Modelling and Searching Networks Lecture 2 – Complex Networks Miniconference on the Mathematics of Computation MTH 707 Modelling and Searching Networks Lecture 2 – Complex Networks Dr. Anthony Bonato Ryerson University

Complex Networks web graph, social networks, biological networks, internet networks, … Networks - Bonato

What is a complex network? no precise definition however, there is general consensus on the following observed properties large scale evolving over time power law degree distributions small world properties other properties depend on the kind of network being discussed

Examples of complex networks technological/informational: web graph, router graph, AS graph, call graph, e-mail graph, bitcoin graph social: on-line social networks (Facebook, Twitter, LinkedIn,…), collaboration graphs, co-actor graph biological networks: protein interaction networks, gene regulatory networks, food networks

Example: the web graph nodes: web pages edges: links one of the first complex networks to be analyzed viewed as directed or undirected Networks - Bonato

Example: On-line Social Networks (OSNs) nodes: users on some OSN edges: friendship (or following) links maybe directed or undirected Anthony Bonato - The web graph

Example: Co-author graph nodes: mathematicians and scientists edges: co-authorship undirected

Example: Co-actor graph nodes: actors edges: co-stars Hollywood graph undirected

Heirarchical social networks social networks which are oriented from top to bottom information flows one way examples: Twitter, executives in a company, terrorist networks

Example: protein interaction networks nodes: proteins in a living cell edges: biochemical interaction undirected Introducing the Web Graph - Anthony Bonato

Bitcoin graph nodes: users edges: transactions or protocols

Properties of complex networks Large scale: relative to order and size web graph: order > trillion some sense infinite: number of strings entered into Google Facebook: > 1 billion nodes; Twitter: > 500 million nodes much denser (ie higher average degree) than the web graph protein interaction networks: order in thousands

Properties of complex networks Evolving: networks change over time web graph: billions of nodes and links appear and disappear each day Facebook: grew to 1 billion users denser than the web graph protein interaction networks: order in the thousands evolves much more slowly

Properties of Complex Networks Power law degree distribution for a graph G of order n and i a positive integer, let Ni,n denote the number of nodes of degree i in G we say that G follows a power law degree distribution if for some range of i and some b > 2, b is called the exponent of the power law Complex Networks

Properties of Complex Networks power law degree distribution in the web graph: (Broder et al, 01) reported an exponent b = 2.1 for the in-degree distribution (in a 200 million vertex crawl) Complex Networks

Interpreting a power law Many low-degree nodes Few high-degree nodes Complex Networks

Binomial Power law Highway network Air traffic network Complex Networks

Notes on power laws b is the exponent of the power law note that the law is approximate: constants do not affect it asymptotic: holds only for large n may not hold for all degrees, but most degrees (for example, sufficiently large or sufficiently small degrees) Complex Networks

Degree distribution (log-log plot) of a power law graph Complex Networks

Power laws in OSNs Complex Networks

Exercise 3.1 Which of the following are power law graphs? High school/secondary school graph. Nodes: students in a high school; edges: friendship links. Power grids. Nodes: generators, power plants, large consumers of power; edges: electrical cable. Banking networks. Nodes: banks; edges: financial transaction.

Graph parameters Wiener index, W(G) average distance: clustering coefficient: Wiener index, W(G) Complex Networks

Examples Cliques have average distance 1, and clustering coefficient 1 Triangle-free graphs have clustering coefficient 0 Clustering coefficient of following graph is 0.75. Note: average distance bounded above by diameter

Properties of Complex Networks Small world property small world networks introduced by Watts and Strogatz in 1998 low distances diam(G) = O(log n) L(G) = O(loglog n) higher clustering coefficient than random graph with same expected degree Complex Networks

Nuit Blanche Ryerson City of Toronto Four Seasons Hotel Frommer’s Greenland Tourism

Sample data: Flickr, YouTube, LiveJournal, Orkut (Mislove et al,07): short average distances and high clustering coefficients Complex Networks

Other properties of complex networks many complex networks (including on-line social networks) obey two additional laws: Densification Power Law (Leskovec, Kleinberg, Faloutsos,05): networks are becoming more dense over time; i.e. average degree is increasing |(E(Gt)| ≈ |V(Gt)|a where 1 < a ≤ 2: densification exponent Complex Networks

Densification – Physics Citations 1.69 Complex Networks

Densification – Autonomous Systems e(t) 1.18 n(t) Complex Networks

Decreasing distances (Leskovec, Kleinberg, Faloutsos,05): distances (diameter and/or average distances) decrease with time (Kumar et al,06): Diameter first, DPL second Check diameter formulas As the network grows the distances between nodes slowly grow Complex Networks

Diameter – ArXiv citation graph time [years] Complex Networks

Other properties Connected component structure: emergence of components; giant components Spectral properties: adjacency matrix and Laplacian matrices, spectral gap, eigenvalue distribution Small community phenomenon: most nodes belong to small communities (ie subgraphs with more internal than external links) …

Exercise 3.2 Compute the average distance of each of the following graphs. A star with n nodes (i.e. a tree of order n with one vertex of order n-1, the rest degree 1) A path with n nodes A wheel with n+1 nodes, n>2.