Complex Networks

Overview of Overview of Complex Networks Outline Complex Networks Complex Networks Principles of Complex Systems Basic definitions Basic definitions Basic definitions Examples of Examples of Course CSYS/MATH 300, Fall, 2009 Complex Networks Complex Networks Properties of Examples of Complex Networks Properties of Complex Networks Complex Networks

Nutshell Nutshell Prof. Peter Dodds Properties of Complex Networks Basic models of Basic models of complex networks complex networks Dept. of Mathematics & Statistics Generalized random Generalized random networks Nutshell networks Center for Complex Systems :: Vermont Advanced Computing Center Scale-free networks Scale-free networks University of Vermont Small-world networks Small-world networks Generalized affiliation Generalized affiliation networks Basic models of complex networks networks References Generalized random networks References Scale-free networks Small-world networks Generalized affiliation networks

References Frame 1/122 Frame 2/122 Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License.

Overview of Overview of net•work |ˈnetˌwərk| Complex Networks Thesaurus deliciousness: Complex Networks noun 1 an arrangement of intersecting horizontal and vertical lines. Basic definitions Basic definitions • a complex system of roads, railroads, or other transportation routes : a network of railroads. Examples of Examples of Complex Networks Complex Networks 2 a group or system of interconnected people or things : a trade network. • Properties of Properties of a group of people who exchange information, contacts, and Complex Networks Complex Networks experience for professional or social purposes : a support network. Nutshell network Nutshell • a group of broadcasting stations that connect for the simultaneous noun broadcast of a program : the introduction of a second TV network | [as adj. ] Basic models of Basic models of complex networks WEB complex networks network television. Generalized random 1 a network of arteries , lattice, net, matrix, mesh, Generalized random • a number of interconnected computers, machines, or operations : networks networks Scale-free networks crisscross, grid, reticulum, reticulation; Anatomy plexus. Scale-free networks specialized computers that manage multiple outside connections to a network | a Small-world networks Small-world networks Generalized affiliation 2 a network of lanes MAZE, labyrinth, warren, tangle. Generalized affiliation local cellular phone network. networks networks • a system of connected electrical conductors. 3 a network of friends SYSTEM, complex, nexus, web, References References webwork. verb [ trans. ] connect as or operate with a network : the stock exchanges have proven to be resourceful in networking these deals. • link (machines, esp. computers) to operate interactively : [as adj. ] ( networked) networked workstations. • [ intrans. ] [often as n. ] ( networking) interact with other people to exchange information and develop contacts, esp. to further one's career : the skills of networking, bargaining, and negotiation. Frame 3/122 Frame 4/122

DERIVATIVES net•work•a•ble adjective Overview of Overview of Ancestry: Complex Networks Ancestry: Complex Networks

Basic deﬁnitions Basic deﬁnitions

Examples of Examples of From Keith Briggs’s excellent Complex Networks First known use: Geneva Bible, 1560 Complex Networks Properties of Properties of etymological investigation: () Complex Networks ‘And thou shalt make unto it a grate like networke of Complex Networks Nutshell brass (Exodus xxvii 4).’ Nutshell Basic models of Basic models of complex networks complex networks Generalized random From the OED via Briggs: Generalized random networks networks Scale-free networks Scale-free networks Small-world networks I 1658–: reticulate structures in animals Small-world networks Generalized afﬁliation Generalized afﬁliation I Opus reticulatum: networks networks I 1839–: rivers and canals References References I A Latin origin? I 1869–: railways

I 1883–: distribution network of electrical cables

[http://serialconsign.com/2007/11/we-put-net-network] I 1914–: wireless broadcasting networks

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Keith Briggs: : Etymology of `network' 6/6/09 9:45 PM gescylde hálgan nette (with a net-work of clouds), Cd. Th. 182, Ii; Exod. 74. [Goth, nati: O. Sax. netti, (fisk-)net: O. Frs. nette: Icel. net; gen. pl. netja : O. H. Ger. nezzi rete.] v. æl-, boge-, breóst-, deór-, drag-, feng-, fisc-, fleóh-, here-, bring-, inwit-, mycg-, searo-, wæl-nett, and next word. Overview of Overview of Ancestry: Complex Networks Key Observation: Complex Networks nette, an; f. The net-like caul :-- Nette (under the heading de membris hominum) disceptum i. reticulum (cf. hoc reticulum, pinguedo circa jecur [fat around liver], 704, 7), Wülck. Gl. 293,6. Nettae oligia, 35, 34. Nytte obligia [binding, bandage], Wrt. Voc. i. 45, 18. Nette, ii. 63, 39 : disceptum, 26,19. [Icel. netja the caul: cf. O. H. Ger. nezzi adeps [fat] intestini; pl. intestina.] v. neta. Basic definitions I Many complex systems Basic definitions Net has many cognatesNet in Germanicand languages and Work a probable one in are Latin. The venerableGermanic words are old words: of neuter gender (except for nót which is feminine); the Latin word is feminine. Note that the normal Latin word for a net is rete, although there may have also been a Vulgar Latin word *tragina for a Examples of can be viewed as complex networks Examples of dragnet. (King Ælfréd, 888) Complex Networks Complex Networks language word I ‘Net’ first used to mean spider web . of physical or abstract interactions. Gothic nati Properties of Properties of Old net(t), ‘Work’ appear to have long meant purposeful action. Complex Networks Complex Networks English netti I I Opens door to mathematical and numerical analysis. Middle net Nutshell Nutshell English Keith Briggs: : Etymology of `network' 6/6/09 9:45 PM Modern net Dominant approach of last decade of a English Basic models of I Basic models of Modern net complex networks complex networks Dutch theoretical-physics/stat-mechish flavor. Generalized random Generalized random Old High nezzi German networks networks Middle Scale-free networks I Mindboggling amount of work published on complex Scale-free networks High netze Small-world networks Small-world networks German Modern Generalized affiliation networks since 1998... Generalized affiliation High Netz networks networks German Old nette References References Frisian I ... largely due to your typical theoretical physicist: Old netti Saxon net Old (netja Norse v.), nót sources Modern I Piranha physicus net Icelandic Bosworth, Joseph: (Northcote Toller, T. ed.) An Anglo-Saxon Dictionary. Dictionary, Supplement and Modern I ‘Network’ = something builtAddenda based Oxford University on Press, 1st the ed. n.d., ideareprinted 1983, of 1966, 1972. [Abbreviations: Wrt. Voc. i: nät Swedish T. Wright, A volume of vocabularies, 1857. Quoted by page and number of gloss. Wrt. Voc. ii: T. Wright, I A second volume of vocabularies, 1873. Quoted by page and line. Ælfc. Gr: Ælfric's Grammar. Ps. Spl.: Hunt in packs. Latin nassa natural, flexible lattice or web.Psalterium Davidis Latino-Saxonicum vetus. Lk. Skt.: The gospel according to St. Luke. Mt. Kmbl.: The gospel according to St. Matthew in Anglo-Saxon and Northumbrian versions. ed. J. M. Kemble. Jn. Skt.: The Gothic word occurs exactly seven times in the bible. It takes the form natja in the dative. Because The gospel according to St. John. Som.: Dictionarium Saxonico-Latino-Anglicum, by E. Somner, Oxford Gothic is the earliest record of Germanic (from about 350), it is worth looking at all these occurrences: 1659. Coll. Monast. Th.: Colloquium ad pueros linguae Latinae locutione exercendos ad Ælfrico I Feast on new and interesting ideas I c.f., ironwork, stonework, fretwork.compilatum. Homl. Th.: The homilies of Ælfric. Wülck. Gl.: Anglo-Saxon and Old English vocabularies, by Thomas Wright, edited by R. P. Wülcker, London 1884. Cd. Th.: Cædmon's metrical paraphrase, by B. Thorpe, London 1832.] Frame 7/122 (see chaos, cellular automata, ...) Frame 8/122 http://keithbriggs.info/network.html Page 2 of 4 Stamm, Friedrich Ludwig: Friedrich Ludwig Stamm's Ulfilas oder die uns erhaltenen Denkmäler der gotischen Sprache: Text, Wörterbuch und Grammatik Ungekürzte Neuauflage, Nachdruck der Ausgabe aus dem Jahre 1872 [Essen]: Magnus-Verlag, [1984]. ISBN: 3884001655

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http://keithbriggs.info/network.html Page 4 of 4 Overview of Overview of Popularity (according to ISI) Complex Networks Popularity according to books: Complex Networks

Basic definitions Basic definitions “Collective dynamics of ‘small-world’ networks” [28] Examples of Examples of Complex Networks Complex Networks Watts and Strogatz Properties of Properties of I Complex Networks The Tipping Point: How Little Things can Complex Networks Nature, 1998 Nutshell make a Big Difference—Malcolm Gladwell [12] Nutshell Basic models of Basic models of ≈ (as of June 5, 2009) I 3752 citations complex networks complex networks Generalized random Generalized random I Over 1100 citations in 2008 alone. networks networks Scale-free networks Scale-free networks Small-world networks Small-world networks Generalized affiliation Generalized affiliation “Emergence of scaling in random networks” [3] networks networks References References I Barabási and Albert Nexus: Small Worlds and the Groundbreaking Science, 1999 Science of Networks—Mark Buchanan

I ≈ 3860 citations (as of June 5, 2009)

I Over 1100 citations in 2008 alone.

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Overview of Overview of Popularity according to books: Complex Networks Numerous others: Complex Networks

Complex Social Networks—F. Vega-Redondo [25] Basic definitions I Basic definitions Examples of I Fractal River Basins: Chance and Self-Organization—I. Examples of Complex Networks Complex Networks Rodríguez-Iturbe and A. Rinaldo [20] Properties of Properties of Linked: How Everything Is Connected to Complex Networks Complex Networks I Random Graph Dynamics—R. Durette Everything Else and What It Nutshell Nutshell Means—Albert-Laszlo Barabási Basic models of I Scale-Free Networks—Guido Caldarelli Basic models of complex networks complex networks Generalized random Generalized random networks I Evolution and Structure of the Internet: A Statistical networks Scale-free networks Physics Approach—Romu Pastor-Satorras and Scale-free networks Small-world networks Small-world networks Generalized affiliation Alessandro Vespignani Generalized affiliation networks networks References I Complex Graphs and Networks—Fan Chung References Six Degrees: The Science of a Connected Age—Duncan Watts [26] I Social Network Analysis—Stanley Wasserman and Kathleen Faust

I Handbook of Graphs and Networks—Eds: Stefan Bornholdt and H. G. Schuster [6] Evolution of Networks—S. N. Dorogovtsev and J. F. F. Frame 11/122 I Frame 12/122 Mendes [11] Overview of Overview of More observations Complex Networks More observations Complex Networks

Basic deﬁnitions Basic deﬁnitions

Examples of Examples of Complex Networks I Web-scale data sets can be overly exciting. Complex Networks I But surely networks aren’t new... Properties of Properties of Complex Networks Complex Networks I Graph theory is well established... Witness: Nutshell Nutshell I Study of social networks started in the 1930’s... Basic models of I The End of Theory: The Data Deluge Makes the Basic models of complex networks complex networks I So why all this ‘new’ research on networks? Generalized random Scientific Theory Obsolete (Anderson, Wired) ( ) Generalized random networks networks Scale-free networks Scale-free networks I Answer: Oodles of Easily Accessible Data. Small-world networks I “The Unreasonable Effectiveness of Data,” Small-world networks Generalized affiliation [13] Generalized affiliation networks Halevy et al. . networks I We can now inform (alas) our theories with a much more measurable reality.∗ References References

I A worthy goal: establish mechanistic explanations. But: ∗If this is upsetting, maybe string theory is for you... I For scientists, description is only part of the battle. I We still need to understand.

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Overview of Overview of Super Basic deﬁnitions Complex Networks Super Basic deﬁnitions Complex Networks

Basic deﬁnitions Basic deﬁnitions

Examples of Examples of Nodes = A collection of entities which have Complex Networks Node degree = Number of links per node Complex Networks Properties of Properties of properties that are somehow related to each other Complex Networks Complex Networks I Notation: Node i’s degree = k . Nutshell i Nutshell I e.g., people, forks in rivers, proteins, webpages, Basic models of I ki = 0,1,2,. . . . Basic models of organisms,... complex networks complex networks Generalized random Notation: the average degree of a network = hki Generalized random networks I networks Scale-free networks (and sometimes z) Scale-free networks Small-world networks Small-world networks Links = Connections between nodes Generalized afﬁliation Generalized afﬁliation networks I Connection between number of edges m and networks References References I Links may be directed or undirected. average degree: 2m Links may be binary or weighted. hki = . I N

Other spifﬁng words: vertices and edges. I Defn: Ni = the set of i’s ki neighbors

Frame 15/122 Frame 16/122 Overview of Overview of Super Basic deﬁnitions Complex Networks Examples Complex Networks

Basic deﬁnitions Basic deﬁnitions

Examples of Examples of Adjacency matrix: Complex Networks Complex Networks Properties of Properties of I We represent a directed network by a matrix A with Complex Networks So what passes for a complex network? Complex Networks Nutshell Nutshell link weight aij for nodes i and j in entry (i, j). Basic models of I Complex networks are large (in node number) Basic models of I e.g., complex networks complex networks   Generalized random I Complex networks are sparse (low edge to node Generalized random 0 1 1 1 0 networks networks Scale-free networks ratio) Scale-free networks  0 0 1 0 1  Small-world networks Small-world networks   Generalized afﬁliation Generalized afﬁliation A =  1 0 0 0 0  networks I Complex networks are usually dynamic and evolving networks   References References  0 1 0 0 1  I Complex networks can be social, economic, natural, 0 1 0 1 0 informational, abstract, ...

I (n.b., for numerical work, we always use sparse matrices.)

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Overview of Overview of Examples Complex Networks Examples Complex Networks

Physical networks Basic definitions Interaction networks Basic definitions Examples of Examples of Complex Networks I The Blogosphere Complex Networks I River networks I The Internet Properties of Properties of Complex Networks I Biochemical Complex Networks I Neural networks I Road networks Nutshell networks Nutshell I Trees and leaves Power grids Basic models of Basic models of I complex networks I Gene-protein complex networks I Blood networks Generalized random Generalized random networks networks networks Scale-free networks Scale-free networks Small-world networks I Food webs: who Small-world networks Generalized affiliation Generalized affiliation networks eats whom networks References References I The World Wide Web (?)

I Airline networks

I Call networks I Distribution (branching) versus redistribution (AT&T) datamining.typepad.com () (cyclical) Frame 19/122 I The Media Frame 20/122 Overview of Overview of Examples Complex Networks Examples Complex Networks

Interaction networks: Basic definitions Basic definitions Examples of Examples of social networks Complex Networks Complex Networks Properties of Properties of I Snogging Complex Networks Complex Networks Nutshell Nutshell I Friendships Basic models of Basic models of Acquaintances complex networks complex networks I Generalized random Generalized random networks networks I Boards and Scale-free networks Scale-free networks Small-world networks Small-world networks directors Generalized affiliation Generalized affiliation networks networks I Organizations References References I facebook () twitter (), (Bearman et al., 2004)

I ‘Remotely sensed’ by: email activity, instant messaging, phone logs (*cough*).

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Maps of Science

Results and Discussion Convergence. The FR algorithm can converge on different visualizations of the same network data. We do not claim Fig. 5 is According to the above mentioned methodology we constructed the only or best possible visualization. It was selected because it a map of science that visualizes the relationships between journals represents a particularly clear and uncluttered visualization of the according to user clickstreams. We first discuss the visual structure connections between journals in M’, and most importantly, its del.icio.us/url/830cfc21af4d02c392aa6ad04e93b93e 11/14/2006 09:04 PM of the map, and then attempt to validate the structural features of main structural features were stable across many different its underlying clickstream model by comparing the latter to journal iterations of the FR algorithm. centrality rankings and an alternative model of journal relations Connections. The journal connections shown in the map are derived from classification data. given by M’, not the FR algorithm. They are thus not artifacts of Overview of the visualization. Overview of A clickstream map of science Clustering. The FR algorithm will pull together small-scale Examples Complex Networks ClickworthyAny interpretation of the visual structure of Science: the map in Fig. 5 clusters of journals that are strongly connected in M’. The Complex Networks will be governed by the following considerations: appearance of small-scale journal clusters is thus directly related to Relational networks popular | recent del.icio.us / url Basic definitions Basic definitions login | register | help I Consumer purchases Examples of Examples of (Wal-Mart: ≈ 1 petabyte = 1015 bytes) Complex Networks Complex Networks Properties of Properties of Complex Networks Complex Networks » del.icio.us history for http://en.wikipedia.org/wiki/Main_PageI Thesauri: Networks of words generateddel.icio.us by meanings search Nutshell Nutshell I Knowledge/Databases/Ideas check url Basic models of Basic models of Metadata—Tagging: del.icio.us ( ) flickr ( ) complex networks complex networks I Generalized random Generalized random networks networks Scale-free networks Scale-free networks Small-world networks Small-world networks Main Page - Wikipedia, the free common tags cloud | list Generalized affiliation Generalized affiliation networks networks encyclopedia References References http://en.wikipedia.org/wiki/Main_Page community daily dictionary education encyclopedia this url has been saved by 16283 people. english free imported info information internet knowledge save this to your bookmarks » learning news reference research resource resources search tools useful web web2.0 wiki user notes wikipedia Figure 5. Map of science derived[5] from clickstream data. Circles represent individual journals. The lines that connect journals are the edges of theBollen clickstream model et in M’.al. Colors correspond to the AAT classification of the journal. Labels have been assigned to local clusters of journals that Nov ‘06 Frame 23/122 correspond to particular scientific disciplines. Frame 24/122 doi:10.1371/journal.pone.0004803.g005 all the knoweldge PLoS ONE | www.plosone.org 5 March 2009 | Volume 4 | Issue 3 | e4803 - Penfoldio related items - show ! facts - mmayes posting history I would not survive without Wikipedia, it knows » first posted by weblook to util everything, and I want to learn everything. - RobRemi Nov ‘06

The internet encyclopedia by faithfulsprig to reference - limannka by Penfoldio to wikipedia The mother of all wikis by vishinator to wikipedia - Username314 by mmayes to research Wikipedia - lancepickens by markcameron1 to system:unfiled by hirewad to system:unfiled ALWAYS useful! - AmritS by yeager.eng to firefox:toolbar Free Online Encyclopedia, User contributed articles by bandaids to reference - Maluvia by ingee to system:unfiled Welcome to Wikipedia, the free encyclopedia that by hdnev6 to system:unfiled anyone can edit. by hopabonk to encyclopedia - sanders_d by TianCaiWuDi to system:unfiled The Free Encyclopedia by k9entertainment to system:unfiled - puffseeker by llshoemaker to wikipedia Very good encycolopedia by tvn to imported kennis - slams by papaluukas to system:unfiled Knowledge by ureshi to system:unfiled - neilbriody by malph to reference encyclopedia Miscellaneous Info - sahing by salar15 to system:unfiled by RobRemi to wikipedia wiki knowledge encyclopedia free online encyclopedia http://del.icio.us/url/830cfc21af4d02c392aa6ad04e93b93e?settagview=cloud Page 1 of 65 Overview of Overview of A notable feature of large-scale networks: Complex Networks Properties Complex Networks

I Graphical renderings are often just a big mess. Basic deﬁnitions Basic deﬁnitions

Examples of Examples of Complex Networks Some key features of real complex networks: Complex Networks Properties of Properties of Complex Networks Complex Networks I Degree I Concurrency ⇐ Typical hairball Nutshell Nutshell distribution Hierarchical Basic models of I Basic models of I number of nodes N = 500 complex networks I Assortativity scaling complex networks Generalized random Generalized random I number of edges m = 1000 networks networks Scale-free networks I Homophily I Network distances Scale-free networks Small-world networks Small-world networks I average degree hki = 4 Generalized afﬁliation Clustering Generalized afﬁliation networks I I Centrality networks

References I Motifs I Efﬁciency References I And even when renderings somehow look good: I Modularity I Robustness “That is a very graphic analogy which aids understanding wonderfully while being, strictly I Coevolution of network structure speaking, wrong in every possible way” and processes on networks. said Ponder [Stibbons] —Making Money, T. Pratchett.

I We need to extract digestible, meaningful aspects. Frame 25/122 Frame 26/122

Overview of Overview of Properties Complex Networks Properties Complex Networks

Basic deﬁnitions Basic deﬁnitions

Examples of Examples of Complex Networks 2. Assortativity/3. Homophily: Complex Networks

1. Degree distribution Pk Properties of Properties of Complex Networks I Social networks: Homophily () = birds of a feather Complex Networks I Pk is the probability that a randomly selected node Nutshell Nutshell I e.g., degree is standard property for sorting: has degree k Basic models of Basic models of complex networks measure degree-degree correlations. complex networks Generalized random Generalized random I Big deal: Form of Pk key to network’s behavior networks [18] networks I Assortative network: similar degree nodes Scale-free networks Scale-free networks I ex 1: Erdös-Rényi random networks have a Poisson Small-world networks connecting to each other. Small-world networks Generalized afﬁliation Generalized afﬁliation distribution: networks networks I Often social: company directors, coauthors, actors. −hki k References References Pk = e hki /k! I Disassortative network: high degree nodes −γ I ex 2: “Scale-free” networks: Pk ∝ k ⇒ ‘hubs’ connecting to low degree nodes. We’ll come back to this business soon... I I Often techological or biological: Internet, protein interactions, neural networks, food webs.

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letter Basic deﬁnitions Basic deﬁnitions

4. Clustering: Examples of Examples of Complex Networks Network motifs in the transcriptional regulationComplex Networks I Your friends tend to know each other. Properties of network of Escherichia coli Properties of Complex Networks 5. Motifs: Complex Networks I Two measures: Nutshell Shai S. Shen-Orr1, Ron Milo2, Shmoolik Mangan1 & Uri Alon1,2 Nutshell I Small, recurring functional subnetworks *P + Basic models of Published online: 22 April 2002, DOI: 10.1038/ng881 Basic models of aj j complex networks complex networks j1j2∈Ni 1 2 [28] I e.g., Feed Forward Loop: Generalized random com Generalized random C = due to Watts & Strogatz . 1 networks networks 1–10 ( − )/ feedforward loop Little is known about the design principles of transcrip- ki ki 1 2 Scale-free networks a tional regulation networks thatScale-free contr networksol gene expression in i 2,11,12 Small-world networks X cells. Recent advances in dataSmall-world collection networks and analysis , X however, are generating unprecedented amounts of informa- Generalized afﬁliation Generalized afﬁliation networks tion about gene regulation networks.networks To understand these 1–10,13 3 × #triangles enetics.nature Y Y complex wiring diagrams , we sought to break down such [19] 2 C2 = due to Newman References networks into basic building blocksReferences. We generalize the notion Z n of motifs, widely used for sequence analysis, to the level of

#triples http://g networks. We deﬁne ‘network motifs’ as patterns of intercon-

nections that recur in many different parts of a network at fre- b crp oup [21] quencies much higher than those found in randomized

I C1 is the average fraction of pairs of neighbors who Shen-Orr, UriGr Alon, et al. networks. We applied new algorithms for systematically araC detecting network motifs to one of the best-characterized regulation networks, that of direct transcriptional interactions in are connected. 3,6

lishing Escherichia coli . We find that much of the network is com- araBAD posed of repeated appearances of three highly significant Interpret C as probability two of a node’s friends Pub motifs. Each network motif has a specific function in determin- I 2 c single input module (SIM) ing gene expression, such as generating temporal expression programs and governing the responses to fluctuating external

know each other. Nature X X signals. The motif structure also allows an easily interpretable view of the entire known transcriptional network of the organ-

Frame 29/122 2002 ism. This approach may helpFrame define the 30/122 basic computational © n elements of other biological networks. Z1 Z2 ... Zn We compiled a data set of direct transcriptional interactions between transcription factors and the operons they regulate (an d operon is a group of contiguous genes that are transcribed into a single mRNA molecule). This database contains 577 interac- argR tions and 424 operons (involving 116 transcription factors); it was formed on the basis of on an existing database (Regu- lonDB)3,14. We enhanced RegulonDB by an extensive literature argI argF argD argE search, adding 35 new transcription factors, including alterna- argCBH tive σ-factors (subunits of RNA polymerase that confer recogni- tion of specific promoter sequences). The data set consists of Overview of e dense overlapping regulons (DOR) established interactions in which aOverview transcription of factor directly Properties Complex Networks Properties binds a regulatory site. Complex Networks The transcriptional network can be represented as a directed X1 X2 X3 ... Xn X1 X2 X3 Xn graph, in which each node represents an operon and edges represent direct transcriptional interactions. Each edge is directed 6. modularity: Basic definitions Basic definitions Z1 Z2 Z3 Z4 ... Zm Fig. 1 Network motifs found in the E. coli transcriptional regulation network. Examples of Symbols representing the motifs areExamples also shown. a, Feedforward of loop: a transcription factor X regulates a second transcription factor Y, and both jointly Complex Networks f regulate one or more operons Z1...ZnComplex. b, Example of Networksa feedforward loop (L-ara- 49 binose utilization). c, SIM motif: a single transcription factor, X, regulates a set crp 53 rpoS ada oxyR ihf lrp hns rcsA nhaR fis 58 of operons Z1...Zn. X is usually autoregulatory. All regulations are of the same 63 46 Properties of sign. No other transcription factor regulatesProperties the operons. of d, Example of a SIM 83 114 system (arginine biosynthesis). e, DOR motif: a set of operons Z1...Zm are each 33 28 Complex Networks Complex Networks 25 11 7. Concurrency: regulated by a combination of a set of input transcription factors, X1...Xn. 97 88 dps DORs are defined by an algorithm that detects dense regions of connections, 1 59 alkA katG proP 67 nhaA osmC with a high ratio of connections to transcription factors. f, Example of a DOR 73 ftsQAZ 105 24 Nutshell Nutshell 50 (stationary phase response). 103 37 89 69 36 Transmission of a contagious element only occurs 45 109 110 Basic models of I Basic models of 57 90 complex networks [16]1Department of Molecular Cell Biology, 2Department of Physics of Complex Systems, Weizmann Institute of Science,complex Rehovot 76100, networks Israel. Correspondence 44 66 34 should be addressed to U.A. (e-mail: [email protected]). 42 Generalized random during contact Generalized random 16 82 4 75 31 networks networks 86 93 91 112 80 0 Scale-free networks 64 naturScale-freee genetics networks • volume 31 • may 2002 48 18 54 Rather obvious but easily missed in a simple model 9 92 Small-world networks I Small-world networks

23 7 104 29 8 61 Generalized afﬁliation Generalized afﬁliation 94 71 41 78 35 networks networks 68 I Dynamic property—static networks are not enough 99 22 19 55 21 77 References References 5 10 111 81 101 30 I Knowledge of previous contacts crucial 3 79 108 51 85 38 52 84 Beware cumulated network data! 98 I 2 6 113 17 43 26 76 70 107 60 39 40 14 74 72 47 62 13 95 27 96 12 100 15 102 65 20 87 106 64 32 56 Clauset et al., 2006 [8]: NCAA football Frame 31/122 Frame 32/122 Overview of Overview of Properties Complex Networks Properties Complex Networks

8. Horton-Strahler stream ordering: Basic deﬁnitions Basic deﬁnitions

Examples of Examples of I Metrics for branching networks: Complex Networks 9. Network distances: Complex Networks I Method for ordering streams hierarchically Properties of Properties of Complex Networks (a) shortest path length d : Complex Networks I Reveals fractal nature of natural branching networks ij [23, 10] Nutshell Nutshell I Hierarchy is not pure but mixed (Tokunaga). Basic models of I Fewest number of steps between nodes i and j. Basic models of I Major examples: rivers and blood networks. complex networks complex networks Generalized random Generalized random networks I (Also called the chemical distance between i and j.) networks Scale-free networks Scale-free networks Small-world networks Small-world networks Generalized afﬁliation Generalized afﬁliation networks (b) average path length hdij i: networks References References I Average shortest path length in whole network.

I Good algorithms exist for calculation.

I Weighted links can be accommodated.

c a

Frame 33/122 Frame 34/122 I Beautifully described but poorly explained.

Overview of Overview of Properties Complex Networks Properties Complex Networks

Basic deﬁnitions Basic deﬁnitions

Examples of Examples of Complex Networks Complex Networks 9. Network distances: Properties of 10. Centrality: Properties of Complex Networks Complex Networks (c) Network diameter d : max Nutshell I Many such measures of a node’s ‘importance.’ Nutshell Basic models of Basic models of I Maximum shortest path length in network. complex networks I ex 1: Degree centrality: ki . complex networks Generalized random Generalized random networks I ex 2: Node i’s betweenness networks Scale-free networks Scale-free networks −1 n −1 (d) Closeness d = [P d / ] : Small-world networks = fraction of shortest paths that pass through i. Small-world networks cl ij ij 2 Generalized afﬁliation Generalized afﬁliation networks networks I ex 3: Edge `’s betweenness References References I Average ‘distance’ between any two nodes. = fraction of shortest paths that travel along `. Closeness handles disconnected networks (d = ∞) I ij I ex 4: Recursive centrality: Hubs and Authorities (Jon [15] I dcl = ∞ only when all nodes are isolated. Kleinberg )

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Basic definitions Basic definitions Overview Key Points (cont.): Overview Key Points: Examples of Examples of Complex Networks Complex Networks Obvious connections with the vast extant field of The field of complex networks came into existence in Properties of I Properties of I Complex Networks graph theory. Complex Networks the late 1990s. Nutshell Nutshell I But focus on dynamics is more of a I Explosion of papers and interest since 1998/99. Basic models of Basic models of complex networks physics/stat-mech/comp-sci flavor. complex networks Generalized random Generalized random I Hardened up much thinking about complex systems. networks networks Scale-free networks I Two main areas of focus: Scale-free networks Specific focus on networks that are large-scale, Small-world networks Small-world networks I Generalized affiliation 1. Description: Characterizing very large networks Generalized affiliation sparse, natural or man-made, evolving and dynamic, networks 2. Explanation: Micro story ⇒ Macro features networks and (crucially) measurable. References References I Some essential structural aspects are understood: I Three main (blurred) categories: degree distribution, clustering, assortativity, group 1. Physical (e.g., river networks), structure, overall structure,... 2. Interactional (e.g., social networks), 3. Abstract (e.g., thesauri). I Still much work to be done, especially with respect to dynamics...

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Overview of Overview of Models Complex Networks Models Complex Networks

Basic deﬁnitions Basic deﬁnitions

Examples of Examples of Complex Networks Complex Networks

Properties of Properties of Some important models: Complex Networks Generalized random networks: Complex Networks Nutshell Nutshell Basic models of I Arbitrary degree distribution Pk . Basic models of 1. generalized random networks complex networks complex networks Generalized random I Create (unconnected) nodes with degrees sampled Generalized random 2. scale-free networks networks networks Scale-free networks from P . Scale-free networks Small-world networks k Small-world networks 3. small-world networks Generalized affiliation Generalized affiliation networks Wire nodes together randomly. networks 4. statistical generative models (p∗) I References Create ensemble to test deviations from References 5. generalized affiliation networks I randomness.

Frame 39/122 Frame 41/122 Overview of Overview of Building random networks: Stubs Complex Networks Building random networks: First rewiring Complex Networks

Phase 1: Basic deﬁnitions Basic deﬁnitions Examples of Examples of I Idea: start with a soup of unconnected nodes with Complex Networks Complex Networks Phase 2: stubs (half-edges): Properties of Properties of Complex Networks Complex Networks

Nutshell I Now ﬁnd any (A) self-loops and (B) repeat edges and Nutshell

Basic models of randomly rewire them. Basic models of complex networks complex networks Generalized random Generalized random networks networks Scale-free networks Scale-free networks Small-world networks Small-world networks Generalized afﬁliation (A) (B) Generalized afﬁliation networks networks Being careful: we can’t change the degree of any I Randomly select stubs References I References (not nodes!) and node, so we can’t simply move links around. connect them. I Simplest solution: randomly rewire two edges at a time. I Must have an even number of stubs.

I Initially allow self- and repeat connections. Frame 42/122 Frame 43/122

Overview of Overview of General random rewiring algorithm Complex Networks Sampling random networks Complex Networks e i i 1 2 1 Basic definitions Basic definitions Examples of Examples of I Randomly choose two edges. Complex Networks Complex Networks (Or choose problem edge and Properties of Phase 2: Properties of a random edge) Complex Networks Complex Networks Nutshell Use rewiring algorithm to remove all self and repeat Nutshell Check to make sure edges I I Basic models of loops. Basic models of are disjoint. complex networks complex networks Generalized random Generalized random e i networks networks i 2 4 Scale-free networks Scale-free networks 3 Small-world networks Phase 3: Small-world networks i Generalized affiliation Generalized affiliation 2 networks networks i Randomize network wiring by applying rewiring 1 Rewire one end of each edge. References I References I algorithm liberally. I Node degrees do not change. [17] I Rule of thumb: # Rewirings ' 10 × # edges . e’ e’ I Works if e1 is a self-loop or 1 2 repeated edge.

I Same as ﬁnding on/off/on/off

i 4-cycles. and rotating them. Frame 44/122 Frame 45/122 i 4 3 Overview of Overview of Scale-free networks Complex Networks Scale-free networks Complex Networks

Basic deﬁnitions Basic deﬁnitions

Examples of Examples of I Networks with power-law degree distributions have Complex Networks Complex Networks become known as scale-free networks. Properties of Properties of Complex Networks Complex Networks I Scale-free refers specifically to the degree Nutshell I Scale-free networks are not fractal in any sense. Nutshell distribution having a power-law decay in its tail: Basic models of Basic models of complex networks I Usually talking about networks whose links are complex networks Generalized random Generalized random −γ networks abstract, relational, informational, . . . (non-physical) networks Pk ∼ k for ‘large’ k Scale-free networks Scale-free networks Small-world networks I Primary example: hyperlink network of the Web Small-world networks Generalized affiliation Generalized affiliation networks networks I Much arguing about whether or networks are References References I One of the seminal works in complex networks: ‘scale-free’ or not. . . Laszlo Barabási and Reka Albert, Science, 1999: “Emergence of scaling in random networks” [3]

I Somewhat misleading nomenclature...

Frame 47/122 Frame 48/122

Overview of Overview of Random networks: largest components Complex Networks Scale-free networks Complex Networks

Basic deﬁnitions Basic deﬁnitions

Examples of Examples of Complex Networks Complex Networks

Properties of The big deal: Properties of Complex Networks Complex Networks Nutshell I We move beyond describing networks to finding Nutshell Basic models of mechanisms for why certain networks are the way Basic models of complex networks complex networks γ = 2.5 γ = 2.5 γ = 2.5 γ = 2.5 Generalized random they are. Generalized random hki = 1.8 hki = 2.05333 hki = 1.66667 hki = 1.92 networks networks Scale-free networks Scale-free networks Small-world networks Small-world networks Generalized affiliation Generalized affiliation networks A big deal for scale-free networks: networks References References I How does the exponent γ depend on the mechanism?

I Do the mechanism details matter? γ = 2.5 γ = 2.5 γ = 2.5 γ = 2.5 hki = 1.6 hki = 1.50667 hki = 1.62667 hki = 1.8

Frame 49/122 Frame 50/122 Overview of Overview of BA model Complex Networks BA model Complex Networks

Basic definitions Basic definitions Examples of I Definition: Ak is the attachment kernel for a node Examples of I Barabási-Albert model = BA model. Complex Networks with degree k. Complex Networks Properties of Properties of I Key ingredients: Complex Networks Complex Networks I For the original model: Growth and Preferential Attachment (PA). Nutshell Nutshell Basic models of A = k Basic models of I Step 1: start with m0 disconnected nodes. complex networks k complex networks Generalized random Generalized random I Step 2: networks networks Scale-free networks I Definition: Pattach(k, t) is the attachment probability. Scale-free networks 1. Growth—a new node appears at each time step Small-world networks Small-world networks Generalized affiliation Generalized affiliation t = 0, 1, 2,.... networks I For the original model: networks 2. Each new node makes m links to nodes already References References present. ki (t) ki (t) Pattach(node i, t) = = 3. Preferential attachment—Probability of connecting to PN(t) Pkmax(t) j=1 kj (t) k=0 kNk (t) ith node is ∝ ki .

I In essence, we have a rich-gets-richer scheme. where N(t) = m0 + t is # nodes at time t and Nk (t) is # degree k nodes at time t.

Frame 51/122 Frame 52/122

Overview of Overview of Approximate analysis Complex Networks Approximate analysis Complex Networks

I Deal with denominator: each added node brings m I When (N + 1)th node is added, the expected Basic definitions Basic definitions new edges. increase in the degree of node i is Examples of Examples of N(t) Complex Networks X Complex Networks ki,N Properties of ∴ kj (t) = 2tm Properties of E(k − k ) ' m . Complex Networks Complex Networks i,N+1 i,N N(t) j=1 P Nutshell Nutshell j=1 kj (t) Basic models of Basic models of complex networks I The node degree equation now simplifies: complex networks I Assumes probability of being connected to is small. Generalized random Generalized random networks networks Scale-free networks Scale-free networks Dispense with Expectation by assuming (hoping) that d ki (t) ki (t) 1 I Small-world networks k = m = m = k (t) Small-world networks Generalized affiliation i,t N(t) i Generalized affiliation over longer time frames, degree growth will be networks dt P 2mt 2t networks j=1 kj (t) smooth and stable. References References d I Approximate ki,N+1 − ki,N with dt ki,t : I Rearrange and solve: d ki (t) dk (t) dt ki,t = m i 1/2 dt PN(t) = ⇒ ki (t) = ci t . j=1 kj (t) ki (t) 2t

where t = N(t) − m0. Frame 53/122 Frame 54/122 I Next ﬁnd ci ... Overview of Overview of Approximate analysis Complex Networks Approximate analysis Complex Networks

Basic deﬁnitions Basic deﬁnitions

Examples of Examples of I Know ith node appears at time Complex Networks 20 Complex Networks Properties of Properties of Complex Networks Complex Networks i − m0 for i > m0 ti,start = Nutshell 15 Nutshell 0 for i ≤ m0 Basic models of Basic models of complex networks k (t) complex networks I So for i > m (exclude initial nodes), we must have Generalized random i Generalized random 0 networks 10 I m = 3 networks Scale-free networks Scale-free networks 1/2 Small-world networks t = Small-world networks Generalized afﬁliation I i,start Generalized afﬁliation t networks networks ki (t) = m for t ≥ ti,start. 1, 2, 5, and 10. ti,start References 5 References

1/2 I All node degrees grow as t but later nodes have 0 larger ti,start which ﬂattens out growth curve. 0 10 20 30 40 50 t I Early nodes do best (First-mover advantage).

Frame 55/122 Frame 56/122

Overview of Overview of Degree distribution Complex Networks Degree distribution Complex Networks

Basic definitions Basic definitions So what’s the degree distribution at time t? I I Examples of Examples of Complex Networks Pr(ki )dki = Pr(ti,start)dti,start Complex Networks I Use fact that birth time for added nodes is distributed Properties of Properties of uniformly: Complex Networks Complex Networks I dti,start Nutshell dti,start Nutshell Pr(ti,start)dti,start ' = Pr(ti,start)dki t Basic models of dki Basic models of complex networks complex networks I Also use Generalized random I Generalized random networks 2 networks Scale-free networks 1 m t Scale-free networks 1/2 2 Small-world networks = dki 2 Small-world networks t m t Generalized affiliation t k (t)3 Generalized affiliation ( ) = ⇒ = . networks i networks ki t m ti,start 2 ti,start k (t) i References I References m2 = Transform variables—Jacobian: 2 3 dki ki (t) dt m2t i,start = − . I 2 3 −3 dki ki (t) ∝ ki dki .

Frame 57/122 Frame 58/122 R EPORTS ing systems form a huge genetic network these two ingredients, we show that they are cited in a paper. Recently Redner (11) has whose vertices are proteins and genes, the responsible for the power-law scaling ob- shown that the probability that a paper is chemical interactions between them repre- served in real networks. Finally, we argue cited k times (representing the connectivity of senting edges (2). At a different organization- that these ingredients play an easily identifi- a paperOverview within the of network) follows a power Overview of

Degreeal level, a large distribution network is formed by the able and important role in the formation of lawComplex with exponent Networks␥cite ϭ 3. Examples Complex Networks nervous system, whose vertices are the nerve many complex systems, which implies that The above examples (12) demonstrate that cells, connected by axons (3). But equally our results are relevant to a large class of many large random networks share the com- complex networks occur in social science, networks observed in nature. monBasic feature definitions that the distribution of their local Basic definitions where vertices are individuals or organiza- Although there are many systems that connectivity is free of scale, following a power tions and the edges are the social interactions form complex networks, detailed topological lawExamples for large ofk with an exponent ␥ between Examples of between them (4), or in the World Wide Web data is available for only a few. The collab- 2.1Complex and 4, which Networks is unexpected within the Complex Networks (WWW),I We whose thus vertices have are a HTML very docu- specificoration prediction graph of movie of actors represents a frameworkProperties of of the existing network models. Properties of ments connected by links−γ pointing from one well-documented example of a social net- TheComplex random Networks graph model of ER (7) assumes Complex Networks pagePr to another(k) ∼ (5,k 6). Becausewith of theirγ = large3. work. Each actor is represented by a vertex, that we start with N vertices and connect each Nutshell Nutshell size and the complexity of their interactions, two actors being connected if they were cast pair of vertices with probability p. In the theI topologyTypical offor these real networks networks: is largely 2together< γ in < the3 same. movie. The probability model,Basic the models probability of that a vertex has k WWW γ ' 2.1 for in-degree Basic models of unknown. that an actor has k links (characterizing his or edgescomplex follows networks a Poisson distribution P(k) ϭ WWW γ ' 2.45 for out-degree complex networks Range true more generally for events with size Ϫ␭ Generalizedk random Generalized random ITraditionally, networks of complex topol- her popularity) has a power-law tail for large e ␭networks/k!, where networks ogy have been described with the random k, following P(k) ϳ kϪ␥actor, where ␥ ϭ Movie actors γ ' 2.3 distributions that have power-law tails. actor Scale-free networksN Ϫ 1 Scale-free networks graph theory of Erdo˝s and Reńyi (ER) (7), 2.3 Ϯ 0.1 (Fig. 1A). A more complex net- Small-world networks Small-world networks ϭ N pk 1 Ϫ p NϪ1Ϫk Words (synonyms) γ ' 2.8 but in the absence of data on large networks, work with over 800 million vertices (8) is the Generalized␭ affiliation ͑ ͒ Generalized affiliation 2 < γ < 3: finite mean and ‘infinite’ variance (wild) networks ͩ k ͪ networks theI predictions of the ER theory were rarely WWW, where a vertex is a document and the tested in the real world. However, driven by edges are the links pointing from one docu- In theReferences small-world model recently intro- References theI computerizationIn practice, of dataγ< acquisition,3 means such variancement to another. is The governed topology of this by graph duced by Watts and Strogatz (WS)The (10), N Internets is a different business... topologicalupper information cutoff. is increasingly avail- determines the Web’s connectivity and, con- vertices form a one-dimensional lattice, able, raising the possibility of understanding sequently, our effectiveness in locating infor- each vertex being connected to its two the dynamical and topological stability of mation on the WWW (5). Information about nearest and next-nearest neighbors. With I γ > 3: finite mean and variance (mild) on October 19, 2007 large networks. P(k) can be obtained using robots (6), indi- probability p, each edge is reconnected to a Here we report on the existence of a high cating that the probability that k documents vertex chosen at random. The long-range degree of self-organization characterizing the point to a certain Web page follows a power connections generated by this process de-

large-scale properties of complex networks. law, with ␥www ϭ 2.1 Ϯ 0.1 (Fig. 1B) (9). A crease the distance between the vertices, Exploring several large databases describing network whose topology reflects the histori- leading to a small-world phenomenon (13), the topology of large networks that span cal patterns of urban and industrial develop- oftenFrame referred 59/122 to as six degrees of separa- Frame 60/122 fields as diverse as the WWW or citation ment is the electrical power grid of the west- tion (14). For p ϭ 0, the probability distri- patterns in science, we show that, indepen- ern United States, the vertices being genera- bution of the connectivities is P(k) ϭ␦(k Ϫ dent of the system and the identity of its tors, transformers, and substations and the z), where z is the coordination number in constituents, the probability P(k) that a ver- edges being to the high-voltage transmission the lattice; whereas for finite p, P(k) still www.sciencemag.org tex in the network interacts with k other lines between them (10). Because of the rel- peaks around z, but it gets broader (15).A vertices decays as a power law, following atively modest size of the network, contain- common feature of the ER and WS models P(k) ϳ kϪ␥. This result indicates that large ing only 4941 vertices, the scaling region is is that the probability of finding a highly networks self-organize into a scale-free state, less prominent but is nevertheless approxi- connectedOverview vertex of (that is, a large k) decreas- Overview of Reala feature data unpredicted by all existing random mated by a power law with an exponent es exponentiallyComplex Networks with k; thus, verticesThings with to do and questions Complex Networks

network models. To explain the origin of this ␥power Ӎ 4 (Fig. 1C). Finally, a rather large large connectivity are practically absent. In

scale invariance, we show that existing net- complex network is formed by the citation contrast, the power-law tail characterizing Downloaded from work models fail to incorporate growth and patterns of the scientific publications, the ver- P(k)Basic for the definitions networks studied indicates that Basic definitions preferential attachment, two key features of tices being papers published in refereed jour- highly connected (large k) vertices have a real networks. Using a model incorporating nals and the edges being links to the articles largeExamples chance of of occurring, dominating theVary attachment kernel. Examples of [3] Complex Networks I Complex Networks From Barabási and Albert’s original paper : connectivity. ThereProperties aretwo of generic aspects of realI net-Vary mechanisms: Properties of worksComplex that are Networks not incorporated in these mod- 1. Add edge deletion Complex Networks els.Nutshell First, both models assume that we start Nutshell with a fixed number (N) of vertices that are 2. Add node deletion thenBasic randomly models connected of (ER model), or re- 3. Add edge rewiring Basic models of connectedcomplex (WS networks model), without modifying complex networks N. InGeneralized contrast, random most real world networks are Generalized random networks I Deal with directed versus undirected networks. networks openScale-free and they networks form by the continuous addi- Scale-free networks tionSmall-world of new networks vertices to the system, thus theImportant Q.: Are there distinct universality classes Small-world networks numberGeneralized of verticesaffiliation N increases throughoutI Generalized affiliation networks networks the lifetime of the network. For example, thefor these networks? actorReferences network grows by the addition of new References actors to the system, the WWW grows expo-I Q.: How does changing the model affect γ? nentially over time by the addition of new Fig. 1. The distribution function of connectivities for various large networks. (A) Actor collaboration graph with N ϭ 212,250 vertices and average connectivity ͗k͘ϭ28.78. (B) WWW, N ϭ Web pages (8), and the research literatureI Q.: Do we need preferential attachment and growth? 325,729, ͗k͘ϭ5.46 (6). (C) Power grid data, N ϭ 4941, ͗k͘ϭ2.67. The dashed lines have constantly grows by the publication of new slopes (A) ␥ ϭ 2.3, (B) ␥ ϭ 2.1 and (C) ␥ ϭ 4. papers. Consequently, a common feature of actor www power I Q.: Do model details matter?

510 15 OCTOBER 1999 VOL 286 SCIENCE www.sciencemag.org I The answer is (surprisingly) yes. More later re Zipf.

Frame 61/122 Frame 62/122 letters to nature

called scale-free networks, which include the World-Wide Web3–5, The inhomogeneous connectivity distribution of many real net- the Internet6, social networks7 and cells8. We ﬁnd that such works is reproduced by the scale-free model17,18 that incorporates networks display an unexpected degree of robustness, the ability two ingredients common to real networks: growth and preferential of their nodes to communicate being unaffected even by un- attachment. The model starts with m0 nodes. At every time step t a realistically high failure rates. However, error tolerance comes at a new node is introduced, which is connected to m of the already-

high price in that these networks are extremely vulnerable to existing nodes. The probability Πi that the new node is connected attacks (that is, to the selection and removal of a few nodes that to node i depends on the connectivity ki of node i such that play a vital role in maintaining the network’s connectivity). Such Πi ¼ ki=Sjkj. For large t the connectivity distribution is a power- error tolerance and attack vulnerability are generic properties of law following PðkÞ¼2m2=k3. communication networks. The interconnectedness of a network is described by its diameter Overview of The increasing availability of topological data on large networks, d, definedOverview of as the average length of the shortest paths between any Preferential attachment Complex Networks Preferentialaided by the computerization attachment of data through acquisition, randomness had led to great twoComplex nodes Networks in the network. The diameter characterizes the ability of advances in our understanding of the generic aspects of network two nodes to communicate with each other: the smaller d is, the Basic definitions structure and development9–16. The existing empirical and theo- shorterBasic definitions is the expected path between them. Networks with a very Examples of retical results indicate that complex networks can be divided into largeExamples number of of nodes can have quite a small diameter; for example, Complex Networks Instead of attaching preferentially, allow new nodes Complex Networks twoI major classes based on their connectivity distribution P(k), the diameter of the WWW, with over 800 million nodes20, is around I Let’s look at preferential attachment (PA) a little more Properties of to attach randomly. Properties of Complex Networks giving the probability that a node in the network is connected to k 19Complex (ref. Networks 3), whereas social networks with over six billion individuals closely. Nutshell otherI Now nodes. add The an firstextra class step of networks: new nodes is characterized then connect by a toP(k) Nutshell PA implies arriving nodes have complete knowledge that peaks at an average k and decays exponentially for large k. The I Basic models of some of their friends’! " friends. Basic models of of the existing network’s degree distribution. complex networks most investigated examples of such exponential networks are the complex networks Generalized random Can also do this at random. 9,10 Generalized random networks randomI graph model of Erdo¨s and Reńyi and the small-world networks 12 Scale-free networks Scale-free networksa I For example: If Pattach(k) ∝ k, we need to determine 11 Small-world networks modelI Assuming of Watts and the Strogatz existing, both network leading is to random, a fairly homogeneous we know Small-world networks E SF the constant of proportionality. Generalized affiliation Generalized affiliation Failure networks network,probability in which of each a random node has friend approximatelyhaving degree the samek numberis networks 10 Attack We need to know what everyone’s degree is... References of links, k Ӎ k . In contrast, results on the World-Wide Web References I 3–5 ! " 6 17–19 (WWW) , the Internet andQ other∝ kP large networks indicate I PA is ∴ an outrageous assumption of node capability. that many systems belong to a classk of inhomogeneousk networks, 8 called scale-free networks, for which P(k) decays as a power-law, I But a very simple mechanism saves the day. . . So rich-gets-richer scheme can now be seen to work thatI is PðkÞϳk Ϫ g, free of a characteristic scale. Whereas the prob- in a natural way. ability that a node has a very large number of connections (k q k ) 6 is practically prohibited in exponential networks, highly connected! "

Frame 63/122 nodes are statistically significant in scale-free networks (Fig. 1). Frame 64/122 We start by investigating the robustness of the two basic con- 4 ¨ ´ 9,10 0.00 0.02 0.04 nectivity distribution models, the Erdos–Renyi (ER) model that d produces a network with an exponential tail, and the scale-free model17 with a power-law tail. In the ER model we first define the N b c nodes, and then connect each pair of nodes with probability p. This 15 Overview of algorithm generates a homogeneous network (Fig. 1), whose con- Overview of Internet WWW Robustness Complex Networks Robustness Complex Networks nectivity follows a Poisson distribution peaked at k and decaying 20 exponentially for k q k . ! " Basic definitions Standard random! " networks (Erdös-Rényi) Basic definitions10 I Attack Attack Examples of versus Examples of Complex Networks Complex Networks 15 Properties of Scale-free networks Properties5 of Complex Networks Complex Networks Nutshell a b Nutshell Failure Failure Basic models of Basic models of I System robustness and system robustness. complex networks complex networks0 10 Generalized random Generalized0.00 random 0.01 0.02 0.00 0.01 0.02 Albert et al., Nature, 2000: networks networks I Scale-free networks Scale-free networks f [2] “Error and attack tolerance of complex networks” Small-world networks Small-world networks Generalized affiliation Generalized affiliation networks Figurenetworks 2 Changes in the diameter d of the network as a function of the fraction f of the References removedReferences nodes. a, Comparison between the exponential (E) and scale-free (SF) network models, each containing N ¼ 10;000 nodes and 20,000 links (that is, k ¼ 4). The blue ! " symbols correspond to the diameter of the exponential (triangles) and the scale-free (squares) networks when a fraction f of the nodes are removed randomly (error tolerance). Red symbols show the response of the exponential (diamonds) and the scale-free (circles) networks to attacks, when the most connected nodes are removed. We determined the f Exponential Scale-free fromdependence of the diameter for different system sizes (N ¼ 1;000; 5,000; 20,000) and Figure 1 Visual illustration of the difference between an exponential and a scale-free found that the obtained curves, apart from a logarithmic size correction, overlap with Frame 65/122 Albert et al., 2000 Frame 66/122 network. a, The exponential network is homogeneous: most nodes have approximately those shown in a, indicating that the results are independent of the size of the system. We the same number of links. b, The scale-free network is inhomogeneous: the majority of note that the diameter of the unperturbed (f ¼ 0) scale-free network is smaller than that the nodes have one or two links but a few nodes have a large number of links, of the exponential network, indicating that scale-free networks use the links available to guaranteeing that the system is fully connected. Red, the five nodes with the highest them more efficiently, generating a more interconnected web. b, The changes in the number of links; green, their first neighbours. Although in the exponential network only diameter of the Internet under random failures (squares) or attacks (circles). We used the 27% of the nodes are reached by the five most connected nodes, in the scale-free topological map of the Internet, containing 6,209 nodes and 12,200 links ( k ¼ 3:4), ! " network more than 60% are reached, demonstrating the importance of the connected collected by the National Laboratory for Applied Network Research http://moat.nlanr.net/ ! nodes in the scale-free network Both networks contain 130 nodes and 215 links Routing/rawdata/ . c, Error (squares) and attack (circles) survivability of the World-Wide " ( k ¼ 3:3). The network visualization was done using the Pajek program for large Web, measured on a sample containing 325,729 nodes and 1,498,353 links3, such that ! " network analysis: http://vlado.fmf.uni-lj.si/pub/networks/pajek/pajekman.htm . k ¼ 4:59. ! " ! "

NATURE | VOL 406 | 27 JULY 2000 | www.nature.com © 2000 Macmillan Magazines Ltd 379 letters to nature called scale-free networks, which include the World-Wide Web3–5, The inhomogeneous connectivity distribution of many real net- the Internet6, social networks7 and cells8. We find that such works is reproduced by the scale-free model17,18 that incorporates networks display an unexpected degree of robustness, the ability two ingredients common to real networks: growth and preferential of their nodes to communicate being unaffected even by un- attachment. The model starts with m0 nodes. At every time step t a realistically high failure rates. However, error tolerance comes at a new node is introduced, which is connected to m of the already- high price in that these networks are extremely vulnerable to existing nodes. The probability Πi that the new node is connected attacks (that is, to the selection and removal of a few nodes that to node i depends on the connectivity ki of node i such that play a vital role in maintaining the network’s connectivity). Such Πi ¼ ki=Sjkj. For large t the connectivity distribution is a power- error tolerance and attack vulnerability are generic properties of law following PðkÞ¼2m2=k3. communication networks. The interconnectedness of a network is described by its diameter The increasing availability of topological data on large networks, d, defined as the average length of the shortest paths between any aided by the computerization of data acquisition, had led to great two nodes in the network. The diameter characterizes the ability of advances in our understanding of the generic aspects of network two nodes to communicate with each other: the smaller d is, the structure and development9–16. The existing empirical and theo- shorter is the expected path between them. Networks with a very retical results indicate that complex networks can be divided into large number of nodes can have quite a small diameter; for example, Overview of Overview of two major classes based on their connectivity distribution P(k), the diameterRobustness of the WWW, with over 800 million nodes20, is around Complex Networks Robustness Complex Networks giving the probability that a node in the network is connected to k 19 (ref. 3), whereas social networks with over six billion individuals other nodes. The first class of networks is characterized by a P(k) that peaks at an average k and decays exponentially for large k. The most investigated examples! " of such exponential networks are the Basic definitions Basic definitions random graph model of Erdo¨s and Reńyi9,10 and the small-world 12 11 a model of Watts and Strogatz , both leading to a fairly homogeneous E SF Examples of Examples of network, in which each node has approximately the same number Failure Complex Networks Complex Networks 10 Scale-free networks are thus robust to random of links, k Ӎ k . In contrast, results on the World-Wide Web Attack Plots of network I (WWW)3–5, the! " Internet6 and other large networks17–19 indicate I Properties of Properties of that many systems belong to a class of inhomogeneous networks, failures yet fragile to targeted ones. 8 Complex Networks Complex Networks called scale-free networks, for which P(k) decays as a power-law, diameter as a function Ϫ g that is PðkÞϳk , free of a characteristic scale. Whereas the prob- of fraction of nodes Nutshell I All very reasonable: Hubs are a big deal. Nutshell ability that a node has a very large number of connections (k q k ) 6 is practically prohibited in exponential networks, highly connected! " Basic models of Basic models of nodes are statistically significant in scale-free networks (Fig. 1). removed complex networks I But: next issue is whether hubs are vulnerable or not. complex networks We start by investigating the robustness of the two basic con- 4 Generalized random Generalized random 9,10 0.00 0.02 0.04 ¨ ´ networks networks nectivity distribution models, the Erdos–Renyi (ER) model that d I Erdös-Rényi versus Representing all webpages as the same size node is produces a network with an exponential tail, and the scale-free Scale-free networks I Scale-free networks 17 model with a power-law tail. In the ER model we first define the N b c scale-free networks Small-world networks obviously a stretch (e.g., google vs. a random Small-world networks nodes, and then connect each pair of nodes with probability p. This Generalized affiliation Generalized affiliation 15 algorithm generates a homogeneous network (Fig. 1), whose con- Internet WWW networks networks nectivity follows a Poisson distribution peaked at k and decaying 20 I blue symbols = person’s webpage) exponentially for k q k . ! " References References ! " 10 Attack Attack random removal I Most connected nodes are either: 15 1. Physically larger nodes that may be harder to ‘target’ 5 I red symbols = a b 2. or subnetworks of smaller, normal-sized nodes. Failure Failure targeted removal 0 10 0.00 0.01 0.02 0.00 0.01 0.02 Need to explore cost of various targeting schemes. f from (most connected first) I

FigureAlbert 2 Changes et in al., the diameter2000 d of the network as a function of the fraction f of the removed nodes. a, Comparison between the exponential (E) and scale-free (SF) network models, each containing N ¼ 10;000 nodes and 20,000 links (that is, k ¼ 4). The blue ! " symbols correspond to the diameter of the exponential (triangles) and the scale-free Frame 67/122 Frame 68/122 (squares) networks when a fraction f of the nodes are removed randomly (error tolerance). Red symbols show the response of the exponential (diamonds) and the scale-free (circles) networks to attacks, when the most connected nodes are removed. We determined the f Scale-free Exponential dependence of the diameter for different system sizes (N ¼ 1;000; 5,000; 20,000) and Figure 1 Visual illustration of the difference between an exponential and a scale-free found that the obtained curves, apart from a logarithmic size correction, overlap with network. a, The exponential network is homogeneous: most nodes have approximately those shown in a, indicating that the results are independent of the size of the system. We the same number of links. b, The scale-free network is inhomogeneous: the majority of note that the diameter of the unperturbed (f ¼ 0) scale-free network is smaller than that the nodes have one or two links but a few nodes have a large number of links, of the exponential network, indicating that scale-free networks use the links available to guaranteeing that the system is fully connected. Red, the five nodes with the highest them more efficiently, generating a more interconnected web. b, The changes in the number of links; green, their first neighbours. Although in the exponential network only diameter of the Internet under random failures (squares) or attacks (circles). We used the Overview of Overview of 27% of the nodes are reached by the five most connected nodes, in the scale-free topological map of the Internet, containing 6,209 nodes and 12,200 links ( k ¼ 3:4), Some problems for! " people thinking about Complex Networks Social Search Complex Networks network more than 60% are reached, demonstrating the importance of the connected collected by the National Laboratory for Applied Network Research http://moat.nlanr.net/ ! nodes in the scale-free network Both networks contain 130 nodes and 215 links Routing/rawdata/ . c, Error (squares) and attack (circles) survivability of the World-Wide " ( k ¼ 3:3). The network visualization was done using the Pajek program for large Web, measured on a sample containing 325,729 nodes and 1,498,353 links3, such that ! " people?: network analysis: http://vlado.fmf.uni-lj.si/pub/networks/pajek/pajekman.htm . k ¼ 4:59. ! " ! " Basic definitions Basic definitions

NATURE | VOL 406 | 27 JULY 2000 | www.nature.com © 2000 Macmillan Magazines Ltd 379 Examples of Examples of How are social networks structured? Complex Networks Complex Networks Properties of Properties of Complex Networks Complex Networks I How do we define connections? Nutshell A small slice of the pie: Nutshell I How do we measure connections? Basic models of Basic models of complex networks complex networks I (remote sensing, self-reporting) Generalized random I Q. Can people pass messages between distant Generalized random networks networks Scale-free networks individuals using only their existing social Scale-free networks Small-world networks Small-world networks Generalized affiliation connections? Generalized affiliation What about the dynamics of social networks? networks networks References I A. Apparently yes... References I How do social networks evolve?

I How do social movements begin?

I How does collective problem solving work?

I How is information transmitted through social networks? Frame 70/122 Frame 71/122 Overview of Overview of Milgram’s social search experiment (1960s) Complex Networks The problem Complex Networks

Basic deﬁnitions Basic deﬁnitions Target person = Examples of Lengths of successful chains: Examples of I Complex Networks Complex Networks 18 Boston stockbroker. Properties of Properties of Complex Networks Complex Networks I 296 senders from Boston and Nutshell 15 Nutshell Omaha. Basic models of From Travers and Basic models of 20% of senders reached complex networks 12 complex networks I Generalized random Milgram (1969) in Generalized random networks networks target. Scale-free networks [24] Scale-free networks ) Sociometry:

Small-world networks L Small-world networks

( 9 Generalized afﬁliation Generalized afﬁliation

chain length ' 6.5. n “An Experimental I networks networks Study of the Small References 6 References Popular terms: World Problem.”

I The Small World 3 Phenomenon; 0 http://www.stanleymilgram.com I “Six Degrees of Separation.” 1 2 3 4 5 6 7 8 9 10 11 12 L Frame 72/122 Frame 73/122

Overview of Overview of The problem Complex Networks Social Search Complex Networks

Basic deﬁnitions Milgram’s small world experiment with e-mail [9] Basic deﬁnitions Examples of Examples of Complex Networks Complex Networks

Properties of Properties of Complex Networks Complex Networks Two features characterize a social ‘Small World’: Nutshell Nutshell Basic models of Basic models of complex networks complex networks 1. Short paths exist Generalized random Generalized random networks networks and Scale-free networks Scale-free networks Small-world networks Small-world networks Generalized affiliation Generalized affiliation 2. People are good at finding them. networks networks References References

Frame 74/122 Frame 75/122 Overview of Overview of Social search—the Columbia experiment Complex Networks Social search—the Columbia experiment Complex Networks

Basic deﬁnitions Basic deﬁnitions

Examples of Examples of Complex Networks Complex Networks

Properties of Properties of I 60,000+ participants in 166 countries Complex Networks Complex Networks Nutshell I Milgram’s participation rate was roughly 75% Nutshell I 18 targets in 13 countries including Basic models of I Email version: Approximately 37% participation rate. Basic models of I a professor at an Ivy League university, complex networks complex networks Generalized random Generalized random I an archival inspector in Estonia, networks I Probability of a chain of length 10 getting through: networks I a technology consultant in India, Scale-free networks Scale-free networks Small-world networks Small-world networks 10 −5 I a policeman in Australia, Generalized afﬁliation Generalized afﬁliation networks .37 ' 5 × 10 networks

and References References I a veterinarian in the Norwegian army. I ⇒ 384 completed chains (1.6% of all chains). I 24,000+ chains

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Overview of Overview of Social search—the Columbia experiment Complex Networks Social search—the Columbia experiment Complex Networks

Basic deﬁnitions Basic deﬁnitions

Examples of Successful chains disproportionately used Examples of Complex Networks Complex Networks Properties of I weak ties (Granovetter) Properties of Complex Networks Complex Networks I Motivation/Incentives/Perception matter. Nutshell I professional ties (34% vs. 13%) Nutshell I If target seems reachable Basic models of I ties originating at work/college Basic models of complex networks complex networks ⇒ participation more likely. Generalized random Generalized random networks I target’s work (65% vs. 40%) networks Small changes in attrition rates Scale-free networks Scale-free networks I Small-world networks Small-world networks ⇒ large changes in completion rates Generalized afﬁliation Generalized afﬁliation networks . . . and disproportionately avoided networks I e.g., & 15% in attrition rate References References ⇒ % 800% in completion rate I hubs (8% vs. 1%) (+ no evidence of funnels) I family/friendship ties (60% vs. 83%)

Geography → Work

Frame 78/122 Frame 79/122 Overview of Overview of Social search—the Columbia experiment Complex Networks Social search—the Columbia experiment Complex Networks

Basic deﬁnitions Basic deﬁnitions

Examples of Examples of Senders of successful messages showed Complex Networks Nevertheless, some weak discrepencies do exist... Complex Networks Properties of Properties of little absolute dependency on Complex Networks An above average connector: Complex Networks Nutshell Nutshell I age, gender Norwegian, secular male, aged 30-39, earning over Basic models of Basic models of I country of residence complex networks $100K, with graduate level education working in mass complex networks Generalized random Generalized random networks media or science, who uses relatively weak ties to people networks I income Scale-free networks Scale-free networks Small-world networks they met in college or at work. Small-world networks Generalized afﬁliation Generalized afﬁliation I religion networks networks

I relationship to recipient References A below average connector: References Italian, Islamic or Christian female earning less than $2K, Range of completion rates for subpopulations: with elementary school education and retired, who uses 30% to 40% strong ties to family members.

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Overview of Overview of Social search—the Columbia experiment Complex Networks Social search—the Columbia experiment Complex Networks

Basic deﬁnitions Basic deﬁnitions

Examples of Examples of Complex Networks Complex Networks Mildly bad for continuing chain: Properties of Basic results: Properties of Complex Networks Complex Networks choosing recipients because “they have lots of friends” or Nutshell I hLi = 4.05 for all completed chains Nutshell because they will “likely continue the chain.” Basic models of Basic models of complex networks I L∗ = Estimated ‘true’ median chain length (zero complex networks Generalized random Generalized random Why: networks attrition) networks Scale-free networks Scale-free networks Small-world networks = Small-world networks Generalized affiliation I Intra-country chains: L∗ 5 Generalized affiliation I Specificity important networks networks References I Inter-country chains: L∗ = 7 References I Successful links used relevant information. (e.g. connecting to someone who shares same I All chains: L∗ = 7 profession as target.) I Milgram: L∗ ' 9

Frame 82/122 Frame 83/122 Overview of Overview of The social world appears to be small... Complex Networks Simple socialness in a network: Complex Networks

Basic deﬁnitions Basic deﬁnitions

Examples of Examples of Complex Networks Complex Networks

Properties of Properties of Complex Networks Complex Networks I Connected random networks have short average Nutshell Nutshell path lengths: Basic models of Basic models of complex networks complex networks hdABi ∼ log(N) Generalized random Generalized random networks Need “clustering” (your networks Scale-free networks friends are likely to Scale-free networks N = population size, Small-world networks Small-world networks Generalized afﬁliation Generalized afﬁliation networks know each other): networks dAB = distance between nodes A and B. References References I But: social networks aren’t random...

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Overview of Overview of Non-randomness gives clustering: Complex Networks Randomness + regularity Complex Networks

Basic deﬁnitions Basic deﬁnitions

B Examples of B Examples of Complex Networks Complex Networks

Properties of Properties of Complex Networks Complex Networks

Nutshell Nutshell

Basic models of Basic models of complex networks complex networks Generalized random Generalized random networks networks Scale-free networks Scale-free networks Small-world networks Small-world networks Generalized afﬁliation Generalized afﬁliation networks networks

References References

A A

= h i dAB = 10 → too many long paths. Frame 86/122 Now have dAB 3 d decreases overall Frame 87/122 letters to nature

removed from a clustered neighbourhood to make a short cut has, at model—it is probably generic for many large, sparse networks most, a linear effect on C; hence C(p) remains practically unchanged found in nature. for small p even though L(p) drops rapidly. The important implica- We now investigate the functional significance of small-world tion here is that at the local level (as reflected by C(p)), the transition connectivity for dynamical systems. Our test case is a deliberately to a small world is almost undetectable. To check the robustness of simplified model for the spread of an infectious disease. The these results, we have tested many different types of initial regular population structure is modelled by the family of graphs described graphs, as well as different algorithms for random rewiring, and all in Fig. 1. At time t ¼ 0, a single infective individual is introduced give qualitatively similar results. The only requirement is that the into an otherwise healthy population. Infective individuals are rewired edges must typically connect vertices that would otherwise removed permanently (by immunity or death) after a period of8 be much farther apart than Lrandom. sickness that lasts one unit of dimensionless time. During this time, The idealized construction above reveals the key role of short each infective individual can infect each of its healthy neighbours cuts. It suggests that the small-world phenomenon might be with probability r. On subsequent time steps, the disease spreads common in sparse networks with many vertices, as even a tiny along the edges of the graph until it either infects the entire fraction of short cuts would suffice. To test this idea, we have population, or it dies out, having infected some fraction of the computed L and C for the collaboration graph of actors in feature population in the process. films (generated from data available at http://us.imdb.com), the electrical power grid of the western United States, and the neural network of the nematode worm C. elegans17. All three graphs are of Table 1 Empirical examples of small-world networks scientific interest. The graph of film actors is a surrogate for a social 18 Lactual Lrandom Cactual Crandom letters to nature network , with the advantage of being much more easily specified...... It is also akin to the graph of mathematical collaborations centred, Film actors 3.65 2.99 0.79 0.00027 traditionally, on P. Erdo¨s (partial data available at http:// Power grid 18.7 12.4 0.080 0.005 removed from a clustered neighbourhood to make a short cut has, at model—it is probably generic for many large, sparse networks C. elegans 2.65 2.25 0.28 0.05 www.acs.oakland.edu/ϳgrossman/erdoshp.html). The graph of ...... most, a linear effect on C; hence C(p) remains practically unchanged found in nature. Characteristic path length L and clustering coefficient C for three real networks, compared the power grid is relevant to the efficiency and robustness of to random graphs with the same number of vertices (n) and average number of edges per 19 for small p even though L(p) drops rapidly. The important implica- We now investigate the functional significance of small-world power networks . And C. elegans is the sole example of a completely vertex (k). (Actors: n ¼ 225;226, k ¼ 61. Power grid: n ¼ 4;941, k ¼ 2:67. C. elegans: n ¼ 282, tion here is that at the local level (as reflected by C(p)), the transition connectivity for dynamical systems. Our test case is a deliberately k ¼ 14.) The graphs are defined as follows. Two actors are joined by an edge if they have mapped neural network. acted in a film together. We restrict attention to the giant connected component16 of this to a small world is almost undetectable. To check the robustness of simplified model for the spread of an infectious disease. The Table 1 shows that all three graphs are small-world networks. graph, which includes ϳ90% of all actors listed in the Internet Movie Database (available at these results, we have tested many different types of initial regular population structure is modelled by the family of graphs described These examples were not hand-picked; they were chosen because of http://us.imdb.com), as of April 1997. For the power grid, vertices represent generators, Overview of transformerOverviews ofand substations, and edges represent high-voltage transmission lines graphs, as well as different algorithms for random rewiring, and all in FigSmall-world. 1. At time t ¼ networks0, a single infective individual is introduced Complex Networks Toytheir model:inherent interest and because complete wiring diagrams were betweComplexen them. NetworksFor C. elegans, an edge joins two neurons if they are connected by either a synapse or a gap junction. We treat all edges as undirected and unweighted, and all give qualitatively similar results. The only requirement is that the into an otherwise healthy population. Infective individuals are available. Thus the small-world phenomenon is not merely a vertices as identical, recognizing that these are crude approximations. All three networks 13,14 show the small-world phenomenon: L Ռ L but C q C . rewired edges must typically connect vertices that would otherwise removed permanently (by immunity or death) after a period of Basic definitions curiosity of social networks nor an artefact of an idealized Basic definitions random random Introduced by Watts and Strogatz (Nature, 1998) [28] 8 be much farther apart than Lrandom. sickness that lasts one unit of dimensionless time. During this time, Examples of Examples of The idealized construction above reveals the key role of short each“Collectiveinfective indiv dynamicsidual can ofinfect ‘small-world’each of its networks.”healthy neighbours Complex Networks Complex Networks1 cuts. It suggests that the small-world phenomenon might be with probability r. On subsequent time steps, the disease spreads Properties of Properties of Small-world networks were found everywhere: Complex Networks Regular Small-world Random Complex0.8 Networks C(p) / C(0) common in sparse networks with many vertices, as even a tiny along the edges of the graph until it either infects the entire Nutshell Nutshell fraction of short cuts would suffice. To test this idea, we have population,I neuralor it networkdies out, ofhaving C. elegans,infected some fraction of the Basic models of Basic models of computed L and C for the collaboration graph of actors in feature population in the process. complex networks complex0.6 networks I semantic networks of languages, Generalized random Generalized random films (generated from data available at http://us.imdb.com), the networks networks Scale-free networks Scale-free networks0.4 electrical power grid of the western United States, and the neural I actor collaboration graph, Small-world networks Small-world networks 17 Generalized affiliation Generalized affiliation network of the nematode worm C. elegans . All three graphs are of networks networks Table 1 IEmpiricafoodl webs,examples of small-world networks L(p) / L(0) scientific interest. The graph of film actors is a surrogate for a social References References0.2 social networks of comic book characters,... 18 I Lactual Lrandom Cactual Crandom network , with the advantage of being much more easily specified...... Film actors 3.65 2.99 0.79 0.00027 0 It is also akin to the graph of mathematical collaborations centred, p = 0 p = 1 0.0001 0.001 0.01 0.1 1 traditionally, on P. Erdo¨s (partial data available at http:// PowerVerygrid weak requirements:18.7 12.4 0.080 0.005 Increasing randomness C. elegans 2.65 2.25 0.28 0.05 p www.acs.oakland.edu/ϳgrossman/erdoshp.html). The graph of ...... CharacteristicI localpath length regularityL and clustering+ randomcoefficientshortC for cutsthree real networks, compared Figure 1 Random rewiring procedure for interpolating between a regular ring Figure 2 Characteristic path length L(p) and clustering coefficient C(p) for the the power grid is relevant to the efficiency and robustness of to random graphs with the same number of vertices (n) and average number of edges per 19 lattice and a random network, without altering the number of vertices or edges in family of randomly rewired graphs described in Fig. 1. Here L is defined as the power networks . And C. elegans is the sole example of a completely vertex (k). (Actors: n ¼ 225;226, k ¼ 61. Power grid: n ¼ 4;941, k ¼ 2:67. C. elegans: n ¼ 282, k ¼ 14.) The graphs are defined as follows. Two actors are joined by an edge if they have Frame 88/122 the graph. We start with a ring of n vertices, each connected to its k nearest numberFrame 89/122of edges in the shortest path between two vertices, averaged over all mapped neural network. 16 acted in a film together. We restrict attention to the giant connected component of this neighbours by undirected edges. (For clarity, n ¼ 20 and k ¼ 4 in the schematic pairs of vertices. The clustering coefficient C(p) is defined as follows. Suppose Table 1 shows that all three graphs are small-world networks. graph, which includes ϳ90% of all actors listed in the Internet Movie Database (available at http://us.imdb.com), as of April 1997. For the power grid, vertices represent generators, examples shown here, but much larger n and k are used in the rest of this Letter.) that a vertex v has kv neighbours; then at most kvðkv Ϫ 1Þ=2 edges can exist These examples were not hand-picked; they were chosen because of transformers and substations, and edges represent high-voltage transmission lines We choose a vertex and the edge that connects it to its nearest neighbour in a between them (this occurs when every neighbour of v is connected to every other between them. For C. elegans, an edge joins two neurons if they are connected by either their inherent interest and because complete wiring diagrams were clockwise sense. With probability p, we reconnect this edge to a vertex chosen neighbour of v). Let C denote the fraction of these allowable edges that actually a synapse or a gap junction. We treat all edges as undirected and unweighted, and all v available. Thus the small-world phenomenon is not merely a vertices as identical, recognizing that these are crude approximations. All three networks uniformly at random over the entire ring, with duplicate edges forbidden; other- exist. Define C as the average of Cv over all v. For friendship networks, these 13,14 q Overview of Overview of curiosity of social networks nor an artefact of an idealized show the small-world phenomenon: L Ռ Lrandom but C Crandom. wise we leave the edge in place. We repeat this process by moving clockwise statistics have intuitive meanings: L is the average number of friendships in the The structural small-world property: Complex Networks Previous work—finding short paths Complex Networks around the ring, considering each vertex in turn until one lap is completed. Next, shortest chain connecting two people; Cv reflects the extent to which friends of v 1 we consider the edges that connect vertices to their second-nearest neighbours are also friends of each other; and thus C measures the cliquishness of a typical Basic definitions Basic definitions clockwise. As before, we randomly rewire each of these edges with probability p, friendship circle. The data shown in the figure are averages over 20 random Regular Small-world Random Examples of Examples of 0.8 C(p) / C(0) Complex Networks and continue this process, circulating around the ring and proceeding outward to realizationsComplex Networksof the rewiring process described in Fig.1, and have been normalized more distant neighbours after each lap, until each edge in the original lattice has by the values L(0), C(0) for a regular lattice. All the graphs have n ¼ 1;000 vertices Properties of Properties of Complex Networks been considered once. (As there are nk/2 edges in the entire graph, the rewiring andComplexan av Networkserage degree of k ¼ 10 edges per vertex. We note that a logarithmic 0.6 But are these short cuts findable? Nutshell process stops after k/2 laps.) Three realizations of this process are shown, for horizontalNutshell scale has been used to resolve the rapid drop in L(p), corresponding to Basic models of different values of p. For p ¼ 0, the original ring is unchanged; as p increases, the theBasiconset modelsof the of small-world phenomenon. During this drop, C(p) remains almost complex networks Nope. complex networks 0.4 graph becomes increasingly disordered until for p ¼ 1, all edges are rewired constant at its value for the regular lattice, indicating that the transition to a small Generalized random Generalized random networks randomly. One of our main results is that for intermediate values of p, the graph is worldnetworksis almost undetectable at the local level. Scale-free networks Scale-free networks L(p) / L(0) Small-world networks Nodes cannot find each other quickly Small-world networks 0.2 a small-world network: highly clustered like a regular graph, yet with small Generalized affiliation Generalized affiliation networks withcharacterisanytic localpath searchlength, like methoda random.graph. (See Fig. 2.) networks References References 0 Nature © Macmillan Publishers Ltd 1998 p = 0 p = 1 0.0001 0.001 0.01 0.1 1 NeedNATURE a| V moreOL 393 | 4 sophisticatedJUNE 1998 model... 441 Increasing randomness p

Figure 1 Random rewiring procedure for interpolating between a regular ring Figure 2 Characteristic path length L(p) and clustering coefficient C(p) for the lattice and a random network, without altering the number of vertices or edges in family ofIrandoL(pmly) =r averageewired graphs shortestdescri pathbed in lengthFig. 1. asHer ae L functionis defined of asp the the graph. We start with a ring of n vertices, each connected to its k nearest number of edges in the shortest path between two vertices, averaged over all I C(p) = average clustring as a function of p neighbours by undirected edges. (For clarity, n ¼ 20 and k ¼ 4 in the schematic pairs of vertices. The clustering coefficient C(p) is defined as follows. Suppose Frame 90/122 Frame 91/122 examples shown here, but much larger n and k are used in the rest of this Letter.) that a vertex v has kv neighbours; then at most kvðkv Ϫ 1Þ=2 edges can exist We choose a vertex and the edge that connects it to its nearest neighbour in a between them (this occurs when every neighbour of v is connected to every other clockwise sense. With probability p, we reconnect this edge to a vertex chosen neighbour of v). Let Cv denote the fraction of these allowable edges that actually uniformly at random over the entire ring, with duplicate edges forbidden; other- exist. Define C as the average of Cv over all v. For friendship networks, these wise we leave the edge in place. We repeat this process by moving clockwise statistics have intuitive meanings: L is the average number of friendships in the around the ring, considering each vertex in turn until one lap is completed. Next, shortest chain connecting two people; Cv reflects the extent to which friends of v we consider the edges that connect vertices to their second-nearest neighbours are also friends of each other; and thus C measures the cliquishness of a typical clockwise. As before, we randomly rewire each of these edges with probability p, friendship circle. The data shown in the figure are averages over 20 random and continue this process, circulating around the ring and proceeding outward to realizations of the rewiring process described in Fig.1, and have been normalized more distant neighbours after each lap, until each edge in the original lattice has by the values L(0), C(0) for a regular lattice. All the graphs have n ¼ 1;000 vertices been considered once. (As there are nk/2 edges in the entire graph, the rewiring and an average degree of k ¼ 10 edges per vertex. We note that a logarithmic process stops after k/2 laps.) Three realizations of this process are shown, for horizontal scale has been used to resolve the rapid drop in L(p), corresponding to different values of p. For p ¼ 0, the original ring is unchanged; as p increases, the the onset of the small-world phenomenon. During this drop, C(p) remains almost graph becomes increasingly disordered until for p ¼ 1, all edges are rewired constant at its value for the regular lattice, indicating that the transition to a small randomly. One of our main results is that for intermediate values of p, the graph is world is almost undetectable at the local level. a small-world network: highly clustered like a regular graph, yet with small characteristic path length, like a random graph. (See Fig. 2.)

Nature © Macmillan Publishers Ltd 1998 NATURE | VOL 393 | 4 JUNE 1998 441 Overview of Overview of Previous work—ﬁnding short paths Complex Networks Previous work—ﬁnding short paths Complex Networks

Basic deﬁnitions Basic deﬁnitions

Examples of Examples of I What can a local search method reasonably use? Complex Networks Complex Networks How to find things without a map? Properties of [14] Properties of I Complex Networks Jon Kleinberg (Nature, 2000) Complex Networks I Need some measure of distance between friends Nutshell “Navigation in a small world.” Nutshell and the target. Basic models of Basic models of complex networks complex networks Generalized random Generalized random networks Allowed to vary: networks Scale-free networks Scale-free networks Small-world networks Small-world networks Some possible knowledge: Generalized affiliation 1. local search algorithm Generalized affiliation networks networks References and References I Target’s identity 2. network structure. I Friends’ popularity

I Friends’ identities

I Where message has been

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Overview of Overview of Previous work—ﬁnding short paths Complex Networks Previous work—ﬁnding short paths Complex Networks

Kleinberg’s Network: Basic definitions Basic definitions Examples of Examples of 1. Start with regular d-dimensional cubic lattice. Complex Networks Complex Networks Properties of Properties of 2. Add local links so nodes know all nodes within a Complex Networks Theoretical optimal search: Complex Networks distance q. Nutshell Nutshell Basic models of I “Greedy” algorithm. Basic models of 3. Add m short cuts per node. complex networks complex networks Generalized random Generalized random networks I Same number of connections at all scales: α = d. networks 4. Connect i to j with probability Scale-free networks Scale-free networks Small-world networks Small-world networks Generalized affiliation Generalized affiliation −α networks 2 networks pij ∝ xij . Search time grows slowly with system size (like log N). References References But: social networks aren’t lattices plus links.

I α = 0: random connections.

I α large: reinforce local connections. α = I d: same number of connections at all scales. Frame 94/122 Frame 95/122 Overview of Overview of Previous work—ﬁnding short paths Complex Networks The problem Complex Networks

Basic deﬁnitions Basic deﬁnitions

Examples of Examples of Complex Networks Complex Networks

Properties of Properties of Complex Networks If there are no hubs and no underlying lattice, how can Complex Networks I If networks have hubs can also search well: Adamic Nutshell search be efficient? Nutshell et al. (2001) [1] −γ Basic models of Basic models of P(ki ) ∝ k complex networks complex networks i Generalized random Which friend of a is closest Generalized random networks networks b where k = degree of node i (number of friends). Scale-free networks to the target b? Scale-free networks Small-world networks Small-world networks Generalized affiliation Generalized affiliation Basic idea: get to hubs first networks networks I What does ‘closest’ mean? (airline networks). References References a I But: hubs in social networks are limited. What is ‘social distance’?

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Overview of Overview of Models Complex Networks Social distance—Bipartite afﬁliation networks Complex Networks

Basic deﬁnitions 1 2 3 4 contexts Basic deﬁnitions

Examples of Examples of One approach: incorporate identity. Complex Networks Complex Networks

Properties of Properties of Identity is formed from attributes such as: Complex Networks Complex Networks Nutshell Nutshell Geographic location I Basic models of Basic models of complex networks individuals complex networks I Type of employment Generalized random Generalized random networks a b c d e networks Religious beliefs Scale-free networks Scale-free networks I Small-world networks Small-world networks Generalized affiliation Generalized affiliation I Recreational activities. networks networks References References b Groups are formed by people with at least one similar d attribute. a unipartite network Attributes ⇔ Contexts ⇔ Interactions ⇔ Networks. e c Bipartite affiliation networks: boards and directors, Frame 99/122 Frame 100/122 movies and actors. Overview of Overview of Social distance—Context distance Complex Networks Models Complex Networks

Basic deﬁnitions Basic deﬁnitions

occupation Examples of Examples of Complex Networks Complex Networks Properties of Distance between two individuals xij is the height of Properties of Complex Networks lowest common ancestor. Complex Networks education health care Nutshell Nutshell Basic models of l=4 Basic models of complex networks complex networks Generalized random b=2 Generalized random high school kindergarten networks networks doctor Scale-free networks Scale-free networks teacher teacher nurse Small-world networks Small-world networks Generalized afﬁliation g=6 Generalized afﬁliation networks networks References j References v i k

xij = 3, xik = 1, xiv = 4.

a b c d e

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Overview of Overview of Models Complex Networks Models Complex Networks

Basic deﬁnitions Basic deﬁnitions

Examples of Examples of Complex Networks Generalized affiliation networks Complex Networks Properties of Properties of I Individuals are more likely to know each other the Complex Networks geography occupation age Complex Networks closer they are within a hierarchy. Nutshell Nutshell 0 100 Basic models of Basic models of I Construct z connections for each node using complex networks complex networks Generalized random Generalized random networks networks pij = c exp{−αxij }. Scale-free networks Scale-free networks Small-world networks Small-world networks Generalized affiliation Generalized affiliation networks networks

References References α = 0: random connections. I a b c d e I α large: local connections. [4] [22] [7] I Blau & Schwartz , Simmel , Breiger , Watts et al. [27]

Frame 103/122 Frame 104/122 Overview of Overview of The model Complex Networks The model Complex Networks

h=1 h=2 Basic definitions Basic definitions Examples of Examples of Complex Networks Complex Networks Properties of Triangle inequality doesn’t hold: Properties of Complex Networks Complex Networks i j i j Nutshell h=1 h=2 Nutshell Basic models of Basic models of complex networks complex networks Generalized random Generalized random h=3 networks networks Scale-free networks Scale-free networks Small-world networks Small-world networks Generalized affiliation Generalized affiliation networks i j,k i, j k networks References References

i, j yik = 4 > yij + yjk = 1 + 1 = 2.

T T ~vi = [1 1 1] , ~vj = [8 4 1] Social distance: 1 2 3 h xij = 4, xij = 3, x = 1. yij = min xij . ij h

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Overview of Overview of The model Complex Networks The model-results—searchable networks Complex Networks

Basic deﬁnitions α = 0 versus α = 2 for N ' 105: Basic deﬁnitions Examples of −0.5 Examples of Complex Networks Complex Networks q ≥ r Individuals know the identity vectors of Properties of −1 Properties of I Complex Networks q < r Complex Networks q

1. themselves, Nutshell 10 Nutshell −1.5 r = 0.05 2. their friends, Basic models of log Basic models of and complex networks complex networks Generalized random −2 Generalized random networks networks 3. the target. Scale-free networks Scale-free networks Small-world networks −2.5 Small-world networks 1 3 5 7 9 11 13 15 I Individuals can estimate the social distance between Generalized afﬁliation Generalized afﬁliation networks H networks their friends and the target. References q = probability an arbitrary message chain reaches a References I Use a greedy algorithm + allow searches to fail target. randomly. I A few dimensions help.

I Searchability decreases as population increases.

I Precise form of hierarchy largely doesn’t matter. Frame 107/122 Frame 108/122 Overview of Overview of The model-results Complex Networks Social search—Data Complex Networks

Basic deﬁnitions Basic deﬁnitions

Examples of Examples of Milgram’s Nebraska-Boston data: Complex Networks Complex Networks Properties of Properties of Model parameters: Complex Networks Complex Networks 12 Nutshell Adamic and Adar (2003) Nutshell 8 10 I N = 10 , Basic models of Basic models of complex networks I For HP Labs, found probability of connection as complex networks 8 Generalized random Generalized random ) I z = 300, g = 100, networks function of organization distance well fit by networks L Scale-free networks Scale-free networks ( 6 n I b = 10, Small-world networks exponential distribution. Small-world networks Generalized affiliation Generalized affiliation 4 networks networks I α = 1, H = 2; I Probability of connection as function of real distance References References 2 ∝ 1/r. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 L I hLmodeli ' 6.7 I Ldata ' 6.5

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Overview of Overview of Social Search—Real world uses Complex Networks Social Search—Real world uses Complex Networks

Basic deﬁnitions Basic deﬁnitions

Examples of Examples of Complex Networks Complex Networks

Properties of Properties of Complex Networks Complex Networks I Tags create identities for objects Nutshell Nutshell http://www.del.icio.us Recommender systems: I Website tagging: Basic models of Basic models of complex networks complex networks I (e.g., Wikipedia) Generalized random I Amazon uses people’s actions to build effective Generalized random networks networks Photo tagging: http://www.flickr.com Scale-free networks connections between books. Scale-free networks I Small-world networks Small-world networks Generalized affiliation Conflict between ‘expert judgments’ and Generalized affiliation I Dynamic creation of metadata plus links between networks I networks information objects. References tagging of the hoi polloi. References

I Folksonomy: collaborative creation of metadata

Frame 111/122 Frame 112/122 Overview of Overview of Nutshell Complex Networks References I Complex Networks

Basic deﬁnitions Basic deﬁnitions L. Adamic, R. Lukose, A. Puniyani, and B. Huberman. Examples of Examples of Complex Networks Complex Networks

Properties of Properties of Complex Networks Search in power-law networks. Complex Networks I Bare networks are typically unsearchable. Nutshell Phys. Rev. E, 64:046135, 2001. pdf () Nutshell I Paths are findable if nodes understand how network Basic models of Basic models of is formed. complex networks R. Albert, H. Jeong, and A.-L. Barabási. complex networks Generalized random Generalized random networks Error and attack tolerance of complex networks. networks I Importance of identity (interaction contexts). Scale-free networks Scale-free networks Small-world networks Nature, 406:378–382, July 2000. pdf ( ) Small-world networks Generalized affiliation Generalized affiliation I Improved social network models. networks networks References A.-L. Barabási and R. Albert. References I Construction of peer-to-peer networks. Emergence of scaling in random networks. I Construction of searchable information databases. Science, 286:509–511, 1999. pdf () P. M. Blau and J. E. Schwartz. Crosscutting Social Circles. Academic Press, Orlando, FL, 1984. Frame 113/122 Frame 114/122

Overview of Overview of References II Complex Networks References III Complex Networks

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