization the Furthermore, . | 9 8 I I n hntewl otne to continues walk the then and ) | 9 8 C [ 9 8 d ) n h trignode starting the and , + ) E S C E 9 ] = nfrl from uniformly 9 3 ∈ % 1 4 8 sbsdo rthaving first on based is E , ( 9 8 # 4 : 9 8 8 given d , ] | I C 9 8 ) 3 1 = d : h trignode starting the , E 0 8 E , ecncluaethe calculate can We . E 8 d , hn u othe to due Then, . E E C : 8 ) h conditional the , hrfr,we Therefore, . ( I E 1 9 8 : − o each for hti a is that E d 9 E d ) 8 (4.2) (4.1) (i.e., and and and

66 lsn rcdr r aacd hnw a solve can we we then if balanced, and are procedure values closing mentioned above the calculate can we let Then, network. input the of balanced the percentage be should methods two the from percentage balanced combined the states simply which desire we network, synthetic the in following: balanced the being triangles of percentage the maintain as correctly balanced be to expect we these Δ of percent what calculate will we Furthermore, utat1from 1 subtract both be edge the if created vertices two between neighbors added been probability have with will edge network each the Furthermore, into them. to assigned randomly signs edge and method estimate to how discuss we Next, methods two these for calculate we values the denote us Let as network. the average to on added methods edge insertion each edge for random the and closure wedge the through construct will BSCL E ; CA80=6;4 G Δ Δ ∈ Δ ooti h ubro omnnihosfor neighbors common of number the obtain To Estimating 9 8 A0=3>< CA80=6;4 G E CA80=6;4 G + 8 and \{ eal festimating of Details . E 8 E . E , and 9 = oeta fe aigtepoaiiyo h xsec o h rtedge ﬁrst the for existence the of probability the having after that Note . 9 } V 3 Δ sacmo egbrbsdo h rbblt hr xssa defrom edge an exists there probability the on based neighbor common a is Δ ; random G ic ehv led odtoe nteeitneo h rtedge ﬁrst the of existence the on conditioned already have we since , CA80=6;4 G 4 A0=3>< G 9 8 a netdit h ewr hr ednt hsnme ftinlsto triangles of number this denote we where network the into inserted was ent httesatn e fegsaecntutdwt h FCL the with constructed are edges of set starting the that note We : hc eoe that denotes which , epciey hc ilb acltdwt epc oboth to respect with calculated be will which respectively, , V = Δ E 8 Δ Δ and Δ A0=3>< ? G CA80=6;4 G Δ 9 8 CA80=6;4 G = E = 9 Δ Δ ol eeuvln otenme ftinlsta get that triangles of number the to equivalent be would , 3 2 , CA80=6;4 CA80=6;4 G G 8 Δ " 3 Δ A0=3>< G 9 + Δ A0=3>< G ent htteepce ubro common of number expected the that note We . 67 V CA80=6;4 Δ G ecn ftetinlsw ls i h wedge the via close we triangles the of percent A0=3>< G V + + n bantebelow: the obtain and and Δ Δ E and , 8 A0=3>< A0=3>< G G and Δ − CA80=6;4 G Δ E Δ A0=3>< G 9 A0=3>< G ecluaetepoaiiythat probability the calculate we , . ilb icse ae.To later. discussed be will Δ 4 CA80=6;4 G 8; emust we , d 4 8; and thus , E (4.3) and ; U to .

67 follows: h oe.W oeta h bv ol require would 2 above the that note We model. the of value average the present we Next in rewrite can therefore We includes where causing 0E6 9 " = $ is ent htw nyne ocompute to need only we that note we First 8 ( = ( 3 + # 2 Δ # 1 ) ) E A0=3>< G to E ersnsteaeaevleo qae ere.Te ehave: we Then degrees. squared of value average the represents ∗ ; ie.Scn,for Second, time). 8 ohv n esopruiyt onc to connect to opportunity less one have to 0E6 # and ecniseduednmcpormigt osrc vector a construct to programming dynamic use instead can we , ( E 3 ; E sue odnt h vrg rage osrce yarnol netdeg in edge inserted randomly a by constructed triangles average the denote to used is ∈ ) 9 + ecnuetefloigapoiaini etettesmainsc htit that such summation the treat we if approximation following the use can we , nta fecuigte.W use We them. excluding of instead Õ \{ E 8 ,E 9 Δ Δ } Δ A0=3>< G A0=3>< G Δ 9 8 A0=3>< 3 A0=3>< G ; Í ( 3 2 8 # ; = " − 1 − sfollows: as ≈ 1 = = 1 = 0E6 3 ) 0E6 E Δ 8 ; 3 2 1 2 ∈ 9 8 A0=3>< Í 8 # = = = ≈ + " ( 3 ( Õ # 3 \{ 9 ( 3 9 E = ( Õ ) # ; 2 1 0E6 E 3 8 0E6 "# ∈ + 8 − ) 1 2 E ,E + 1 − 0E6 ; 68 + ∈ 3 9 cosalpsil nree ar fvrie as vertices of pairs unordered possible all across ( 1 ( + } 0E6 ( 3 3 9 3 3 ) $ Õ 0E6 # \{ ahrta trtn vrtense u of sum nested the over iterating than rather , ( 2 ; # 2 2 2 3 # Õ 3 ( 8 ) 2 − ) ( E = + · · · + − 3 ) 2 # 8 − E # ( 8 ∗ " 1 3 ,E ; " 3 ( and 1 9 1 3 0E6 ; 3 0E6 efruaeti daa h following: the as idea this formulate We . − ) 0E6 9 9 ) ) − ) Õ } = 1 # # iet opt,btuigtefc that fact the using but compute, to time 8 Õ 8 0E6 + 3 ) = ( ( − 3 0E6 ( 1 3 3 1 3 9 1 3 ; 2 Δ ( 2 # ) ) ( ) 2 3 " 9 ( 3 2 Õ 9 8 A0=3>< ( − ) = odntsteaeaedge and degree average the denotes to 3 ( " ; # ; " 8 3 2 + − − ) ) 1 # 1 ne(hc a eperformed be can (which once 1 3 3 ) ) 8 1 3 + · · · + 9 s 3 where # ) B 8 represents (4.4)

68 egs(.. omnnihos ob n ftefloigformations: following the of one be to neighbors) common (i.e., wedges ol edt aeasg uhta hr r nee ubro eaielnsi h resulting the in links negative of number even edge an third are added there the that triangle, such balanced sign a a to have close to to need edges would existing two with wedge a for where probability link positive corrected randomly, are: types wedge the all for use we (where Below, distribution positive). sign original then the and by network, created parameter were the original wedges with the distribution of this distribution maintain correctly sign to the attempt match we perfectly to FCL signs from edges edge with model select our and initialize first we that Note respectively. negative, and positive are Í rage hnadn h edge the adding when triangles using vector the in filling ieisedof instead time vertices of number {− # 9 , = Estimating h xetdnme fblne rage htwudb rae fteedge the if created be would that triangles balanced of number expected The −}) 8 + 1 where , 3 9 ecncntutti vector this construct can We . Δ 9 8 A0=3>< $ {+ Δ ( A0=3>< G , # # −} a eotie i h xetdnme fwde fdffrn ye n the and types different of wedges of number expected the via obtained be can , 3 hrfr h eo prxmto for approximation below the Therefore . ) . sue orpeettewdefre yedges by formed wedge the represent to used is efrhraayebyn u aclto of calculation our beyond analyze further We : Δ Δ Δ Δ Δ B 9 8 A0=3>< 8 9 8 9 8 9 8 9 8 A0=3>< A0=3>< A0=3>< A0=3>< − Δ 1 A0=3>< G = 4 9 8 B +− U 8 −− +− ++ = n eefre ihawdeo type of wedge a with formed were and +( + as: orpeettenme fwde htwudb lsdinto closed be would that wedges of number the represent to U = Δ = 3 ≈ = 1 Δ 8 s [[ hc a epromdi iertm nrlto othe to relation in time linear in performed be can which , − ( 0E6 9 8 A0=3>< 0E6 trigwith starting 1 9 8 A0=3>< U Δ − ) 9 8 A0=3>< ( Δ ( [ 3 3 )( 9 8 A0=3>< ) 69 2 ++ "# 1 − ) −+ − ( + = 0E6 ( [ # −+ ) 1 B [ Δ 8 − ( − ( 9 8 A0=3>< = 1 + 3 U 1 − U ) 3 ) ) Δ # Δ [ # Õ 8 Δ ) 9 8 A0=3>< = 9 8 A0=3>< [ when − Δ 1 A0=3>< G 1 stepoaiiyo ikbeing link a of probability the is 9 8 A0=3>< 3 8 B 8 8 −− +− U = ec easm htall that assume we hence ; Δ a epromdin performed be can 4 # 8; A0=3>< G − ({+ {+ and 1 , , n hnrecursively then and −} +} 4 9 ; yeaiigthe examining by h definitions The . , n hi signs their and {+ 4 9 8 , −} sinserted is , {− $ (4.5) (4.6) , ( +} # ) ,

69 ecnaefrbt h eg lsn n h admeg neto,wihaedntdas denoted are which insertion, this edge examine random will the we and Therefore, closing [ wedge edges. positive the of both percentage for the percentage estimate to want we Here Learning 4.1.4.3 calculate also can we Then of formulation the simplify can we (4.4), Eq. in approximation the as Similarly, as edges neighbor the common with common wedge selected one a the forms denote this which us discover Let to insertion. both edge from closure wedge coming the links for the neighbor of one used explicitly of already degree have the we discount however must we that Note above). of created being neighbors be would triangle common that one triangles other of least number by expected at randomly the have add we also to that need such then We vertices time. wedge select each this created to for guaranteed idea are main we The that procedure. is closing closure wedge the using when percentage balanced the of calculation the edges. for all extended across be averaging also can This theory. balance to according triangle, CA80=6;4 Estimating and [ A0=3>< Δ Δ Δ CA80=6;4 G 9 8 CA80=6;4 CA80=6;4 G Δ U 9 8 CA80=6;4 epciey hn fw aeteetovle,w a orcl maintain correctly can we values, two these have we if Then, respectively. , iial,w ilcluaeteepce oa ubro rage and triangles of number total expected the calculate will we Similarly, : = = ≈ Δ 1 1 CA80=6;4 G ≈ 1 + + + 1 E ; 3 + 0E6 ∈ 0E6 8 + 3 \{ 4 3 9 iia to similar Õ ( 82 8 ( E − 3 3 3 8 ,E ) and 9 2 2 3 "# − ) 9 8 − " ,E E − 2 3 2 8 4 0E6 3 } 8 ( " 9 2 and # − 9 Δ 70 . ( + − 3 A0=3>< G 3 ( 1 3 8 9 E 1 2 ) − + 9 ) " E smlrt h admeg neto case insertion edge random the to (similar 1 1 # Õ 8 ; = ) ∈ − hc eut ntefollowing: the in results which , 3 1 + 1 ; \{ 0E6 Õ ( E E 3 8 ( 8 ,E 8 ( 3 and − 3 9 0E6 9 ,E 2 1 − − ) 2 )( } E 1 2 ( B 3 )( 9 " 0E6 8 3 ) y1 ic nti method, this in since 1, by 3 − ; ( ; # 3 2 ( − 3 ; " + 1 ) − ) 8 1 ) ) Δ 9 8 CA80=6;4 Δ A0=3>< G as: E by 2 ,

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81 aac hoyi tl plcbetwrsabpriesetting. bipartite a towards applicable whether still unknown is is it theory types, balance node negative different two no have of they are number and there networks even since bipartite mentioned, an signed previously in having as triangles Nevertheless, to unbalanced. (due are balanced (d) are and (c) (b) while and links), (a) in observe triangles seen further that can As We 2.2 network. nodes. Figure three signed in the unipartite between a configurations possible in four 3) are length there of 2.2, cycles Figure almost (or but triangles 55], of 2, form [74, works the tasks many in analysis towards always network applied then signed and investigated recent usually is In maintaining theory in balance involved tensions connections. social unnatural the seemingly to due and balanced) “unbalanced” be (to change to those likely and stable), more (i.e., are “balanced” that socially are that networks signed 29], in [30, settings theory the balance discusses is which theories social studied fundamentally most the of one networks signed In Networks Bipartite Signed in Butterflies Signed 4.2.1.2 setting. investigate network next bipartite signed we the structures, to 29] social [30, Although real-world theory balance different datasets. vastly Congress U.S. representing two are the datasets to compared these as links the negative between imbalance and significant positive a of is there number dataset Bonanza the in that note We datasets. collected in those and “buyers” missing connection the leave and respectively, them, otherwise. between links negative or positive set as the these by In voted House). were U.S. that as bills by to the representatives refer represent will we we datasets (which these Representatives of of each House U.S. the for other datasets the separate two and collected we specifically, More Congress. States 7 et ewl hrfr nrdc o epa oetn h sg fblneter othe to theory balance of usage the extend to plan we how introduce therefore will we Next, in nodes the to refer will we section this of rest the throughout simplicity for that Note https://www.govtrack.us/data/ U ( facnrspro oe Ya r“a”frtebl,te erepresent we then bill, the for “Nay” or “Yea” voted congressperson a If . U ( s“elr” nTbe41 erpr oebscsaitc forthree our of statistics basic some report we 4.10 Table In “sellers”. as 82 7 n o h ..Senate U.S. the for one ; U n h eaosor senators the and U as

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83 a ese stecmlmn i enda wpigln in nasge ewr)o the of network) signed a in signs link swapping as defined class (if isomorphism class complement The the as sellers. seen two the be for can structure single positive similar a this and with where buyer link, one matter negative links, positive no and two that with buyer observe one have can always we We placed, is the link link. all negative negative encompasses single that class a isomorphism negative with butterfly of butterflies signed number signed the odd have an we have 4.3(F) they Figure (since In butterflies links). signed of classes unbalanced two the are ik,w bev nFgr .()ta ohsleshv igepstv n igengtv link, negative single of and class positive isomorphism single the a positive have define two sellers and has both buyer that other 4.3(B) the Figure while in links, observe positive we two links, has buyers in the link negative of one one and When positive cycle. one have their sellers and buyers all class where the butterflies see signed we of represent 4.3(D) links, Figure negative In two classes. and positive isomorphism two three have having we butterflies which signed the leaves This definition. the in h indbprientok nohrwrs o xml,“ example, for words, other In to signs network. negative bipartite and signed positive the the reassigning randomly when the class isomorphism extending given the this, into quantify fall To “ calculate frequently. we [2], less in appear taken approach should unbalanced) while (being frequently (G) appear and should (E) (F) through (A) theory classes balance isomorphism Remember, balanced that underrepresented. suggest would or and overrepresented networks are they bipartite signed whether test in found to “ being wanted column butterflies in signed (given these dataset of each significance in the count each analyzed butterfly for signed total butterflies the signed isomorphism of each up of percentage takes the number calculated class further the We 4.3. counting Figure in after shown as analysis class our isomorphism report we 4.11 Table In Analysis Butterfly Signed 4.2.1.4 the as represent we which other, the to negatively class and seller, one to positively connected buyers (+ (+ , , + + , , − + , , −) −) hw nFgr 4.3(C). Figure in shown n endi iia a,btwt wpigtepstv n eaielinks negative and positive the swapping with but way, similar a in defined and (+ % , − steepce ecnaeo oa indbtefle to butterflies signed total of percentage expected the as ” , − , +) 84 ial,tels yeo indbtefl a both has butterfly signed of type last the Finally, . % o h smrhs class isomorphism the for ” (+ , − , + % , −) ) et we Next, ”). (+ sue to used is , − , − , −)

84 Table 4.11: Signed butterﬂy statistics on signed bipartite networks.

U.S. House of Signed Butterﬂy Bonanza U.S. Senate Representatives Isomorphism Classes Count % % B Count % % B Count % % B () (+, +, +, +) 2554388 0.986 0.922 386 13404168 0.262 0.094 4142 227660420 0.244 0.085 17459 () (+, −, −, +) 3830 0.001 7.8e-04 40 5595440 0.110 0.122 -277 103731010 0.111 0.123 -1137 () (+, +, −, −) 726 2.8e-04 7.8e-04 -29 9404006 0.184 0.122 1349 173875858 0.186 0.123 5843 () (+, −, +, −) 456 1.7e-04 7.8e-04 -35 5537080 0.108 0.122 -302 101409932 0.109 0.123 -1368 () (−, −, −, −) 20 7.7e-06 1.7e-07 30 6815324 0.133 0.040 3414 137478104 0.147 0.045 15104 Balanced 2559420 0.988 0.924 40756018 0.797 0.500 744155324 0.797 0.500 () (+, +, +, −) 30685 0.012 0.076 -390 6225745 0.122 0.302 -2811 109763190 0.118 0.289 -11565 () (+, −, −, −) 100 3.9e-05 3.2e-05 2 4118075 0.081 0.197 -2099 79053742 0.085 0.210 -9430 Unbalanced 30785 0.012 0.076 10343820 0.203 0.500 188816932 0.203 0.500

85 indbtefl.Teeoe indctrilrcntk noeo ih ieetfrs since forms, different eight of becoming one to on link take one can just caterpillar missing signed are a that Therefore, 3 butterfly. length signed of a paths as define we caterpillar” “signed A Networks Bipartite in Caterpillars Signed 4.2.1.5 in tasks numerous advance when to theory applicable networks. them bipartite balance making signed to thus adhere butterflies, signed networks of bipartite terms signed in defined 2) and networks; bipartite signed for datasets. three all i.e., across link, negative sellers and one buyers and all positive where class one the have example, For overrepresented. found be always not to appear links negative two and positive signed two involving in classes isomorphism theory the However, balance networks. bipartite of applicability the strengthening further overrepresented, significantly the datasets all across Similarly, representation. the “ for significantly except are columns underrepresented, datasets three comparing the across (i.e., butterflies signed network unbalanced given all that the is observation in second expected ratio than sign balanced link more the significantly on are based they Furthermore, balanced. indeed are networks “ negative) expected. (or positive a [2], “ in on as (based number expected calculated our from “ value independent the the Finally, be would (i.e., link signs positive link single a assigned having of randomly probabilities with network signed a class in in appearing butterfly link signed a in link positive single a having (+ , − nsmay u nig ugs ht )w a s indbtefle oetn aac theory balance extend to butterflies signed use can we 1) that: suggest findings our summary, In bipartite signed three our in butterflies signed of majority large the that observe first We , − , −) scluae by calculated is B sue odnt h ubro tnaddvain h culcutdiffers count actual the deviations standard of number the denote to used is ” 1 4 (+ (|E , B − au infisapaigsgicnl oe(rls)than less) (or more significantly appearing signifies value ” + , − /E)×(|E × |/|E|) , −) utrisi oaz,weei hw iia over minimal a shows it where Bonanza, in butterflies (+ 86 |E , (+ − + , , − + |/|E| + |/|E|) % , , + −) )frec indbtefl yeadjust and type butterfly signed each for ”) , +) n he eaielns(i.e., links negative three and ) sls omnyfudta expected than found commonly less is , (+ 3 and , ic hr r emttosof permutations 4 are there since , − , (− − , , −) − , n h rbblt feach of probability the and − , −) indbtefle are butterflies signed % and ” |E % − .The ). |/|E| ).

86 )arno akbsdapoc htitgae n-oepoeto ewrsfor networks projection theory. one-mode balance using integrates constructed that approach based walk and random butterflies; a signed balanced 3) low-rank more creating a favor signs extend predicted 2) the ensure caterpillars/butterflies; to method signed modeling 1) uses propose: that we method specifically, More classification supervised group. a each from networks bipartite signed approximation to prediction low-rank specific sign representative method 2) a develop we methods; Therefore supervised methods. based 1) propagation 3) and groups: methods; networks main signed three unipartite into in signed categorized signs in be link butterflies predicting can signed for of Methods majority balanced. large are the no networks that are bipartite validated there we since 4.2.1.2, applicable, longer Section no In are methods triangles. these of many setting, bipartite signed the in matrix adjacency following: the as networks bipartite signed undirected in prediction sign of problem the define formally we notations, and definitions aforementioned the With Networks Bipartite Signed for Prediction Sign 4.2.2 signed unbalanced) (or balanced balance a and into links close to negative butterfly. link of positive) number (or negative odd a an suggest would having theory signed when a path” an “unbalanced define we an Similarly, link as negative) butterfly. (or caterpillar signed positive unbalanced) a (or suggest balanced would a theory into balance links, it negative and transforming path” of “balanced number even a is an as that contains this 4) caterpillar refer length we signed of cycle a a If into unbalanced. closed have or (i.e., types balanced butterfly caterpillar either signed all a that into Note transformed negative. be or to positive potential ether the being links three of composed is it {U inpeito nsge ewrshsbe rvosysuid[1 2,15 2,4.However, 4]. 126, 125, 124, [61, studied previously been has networks signed in prediction Sign network bipartite signed undirected an Given U × ( }\{E + E ∪ B − ∈ } . R |U |×|U ( | ese opeittesgso oln pairs link no of signs the predict to seek we , 87 G = (U , U ( , E + , E − ) ersne sabi- a as represented U ( 1 8 and B , 9 ∈ ) U (

87 ecntutd hsi ncmaio ool sn h ere namto iia nntr to nature in similar method and a in with degrees model the attachment using preferential only signed to a would comparison that in butterflies is signed This of or types constructed. positive the be be considering to likely when is theory sign balance link to their according whether negative to as informaiton of amount vast a provide would h onsfrec fte8psil indctrilr hthave that features caterpillars the signed that possible is 8 expectation The the of each for both counts for degrees the vector negative feature and the positive Thus, the the butterfly). includes signed unbalanced or caterpillar two balanced signed the the a transforming connection are to seller they and buyer caterpillars the be signed would they many (i.e., how of endpoints about information or distributions degree nodes signed directed on [4]. prediction in pair. work the seller learned networks following and unipartite model we buyer regression unknown what logistic an extrapolate a for can use sign negative we we specifically, or positive More model, a trained predict a to data having training after the from Then, seller). and buyer a signed (i.e., theory balance caterpillars). on based features neighborhood local sign or degrees) the negative predicting and we positive of Here problem the 128]. formulating 127, buyer by 61, a setting [4, between bipartite problem signed classification the supervised to a idea of the terms extend in task the networks frame unsigned and to signed both is in signs link or links predicting towards approach common One Classifier Based Caterpillars Signed 4.2.2.1 etr xrcin h w ieetst ffaue eeaut r ihrbsdo h two the on based either are evaluate we features of sets different two The Extraction. Feature (between links signed known of consisting dataset training a construct we model our train To x 9 8 B2 sSdadSs,respectively. SCsc, and SCd as 1 8 n seller a and B 9 x yetatn etrsfo ihrteidvdas(.. their (i.e., individuals the either from features extracting by 9 8 B2 ilb oeifraieta hs of those than informative more be will x 3 ednt h uevsdcasfir htuse that classifiers supervised the denote We . 88 1 8 and B 9 ncomparison, In . 1 8 and x 9 8 3 B o h pair the for 8 steendpoints. the as x 9 8 3 eas they because x 9 8 B2 contains ( 1 8 B , x 9 9 8 3 )

88 ugse mlctsge ik htwudcntuttems aacdsge utriswt the with from butterflies derived signed sign. balanced sellers link most and suggested the buyers construct would of that pairs links additional signed implicit of suggested inclusion signed the the through model model this to modify approach matrix factorization biadjacency matrix the basic using network a bipartite introduce first we bipartite signed Thus, in balanced being networks. link butterflies select signed to more towards structured push are explicitly none would 130], that 132, signs [131, networks signed on focused for some have works Although popularity these 131]. of gaining 130, [129, been predictions have network related approaches link involving factorization applications numerous matrix low-rank the years recent In Prediction Sign Low-Rank 4.2.2.2 where matrices oblneter ie,hvn oesge utrisblne hnublne) hrfr we Therefore unbalanced). than balanced butterflies adhere signed that more signs having link (i.e., predict to theory going balance actually to are explicitly links not non-existing does of it signs links, the existing whether the control predict accurately effec- is can model that this representation although However, a pairs. learning seller tively and buyer unknown of sign the predict to seller (4.11). Eq. in objective Stochastic to the use minimize associated we to [131] value (SGD) in loss work Descent the higher Gradient Following a process. is training the there during then minimization signs the drive differing have values predicted and real the then negative) both or (i.e., sign link problem: optimization following the solve to respectively, sellers, and buyers of set the E = ai arxFcoiainModel: Factorization Matrix Basic hsalw st hnuiietelandlwdmninlrpeettosfrec ue and buyer each for representations low-dimensional learned the utilize then to us allows This {( u 1 8 8 > B , U v 9 9 = )| sue omdlteln inbtenbuyer between sign link the model to used is B [ B u ≠ 9 8 1 , n h rdce iksg (i.e., sign link predicted the and ) 0 u } 2 min U ntrso h iksg rdcints ewudlk odsoe w latent two discover to like would we task prediction sign link the of terms In . , . . . , , V B 9 8 ( 1 ( u 8 u Õ ,B = 8 > 9 )∈E v ∈ ] 9 ) max spstv,adi vr1te hr sn os oee,when However, loss. no is there then 1 over if and positive, is R 3 × = 0 , 1 and h e feitn de in edges existing of set The − B V 89 9 8 B = ( hnw nrdc o ecnsuccessfully can we how introduce we Then . u u [ 8 > v 8 > v 1 v , 9 ) v 9 r ftesm in(.. ohpositive both (i.e., sign same the of are ) 2 2 , . . . , 1 + 8 oseller to _ v | U = ( | 2 ∈ ] | + B 9 R V oeta hntereal the when that Note . B 3 | × 2 r eoe nteset the in denoted are = ( fdimension of (4.11) 3 for

89 naacdsge utriscetdsmlaeul)i h ugse indln eet eadded be to were between link signed suggested the if simultaneously) created butterflies signed unbalanced of Proof. that such 3. Theorem of. endpoints the and in involved jointly they’re caterpillars signed of types ( pair seller and buyer each for exist. to were they if butterflies would signed but balanced network, into bipartite signed caterpillars the signed in many exist convert not link do learning currently model that pairs the seller encourage and further buyer to for signs is selected have the we predicting approach on The error the signs. minimizes link model low-dimensionalexisting the learning that on such focus seller and only buyer can each for it representations Instead relationships. balanced non-existing favor the to enforce explicitly signs not link does (4.11) Eq. in given approach factorization matrix basic to balanced. framework are basic links this missing to the extension between butterflies an signed present more will ensure we further Next, MF. as simply method this denote fi a omdb nublne ahi ol eur h lsn ikt engtv ofr a form to negative be to link closing that the follows require it would but Therefore, butterfly, it butterfly. signed path balanced a balanced unbalanced is a an one be by if to formed close caterpillar, was to signed it link a if positive of a definition suggest would By it then 3. path, length balanced of paths unbalanced of number the by seller and as represented network signed a in nodes of pairs R h B |U|×|U| ) | S h rtse scluaigwehrblneter ol ugs oiieo eaielink negative or positive a suggest would theory balance whether calculating is step first The Theory: Balance with Factorization Matrix ˆ BB BB 0 | diinlblne indbtefle rae atrsbrcigtenme fpotential of number the subtracting (after created butterflies signed balanced additional ) fw let we If 1 tr h ubro aacdadublne ah flength of paths unbalanced and balanced of number the store 8 B B86= 9 0 and ) eosreta hsrpeet h ubro fblne ah flnt subtracted 3 length of paths balanced of of number the represents this that observe we , B ie indudrce idaec matrix biadjacency undirected signed a Given ( i S B ˆ ent hti 13 thsbe shown been has it [133] in that note We . 9 8 A 9 hr edefine we where , ) = ugsstesg fanneitn ikin link non-existent a of sign the suggests h B B 0 ) 0 i 1 eteajcnymti in matrix adjacency the be 8 , B 9 ,ta urnl onthv ikbtente,bsdo the on based them, between link a have not do currently that ), B as B 9 8 B86= = 0 90 A if ( hs since Thus, . BB B speiul icse,teaforementioned the discussed, previously As 9 8 ) ≠ B 0 ) R and = B A |U|×|U| hntematrix the then , ; B86= A B = B 9 8 3 9 8 htwudrsl nantgain net a in result would that M = M ecnosrethat observe can We . = ; ; 1 BB ; − epciey ewe all between respectively, , − when M ) M B * ; * ; where , 9 8 B S ˆ 9 8 nedrepresents indeed o oebuyer some for = = BB 0 . M ) ; B , M

A * ; B 3 1 is = ∈ 8

90 ffcieycut o ahnd pair using node by each theory for balance counts by effectively implied are links negative again and We positive respectively. implicit links, these negative that and note positive implicit the of inclusion the through butterflies where in where indbtefle ntesge iatt ewr.W en hs esa follows: as sets these define We network. balanced bipartite of signed gain the net in highest butterflies the signed create would that theory balance by suggested respectively, links, matrix dense very potentially the using avoid to with product element-wise difference the their taking after of value absolute for the similarly equaling and formed being butterflies, butterflies signed balanced balanced of more gain of net creation the the promote would that sign the ( 8.4, S ˆ efruaeorojc hticroae aac hoya follows: as theory balance incorporates that object our formulate We sn hoe ecncntutadtoa sets additional construct can we 3 Theorem Using that Note . | M min + U C>? U ; , U V − and ( : ( M 1 1 ( 8 8 S ˆ ,B Õ ,B S * ; ˆ Õ ) V 9 9 |) a lob acltd(oeie oeecety sn h following: the using efficiently) more (sometimes calculated be also can )∈ and )∈E r sdt oto h ee twihw noprt h oeigo signed of modeling the incorporate we which at level the control to used are ti hnes oetn ool h ue n elrpairs seller and buyer the only to extend to easy then is It . E ˆ 8 + max 1>CC>< max E E ˆ ˆ 0 8 8 0 − + , : , 1 = = 1 ( − S ˆ − {( {( ) B r sdt eoethe denote to used are S 1 ˆ 1 S 9 8 ˆ 9 8 8 8 9 8 B , B , ( ( u ( u = 9 8 > 9 1 8 > | ) | ) v 8          v B , S S 9 ˆ 0 ˆ 9 ) 9 8 9 8 BB ) 9 B ) 2 > 2 < htzrsottepista aea xsiglink. existing an have that pairs the out zeros that httentgi fttlblne indbutterflies signed balanced total of gain net the what > + + 0 0 B _ 91 B V and and 9 8 | ( o presge iatt networks. bipartite signed sparse for U 1 8 | ,B S S 2 ˆ ˆ Õ E otherwise if 9 8 9 8 9 | + )∈ 8 : + B ∈ ∈ ags n mletvle,respectively, values, smallest and largest E and 9 8 V ˆ 8 C>? − 1>CC>< | = 2 max E 0 : 8 − ( S ˆ fipii oiieadnegative and positive implicit of )} 0 : , ( 1 S ˆ )} − S ˆ 1 9 8 8 and ( u 8 > v B 9 9 ) in 2 BB S ˆ ) which , B (4.13) (4.12)

91 oepoetosaetpclyue o ohaayi n iigt ov aiu ak 15 136, [135, tasks various solve to aiding and analysis both for used typically are projections mode propagate and promote further can network. that the throughout matrix relations transition matrix balanced adjacency signed signed a a defining of by construction followed the theory, be will balance step promoting be first walker The random the theory. have furthermore to but matrix, only not transition to proper approach a based walk allow random the signed a handling develop and in setting, aid bipartite the to with theory, faced issues balance using constructed are which matrices, adjacency projection a towards networks bipartite signed for relations. method balanced more based having walk solution random the using be guide also indeed to should butterflies, we theory therefore signed balance and the balance of of levels analysis high our showing to are networks due bipartite Thus, signed unipartite links. for unknown of methods sign when prediction the performance higher predicting sign obtaining towards random previous component key proposed in the our is seen against theory balance comparison as networks, signed a Furthermore as method this method. use based later walk will We node. stay probabilistically same will the walker the at where a walk, have random “lazy” not a do considered is networks unsigned networks in bipartite problem bipartite this handling that of way One is [134]. converge problem not do such thus and distribution One stationary from them methods. prevent typical that the challenges using multiple directly pose unipartite networks unsigned in bipartite tasks signed many related However, ranking seen and have networks. prediction [44] link restart solve to with applied walk been random and variants the as such methods, based propagation Typical Prediction Sign Based Walk Random 4.2.2.3 MFwBT. factorization as matrix theory this balance denote using We method 3). length of signed paths of unbalanced count or majority balanced the of to being (according caterpillars sign suggested the with link the including once be would osrcigteoemd daec matrices: adjacency one-mode the Constructing the integrates that approach based walk random a present we Here 92 nusge iatt ewr nlssone- analysis network bipartite unsigned In U A and ae nbalance on based U ( one-mode

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112 meddsaeta h oln ar,adten ikpie sr hudb lsrta users than closer be should users paired link no the the in and closer users pairs, linked link positively no have the to is than term space second embedded this of goal The objective. overall the by controlled is term This theory. balance structural extended classifier, the into input For users l for users. embeddings of final pair the the between use link we no or negative, positive, a was there aro oeebdig r rmueswt oiie eaie rn ikbtente.More them. set between a then link and no users or of mini-batch negative, a positive, construct a we specifically, with users from are embeddings node of pair ( 10 D 2 1 5 6 4 3 7 B loih 5.2: Algorithm 9 8 8 D , odnt h egtascae ihclass with associated weight the denote to for h z if Output: Input: 8 8 ( L 9 0 [ ← B , ) D for h h > 8 ← * 8 8 ) U ∈ 1 h ( ( hc eoe h aro sr ( users of pair the denotes which aesL egtmatrices weight L; layers G ; 1 1 for 8 x then ) ) = o-iesoa representations Low-dimensional 8 ( = , ← ! ← ∀ 2 do D ) h h (U D 8 , ! . . . * 8 8 f 8 f h U ∈ ( ( * 8 U ∈ indGahCnouinlNtok(GN medn generation. embedding (SGCN) Network Convolutional Graph Signed ,

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) ) W W D h ) h niiilse oerpeetto { representation node seed initial an ; 8 9 : * ∈N U ∈ ∈N Í ( Í ( ; ; ) 8 ) + 8 − h h 9 |N h 9 |N h ∈N W ∈N ( Í 9 Í : ( 8 0 + 8 0 − ) 8 | 8 ) + + | , ( , ; h h ) h h D 8 ( * 9 |N 8 9 and ( 0 |N 8 0 ( ) ( ; ) 8 i ; and + 8 z − + i − ! 8 D | 1 W ! | , 1 8 ) ∀ ) , , D 113 * B D , D : eitoueascn emta sfuddon founded is that term second a introduce We . : 8 9 ( ∈N 9 ln with along ) ∈N ; Í U ∈ Í ) octntdtgte ie.[ (i.e,. together concatenated , 8 − ∀ 8 − ; h h { ∈ * : |N : |N ( ( ; 8 1 ; − 8 − − − ! , . . . , | 1 | 1 M x ) ) _ 8 , , , B hc otistilt fteform the of triplets contains which , h oblnetecnrbto towards contribution the balance to h ∀ 8 * 8 D {+ ∈ } ( 8 ( ; o-ierfunction non-linear ; ; − U} ∈ − 1 1 ) , ) i − i ! ! ubro aggregation of number ; , ? } o eoigwhether denoting for z 8 , z 9 f ) euse We ]).

113 ypromn h otfnaetlsge ewr nlssts,nml iksg rdcin[4], prediction sign link namely task, analysis network signed fundamental most the performing by path longer embeddings? node with the along learning process in aggregation increase performance the a into provide and information theory balance representations? of low-dimensional introduction meaningful the learning following Does of the (2) answer capable to SGCN seek Is We (1) con- graph representations. questions: signed node proposed learning the in of (SGCN) effectiveness network the volutional evaluate experimentally we section, this In Experiments 5.1.3 that [22] in as such descent). work gradient previous the to batch update compared performed effectively (as to setting mini-batch show a been using has GCN it a since of [75], parameters utilize in we presented parameters, as the updating updating style For SGD model. same our the of parameters the on regularization for use we user pair linked every for where respectively, users, linked negatively classes), link no and negative, positive, classifier, MLG the aigangtv ikbtente.Teoealojciei omlzdi h following: the in formalized is objective overall The them. between link negative a having \ , oadestefis usin ecnuteprmnst esr h ere medn quality embedding learned the measure to experiments conduct we question, first the address To ersnstewih arcsue ntelyr forSGCN, our of layers the in used matrices weight the represents D : admy(n ieeti aheoh hthsn ikto link no has that epoch) each in different (and randomly L( \ , \ , "! l B sue o h egtascae ihteclass the with associated weight the for used is ) = + + − + _ '46 |M M 1 " |M (− 1 ( ( D \ 8 , (+ 1 ? , ,D ) Õ , | 9 \ , ? M ,B ( ) ∈M D | )∈M "! 8 ( (+ ,D Õ ∈M D (− 8 , 9 ? ,D Õ ,D ) l , ) (+ ? 9 : 114 and ) B ,D ) , ? log : max ) ) M max @ ∈{+ (− 0 Í , , ? , exp 0 (|| ) − , , r h esfrtepiso oiieand positive of pairs the for sets the are ? z (|| } 8 ([ exp z − 8 z ( z 8 − D , : \ ([ 8 z z D , || D "! 9 z 2 9 2 8 ] 8 h term The . || \ 9 , || − 2 ) 2 B B z "! efrhrsml another sample further we || − (with 9 eoe h aaeesof parameters the denotes ] z \ 8 @ "! z − ) 8 z − B 9 {+ ∈ z || ) '46 : 2 2 || ) 2 2 , # ) ( − \ , , ? \ , } o the for "! (5.7) )

114 a td h ffcieeso u indGN(GN nlann oerpeettosi signed in representations node learning in (SGCN) GCN signed we our that of such effectiveness methods the study embeddings can network signed state-of-the-art existing some present we Here Comparison Performance 5.1.3.1 model. along our search of grid hyperparameters a the used tune we to that data Note training the training. on as validation 80% cross remaining with the and test, choose as randomly we data dataset, the each of For 20% We performance. better (AUC). mean Curve both characteristic AUC and operating F1 receiver higher the that Under note Area and F1 both negative utilize than links we positive more links), many are since of there evaluation, (i.e., set For unbalanced the are data. links training as negative the and together positive from users the edges labeled two the the using of trained is embeddings model final The regression the logistic features. concatenate a employ we we based case specifically sign our (more the users In of predict pair model). to the used from is features input classifier or of binary positive set a being a Thus, signs on their users. predict to of wish pairs we those set, between training negative the of out held been had that network 4.2. basic Table some in presented with previously 5.2 as We same Table the in are links. datasets Bitcoin-OTC few and Epinions Bitcoin-Alpha very and while statistics, Slashdot had the that of Epinions) variants new and the (Slashdot summarize networks users larger out two filtered further the have from and networks randomly signed datasets undirected these the of on experiments each for our that perform we note We Epinions. and datasets, Slashdot, network Bitcoin-OTC, signed Bitcoin-Alpha, real-world graph i.e., four signed on using experiments representations our learning conduct of we study networks, our convolutional For evaluation. for used metrics the and the (i.e., theory balance of theory). use network make social not signed do fundamental that paths or longer step) the aggregation exploit single not a do performing that only framework (i.e., our of variants answer investigate To we question, methods. second baseline the state-of-the-art embedding network signed the against compare and h rbe fpeitn h in flns[]i htgvnasto xsiglnsi h signed the in links existing of set a given that is [4] links of signs the predicting of problem The Settings: Experimental et ent h aaesue,teln inpeito problem, prediction sign link the used, datasets the note we Next, 115

115 yeprmtr 8]froreprmns )FrSD,w s h uhr implementation authors the use we SIDE, For SiNE 4) for experiments; code our released for authors [87] the hyperparameters used We 3) (i.e., 64; matrix be feature to input representation the as model SSE the of embedding final the use we instead therefore follows: as are baselines The network signed network embeddings. towards signed efforts non-dedicated previous the over since superiority methods their shown unsigned has work include embedding not do we succinctness For networks. - ) 4 3 oefia oe r h olwn:1 noreprmnsw onthv oeattributes, node have not do we experiments our in 1) following: the are notes final Some model: our of variants two following the evaluate to propose we Furthermore, oalorSC ains )fralebdigmtosw xdtefia low-dimensional final the fixed we methods embedding all for 2) variants; SGCN our all to • • • • • https://datalab.snu.ac.kr/side/resources/side.zip swang187/codes/SiNE.zip http://www.public.asu.edu/ ersnainfrec sri ae npoaaigifrainaogtepstv links final positive the the words, separately. along twice, other links information negative In propagating the and on twice, twice. based but is layer, user first each instead for the theory, representation of balance aggregation of naïve use make the not performs does it SGCN-1 to similar method This SGCN-1+: theory balance of use make paths). yet balanced not defined does our and (i.e, links negative the only from therefore positive and layer the aggregation separates single first the of use makes only method This SGCN-1: indirect obtain formulation. to likelihood used a is for theory, connections balance utilizing method, based walk random A structural [89]: extended SIDE utilized that framework learning deep a theory. balance is method This [87]: SiNE top- the utilize We the on matrix. based Laplacian algorithm the clustering of spectral version A signed proposed [153]: (SSE) Embedding Spectral Signed orsodn otesals ievle steebdigvcosfrec node. each for vectors embedding the as eigenvalues smallest the to corresponding 116 3 n s hi suggested their use and 3 >DC eigenvectors 4 and

116 o h als emk h olwn observations: following the make we respectively. 5.4, tables, and the 5.3 Tables For in demonstrated are F1 and AUC of terms in results comparison The Results: Comparison 64. size of were embeddings ﬁnal the that such 32, to set each were set representations we a hidden models “foe” our used and for we 5) and parameters settings; unsuggested default code’s the their around for search but grid [89], from settings hyperparameter suggested the • efrac ihtebs efrac rmtebslns hsosrainsget htit that suggests observation This baselines. the from performance best the comparable with obtains performance links negative and positive from aggregation step one only with SGCN-1 al .:Saitc ftosge ewr aae ainsfrSGCN. for variants dataset network signed two of Statistics 5.2: Table Embedding Embedding SGCN-1+ SGCN-1+ SGCN-2 SGCN-1 SGCN-2 SGCN-1 Method Method SIDE SIDE SiNE SiNE SSE SSE al .:Ln inpeito eut ihAUC. with results prediction sign Link 5.3: Table Epinions Network Slashdot al .:Ln inpeito eut ihF1. with results prediction sign Link 5.4: Table Bitcoin-Alpha Bitcoin-Alpha 0 0 0.912 0.910 0.738 0.888 0.898 0.785 0.780 0.630 0.778 0.764 . . Users # 917 796 16,992 33,586 Bitcoin-OTC Bitcoin-OTC Positive # 117 276,309 295,201 Links 0 0 0.817 0.818 0.618 0.814 0.803 0.923 0.918 0.750 0.878 0.923 . . 823 925 Negative # Slashdot Slashdot 100,802 0 0 0 50,918 0.784 0.547 0.792 0.769 0.864 0.853 0.646 0.854 0.820 Links . . . 804 804 865 Epinions Epinions 0 0 0.722 0.663 0.571 0.849 0.822 0.893 0.851 0.711 0.914 0.901 _ . . 864 933 = 5 n h “friend” the and

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132 inlVt rdcin”I rceig fte21 EEAMItrainlCneec on Conference International IEEE/ACM 2019 2019. (ASONAM). the Mining of and Analysis Proceedings Networks Social In in Advances Prediction.” (ASONAM). Vote Mining sional IEEE/ACM and 2018 Analysis the Networks of Social Proceedings in In Advances 2018. on Networks.” Conference Signed International in Predictions Polarity Interaction 2020. Mining. and Analysis Network Social science Networks.” Signed political in in Predictions and [184]; community glasses analyzing spin in for frustration ecology modeling in for physics [182]; in systems [183]; structure molecular Möbius studying (i.e., with chemistry in used networks are signed specifically, graphs) More networks. signed from benefit can polarity interaction users. between the directly predict polarity both the predict to simultaneously able but is content, that and users model between themselves. joint scores users a the present by we generated 6.1 were Section “items”) in (i.e., Thus, content the utilize not if does two naturally the This between independently. linkage tasks the two these tackled have them most methods Furthermore, these yet. of interactions/links many logged not have and system the defined join is first which users 179], when networks [181, problem signed cold-start between with the associated signs as such challenges link polarities, some interaction/link and still predicting for are 179] there However, 178, 180]. 177, both 4, [176, for 131, scores [61, algorithms users polarity been have interaction there predicting recently of although tasks addition, the In recom- and 130]. 174] 132, 173, [111, [175, towards propagation mendation used information been in also applications have traditional networks these signed Similarly, improving propaga- [172]. information recommendation as and such [171] networks, tion these harnessing mining through data improved many are are that there analysis applications network unsigned traditional In problems. application world chapter this In 3 2 1 nadto otetaiinlapiain,teeaeas ltoatssi te oan that domains other in tasks plethora a also are there applications, traditional the to addition In ye er,HmdKrm* ao rohue n iin ag MliFco Congres- “Multi-Factor Tang. Jiliang and Brookhouse, Aaron Karimi*, Hamid and Derr*, Link Tyler Joint Opinions: Power “Opinions Tang. Polarity Jiliang Interaction and and Wang, “Link Zhiwei Derr, Tang. Tyler Jiliang and Dacon, Jamell Wang, Zhiwei Derr, Tyler 1 , 2 , 3 eivsiaeapyn indntok oudrtnigadsligreal- solving and understanding to networks signed applying investigate we , INDNTOKAPPLICATIONS NETWORK SIGNED HPE 6 CHAPTER 133

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177 BIBLIOGRAPHY 178

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