A Short Introduction to Networks

Transcription

1 2015 Aaron Clauset Aaron Assistant Professor of Computer Science University of Colorado Boulder External Faculty, Santa Fe Institute 051 A Short Introduction to Networks

2 what are networks?

3 what are networks? an approach a representation structure for complexity structure above individuals / components structure below system / population }networks system / population individuals / components

4 these lectures build intuition expose key concepts highlight some big questions teach a little math provide examples give pointers to further study not a substitute for technical coursework About 5,280,000 results (0.06 sec) it s a big field now

5 3 Network Analysis and Modeling Instructor: Aaron Clauset This graduate-level course will examine modern techniques for analyzing and modeling the structure and dynamics of complex networks. The focus will be on statistical algorithms and methods, and both lectures and assignments will emphasize model interpretability and understanding the processes that generate real data. Applications will be drawn from computational biology and computational social science. No biological or social science training is required. (Note: this is not a scientific computing course, but there will be plenty of computing for science.) Full lectures notes online (~150 pages in PDF)

6 3 Software R Python Matlab NetworkX [python] graph-tool [python, c++] GraphLab [python, c++] Standalone editors UCI-Net NodeXL Gephi Pajek Network Workbench Cytoscape yed graph editor Graphviz Data sets Mark Newman s network data sets Stanford Network Analysis Project Carnegie Mellon CASOS data sets NCEAS food web data sets UCI NET data sets Pajek data sets Linkgroup s list of network data sets Barabasi lab data sets Jake Hofman s online network data sets Alex Arenas s data sets

7 3 Mark Newman Professor of Physics University of Michigan External Faculty Santa Fe Institute

8 3

9 3

10 outline 1. what is a network? 2. representing a network 3. four basic concepts degrees paths position communities 4. network randomness 5. questions

11 vertices edges what is a vertex? distinct objects (vertices / nodes / actors) when are two vertices connected? pairwise relations (edges / links / ties)

12 biological social transportation informational telecommunications network vertex edge Internet(1) computer IP network adjacency Internet(2) autonomous system (ISP) BGP connection software function function call World Wide Web web page hyperlink documents article, patent, or legal case citation power grid transmission generating or relay station transmission line rail system rail station railroad tracks road network(1) intersection pavement road network(2) named road intersection airport network airport non-stop flight friendship network person friendship sexual network person intercourse metabolic network metabolite metabolic reaction protein-interaction network protein bonding gene regulatory network gene regulatory effect neuronal network neuron synapse food web species predation or resource transfer

13 social networks vertex: a person edge: friendship, collaborations, sexual contacts, communication, authority, exchange, etc. high school friendships

14 information networks vertex: books, blogs, webpages, etc. edge: citations, hyperlinks, recommendations, similarity, etc. political books political blogs

15 communication networks ISP network vertex: network router, ISP, address, mobile phone number, etc. edge: exchange of information IP-level Internet Enron

16 transportation networks vertex: city, airport, junction, railway station, river confluence, etc. global shipping edge: physical transportation of material global air traffic US Interstates

17 biological networks core metabolism vertex: species, metabolic, protein, gene, neuron, etc. edge: predation, chemical reaction, binding, regulation, activation, etc. grassland foodweb

18 what s a network? pop quiz

19 what s a network? Andromeda galaxy

20 what s a network? cauliflower fractal

21 what s a network? diamond lattice

22 representing networks

23 a simple network undirected unweighted no self-loops

24 a simple network adjacency matrix A undirected unweighted no self-loops adjacency list A 1 {2, 5} 2 {1, 3, 4} 3 {2, 4, 5, 6} 4 {2, 3} 5 {1, 3} 6 {3}

25 a less simple network Directed edge 1 2 Weighted edge 3 4 Multi-edge 5 6 Self-loop Weighted node undirected unweighted no self-loops

26 a less simple network adjacency matrix Directed edge Weighted edge 4 A {1, 1, 2} {2, 1} {2, 1} {1, 1, 2} Multi-edge 5 Weighted node 6 Self-loop adjacency list A 1!{(5, 1), (5, 1), (5, 2)} 2!{(1, 1), (2, 1 2 ), (3, 2), (3, 1), (4, 1)} 3!{(2, 2), (2, 1), (4, 2), (5, 4), (6, 4)} 4!{(2, 1), (3, 2)} 5!{(1, 1), (1, 1), (1, 2), (3, 4)} 6!{(3, 4), (6, 2)}

27 directed networks citation networks foodwebs* epidemiological others? directed acyclic graph directed graph WWW friendship? flows of goods, information economic exchange dominance neuronal transcription time travelers

28 bipartite networks } bipartite network no within-type edges authors & papers actors & movies/scenes musicians & albums people & online groups people & corporate boards people & locations (checkins) metabolites & reactions genes & substrings words & documents plants & pollinators

29 bipartite networks } red-red only 5 bipartite network one-mode }projections no within-type edges 3 blue-blue only authors & papers actors & movies/scenes musicians & albums people & online groups people & corporate boards people & locations (checkins) metabolites & reactions genes & substrings words & documents plants & pollinators

30 temporal networks t t +1 t +2 t +3 any network over time discrete time (snapshots), edges (i, j, t) continuous time, edges (i, j, t s, t)

31 describing networks what networks look like

32 describing networks what networks look like questions: how are the edges organized? how do vertices differ? does network location matter? are there underlying patterns? what we want to know what processes shape these networks? how can we tell?

33 describing networks degree

34 describing networks 3 degree: number of connections k k i = X j A ij 2 1

35 describing networks 3 degree: number of connections k k i = X j A ij 2 1 number of edges m = 1 2 nx k i = 1 2 i=1 nx i=1 nx A ij = 1 2 j=1 nx j=1 nx i=1 A ji

36 describing networks 3 degree: number of connections k k i = X j A ij 2 1 number of edges m = 1 2 nx k i = 1 2 i=1 nx i=1 nx A ij = 1 2 j=1 nx j=1 nx i=1 A ji mean degree hki = 1 n nx k i = 2m n i=1

37 describing networks 3 degree: number of connections k k i = X j A ij 2 degree sequence 1 {1, 2, 2, 2, 3, 4} degree distribution Pr(k) = apple 1, 1, 2, 3, 3, 1, 4,

38 degree distributions Zachary karate club Zachary, J. Anthropological Research 33, (1977).

39 degree distributions Zachary karate club Pr(k) degree, k Zachary, J. Anthropological Research 33, (1977).

40 degree distributions political blogs Adamic and Glance, WWW Workshop on the Weblogging Ecosystem (2005) Karrer and Newman, Physical Review E 83, (2010)

41 degree distributions Pr(k) Pr(k) Pr(K k) umm political blogs * degree, k Adamic and Glance, WWW Workshop on the Weblogging Ecosystem (2005) Karrer and Newman, Physical Review E 83, (2010)

42 degree distributions Pr(k) Pr(K k) Pr(k) political blogs * degree, k Adamic and Glance, WWW Workshop on the Weblogging Ecosystem (2005) Karrer and Newman, Physical Review E 83, (2010)

43 describing networks 3 degree: number of connections k k i = X j A ij 2 1 when does node degree matter?

44 network degrees spreading processes on networks biological (diseases) SIS and SIR models social (information) SIS, SIR models threshold models Susceptible-Infected-Susceptible S I Susceptible-Infected-Recovered threshold S I R

45 network degrees 2004 relationship network in Jefferson High this subgraph is 52% of school who are most important disease spreaders? Bearman, et al., Amer. J. Sociology 110, (2004)

46 network degrees 2007 amazon.com viral marketing viral trace for Oh my Goddess! community very high degrees! most attempts to influence fail Leskovec et al., ACM Trans. on the Web 1, article 5 (2007)

47 network degrees cascade epidemic branching process spreading process R 0 = net reproductive rate = average degree hki R 0 = caveat: ignores network structure, dynamics, etc.

48 network degrees R 0 < 1 R 0 =1 R 0 > 1 sub-critical small outbreaks critical outbreaks of all sizes super-critical global epidemics

49 network degrees vaccination disease R0 minimum Measles % Chicken pox % Polio % Smallpox % H1N1 influenza all super-critical data from Lauren Ancel Meyers (UT Austin)

50 network degrees vaccination disease R0 minimum Measles % Chicken pox % Polio % Smallpox % H1N1 influenza data from Lauren Ancel Meyers (UT Austin)

51 Volz, J. Math. Bio. 56, (2008) Bansal et al., J. Royal Soc. Interface 4, (2007) Karrer and Newman, Phys. Rev. E 82, (2010) Salathe and Jones, PLoS Comp. Bio. 6, e (2010) network degrees bigger cascades low overlap among neighbors more expander-like higher transmission probability lower activation threshold

52 network degrees bigger cascades low overlap among neighbors more expander-like higher transmission probability lower activation threshold Volz, J. Math. Bio. 56, (2008) Bansal et al., J. Royal Soc. Interface 4, (2007) Karrer and Newman, Phys. Rev. E 82, (2010) Salathe and Jones, PLoS Comp. Bio. 6, e (2010) smaller cascades larger overlap among neighbors more triangles smaller "communities" more spatial-like organization lower transmission probability higher activation threshold

53 describing networks from degrees to components

54 network terminology component: a group of connected nodes

55 network components the percolation game: choose X random pairs connect them repeat (count components) 15 hki =0.000 # nodes in component size, s

56 network components the percolation game: choose X random pairs connect them repeat (count components) 15 hki =0.533 # nodes in component size, s

57 network components the percolation game: choose X random pairs connect them repeat (count components) 15 hki =1.066 # nodes in component size, s

58 network components the percolation game: choose X random pairs connect them repeat (count components) 15 hki =1.600 # nodes in component size, s

59 network components the percolation game: choose X random pairs connect them repeat (count components) 15 hki =2.133 # nodes in component size, s

60 network components the percolation game: choose X random pairs connect them repeat (count components) 15 hki =2.133 # nodes in component size, s

61 the giant component add edges randomly at first, components are small and disconnected at critical value, these components begin linking beyond, all nodes in single giant component all components: small + disconnected Size of the giant component S one giant component, some tiny components phase transition Mean degree c

62 network components component = connected group component dynamics are independent (no information flow) phase transition in global connectivity: from smoothly increasing mean degree open questions: other network properties + phase transitions adaptive wiring local vs. global connectivity rules

63 network degrees how could we halt the spread? break network into disconnected pieces or

64 network degrees two networks... Error and attack tolerance of complex networks Réka Albert, Hawoong Jeong & Albert-László Barabási 2000 homogeneous in degree a heterogeneous in degree b Exponential Albert et al., Nature 406, (2000)

65 network degrees two networks a 0.4 degree distributions 10 0 Probability Probability Degree Exponential b Degree Albert et al., Nature 406, (2000)

66 network degrees strategy: delete vertices 1. uniformly at random ( failure ) 2. in order of degree ( attack ) a hom. E SF het. 12 a 10 Failure Attack Exponential b diameter Albert et al., Nature 406, (2000) fraction of vertices deleted

67 network degrees what promotes spreading? high-degree vertices* centrally-located vertices homogeneous in degree a heterogeneous in degree b Exponential Albert et al., Nature 406, (2000)

68 network degrees strategy: delete vertices 3. build fire breaks patient 0 vaccinated = deleted ( fire break )

69 network degrees patient 0 vaccination strategies the front line (hospitals) high degree nodes the vulnerable (old/young) effective buffer

70 network degrees but, in social networks

71 network degrees 2007 classic information marketing message saturation degree is most important broadcast influence Watts and Dodds, J. Consumer Research 34, (2007)

72 network degrees 2007 network (decentralized) marketing high-degree = opinion leader high-degree alone = irrelevant a cascade requires a legion of susceptibles (a system-level property) network influence Watts and Dodds, J. Consumer Research 34, (2007)

73 network degrees ( ) The only thing worse than being talked about is not being talked about. "influence" not really about the influencer as much about the susceptibles warning: there is no quantitative, agreed-upon, measurable definition of influence

74 network degrees how to start a social movement?

75 network degrees how to start a social movement? 2010, Derek Sivers

76 network degrees degrees: first-order description of network structure direct implications for spreading processes cascades require both susceptible population and spreaders open questions: impact of degrees on other dynamics feedback from dynamics to degree [adaptive behaviors like self-quarantine, evangelism] when does degree not matter Pr(K k) degree, k

77 describing networks path

78 describing networks a b path: number of hops between two nodes `a!b =2

79 describing networks a b path: number of hops between two nodes `a!b =2 the longer the path, the weaker the coupling

80 network paths present

81 network paths 1967 Milgram, Psychology Today 1, (1967)

82 network paths degrees of separation h`i =6 Boston Omaha Milgram, Psychology Today 1, (1967)

83 network paths 1998 Watts and Strogatz, Nature 393, (1998)

84 network paths all links local most nodes far away high clustering Watts and Strogatz, Nature 393, (1998)

85 network paths most links local some links random most nodes near high clustering short paths can be found Watts and Strogatz, Nature 393, (1998)

86 network paths all links random Erdos-Renyi graph most nodes near short paths hard to find no clustering Watts and Strogatz, Nature 393, (1998)

87 it s a small world after all big world high clustering small world high clustering small world low clustering Watts and Strogatz, Nature 393, (1998)

88 it s a small world after all ,836 geo-located users most links local remaining links span all scales high clustering small diameter Liben-Nowell et al., PNAS 102, (2005)

89 network paths path = sequence of edges a b path length related to coupling strength many short paths = global coupling social networks have short paths and many loops nearly all networks are "small world" open questions: what processes shrink big / grow small worlds? social information filtering

90 describing networks position

91 describing networks position = centrality: measure of positional importance geometric connectivity harmonic centrality closeness centrality betweenness centrality degree centrality eigenvector centrality PageRank Katz centrality many many more Boldi & Vigna, arxiv: (2013) Borgatti, Social Networks 27, (2005)

92 describing networks position = centrality: harmonic, closeness centrality importance = being in center of the network harmonic c i = 1 n 1 X j6=i 1 d ij length of shortest path Boldi & Vigna, arxiv: (2013) Borgatti, Social Networks 27, (2005) distance: d ij = `ij if j reachable from i 1 otherwise

93 describing networks position = centrality: PageRank, Katz, eigenvector centrality importance = sum of importances * of nodes that point at you I j I i = X j!i k j or, the left eigenvector of Ax = x Boldi & Vigna, arxiv: (2013) Borgatti, Social Networks 27, (2005) *modulo several technical details

94 network position an example Giovanni de Medici

95 network position 1993 Duomo Padgett and Ansell, Amer. J. Sociology 98, (1993) Palazzo Medici Giovanni de Medici

96 network position: closeness Peruzzi Strozzi Bischeri Lamberteschi Castellani Ridolfi Tornabuoni Guadagni nodes: Florence families edges: inter-family marriages Barbadori Medici Albizzi which family is most central? Acciaiuoli Salviati Ginori Pazzi

97 network position: closeness Peruzzi Strozzi Bischeri Lamberteschi Castellani Ridolfi Tornabuoni Guadagni nodes: Florence families edges: inter-family marriages Barbadori Medici Albizzi Tornabuoni? Acciaiuoli Salviati Ginori Pazzi

98 network position: closeness Peruzzi Strozzi Bischeri Lamberteschi Castellani Ridolfi Tornabuoni Guadagni nodes: Florence families edges: inter-family marriages Barbadori Medici Albizzi Tornabuoni? Acciaiuoli Salviati Pazzi Ginori C Tornabuoni =

99 network position: closeness Peruzzi Strozzi Bischeri Lamberteschi Castellani Ridolfi Tornabuoni Guadagni nodes: Florence families edges: inter-family marriages Barbadori Medici Albizzi Tornabuoni? Acciaiuoli Salviati Pazzi Ginori C Tornabuoni =

100 network position: closeness Peruzzi Strozzi Bischeri Lamberteschi Castellani Ridolfi Tornabuoni Guadagni nodes: Florence families edges: inter-family marriages Barbadori Albizzi Tornabuoni? Medici Acciaiuoli Salviati Pazzi Ginori C Tornabuoni = =

101 network position: closeness Peruzzi Strozzi Bischeri Lamberteschi Castellani Ridolfi Tornabuoni Guadagni nodes: Florence families edges: inter-family marriages Barbadori Medici Albizzi Strozzi? Acciaiuoli Salviati Ginori Pazzi

102 network position: closeness Peruzzi Strozzi Bischeri Lamberteschi Castellani Ridolfi Tornabuoni Guadagni nodes: Florence families edges: inter-family marriages Barbadori Albizzi Strozzi? Medici Acciaiuoli Salviati Pazzi Ginori C Strozzi =4 1 1 =

103 network position: closeness Peruzzi Strozzi Bischeri Lamberteschi Castellani Ridolfi Tornabuoni Guadagni nodes: Florence families edges: inter-family marriages Barbadori Medici Albizzi Medici? Acciaiuoli Salviati Ginori Pazzi

104 network position: closeness Peruzzi Strozzi Bischeri Lamberteschi Castellani Ridolfi Tornabuoni Guadagni nodes: Florence families edges: inter-family marriages Barbadori Albizzi Medici. Medici Acciaiuoli Salviati Pazzi Ginori C Medici =6 =

105 network position: closeness Pe B Str Gu L Medici 9.5 Guadagni 7.92 Albizzi 7.83 Strozzi 7.67 C R T Ridolfi 7.25 Bischeri 7.2 B Ac Medici Sal Alb Gi Tornabuoni 7.17 Barbadori 7.08 Peruzzi 6.87 Castellani 6.87 Salviati 6.58 P Acciaiuoli 5.92 Ginori 5.33 Lamberteschi 5.28 Pazzi 4.77

106 network position actually, it s complicated... Padgett and Ansell, Amer. J. Sociology 98, (1993)

107 network position most centralized vast wilderness of in-between most decentralized

108 network position most centralized vast wilderness of in-between most decentralized

109 network position most centralized vast wilderness of in-between most decentralized

110 network position most centralized vast wilderness of in-between most decentralized

111 network position positions: geometric description of network structure core vs. periphery Pe B Str Gu L centrality = importance, influence C B R T Alb Medici Ac Sal Gi open questions: P position and dynamics what does position predict? when does position not matter?

112 describing networks community structure

113 describing networks community structure: a group of vertices that connect to other groups in similar ways D t assortative community structure (edges inside the groups) YQKDAPNY DPNY PNY 13

114 community structure internal edges bridge edges community structure: a group of vertices that connect to other groups in similar ways D t assortative community structure (edges inside the groups) YQKDAPNY DPNY PNY 13

115 C D community structure community structure: a group of vertices that connect to other groups in similar ways mixing matrix 9 s D t 600 YQKDAPNY DPNY PNY 13

116 community structure assortative edges within groups disassortative edges between groups ordered linear group hierarchy core-periphery dense core, sparse periphery

117 community structure enormous interest, especially since 2000 dozens of algorithms for extracting various large-scale patterns hundreds of papers published spanning Physics, Computer Science, Statistics, Biology, Sociology, and more this was one of the first: assortative ordered disassortative core-periphery citations on Google Scholar PNAS 2002

118 network communities 1983 most new job opportunities from weak ties within-community links = strong bridge links = weak Granovetter, Amer. J. Sociology 78, (1973) Granovetter, Sociological Theory 1, (1983)

119 network communities 1983 most new job opportunities from weak ties within-community links = strong bridge links = weak why? information propagates quickly within a community, but slowly between communities Granovetter, Amer. J. Sociology 78, (1973) Granovetter, Sociological Theory 1, (1983)

120 network communities blogs 759 liberal 735 conservative liberal conservative Adamic and Glance, WWW Workshop on the Weblogging Ecosystem (2005)

121 network communities Hierarchical block structures and high-resolution model selection in large networks Tiago P. Peixoto ta-an] 25 Mar 2014 Institut für Theoretische Physik, Universität Bremen, Hochschulring 18, D Bremen, Germany Discovering and characterizing the large-scale topological features in empirical networks are crucial steps in understanding how complex systems function. However, most existing methods used to obtain the modular structure of networks suffer from serious problems, such as being oblivious to the statistical evidence supporting the discovered patterns, which results in the inability to separate actual structure from noise. In addition to this, one also observes a resolution limit on the size of communities, where smaller but well-defined clusters are not detectable when the network becomes large. This phenomenon occurs not only for the very popular approach of modularity optimization, which lacks built-in statistical validation, but also for more principled methods based on statistical inference and model selection, which do incorporate statistical validation in a formally correct way. Here we construct a nested generative model that, through a complete description of the entire network hierarchy at multiple scales, is capable of avoiding this limitation, and enables the detection of modular structure at levels far beyond those possible with current approaches. Even with this increased resolution, the method is based on the principle of parsimony, and is capable of separating signal from noise, and thus will not lead to the identification of spurious modules even on sparse networks. Furthermore, it fully generalizes other approaches in that it is not restricted to purely assortative mixing patterns, directed or undirected graphs, and ad hoc hierarchical structures such as binary trees. Despite its general character, the approach is tractable, and can be combined with advanced techniques of community detection to yield an efficient algorithm that scales well for very large networks. not take into account the statistical evidence associated with this deviation, and as a result it is incapable of separating actual structure from those arising simply of The detection of communities and other large-scale statistical fluctuations of the null model, and it even finds structures in networks has become perhaps one of the high-scoring partitions in fully random graphs [13]. This largest undertakings in network science [1, 2]. It is problem not specific to modularity and is a inferred characterfigure 5. The blog and Glance Left:is Topmost partition of the hierarchy with motivated by political the desire to network be able oftoadamic characterize the [67]. Peixoto, Physical Review X 4, (2013) istic with shared bybundling the vastfollowing majoritythe of inferred methodshierarchy proposed[68] the nested model. Right: The same network, using a circular layout, edge I. INTRODUCTION

122 network communities 2004 amazon.com co-purchasing network Clauset et al., Physical Review E 70, (2004)

123 network communities 2004 amazon.com co-purchasing network find partition that maximizes modularity Q of those groups n = 409,687 items m = 2,464,630 edges Clauset et al., Physical Review E 70, (2004)

124 network communities purchases = interests interests = clustered clustered network Clauset et al., Physical Review E 70, (2004)

125 network communities political books on amazon (2004) 2004 Valdis Krebs

126 network communities Efficiently inferring community structure in bipartite networks Daniel B. Larremore, 1,2 Aaron Clauset, 3,4,5 and Abigail Z. Jacobs Communicable Disease Dynamics, Harvard School of Public Health, Boston, Massachusetts IMDB actor-movie bipartite network Actors (53158) 1 Czech, Danish, German 2 French, Dutch, Cantonese, Greek, Filipino, Finnish Movies (39768) Japanese, Russian Adult 6 }English Larremore et al., Physical Review E 90, (2014)

127 network communities community = vertices with same pattern of intercommunity connections network macro-structure finding them like network clustering allow us to coarse grain system structure [decompose heterogeneous structure into homogeneous blocks] constrains network synchronization, information flows, diffusion, influence

128 network communities community = vertices with same pattern of intercommunity connections network macro-structure finding them like network clustering allow us to coarse grain system structure [decompose heterogeneous structure into homogeneous blocks] constrains network synchronization, information flows, diffusion, influence open questions: what processes generate communities? what impact on dynamics? network function?

129 basic network structure paths: 'distance' across the network node degree: individual connectivity communities and groups 9 D position: centrality and status many others... 0 APNY

130 advanced networks questions how does network structure shape system dynamics? especially spreading processes: information, disease, etc. where does large-scale structure come from? communities, hierarchy, core-periphery, etc. how does network metadata relate to network structure? edge weights, node attributes, time, etc. how do system dynamics shape network structure? when edges rewire and vertices come and go how does network structure influence network function? robustness, resilience, adaptability, design

131 advanced networks methods random graph models Erdos-Renyi networks, configuration model, stochastic block models is a network pattern real? null models for networks (tricky ) network time series analysis temporal networks, non-stationary processes, anomalies network predictions predicting missing / spurious / future connections, anomalies & change points, role identification, synchronization, etc. micro-macro tests of micro-level processes that produce macro-level patterns and much more see NetSci conference (annually, in early June) for latest / greatest

132 advanced networks methods random graph models Erdos-Renyi networks, configuration model, stochastic block models is a network pattern real? null models for networks (tricky ) network time series analysis temporal networks, non-stationary processes, anomalies network predictions predicting missing / spurious / future connections, anomalies & change points, role identification, synchronization, etc. micro-macro tests of micro-level processes that produce macro-level patterns and much more see NetSci conference (annually, in early June) for latest / greatest

133 fooled by randomness observe pattern X is X interesting? answer NO, if X random model can be produced by a simple simple random graph models: edges exist probabilistically Pr(i! j i, j ) may depend on node attributes (like degree, community, age/sex/location, etc.)

134 network null models feature degree distribution real-world networks configuration SBM DC SBM HRG heavy-tailed G(n, p) triangle density social: high non-social: low diameter small large-scale structure communities, core-periphery, hierarchies

135 network null models feature real-world networks configuration SBM DC SBM HRG G(n, p) degree distribution heavy-tailed Poisson triangle density social: high non-social: low O(n 1 ) diameter small O(ln n) large-scale structure communities, core-periphery, hierarchies none

136 network null models feature degree distribution real-world networks configuration SBM DC SBM HRG G(n, p) heavy-tailed Poisson specified triangle density social: high non-social: low O(n 1 ) O(n 1 ) diameter small O(ln n) O(ln n) large-scale structure communities, core-periphery, hierarchies none coreperiphery

137 network null models feature real-world networks configuration SBM DC SBM HRG G(n, p) degree distribution heavy-tailed Poisson specified Poisson* specified heavytailed triangle density social: high non-social: low O(n 1 ) O(n 1 ) low* low* high or low O(ln n) O(ln n) diameter small small* small* small large-scale structure communities, core-periphery, hierarchies none coreperiphery communit ies, coreperiphery communi ties, coreperiphery hierarchies, coreperiphery SBM: Holland et al., Social Networks 5, 109 (1983) DC SBM: Karrer and Newman, Physical Review E 83, (2010) HRG: Clauset et al., Nature 453, (2008)

138 frequently asked questions

139 frequently asked questions is my system a network? instead ask: what are the vertices? when are two vertices connected? what else do we know about the vertices and edges? does a network representation buy me anything new? is the network disorganized?

140 frequently asked questions is my network scale-free? is it a small world? small-world network scale-free network

141 frequently asked questions is my network scale-free? is it a small world? your network (system) almost surely is not scale free it almost surely is a small world (short paths) small-world network scale-free network

142 frequently asked questions are the small world or "scale free" network models useful? small-world network scale-free network

143 frequently asked questions are the small world or "scale free" network models useful? not as models of real systems only as tools for building intuition small-world network scale-free network

144 frequently asked questions are social networks complex adaptive systems? it depends. maybe? human nature is probably stable, but culture, technology and institutions are highly non-stationary context matters

145 frequently asked questions are social networks different from other networks? yes. homophily (attributes correlate across edges) = birds of a feather (age, politics, diet, interests, attitudes, preferences, race, etc.)

146 parting thoughts on networks networks are cool!

147 parting thoughts on networks networks are cool! but also complicated objects = enormous structural diversity still figuring out how to describe structure well we have only scratched the surface auxiliary data (weights, attributes, time) multiplex networks stronger tools for testing hypotheses applications abound [new ideas often come from these] structure + dynamics = function how does structure constrain dynamics, robustness, etc. to what degree does structure = function? formation mechanisms where does structure come from?

148 fin 3

149 selected references The structure and function of complex networks. M. E. J. Newman, SIAM Review 45, (2003). The Structure and Dynamics of Networks. M. E. J. Newman, A.-L. Barabási, and D. J. Watts, Princeton University Press (2006). Hierarchical structure and the prediction of missing links in networks. A. Clauset, C. Moore, and M. E. J. Newman, Nature 453, (2008). Modularity and community structure in networks. M. E. J. Newman, Proc. Natl. Acad. Sci. USA 103, (2006). Why social networks are different from other types of networks. M. E. J. Newman and J. Park, Phys. Rev. E 68, (2003) Random graphs with arbitrary degree distributions and their applications. M. E. J. Newman, S. H. Strogatz, and D. J. Watts, Phys. Rev. E 64, (2001). Comparing community structure identification. L. Danon, A. Diaz-Guilera, J. Duch and A. Arenas. J. Stat. Mech. P09008 (2005). Characterization of Complex Networks: A Survey of measurements. L. daf. Costa, F. A. Rodrigues, G. Travieso and P. R. VillasBoas. arxiv:cond-mat/ (2005). Evolution in Networks. S.N. Dorogovtsev and J. F. F. Mendes. Adv. Phys. 51, 1079 (2002). Revisting scale-free networks. E. F. Keller. BioEssays 27, (2005). Currency metabolites and network representations of metabolism. P. Holme and M. Huss. arxiv: (2008). Functional cartography of complex metabolic networks. R. Guimera and L. A. N. Amaral. Nature 433, 895 (2005). Graphs over Time: Densification Laws, Shrinking Diameters and Possible Explanations. J. Leskovec, J. Kleinberg and C. Faloutsos. Proc. 11th ACM SIGKDD Intl. Conf. on Knowledge Discovery and Data Mining The Structure of the Web. J. Kleinberg and S. Lawrence. Science 294, 1849 (2001). Navigation in a Small World. J. Kleinberg. Nature 406 (2000), 845. Towards a Theory of Scale-Free Graphs: Definitions, Properties and Implications. L. Li, D. Alderson, J. Doyle, and W. Willinger. Internet Mathematics 2(4), A First-Principles Approach to Understanding the Internet s Router-Level Topology. L. Li, D. Alderson, W. Willinger, and J. Doyle. ACM SIGCOMM Inferring network mechanisms: The Drosophila melanogaster protein interaction network. M. Middendorf, E. Ziv and C. H. Wiggins. Proc. Natl. Acad. Sci. USA 102, 3192 (2005). Robustness Can Evolve Gradually in Complex Regulatory Gene Networks with Varying Topology. S. Ciliberti, O. C. Martin and A. Wagner. PLoS Comp. Bio. 3, e15 (2007). Simple rules yield complex food webs. R. J. Williams and N. D. Martinez. Nature 404, 180 (2000). A network analysis of committees in the U.S. House of Representatives. M. A. Porter, P. J. Mucha, M. E. J. Newman and C. M. Warmbrand. Proc. Natl. Acad. Sci. USA 102, 7057 (2005). On the Robustness of Centrality Measures under Conditions of Imperfect Data. S. P. Borgatti, K. M. Carley and D. Krackhardt. Social Networks 28, 124 (2006).

150 describing networks homophily and assortative mixing like links with like

151 assortative mixing homophily and assortative mixing like links with like ~v i ~v j assortativity coefficient quantifies homophily r vertex attributes three types: scalar attributes vertex degrees categorical variables Newman, Phys. Rev. E 67, (2003).

152 assortative mixing ~v i ~v j homophily and assortative mixing like links with like scalar attributes: mean value across ties µ = 1 2m = 1 2m X X A ij v i i j X k i v i i Newman, Phys. Rev. E 67, (2003).

153 assortative mixing ~v i ~v j! µ = 1 X k i v i 2m i homophily and assortative mixing like links with like scalar attributes: covariance across ties cov(v i,v j )= P ij A ij(v i µ)(v j µ) P = 1 2m = 1 2m ij A ij X A ij v i v j µ 2 ij X ij A ij k i k j 2m v i v j Newman, Phys. Rev. E 67, (2003).

154 assortative mixing ~v i ~v j homophily and assortative mixing like links with like assortativity coefficient (scalar) r = cov(v i,v j ) var(v i,v j ) P ij = (A ij k i k j /2m) v i v j P ij k i ij k i k j /2m [this is just a Pearson correlation across edges] 1 apple r apple 1 Newman, Phys. Rev. E 67, (2003).

155 assortative mixing 40 ~v i ~v j age of wife [years] age of husband [years] (top) scatter plot of ages of 1141 married couples at time of marriage [1995 US National Survey of Family Growth] number r =0.574 strongly assortative (bottom) histogram of age differences (M-F) for same data age difference [years] Newman, Phys. Rev. E 67, (2003).

156 assortative mixing k i k j homophily and assortative mixing like links with like degree: just another scalar * Newman, Phys. Rev. E 67, (2003). * the assortativity coefficient formula simplifies somewhat in this case. see the Ref in the left corner for more details

157 assortative mixing k i k j social technological biological degree network type size n assortativity r error σ r physics coauthorship undirected biology coauthorship undirected mathematics coauthorship undirected film actor collaborations undirected company directors undirected student relationships undirected address books directed power grid undirected Internet undirected World-Wide Web directed software dependencies directed protein interactions undirected metabolic network undirected neural network directed marine food web directed freshwater food web directed Newman, Phys. Rev. E 67, (2003).

158 assortative mixing homophily and assortative mixing like links with like matrix sum marginals categorical variables: let e ij be fraction of edges connecting vertices of type i to vertices of type j X e ij =1 ij X e ij = a i j i X e ij = b j Newman, Phys. Rev. E 67, (2003).

159 assortative mixing homophily and assortative mixing like links with like categorical variables: * assortativity coefficient P i r = e P ii P i a ib i 1 i a ib i = Tr e e2 1 e 2 Newman, Phys. Rev. E 67, (2003). * this equation is equivalent to the popular modularity measure Q used to score the strength of community structure

160 assortative mixing 1992 study of heterosexual partnerships in San Francisco* (bipartite network) men women black hispanic white other a i black hispanic white other b i r =0.621 strongly assortative Newman, Phys. Rev. E 67, (2003). *Catania et al., Am. J. Public Health 82, (1992).

161 assortative mixing ~v i ~v j homophily and assortative mixing like links with like random graphs tend to be disassortative r apple 0 because the mixing is uniform social networks (apparently) highly assortative, in every way (attribute, degree, category) extremal values r { 1, 1} suggest underlying mechanism on that variable Newman, Phys. Rev. E 67, (2003).

162 describing networks motifs

163 describing networks motifs: small subgraphs (of interest), which we then count compare counts against null model (random graph model) Milo et al., Science 298, (2002).

164 describing networks motifs: small subgraphs (of interest), which we then count compare counts against null model (random graph model) efficient counting is tricky (combinatorics + graph isomorphism) choice of null model key lots of work in this area, mainly in molecular biology and neuroscience see Sporns and Kotter, PLoS Biol. 2, e369 (2004) Matias et al., REVSTAT 4, (2006) Wong et al., Brief. in Bioinfo. 13, (2011)