Centralitiesfor Net works with Consu mable Resources

Hayato Ushiji ma- M wesig wa ∗ 1 , Zadid Khan 2 , Mashrur A. Chowdhury 2 , and Ilya S afr o 1

1 School of Co mputing, Cle mson University, Cle mson, South Carolina, US A 2 Depart ment of Civil Engineering, Cle mson University, Cle mson S C, US A

A bstract Identification ofinfluential nodesis ani mportant stepin understanding and controlling the dyna mics ofinfor mation, tra ffic and spreading processesin networks. As a result, a nu mber of centrality measures have been proposed and used across different application do mains. At the heart of many of these measures,lies an assu mption describing the mannerin which tra ffic (of infor mation, social actors, particles, etc.) flo ws through the net work. For exa mple, so me measures only count shortest paths while others consider rando m walks. This paper considers a spreading process in which a resource necessary for transit is partially consu med along the way while being refilled at special nodes on the net work. Exa mplesinclude fuel consu mption of vehicles together with refueling stations, infor mation loss during disse mination with error correcting nodes, and consu mption of a m munition of military troops while moving. We pro- pose generalizations of the well-kno wn measures of bet weenness, rando m walk bet weenness, and Katz centralities to take such a spreading process with consu mable resourcesinto account. In order to validate the results, experi ments on real- world net works are carried out by developing si mulations based on well-kno wn models such as Susceptible-Infected- Recovered and congestion with respect to particle hopping fro m vehicular flo w theory. The si mulation-based models are sho wn to be highly correlated to the proposed centrality measures. Re pro d uci bility: Our code, and experi ments are available at h ttps://github.co m/h m wesig wa/soc ce ntrality

Key words: net work centrality, Katz centrality, bet weenness centrality, rando m- walk bet ween- ness centrality, consu mable resources

1 Introduction

Spreading processes are ubiquitous throughout science, nature, and society[85, 88, 70]. Thesein- clude, spreading ofinfectious diseases[43], co mputer viruses[45], cascadingfailures[63], tra ffic con- ar Xiv:1903.00642v1gestion[54], [cs.SI] 2 Mar 2019 opinion spreading[58, 8], and reaction-diffusion processes[20]. Understanding a nodes’ spreadinginfluenceisfunda mentalfor a wide variety of applications such as epide miology[23, 43], viral marketing[95, 52], collective dyna mics[1, 10, 5] and robustness of net works[2, 65, 19] and so forth. Whereas many centrality measures were originally developedforsocial net works,so me ofthe m have subsequently been adapted to quantify thei mportance of nodesin epide miological spreading

∗ corresponding author: [email protected]

1 processes[48, 28, 80, 4]. Thisis partly due to thefact that most centralilty measures have si mple assu mptions, thus these measures are oftenintuitive andinterpretablefor a given application. More- over, most popular centrality measures are based on variants of paths and eigenvector co mputations which explain paradig msin spreading models. Popular centrality measuresinclude degree, closeness, bet weenness, current-flo w, Page Rank, eigenvector and Katz centralities[25, 66, 15, 68, 11, 12, 42]. All these measures make ani mplicit assu mption about the processin which a co m modity (e.g.,in- for mation, vehicles, orinfection) flo wsin the net work. Typically, closeness and bet weenness assu me flow on geodesic paths, while Page Rank, eigenvector and Katz centrality model flow via rando m walks. The extent to which a centrality measure can beinterpretedfor a given application depends on whether or not the assu med flow characteristics are a good representation of whatis actually flo wingin the net work. In this work, we consider a flow processin which a resource essentialfor flowis consu med along the way and can be refilled at specially assigned nodesin order to ensure that a flow processis not ter minated. For exa mple, a vehicle consu mesfuel asit travelsin a net work that has refueling station nodes. In another do main,infor mation requires updating or refreshing whileit moves over the net work. For exa mple, in realinfor mation and social networks, ru mors and gossips often die outif not refreshed[62] andforgetting rates are consideredin models[99]. Not much of the existing work models well-kno wn concepts on net works by takinginto account consu mable resources. For si mplicity, we model the resource consu mption as a discrete process thatli mits the nu mber of steps the flo w process can take without refilling the resource. For a graph underlying net work of i nterest, G = ( V, E ), the para meter κ represents the nu mber of steps a process can take without a consu med resource being refilled, and Ω ⊂ V represents the refilling nodes. One of the most intuitive andi mportant modern applications of this processis thein- motion recharging of electric vehicle batteries thatis anticipated to be broadlyi mple mentedin future. We borro w ter minology fro m the charge level of batteries for electric vehicles and refer to the currently available resource as the state of charge (S O C ). T h us, κ represents the full S O C value. In the next section, alist of related real- word applicationsis given.

Our contribution Inthis work, westudy a process where a co m modity(such asinfor mation, andtra ffic)is flowing(or spreading)in a net work while consu ming a resource necessaryfor flo w, and being refilled at special nodes. We give alist of potential applications that have a si milar flo w (or spreading) process. In order to esti mate a nodes’ spreading influence, we generalize the measures of Katz, bet weenness, and rando m- walk bet weenness centralities (includingits generalizationfor directed graphs) and sho w ho w they can be co mputed. Lastly, we present different models to si mulate the spreading processes and sho w that the generalized centrality measures are highly correlated to the si mulation-based m o d els.

2 Applications

Transportation net works: A natural and motivating application of such a processisintransportation net works,in particular, road net works. Let a nodein a road net work represent a road seg ment. An edge bet ween t wo road seg ments existsif they are physically adjacent to each other (as prolongation of each other or with a realintersection). In this case, Ω ⊂ V represents the nodes with refueling stations, and κ - the maxi mu m distance a vehicle can travel without refueling. In particular, we can also consider electric vehicle road net works equipped with wireless charging lanes, where a whole lane can be turnedinto a charginginfrastructure. This technology has seen tre mendous gro wth over

2 the last couple of years with test sites already in place [36]. Ho wever, setting up this technology will co me with a heavy price tagfor a city with ali mited budget, thus tools must be developed to analyze these net works beforehand. While several research studies have carried out foridentifying the opti mallocationsfor the deploy ment of wireless charginglanes[74, 17, 90, 46, 47], these studies often make different assu mptions while solving different opti mization objectives and constraints. Having anindependent tool to analyze the net workis thus also necessaryin orderfor deploy ment strategies to be co mpared. Peer-to- Peer Net works: Peer-to- Peer ( P2 P)infor mation exchange syste ms[6] have gained pop- ularity over the past t wo decades. One of the challengesin P2P syste msis searchingfor content on the net work. Gnutellais a popular open, and decentralized file-sharing protocolin P2P net works [94]. The Gnutella protocol works asfollo ws[75]: 1. A node (co mputer) v connects to the Gnutella network by connecting to a set of one or more n o d es, U , alreadyin the net work. Then v announcesits existence to all nodesin U . 2. The nodesin U announce to all their neighbors that v has joined the net work, which also announce to their neighbors, and soforth. 3. Once all nodes are aware of v ’s existence,it can make a query on the net work. Popular methods of message propagationfor a given queryissued by a nodeinclude such methods as flood-based, and rando m walks routing algorith ms[89]. A global ti me-to-live ( T TL) para meter represents the maxi mu m nu mber of steps (also known as hops) a query can take beforeit gets discarded. In a flood-based routing algorith m, a querying node contacts allits neighbors, who then contact all their neighbors, and so forth. The process stops after each message has taken T TL nu mber of steps. This si mplistic method produces a huge overhead by contacting many nodes. In the rando m walks routing algorith m, the querying node rando mly chooses a subset ofits neighbors and sends each of the m k messages, for so me k . Each ofthese messages startsits own rando m walk in the net work thatis ter minated after T TL steps. Other ter mination conditions exist, ho wever, they are notrelevantforthis work. Therando m walksrouting algorith m greatlyreducesthe message passing throughout the net work, with other advantages such aslocalload balancing, since no nodes arefavored over others during message propagation. Ho wever, depending on the net work topology, successrates could varysignificantly. Theset worouting algorith ms areso meti mesreferredto as blind search methods. On the other hand,infor med search methodsinclude methods thatfor exa mple, take advantage of previous queries making better decisions for message passing, and in an ideal scenario, a message could then take the shortest path to a target node. In the Gnutella network, once a node receives a message,it first reduces the T TL counter of the message before for wardingit. The T TL para meterisintuitively equivalent to the S O C para meter κ in this paper while the set of nodes in Ω represent nodes that reset the TTL counter before for warding the message. These,for exa mple, could be co mpro mised nodes. In the analysis of the Gnutella net work, aninteresting questionis deter mining the mosti mportant nodesin the net work, which could be the nodes that receive the most tra ffic. Social Net works (online): The popularity and co mplexity of online social net works ( OS N) have seen atre mendous growthinthelastt wo decades and willcontinueto grow. In OS Nssuch as T witter or Facebook, a user sharesinfor mation which can be vie wed by other users he/sheis connected to. Centrality measures are often used toidentifyinfluential users within an OS N. Ho wever, in most centrality measures, all the usersin the net work are assu med to exhibit si milar behavioralfeatures. In other words, they all have the sa me desire and motivation to share knowledgein the network. In most cases, all the users are assu med to be active users, users that are willing and motivated to share their knowledge. Many studies have been carried and show that this is not the case.

3 In fact, most users, while being beneficiaries of the content being shared, actually do not share infor mation the mselves. The t wo different types of users are often referred to as posters a n d l urkers [60, 51, 72, 78]. Posters are defined in [78] as the active users who share their experiences and create content on theinternet whilelurkers are defined as the passive users who do not necessarily create any content. The analysis oflurkersin social net worksis no w an active area of research[35]. In order to have a better understanding of online social networks, it is i mportant to understand ho winfor mation and contentis shared over the net work. It has been reported thatlurkers are the majorityin many online co m munities. The percentages oflurkersin an online co m munity varies across different studies with so me giving esti mates as high as 90 percent of the total users [91]. So meti mes referred to as the participation inequality principle [35], only a s mall percentage of users actually contribute to the online content while the rest never do. Lurkers are not registered users who do not use their account, they can shareinfor mationin subtle ways. For exa mple, Facebook has the ”like”feature and thus user’s contacts can see theinfor mation he/sheliked. In this work, we takethe postersto be me mbers ofthe set Ω and assu methat a piece ofinfor mationis coupled with a mo mentu m or penetrating po wer . Followingthe analogy of a battery chargein E V,the mo mentu m of a piece ofinfor mation has the power to drive theinfor mationfor ali mited nu mber of steps, κ , beforeit dies out. Ho wever,ifit reaches a nodein Ω,it regainsits mo mentu m. Personalized web ranking: Consider a person surfing the web at rando m, however, with a topic ofinterestin mind. The surfer begins at a web page ω 1 and perfor ms a rando m walk on the web graph. At each ti me step, the person proceedsfro m the current page u to a rando mly chosen web page that u islinked to. If after κ steps, the surfer has notfound a web page ofinterest, the surfer starts the process again fro m ω 1 . However,if the surfer discovers a web page ω 2 that is relevant to the topic of interest, ω 2 beco mes the ne w restarting point and the process continues. If Ωis the set of web pages known beforehand, then centrality measure defined by such a process can be used to rank the web pages based on the onesin Ω, giving a personalized page ranking strategy. In this class of applications, we can also mention rando m walk based si milarity measures on graphs and hypergraphs[79, 24, 16] that would benefitfro mintroducing resource consu mption restrictionsfor the distance of a rando m walk.

3 Related Work

Centrality as a way of analyzing social net works dates back to Bavelas[7]. Since then, various meth- ods of centrality have been proposed to quantify thei mportance ofindividualsin social net works. These measures have also been effectively used astoolsto study net worksin other diverse fields such as physics, biology, and engineering. Since our initial motivation for this research was related to the analysis of road net works, in this section, we first highlight ho w previous studies have used centrality measures to analyze road networks. Next, we briefly introduce and su m marize studies on electric vehicle road networks. Lastly, based on the potential applications for the proposed centrality measures, we su m marize different existing approaches for possible applications. We briefly su m marize aninco mpletelist of studiesin which centralities have been used to study and analyze road networks. A road network pattern can be vie wed as the geographicallayout and structure of a network. A road network can be laid out in different patterns (for exa mple, see the book[82]for moreinfor mation on road net work patterns) which can affect tra ffic perfor mance, travel behavior, and tra ffic safety. In Zhang et al.,[98], the bet weenness centralityis co mputed to analyze and classify road net work patterns. In particular,itis used to define a measure that can quantitatively distinguish bet ween different pattern types. In Wang et al., [93], the authors use

4 centrality measures to analyze road net worksin urban areas and apply their findings to mitigate congestion. More specifically, they use large-scale mobile phone data, with detailed Geographic Infor mation Syste m data to detect types of road usage and deter mine the origins of the drivers. Thisinfor mationis used to build a bipartite net work with nodes representing road seg ments and driver sources , whichis then called the net work of road usage . Here, a driver sourceis a zone where the mobile phone userlives. This can belocated using the mobile phone data. Given alist of all driver sources,for each road seg ment r , the authors calculated thefraction of tra ffic flo w on r t h at was generated by each driver source. They then ranked the driver sources by their contribution to the tra ffic flow. Based on thisinfor mation, an edgein the network of road usage exists between a road seg ment r and the top-ranked source nodes that produce 80 % of r ’s tra ffic flo w. Finally, the bet weenness centrality of a road seg ment r in the road net work, along with the degree centrality of r in the network of road usage were used to classify and group the road seg mentsin the network. Experi ments carried outin the San Francisco Bay area and Boston area provides evidence to show thatthe findings could enable cities totailortargeted strategiesto reduce the average daily co m mute ti m e. In Scheurer et al.,[77],the authors used a wide variety of centrality measuressuch as bet weenness, closeness and degree centrality toidentify the positive and negative points of the public transporta- tion net works fro m different perspectives such as coverage, connectivity and service levels. The collective hu man spatial move ment behavior is explored in [40]. The authors use, a mong others, Page Rank and bet weenness centrality coupled with agent-based si mulations to study the move ment of pedestrians in London street network. In [3], the authors propose an esti mation method for mobility prediction in transportation net works based on the bet weenness centrality carrying out experi ments on theIsraeli transportation net work. Other studies on transportation net works, where centrality plays a crucial rolein the analysisinclude[37, 39, 71, 38, 21, 69]. As cities move to wards reducing their carbonfootprint, E Vs offer the potential to reduce both petroleu mi mports and greenhouse gas e missions. Ho wever, batteriesin these vehicles have ali mited travel distance per charge. This resultsin a major obstaclefor E V widespread adaptation, na mely, range anxiety , the persistent worry about not having enough battery power to co mplete a trip. The e mergence of E V wireless charging technology where a wholelane can be turnedinto a charging infrastructure providesitself as a potential solution to range anxiety. For a more detailed study of the design, application andfuture prospects of this technology, the readeris encouraged to see,for exa mple,[73, 9, 55, 59, 18, 27, 92, 67, 97, 29]. With a heavy pricetag, a deploy ment ofthistechnology without a careful study canlead toine fficient use ofli mited resources. One of the main purposes of this paperis to provide a tool to study and analyze road networks with a given deploy ment of wireless charginglanes. In these E V road networks, we assu me thatin orderfor an E V to travel between any two nodes,itis possible that a vehicle may need to detour to get charged to arrive atits destination. We envision that our modified version of bet weenness centrality can be usedin studying these E V road net worksin si milar ways as the studiesin the preceding paragraph. Studies analyzing social net works whose users can be categorized as posters and lurkers have recently been gaining attention. In [86, 87], the authors propose centrality measures for ranking lurkersin social net works. In these works, no prior kno wledge of whether a useris alurker/poster or notis assu med. The authors define a topology-drivenlurkingfra me work to model the relationships fro minfor mation-producer toinfor mation-consu mer. As a result,lurkers are ranked based on only the topology of the network. In our applications of the proposed centrality measures, we assu me prior knowledge of whether or not a useris a poster orlurker. The main basisfor this assu mption is that the network topology may not be a related to a user’s desire to share infor mation. For exa mple, two users on Facebook may have the sa me nu mber of connections, however, have very different desires to share or postinfor mation.

5 4 Graph Model

L et G = ( V, E ) be an un weighted (directed or undirected) graph underlying a net work ofinterest. The underlying assu mption is that the co m modity (such as infor mation and moving vehicle) is flo wing (or spreading) on G while consu ming a resource necessaryfor flow. The flowisli mited to the nodes within a geodesic distance of at most κ e d g es, κ ∈ N . In addition, there exists a subset of nodes, Ω ⊂ V that refill the resource. In other words,if the co m modity passes through a node u ∈ Ω,it canthenspreadfurtherto nodesthat are at most κ edges,fro m u . This processis modeled by a directed graph with adjacency matrix B κ (see belo w). L et A be the adjacency matrix of G , and |V | = n . Define the state space of a co m modity traversi ng G as the set V : = { (u, i )|u ∈ V, 0 ≤ i ≤ κ } , in which the state ( u, i ) represents the event that the co m modityis at node u ∈ V with the current level of S O C at i. Thetransitionfro m onestateto anotheris modeled by a directed graph G = ( V , E ), w here E = E 1 ∪ E 2 wit h

E 1 : = (u, i ), (v, κ ) | (u, v ) ∈ E, v ∈ Ω , a n d E 2 : = (u, i ), (v, j ) | (u, v ) ∈ E, i = j + 1 , v∈ / Ω .

The set E 1 represents a transition where the current S O C is increased to κ (r e fill e d), w hil e E 2 represents a transition where the current S O C is reduced by 1 (consu med). The adjacency matrix of G , denoted as B κ ,is defined asfollo ws. Let J Ω be a diagonal n × n matrix given by 1 , if i ∈ Ω [J ] = Ω ii 0 , ot her wise.

For ease of exposition, whereitis clear, we drop the subscriptin J Ω and si mply write J . If I t h e n × n identity matrix define the n (κ + 1) × n (κ + 1) block matrix B κ as   AJ A (I − J ) 0 ...... 0    A J 0 A (I − J ) 0 ...... 0     ..   A J 0 0 . 0 . . . 0   . .  B =  . .. .  ( 1) κ  . . .     . .. .   . . .   A J 0 0 . . . 0 ... A (I − J )  A J 0 0 . . . 0 . . . 0

Thenthe block matrix B κ defines a directed state space graph G = ( V , E )that modelsthe underlying flow process. In orderfor a co m modityto flowfro m node s t o t,it may be necessary to traverse one or more nodesin Ω. With thisin mind, we define afeasible walk to represent the walksin G t h at the co m modity canfully traverse. Definition 1. A w al k w i n G is a f e asi bl e w al k if a co m modity starting with full S O C can traverse w . The proposed centrality measuresin thefollo wing sections are based on co mputing thefeasible walksin the net work.

6 5 Katz Centrality

The centrality measure proposed by Katz[42] was originallyintendedtorank a node(i.e., an actorin asocial syste m)influence within asocial net work accordingtothe nu mber ofits contacts considering different pathlengths to other nodes. Thus, the model takesinto account not only thei m mediate neighbors of a node but alsoits neighbors of second-order, third-order and so on. The co mputation of Katz centralityis based on rando m walks e manatingfro m a node. In this section, we propose S O C - Katz centrality, that only takes feasible walksinto account.

Counting Nu mber of Feasible Walks For an adjacency matrix A of a graph, the ij th entry of the matrix A k , k ∈ N , counts pathsfro m i t o j of length k . However, in our resource consu mption model, not all of these walks are in fact feasible . Thisleads to aninteresting question of finding a matrix that represents the nu mber of i-j f e asi bl e w al ks. k Consider the matrix B κ . Assu me that theindex of nodesin V and matrices I, J, A, B κ start at k 0. For i, j ∈ V , wit h 0 ≤ i < n , 0 ≤ i < n (κ + 1), and j ≡ i ( mod n ), t h e ij th entry of B κ gives the nu mber of walksfro m node i t o j co mpleting with a different S O C. The destination S O Cis given by the value i / n . Thisi mplies that the nu mber of walksfro m i t o j ending with non-negative S O C is given by the su m mation at each S O Clevel. L et I κ be an n (κ + 1) × n block matrix with κ + 1 blocks ofidentity matrix I given by     I I      I   0  I κ =  .  , a n d Z κ =  .  . ( 2)  .   .  I 0

T k Then the matrix I κ B κ I κ , is an n × n matrix whose ij th entry gives the nu mber offeasible walks fro m i t o j of length k . Let S be the n (κ + 1) × n (κ + 1) matrix with ij ter m given by

∞ k k s i j = α [B κ ]i j . ( 3) k = 1 T h us, 2 2 i i S = I n ( κ + 1 ) × n ( κ + 1 ) + α B κ + α B κ + · · · + α B κ + . . . − 1 . = ( I n ( κ + 1 ) × n ( κ + 1 ) − α B κ ) T he n, if W is the n × n matrix given

T − 1 W = Z κ (I n ( κ + 1 ) × n ( κ + 1 ) − α B κ ) I κ ( 4) The centralityis then given by C = W 1 , where 1 is the colu mn vector consisting of all 1’s. For the standard centrality measure, the Katz centralityis co mputed by ( I − α A )− 1 . The para meter α , als o known as the da mping factor , must be chosen carefully such that 0 < α < 1 / λ m a x , where λ m a x is thelargest eigenvalue of B κ .

Le m ma 5.1. 1 / λ m a x (A ) ≤ 1 / λ m a x (B κ ) . T k Pr o of. If A = Z m B m I m , then A is the n × n matrix that counts the walks of length k in the k k k k ∞ bounded- walk graph. Clearly,[ A ]i j ≥ [A ]i j f or all i, j . Thisi mpliesthatifthesequence { α A } k = 1 k k ∞ converges, then { α A } k = 1 converges. Thus, λ m a x (A ) > λ m a x (B )

7 W he n α → 0, then only walks of very shortlength are takeninto account and degree centrality usually perfor ms well[13, 48]. Ho wever, asthe value of α increases, eigenvector and Katz outperfor m other measures[57]. Due tole m ma 5.1, we are able to takelarger values of α co mpared to the standard Katz centrality measure.

6 Bet weenness Centrality

For a graph G , let σ s t be the total nu mber of shortest pathsfro m nodes s t o t, w hil e σ s t (v ) be the total nu mber ofshortest pathsfro m s t o t that passthrough v . Thenthe(unnor malized) bet weenness ce ntrality of v , B C (v ),is given by

σ (v ) B C (v ) : = s t . ( 5) σ s t s = v = t

The decision of whether or not toinclude the end-points of a path tofall on that pathis usually made according to specific applications and goals. Thisis because the only difference this makesis an additive constant to B C (v ). In this paper, we will generallyinclude the end-points. ∗ ∗ L et σ s t bethetotal nu mber of shortest feasible walks (see Definition 1)fro m s t o t, wit h σ s t (v ) b e the total nu mber of shortestfeasible walksfro m s t o t that pass through v . Thenthe(unnor malized) S O C -bet weenness centrality of v , B C ∗ (v ),is given by

σ ∗ (v ) B C ∗ (v ) : = s t . ( 6) σ ∗ s = v = t s t

The co mputation of B C ∗ depends on counting the nu mber of shortestfeasible walksfor each pair s, t ∈ V .

6.1 Counting Shortest Feasible Walks

L et d G (u, v ) for u, v ∈ V be the geodesic distance fro m u t o v . Theter m σ s t (v ) for v ∈ V can be calculated as 0 , if d G (s, t ) < d G (s, v ) + d G (v, t ) σ s t (v ) = ( 7) σ s v · σ v t , ot her wise. This property, ho wever, does not hold for counting shortest feasible walksin G . Thus,in order to count feasible walksin G , we turn to the directed graph G .

Le m ma 6.1. Let w = ( s = u 0 , u 1 ,..., u k = t) be an s -t w al k i n G of le ngth k , with u i ∈ V . w is a f e asi bl e w al k i n G if and only if there exists a walk in G with a node sequence of (s, κ ) = (u 0 , i 0 ), (u 1 , i 1 ),..., (u k , i k ) = ( t, i k ) for so me 0 ≤ i0 ,...,i k ≤ κ .

Pr o of. For walk w = ( s = u 0 , u 1 ,..., u k = t) i n G , let i0 , i 1 ,...,i k be the S O C value at nodes u 0 ,..., u k respectively during the walk. Since, w is afeasible walk, i0 = κ and node( u j + 1 , i j + 1 )is ad- jacent to ( u j , i j ) i n G ,for 0 ≤ j ≤ k − 1. Thereforethe nodesequence( s, κ ) = ( u 0 , i 0 ), (u 1 , i 1 ),..., (u k , i k ) = (t, i k )is a walkin G . On the other hand,if( s, κ ) = ( u 0 , i 0 ), (u 1 , i 1 ) ,..., (u k , i k ) = ( t, i k )is a walk i n G , then u j + 1 is adjacent to u j i n G , for 0 ≤ j ≤ k − 1. T h us, w = ( s = u 0 , u 1 ,..., u k = t) is a w al k i n G .

8 For 0 ≤ i ≤ κ , a walkfro m( s, κ ) ∈ V t o ( t, i ), can be vie wed as afeasible walkfro m s t o t i n G . Ho wever, a shortest pathfro m ( s, κ ) ∈ V t o ( t, i )is not necessarily a shortest feasible walk fro m s t o t i n G . In order to count shortestfeasible walksin G , weintroduce aset of du m my nodesinto G and call the ne w graph G = ( V , E ) where

V : = V ∪ { (u, ) | u ∈ V } E : = E ∪ { (u, i ), (u, ) | (u, i ) ∈ E }

N ot e: can be viewed as a string or marker andis not a variable. The nodes ( u, i ) for 0 ≤ i ≤ κ represent node u ∈ V at different states i. Ho wever, for a shortest feasible walk fro m s t o t, we are interested in arriving at t at any state, thusintroducing a du m my node ( t, ) to capture all final states. The followingle m ma shows how adding these du m my nodes, si mplifies the process representing shortest feasible walks.

Le m ma 6.2. Let w = ( s = u 0 , u 1 ,..., u k = t) be an s -t w al k i n G of le ngth k , with u i ∈ V . w is a s hortest feasible walk in G if and only if there exists a shortest path , i n G with a node sequence of (s, κ ) = ( u 0 , i 0 ), (u 1 , i 1 ),..., (u k , i k ) = ( t, i k ), (u k + 1 , i k + 1 ) = ( t, ) for so me 0 ≤ i0 ,...,i k ≤ κ ; k ∈ N .

Pr o of. If w is a shortest feasible walkin G ,fro m Le m ma 6.1,itfollows that there exists a walk w i n G and subsequentlyin G wit h w = (( s, κ ) = ( u 0 , i 0 ), (u 1 , i 1 ),..., (u k , i k ) = ( t, i k )). Suppose w is not a pathin G . Then there exists a node ( u j , i j )for so me j ∈ N visited more than once. This for ms a cycle C wit hi n w . Define w as a node sequencein G where the cycle C i n w is replaced wit h ( u j , i j ). Itis easy to see that w is a walkin G with length strictly less than w . By Lemma 6.1, this i mplies that there exists a feasible walk in G with length less than w contradicting the assu mption that w is a shortest feasible walkin G . T h us, w and subsequently the node sequence (s, κ ) = ( u 0 , i 0 ), (u 1 , i 1 ),..., (u k , i k ) = ( t, i k ), (u k + 1 , i k + 1 ) = ( t, )for so me 0 ≤ i0 ,...,i k ≤ κ , for m a shortest pathin G . On the other hand, suppose w is a shortest pathfro m ( s, κ ) to ( t, ) i n G of length k + 1 for so me k ∈ N . Le m ma 6.1,i mplies that, there exists an s -t f e asi bl e w al k w i n G of length k . Bythe sa mele m ma, the existence of a shorter s -t feasible walk in G wouldi mply the existence of a walk fro m ( s, κ ) to ( t, ) i n G withlengthless than k +1, contradicting the shortest path assu mption.

For a set A ⊂ V a n d s, t ∈ V , define σ s t (A ) asthe nu mber of s -t shortest paths that pass through one or more nodesin A .

Le m ma 6.3. F or s, t, v ∈ V , let γ, τ ∈ E wit h γ = ( s, κ ), τ = ( t, ) a n d A v = { (v, i )|0 ≤ i ≤ κ } t he n:

∗ 1. σ s t = σ γ τ ∗ 2. σ s t (v ) = σ γ τ A v Pr o of. For the first part, Le m ma 6.2 gives a one-to-one mapping between set of shortest feasible ∗ walks i n G and set of shortest pathsin G whose cardinalities are given by σ s t a n d σ γ τ res pecti vel y. For the second part, σ γ τ (A v ) counts the nu mber of shortest pathsin G that passthrough a nodein A v . Applyingle m ma 6.2, any shortest path that passes through a nodein A v is a shortest feasible w al k i n G that passes through v . Duetothe abovele m ma, we can co mpute B C ∗ (v ) as f oll o ws:

9 Theore m 6.4. For graph G = ( V, E ), and directed graph G , let S = { (s, i ) ∈ E |s ∈ V, i = κ } a n d T = { (t, i ) ∈ E |s ∈ V, i = } , A v = { (v, i )|0 ≤ i ≤ κ } , with v ∈ V , then σ A B C ∗ (v ) = γ τ v . ( 8) σ γ τ γ, τ ∈ E γ ∈ S, τ ∈ T We nowshow howto co mpute B C ∗ (v ) without explicitly constructing G . In our co mputations, κ the value σ γ τ A v is approxi mated by σ γ τ (v, i ) . i= 0

6.2 Co mputing S O C- Bet weenness Centrality In order to co mpute B C ∗ , we build on Brandes’ algorith m[14]for co mputing B C (v ). We first give a su m mary of Brandes’ algorith m. T h e pair-dependency is defined as the ratio

σ s t (v ) δ s t (v ) : = ( 9) σ s t of a pair s, t ∈ V on an inter mediary node v ∈ V . In order to eli minate the need for explicit su m mation of all pair-dependencies, Brandesintroduces the notion of dependency of a vertex s ∈ V on a single vertex v ∈ V , defined as

δ s • (v ) : = δ s t (v ) ( 1 0) t ∈ V and shows that the dependency of s ∈ V on any v ∈ V obeys the follo wing recursive relation

σ s v δ s • (v ) = · ( 1 + δ s • (w )) . ( 1 1) σ s w w :v ∈ P s ( w ) w here P s (v )is the set of predecessors of a vertex v during a breadth-first search ( BFS)fro m source s ∈ V . Itis given by

P s (v ) : = { u ∈ V : { u, v } ∈ E, d G (s, v ) = d G (s, u ) + 1 } ( 1 2) w here d G (s, v )is the geodesic distancefro m s t o v . In su m mary, Brandes’ algorith mfor co mputing B Cis asfollo ws: for each source node, s ∈ V , i perfor m BFS co mputing nu mber of shortest paths to every other node, t ∈ V

ii back propagation: co mpute δ s • (v ) for v ∈ V in order of non-increasing distancefro m s . One major difference between equation (8) and the standard co mputation of B C is that equation (8)is constrained by thefact that the source and target nodes must be chosenfro m sets S a n d T . Therefore, the recursive relation given by equation (11) can not be used asitis because not all nodes are target nodes. For T ⊂ V , consider the function

T δ s • (v ) : = δ s t (v ), ( 1 3) t ∈ T t he n

1 0 Le m ma 6.5. T σ s v T δ s • (v ) = · (✶ T (w ) + δ s • (w )) , ( 1 4) σ s w w :v ∈ P s ( w ) w here ✶ T (w )is theindicatorfunction such that ✶ T (w ) = 1if w ∈ T and 0 other wise. Pr o of. For each ter m on the right side,if w ∈ T , then the su m mandfollowsfro m equation (11). Consider the caseif w /∈ T . Extend the definition of the pair-dependency toinclude an edge e s uc h t h at, δ s t (v, e ) : = σ s t (v, e )/ σ s t w here σ s t (v, e )is the nu mber of shortest s -t paths that contain both v a n d e . Then Brandes showedthat

δ s • (v ) = δ s t (v, { v, w } )

w :v ∈ P s ( w ) t ∈ V a n d σ s v , if t = w σ s w δ s t (v, { v, w } ) = σ ( v ) σ s v · s t , ot her wise σ s w σ s t It then follo ws that T δ s • (v ) = δ s t (v, { v, w } ).

w :v ∈ P s ( w ) t ∈ T So for w /∈ T , then t = w a n d

σ s v σ s t (w ) δ s t (v, { v, w } ) = · σ s w σ s t w :v ∈ P ( w ) t ∈ T w :v ∈ P ( w ) s s σ = s v · δ s • T (w ) σ s w w :v ∈ P s ( w )

The recursive relationin equation 14is used to co mpute the S O C-bet weenness centrality. Algo- rith m (1) describes this co mputationin detail.

1 1 Algorith m 1 S O C- Bet weenness Centrality 1: I n p ut: G = ( V, E ), V , Ω , κ 2: O ut p ut: b c [v ], v ∈ V 3: b c [ν ] ← 0 , ν ∈ V 4: Σ ← { (u, i ) ∈ V |u ∈ V , i = κ } 5: f o r s ∈ Σ d o 6: S ← e mpty stack; 7: P [ω ] ← e mpty list, ω ∈ V ; 8: σ [t] ← 0 , t ∈ V ; σ [s ] ← 1 9: d [t] ← − 1 , t ∈ V ; d [s ] ← 0 1 0: Q ← e mpty queue; 1 1: enqueue s → Q ; 1 2: w hil e Q not e mpty d o 1 3: dequeue ( v, i ) ← Q ; 1 4: push ( v, i ) → S ; 1 5: if i = t he n 1 6: f o r neighbor w ofv d o 1 7: if w ∈ Ω t he n c urre nt no de ← (w, κ ) 1 8: els e 1 9: if i ≥ 0 t he n c urre nt no de ← (w, i − 1) 2 0: els e c urre nt no de ← − 1 2 1: if c urre nt no de = − 1 t he n 2 2: if d [c urre nt no de ] < 0 t he n w found for firstti me ? 2 3: enqueue c urre nt no de → Q ; 2 4: d [c urre nt no de ] ← d [( v, i )] + 1; 2 5: if d [c urre nt no de ] = d [( v, i )] + 1 t he n shortest path to w vi a v ? 2 6: σ [c urre nt no de ] ← σ [c urre nt no de ] + σ [( v, i )]; 2 7: append ( v, i ) → P [c urre nt no de ]; 2 8: δ [ν ] ← 0 , ν ∈ V ; S return verticesin order of non-increasing distancefro m s 2 9: χ [( v, i )] ← 0 , (v, i ) ∈ V 3 0: χ [( v, i )] ← 1 , (v, ) ∈ V 3 1: w hil e S not e mpty d o 3 2: pop ( w, i ) ← S ; 3 3: if χ [( w, i )] = 1 t he n 3 4: f o r (v, j ) ∈ P [( w, i )] d o χ [( v, j )] ← 1 σ [ ( v, j )] 3 5: if i = t he n δ [( v, j )] ← δ [( v, j )] + σ [ ( w,i )] · ( 1 + δ [( w, i )]); σ [ ( v, j )] 3 6: els e δ [( v, j )] ← δ [( v, j )] + σ [ ( w,i )] · δ [( w, i )]; 3 7: if (w, i ) = s t he n b c [( w, i )] ← b c [( w, i )] + δ [( w, i )]

1 2 7 Rando m- Walk Betweenness Centrality

A co m mon criticis mfor bet weenness centralityisthatit does nottake non-shortest pathsinto account andis thereforeinappropriatein cases whereinfor mation spreadis governed by other rules[13]. As a result, variants of bet weenness centrality have been proposed such as bet weenness measures based on network flow[26], and rando m- walk betweenness centrality ( R W B C)[66, 15]. In so me sense, as suggested by New man, “R WBC and BC can be viewed as being on opposite ends of a spectru m of possibilities, one representinginfor mation thatis moving at rando m and has noidea of where itis going and the other knowing precisely whereitis going”. So me real- world situations mi mic these extre mes[66, 25], ho wever, others such as the s mall- world experi ment[49]fall so me wherein bet wee n. In a network where the flow processis coupled with a S O C constraint,itis therefore natural to also propose a variant of R WBC for such networks. If we consider an undirected connected graph, for any pair of nodes s, t , a rando m walk starting at s will eventually arrive at t wit h hi g h probability. Ho wever,in a net work where the flo wis coupled with a S O C constraint, andlike wise, a directed net work thatis not strongly connected, not every rando m walk starting at s has a positive probability of arriving at t. Withthisin mind,the proposed variant of R WB C only considers walks that arrive at the destination node. For exa mple,if a node does not have enough S OC to travel fro m s t o t via any walk, then the pair s -t does not contribute to centrality score. Consider R W B C proposedin[66]. Unlike the standard bet weenness centrality measurethat only considers shortest paths bet ween a pair of nodes, R W B C takes all pathsinto account while giving morei mportancetoshorter paths. R W B C of a node i is defined asthe n et nu mber ofti mes arando m walk passes through i. B y n et , authors meant thatif a walk passes through i andlater passes back throughitin the opposite direction, the two would cancel out and thereis no contribution to the bet weenness. R W B C was originally proposedfor undirected graphs. In this section, we first generalize R W B C to directed graphs. In a directed graph G = ( V , E ),for any pair of nodes s, t ∈ V ,itis not guaranteed that everyrando m walkfro m s will eventually arrive at t. We generalize R W B Cfor directed graphs to onlyinclude rando m walksfro m s t o t. Let G s, t = ( V s, t , E s, t ) be a subgraph of G such that every nodelies on a walkfro m s t o t. R WBCis adjustedfor G s, t as follo ws. Let A adjacency matrix with D the out-degree diagonal matrix, where D is defined as d e g (v ) if i = j [D ] : = + i i j 0 , ot her wise , w here d e g + (v i )is the out-degree of node v i . Define the transition matrix of G s, t as M : = D − 1 A ( 1 5) r For a walk starting at s , the probability that it is at j aft er r steps is given by [ M ]s j . The r − 1 − 1 probability that the walk continues further to an adjacent vertex i is [ M ]s j d j , where d j is the out-degree at j . Thus,the expected nu mber ofti mes a walkfro m s t o t uses the directed edge ( j, i ) − 1 − 1 is given by[( I − M t ) ]s j d j , whichis the s -j th entry of the matrix given by − 1 − 1 − 1 (I − M t ) D t = ( D t − A t ) , ( 1 6) w here D t a n d A t is the matrix derivedfro m deleting row and colu mn t. Addthe zero colu mn back − 1 t o ( D t − A t ) and call this matrix T . Let s be the vector given by   1 , if i = s s : = − 1 , if i = t i  0 , ot her wise

1 3 Let the vector f be defined as f : = s T T then, the ith entry of f , f i , represents the expected nu mber of walksfro m s t o t that pass through no de i. If D f is the diagonal matrix with f i at the ith diagonal position then the matrix

F : = D f A gives a matrix whose i-j value represents the expected nu mber of ti mes a rando m walkfro m s t o t uses edge ( i, j ). The net flow ofrando m walkthroughthe ith vertexisfor a given s -t pair is given b y 1 I ( s t ) = | F − F |. ( 1 7) i 2 i, j j,i ( i, j ) ∈ E s , t

The expressionin (17)is used to co mpute the centrality scores of a directed graph G s, t arisi n g fro m rando m walks starting at node s t o t. Note thatfor every node v i n G s, t , v must be at a finite distance fro m s a n d t. The centralityfor each nodein G is then given by the su m of theindividual scores for each source-target pair. Let Yˆ be the vector of centrality scores of G , the S O C -R WBCis given by the vector ˆ T Y · I κ ( 1 8) w here I κ is the block matrix definedin (2).

8 Co mputational Experi ments

In the preceding sections, mathe matical modelsfor the three proposed centrality measures are given. The question then arises, “ Ho w good are these centrality measures?” We tackle this questionfro m three different perspectives. First, us abilit y : how can we meaningfully use the proposed centrality measures. Second, robust ness : ho w robust are the centrality measures with respect to their para me- t ers. L astl y, n ovelty : ho w are the proposed centrality measures differentfro m their well-established predecessors. Experi mentsin this section are carried on the graph datasets givenin Table 1.

Table 1: Experi mental graph datasets: d mi n , d a v g , d m a x represents the mini mu m, average and max- i mu m degree.

Gra p h Nodes Edges d mi n d a v g d m a x Refere nce RouterNetwork 2114 6632 1 6 109 [76] Minnesota State Road 2642 3303 1 2 5 [22] Net work GnutellaNetwork 6301 2077 1 7 97 [75] Collaboration Network 5242 14496 0 5 81 [53]

8.1 Usability In Borgatti[13], the expected centralityis defined as a centrality score given by a closed-for m ex- pression, and realized ce ntrality as the actual centrality score observedin the context of a particular flo w process. Therefore, one can vie w a centrality measure as afor mula-based prediction of a flo w process through a node. Itis thereforei mportant to co mpare the predictions given by the closed- for m expression with the actualfrequency of tra ffic observed flo wing through a node across multiple

1 4 instances. For exa mple, in order to test whether betweenness centrality is a good prediction of observed tra ffic through a node, the expected bet weenness centralityis co mpared with the realized bet weenness centrality, where the actualfrequency of tra ffic through a given nodeis referred to as the realized bet weenness centrality while thefor mula-based centralityis referred to as the expected bet weenness centrality. In this section, we co mpare the expected centrality with the realized central- ity values for the proposed centrality measures. Given that realized centrality scores are achieved by runninglong si mulations, we show the usability of the proposed measures as way to e fficiently esti mate the outco me of these co mputationally expensive si mulations. In order to observe realized centrality values, si mulations for each of the three flo w process are developed. In general, centrality measures are pri marily used either as ranking algorith ms or as methods for identification of influential nodes. We therefore co mpare the expected and realized centrality values using Kendall’s Tau[44] rank correlation coe fficient. Kendall’s Tauis given by 2 τ : = s g n (y − y )( z − z ) , n (n − 1) i j i j i < j where, for each node i, we denote the node’s spreadinginfluence andits centrality measure by y i a n d z i , respectively. The s g n (y )is a piece wisefunction such that s g n (y ) = 1if y > 0, − 1 if y < 0 and 0if y = 0. The values of τ belong to the range[ − 1 , 1], wherelarger values of τ correspond to a higher correlation bet ween the expected and realized centralities.

8. 1. 1 S O C Katz Centrality In order to co mpute the realized centrality values with respect to S O C - Katz centrality, we turn to the susceptible-infected-recovered (SI R) spreading model (also called susceptible-infected-re moved model)[31]. Kle m m et. al[50] suggested that the eigenvector centrality can be usedfor esti mating a spreadinginfluence of the nodesin the SI R model, by[57] defining the dyna mical-sensitive ( DS) centrality and sho wing thatit more accuratelylocatesinfluential nodesin the SI R model. The DS centralityis very closely related to the Katz centrality. In the SI R model, a node can bein one offollowing states: (i) susceptible, nodes can beco me infected, (ii)infected, nodes areinfected and caninfect susceptible nodes, and (iii) recovered, nodes have recovered and developedi m munity, thus cannot beinfected again. In order to esti mate the spreadinginfluence of a node v ,initially, all nodes are susceptible and v isinfected. At each step, an infected node tries toinfectits susceptible neighbors and succeeds with probability α . Theinfected node enters the recovered state with probability µ . In this work, we set µ = 1,i.e., theinfection can be trans mitted only once. The process stopsif no ne winfections arefor med or after a fixed nu mber of steps. We generalize the SI R spreading process to acco m modate the S O C para meter. Define an edge ( i, j ) as active if node i i nfecte d j via edge ( i, j ). A stopping criteria for the SI R spreading processfor a fixed nu mber of steps κ can be vie wed asfollo ws: Let u b e t h e i niti all y infected node, then an infected node v cannot infect its susceptible neighbors if there exists a path oflength κ fro m u t o v consisting of only active edges ,i.e., theinfection dies out after κ st e ps. In order to generalize the SI R model to acco m modate a flow process based on S O C , we modify the above stopping criteria to: an infected node v cannot infect its susceptible neighbors if there exists a path oflength κ fro m eit her u , or non-susceptible w ∈ Ω t o v consisting of only active edges.

1 5 If the set Ω ⊂ V , is e mpty, then S O C - Katz centrality is equivalent to the DS centrality which is sho wnin[57] to be highly correlated to the nodes’ spreadinginfluence according to the SI R model. Experi ments are carried out to sho w that the above generalized SI R spreading processis highly correlatedtothe proposed S O C - Katz centrality. Forthis experi ment, we usethe net workrepresenting the Internet at the major routerlevel[76, 83] consisting of 2114 nodes and 6632 edges. The nodes and edges represent routers, and the connections bet ween the m, respectively. We set κ = 5, a n d α = 0 .03, and vary the size of the set Ω ⊂ V such that the ratio |Ω |/ |V | rangesfro m 0.1 to 0.9. For each value of |Ω |/ |V |, the set Ωis chosen at rando m, and the corresponding spreadinginfluenceis esti matedfor each node byrunningthe generalized SI R model 10 4 ti mes. Thisis repeated 30 ti mes. The box-plotin Figure 1 shows the correlation between the spreadinginfluence as a result of the generalized SI R model co mpared to S O C - Katz centrality. The results sho w Kendall Tau correlation valuesin the range (0 .9 4 5 , 0 .970) suggesting that the t wo processes are very highly correlated. τ K e n d all

|Ω |/ |V | ( %)

Figure 1: Co mparison of nodes’ spreading influence according to the generalized SI R model and S O C Katz centrality on the routers net work. Each boxplot represents 30 rando m choices of the set Ω with spreading probability α = 0 .0 3, a n d κ = 5

8. 1. 2 S O C -( Rando m- Walk) Bet weenness Centrality To de monstrate the S O C -R WBC and S O C -bet weenness centralities, we experi ment with t wo net- works, na mely, a co mputer network androad network. The co mputer networkis generatedfro mthe P2P net work, Gnutella[53, 75], and consists of 6301 nodes and 20777 edges. The road net work[22] represents Minnesota state roads and consists of 2642 nodes and 3303 edges. We si mulate tra ffic on both net works. The realized centralities are co mputed using the particle hopping. The particle hopping model is a method usedin vehicular flow theory[64]. In this model, a section of a roadis represented by

1 6 a node and a vehicle as a particle where each node can only be occupied by one particle at a given ti me. This modelisso meti mesreferredto as cellular auto mata and gives a mini mal modelfortra ffic flow behaviors[34]. The flow of packets through theinternet have also been modelled by cellular auto mata[56, 33]. In[32], thefraction of ti me steps that nodeis occupied by a particleis referred to as the occupation ratio . For the application to electric vehicles, the value κ represents the nu mber of steps the car can travel beforeits battery runs out of charge. For a message or vehicle being propagatedfro m node s t o t, wesi mulatethetra ffic onthe nodes when arouting algorith m propagatesthe message or vehicle in one of the t wo cases, (i) via a shortestfeasible walk, and (ii) a rando mfeasible walk. We add the condition that the routing algorith misinfor med and takes the current κ counter of the message or vehicle, and target t,into account before deciding which neighbor to directit to. In other words, if a message or vehicle cannot be successfully propagated toits destination due to the value of κ , then the message or vehicleis not propagated at all and therefore does not contribute to the tra ffic of the net work. Experi ments are carried on the Gnutella net work, where we set T T L = 4 and record the occu- pation ratio based on the corresponding routing algorith ms. The occupation ratiois then co mpared to the proposed centrality measures. The results for S O C -bet weenness centrality and S O C -R WBC are presentedin Figure 2 and 3 respectively. The results show Kendall Tau valuesin the range ( 0 .7 9 , 0 .82) for a ratio |Ω |/ |V | of 0.2 and (0 .8 6 , 0 .88) for a ratio |Ω |/ |V | of 0.9 for S O C -bet weenness centrality. Si milar correlation scores and trends for S O C - R WBC are observed suggesting a high correlation bet ween the expected centralities and realized centrality measures. τ K e n d all

|Ω |/ |V | ( %)

Figure 2: Correlation scores for expected versus realized centrality for S O C -bet weenness centrality for the Gnutella net work. Each boxplot represents 30 rando m choices of the set Ω, with κ = 4

For experi ments on the Minnesota state road network, we set κ = 20. We choose a relatively larger value of κ for the road network experi ments because we assu me that electric vehicles can

1 7 τ K e n d all

|Ω |/ |V | ( %)

Figure 3: Correlation scoresfor expected versus realized centralityfor S O C R W B Cfor the Gnutella net work. Each boxplot represents 30 rando m choices of the set Ω, with κ = 4 travel a relativelylong distanceifit startsfully charged. Asin the Gnutella experi ments, we record the occupation ratio. The results for S O C -bet weenness centrality are presented in Figure 4. The results show Kendall Tau valuesin the range (0 .7 9 , 0 .86) for a ratio |Ω |/ |V | = 0.2 and (0 .8 3 , 0 .8 7) for a ratio |Ω |/ |V | = 0.9 for S O C -bet weenness centrality.

8.2 Robustness and Novelty The para meter κ is application dependent, so it is i mportant to understand how the proposed centrality measures behave for different values of κ . Fro m the mathe matical expressions of our novel centrality measures, it is clear that for a large enough κ , the proposed centrality measures would beco meidentical to their baseline centrality measures asin this case, theli mitation of S O C - dependent distanceis gradually vanishing. In this section, we carry out experi ments to understand ho w the proposed measures co mpare to their baseline measures while varying the para meter κ . The goal ofthe experi mentsisto quantify whatis not captured when usingthe well-established centrality measuresfor given values of κ . Thus, de monstrating the robustness of the results and novelty of the meas ures. The first set of experi ments is carried out on toy graphs to illustrate the difference in central nodes when using the proposed centrality measures versus their baseline measures. The first toy graphis a graphfor med by connectingtwo 5 × 5 grid graphs by a path oflength 5. Thesecondis a 1 0 × 10 grid graph. The second set of experi ments uses real- world datasets. In these experi ments, wefocus on the Minnesota state road network and a collaboration network constructed using the scientific collaboration data[53], consisting of 5242 nodes and 14496 edges. The experi ments ontoy graphs are usedto visuallyillustratetothe readerthe difference bet ween

1 8 τ K e n d all

|Ω |/ |V | ( %)

Figure 4: Correlation scores for expected versus realized centrality for S O C -bet weenness centrality forthe Minnesota stateroad net work. Each boxplotrepresents 30rando m choices oftheset Ω, with κ = 2 0 the proposed centrality measures versus their baseline measures. Thus, providing an intuition on how the measures work. The difference between S O C -B C and B Cfors mall values of κ is ill ustrate d in Figure 5. In this experi ment, we set κ = 4. We use a color spectru mfro mredto yellow, showing the most central to theleast central nodes respectively. The graphin Figure 5 (a) represents the standard B C. As expected, the nodes along the bridge are the most central nodes. The graphsin (b)-(f)show differentscenarios wherethe nodesin Ω are marked with a blue-edge dia mond-shaped node. As we can seein the Figures (d) and (f), depending on the value of κ and nodesin Ω, the centrality scores can be significantly differentfro m the standard B C, where the most central nodes based on BC, are now a mongtheleast central nodes based on S O C - B C. Differences bet ween S O C - Katz and Katz centrality areillustratedin Figure 6. Inthis experi ment, we set κ = 4. The graphin Figure 6 (a) represents the standard Katz centrality. As expected, the nodes towards the center of the grid are the mosti mportant nodes according to this model. The graphs in (b) - (f) show different scenarios where the nodes in Ω are marked with a blue-edge dia mond-shaped node. As we can seein the Figure (e), depending on the value of κ , even with a relativelylarge ration of |Ω |/ |V |, (0.5for (e)), the centrality scores can still be significantly different than the scoresfro m their baseline models. A more co mprehensive study co mparing S O C - Katz with Katz on the grid graphis shownin Figure 7. For each value of κ , wit h 2 ≤ κ ≤ 16, we run 30 experi ments. Each experi ment consists of choosing nodes at rando m to bein the set Ω. The boxplotsin blue represent experi ments with |Ω |/ |V | = 0 .1, w hil e |Ω |/ |V | = 0 .2 are representedin red. We choose the values 0.1 and 0.2 for the ratio |Ω |/ |V | becausein most applications the set Ω will be considerably s maller than V . For exa mplein social net works, the percentage oflurkersin an online co m munityis esti mated to range

1 9 (a) Standard B C ( b) |Ω | = 0 , κ = 4

( c) |Ω | = 1 , κ = 4 ( d) |Ω | = 2 , κ = 4

( e) |Ω | = 3 , κ = 4 (f ) |Ω | = 8 , κ = 4

Figure 5: Co mparison of BC and S OC-BC. The blue dia mond-shaped nodes represent nodesin Ω. In (a), the nodes connecting the two co mponents of the graph, which can be vie wed as a ”bridge” bet ween t wo co m munities, have high centrality scores because they are essentialfor flo wfro m one co mponent to another. Whereasin (b)- (f), since the flowisli mited tojust 4 steps, the majoring of the flowin the graph would be within the two2 0 co mponents thus the bridge nodes are nolonger as i mportant. fro m 50 to 90 percent of the total me mbership[41, 61, 81]. As expected, the results show that as κ increases, the correlation bet ween S O C - Katz and Katz rankingincreases. Itisinteresting to observe that since Katz centralityis based oninfinite-length rando m walks e manatingfro m a node,itis not clear what value of κ would make S O C - Katzidentical to the standard Katz centrality for a given graph. Ho wever, thisis not the case with bet weenness centrality whichis based on shortest paths. For a given graph, setting κ to thelongest shortest path would make S O C - B Cidentical to B C. The second set of experi mentsin this sectionis carried out on real- world graphs, the Minnesota state road network, and the collaboration network. In the Minnesota state road network, first, we co mpare S O C -B C with B C, while varying the value of κ , second, we co mpare S O C -R WBC and R W B Cfor a given source-target pair. The co mparison of S O C -BC and BCis shownin Figure 8. The para meter κ is variedfro m 2to 16. For each value of κ , we perfor m 30 experi ments where each experi ment consists of sa mpling a set of nodes Ω ⊂ V for a fixed ratio |Ω |/ |V |. We fixtheratioto 0.1 and 0.2, represented by blue and red boxplots respectively. Given that the Minnesota state road net work has an averageshortest pathlength of approxi mately 35.4,theresultssho wthatfor values of κ , s maller than the average shortest pathlength, we can get significant differences bet ween S O C - B C and B C ranking. Thus, we find where the standard B C may potentiallyfailinidentifying central nodes. We visually de monstrate a si milar result for S O C -R WBC and R WBCin Figure 9. Inthis experi ment, we pick a pair of nodes representing a source and target and then co mpute the R WB C scores contributed by the t wo nodes referring to the m as s -t- R WBC. Giventhat R WBCisidentical to the current flo w bet weenness centrality[15], one can think of this experi ment asinjecting a unit of currentfro m the source flo wing to target and measuring thefraction of current flo wing through each node. The graph on Figure 9 (a) represents the s -t- R W B C scores for the source-target pair represented by nodesin black. The s -t- R W B C valuesidentical to zero are represented with nodes with negligible size. As expected, the results sho w higher s -t- R WB C valuesfor nodes close to the source andtarget. We perfor m a si milar experi mentfor S O C -R WBC with κ = 20, while the distance fro m source to targetislarger than 20. Thisi mplies that every rando m walkfro m source to target must pass through atleast one nodein Ω. The resultsin Figure 9 (b) show the s -t-S O C -R WBC valuesfor the given source-target pair. In the case of S O C - R W B C,the higher central nodes are now the nodes close to the nodesin Ω. This exa mple de monstrates how S O C -R WBC can be usedto identify congested nodesin a road net work thatis equipped with wireless charginglanes. A co mparison of S O C - Katz versus standard Katz with variations of the para meter κ on the Collaboration net workis sho wnin Figure 10. In general,for different values of κ , the correlation of S O C - Katz and Katzis high, generally above 0.8. Ho wever, once we plotthe different ranking we see significant differences with the node rankings of the t wo measures. Thus, Kendall Tau correlation does not give a co mplete picturefor this net work. In particular, a node thatis ranked highly with thestandard Katz centrality can have asignificantlylessrankin aranking with S O C - Katz. Ho wever, conversely, highly ranked nodes with S O C - Katz generally tend to also be highly ranked with respect to the standard Katz. With respect to the application to posters andlurkersin social net works, this follo ws theintuition that a user with alarge nu mber of neighbors(friends) can still be non-influential if the user together with all his/her neighbors (friends) arelurkers. On the other hand, a highly influential node would generally have many neighbors (friends).

2 1 (a) Standard Katz ( b) |Ω |/ |V | = 0 .1 , κ = 4

( c) |Ω |/ |V | = 0 .2 , κ = 4 ( d) |Ω |/ |V | = 0 .3 , κ = 4

( e) |Ω |/ |V | = 0 .4 , κ = 4 (f ) |Ω |/ |V | = 0 .5 , κ = 4

Figure 6: Co mparison of Katz and S O C- Katz; The blue dia mond-shaped nodes represent nodesin Ω. In(a), thestandard Katz centralitysho wsthatthe nodes more centrallylocated onthe grid have a higheri mportance. For exa mple, with respect toinfor mation spreadingin a net work, thisi mplies that these nodes are the mostinfluentialininfor mation spreading. However as observedfro m (b) to (f)if theinfor mation spread has fixed travel distance,2 2 thenjust the connectivity structure of the net workis not enough to conclude about the mostinfluential nodes. τ K e n d all

κ

Figure 7: S O C- Katz Vs. Katzfor 10 × 10 Grid graph. τ K e n d all

κ

Figure 8: S O C- B C Vs BCfor Minnesota State Road Network

2 3 0. 2 7

0. 2 4

0. 2 1

0. 1 8

0. 1 5

0. 1 2

0. 0 9

0. 0 6

0. 0 3

0. 0 0

( a)

0. 2 7

0. 2 4

0. 2 1

0. 1 8

0. 1 5

0. 1 2

0. 0 9

0. 0 6

0. 0 3

0. 0 0

( b)

Figure 9: Co mparison of R WBC with S OC-R WBCfor asingle s -t pair over the Minnesota state road net work. The top-left and center black nodes represent the source and target nodes respectively. Nodes with s -t-centrality scores equal to 0 have have negligible node sizes. (a) R W B Cfor a given s -t pair. (b) S OC- R WBCfor a given s -t pair. Nodes with a triangular marker represent nodesin Ω.

2 4 4500 4500

4000 4000

3500 3500

3000 3000

2500 2500

2000 2000

1500 1500 S O C- Katz S O C- Katz

1000 1000 κ = 2 κ = 3 5 0 0 5 0 0

0 0 0 500 1000 1500 2000 2500 3000 3500 4000 4500 0 500 1000 1500 2000 2500 3000 3500 4000 4500 K atz K atz

4500 4500

4000 4000

3500 3500

3000 3000

2500 2500

2000 2000

1500 1500 S O C- Katz S O C- Katz

1000 1000 κ = 4 κ = 8 5 0 0 5 0 0

0 0 0 500 1000 1500 2000 2500 3000 3500 4000 4500 0 500 1000 1500 2000 2500 3000 3500 4000 4500 K atz K atz

Figure 10: Co mparison of noderankings based on S O C- Katz and Katzfor different values of κ wit h |Ω |/ |V | = 0 .1. Evenif the above ranking give high Kendall Tau correlation, we notice that with the introduction of a ranking based S O C- Katz, highly ranked Katz nodes can significantlyloose their ranking, ho wever,lessi mportant nodes do not significantlyincrease with rank

2 5 9 Conclusion

An esti mation of node spreadinginfluencein a net workis ani mportant step to wards understanding and controlling the spreading dyna mics over the net work. Centrality measures are traditionally used toidentifyinfluential nodesin a net work. In this work, we extend the well-kno wn measures of Katz, bet weenness, and rando m- walk bet weenness centralities to models that acco m modate a resource, necessaryforthespread, being consu med alongthe way. We present algorith msto co mputethe pro- posed centrality measures and carry out experi ments onreal- world net works. Lastly, we de monstrate si mulation models that describe the flow process and show that they are highly correlated to the proposed centralities. In order to ans wer the question, “ Ho w good are these centrality measures?”, we analyze the centrality measures fro m three different perspectives, na mely, usability, robustness, and novelty. Fro m the usability perspective, a mong other experi ments, we de monstrate how the centrality measures can be used toidentify congested nodesin co mputer and road net works. Fro m the robustness and novelty perspective,in the application of posters andlurkersin a social net work, we sho wed that the proposed extension of Katz centrality follo ws theintuition that a user with a large nu mber of neighbors (friends) can still be non-influentialif the user together with all his/her neighbors (connections) arelurkers. On the other hand, a highlyinfluential node would generally have many neighbors (connections). The proposed measures takeinto account a spreading process that depends on a resource, such that the spreading process would bei mpossible without. Our nu merical experi ments de monstrate that the proposed measures differ significantly fro m the original measures when the resource is li mited. Onthe other hand,they beco meidenticaltothe original measures asthe quantity ofresource available tends toinfinity. As a result, the proposed measures give a ne w tool and perspective to different application do mains. For exa mple, in the application of a road network equipped with wireless charging lanes, an opti mal place ment of these lanes with respect to tra ffic distribution, could be one where the distribution of centrality scores of all nodesis takeninto account. In another do main, high centrality nodes can be consideredfor targeted attack ori m munization strategies. For a given application, the choice of which centrality measure to use to dra w a conclusion about the networkis extre melyi mportant as using a wrong measure canlead to meaningless results. The measures of S O C -R WBC and S O C - Katz are both based on rando m walks on the network. Itis, ho wever,i mportant to note that,just as the standard measures, the rando m walks associated with S O C - R WB C have a fixedsource andtarget node, whiletherando m walks associated with S O C - Katz only have a fixed source node. Thus S O C - Katz is more suitable for applications where the flow process does not have a specified destination,for exa mple, a disease spread. The S O C -R WBC and subsequently S O C - B C aresuitablefor applications wherethe flo w process has aspecified destination, for exa mple vehicle flo w. There are nu merous future research directions associated with the resource consu mption based centralities. For exa mple, we propose to explore other funda mental net work properties such as connectedness, clustering, and net work robustnessin the context of consu mable resource net works. In this work, there exists a set of nodes that facilitate flo w in the net work, conversely, proble ms in net workinterdiction[96] deal with theidentification of nodes that hinder flo w. Aninteresting directionis to explore the relationship bet ween these t wo proble msin more detail. Another highly relevant direction for the future work is to consider a distribution of resource consu mption based centralitiesin realistic net work generation[30, 84]. Thisis particularlyi mportantfor the si mulation and verification studies. To the best of our kno wledge, no generating model currently considers a distribution of resource consu mption based centralities. Also, the resource consu mption models can be generalizedfor clusters and co m munities. Moreover, one of the natural extensions of this workis introducing resource consu mption ele ment to node and edge si milarity measures.

2 6 Ackno wledge ments

We thank the anony mous revie wers whose co m ments and suggestions helped toi mprove and clarify this manuscript. This research is supported by the National Science Foundation under A ward #1647361. Any opinions, findings, conclusions or reco m mendations expressedin this material are those of the authors and do not necessarily reflect the vie ws of the National Science Foundation.

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