L Modeling and Simulating Social Systems with MATLAB

Transcription

1 L Modeling and Simulating Social Systems with MATLAB Lecture 6 Introduction to Graphs/Networks Karsten Donnay and Stefano Balietti Chair of Sociology, in particular of Modeling and Simulation ETH Zürich

2 Schedule of the course Introduction to MATLAB Working on projects (seminar thesis) Introduction to social-science modeling and simulations Handing in seminar thesis and giving a presentation 2

3 Schedule of the course Introduction to MATLAB Working on projects (seminar thesis) Different ways of Representing space Dynamical Systems (no-space) Cellular Automata (grid) Networks (graphs) Continuous Space ( ) Handing in seminar thesis and giving a presentation 3

4 Goals of Lecture 6: students will 1. Consolidate knowledge acquired during lecture 5, through brief repetition of the main points and exploration of the additional material (Prisoner Dilemma Tournament). 2. Discover the origin of the Graph Theory 3. Learn how to define rigorously a Graph in terms of its main mathematical properties 4. Apprehend which statistical properties characterize different network topologies. 5. Learn how to translate a graph into a convenient data structure 6. Be introduced to software for visualizing complex graphs. 4

5 Repetition: Game Theory Game Theory: powerful analytical framework to formalize the decision process of interacting individuals Nash equilibrium (NA): Every player adopt his best strategy given the strategy of the other players. NA does not mean that the solution found is (Pareto) efficient. Players can get stuck in local optima (eg. Prisoner Dilemma) with low payoff for all of them. Evolutionary Learning: In iterated games mechanisms of Selection, Replication, Mutation creates evolutionary dynamics. 5

6 Repetition: Programming a Simulator 5 phases (at least). They can be separated in the code with the %% (double percent) which creates an executable cell tournament.m -> B iteratedpd.m -> A play.m -> C Initialization Time loop Time and Agents loops are inverted Ah, if you played the tournament you noticed that TIT for TAT was outperformed by GRIM. Why? end Agents loop end Save data A B Update state C 6

7 Seven Bridges of Königsberg Graph Theory was born in 1736, when Euler posted the following problem: Is it possible to have a walk in the city of Königsberg, that crosses each of the seven bridges only once? 7

8 Seven Bridges of Königsberg (II) In order to approach the problem, Euler represented the important information as a graph: Source: wikipedia.org 8

9 Definition of Graph A graph consists of two entities: Source: Batagelj Nodes (vertices): N Links: L Edge: undirected link Arc: directed link The graph is defined as G = (N,L) 9

10 Properties of Links and Nodes A link can either be encoded as a: boolean flag (connection vs. no connection), or value or weight (distance, traveling time, etc.) A node can also contain information ( attributes ) When a Graph is enriched with extra information encoded either in the nodes or in the links, we call it Network. 10

11 Graphs - Examples NODES LINKS Protein interaction Proteins Metabolic reactions Internet Routers Communication channels Social networks Individuals Social relations WWW Web pages Hyperlinks Scientific Coauthorship Networks Authors Papers 11

12 Graphs - examples Internet Map [lumeta.com] Food Web [Martinez 91] Friendship Network [Moody 01] Protein Interactions [genomebiology.com] 12

13 Mathematical Description of a Graph A node can be characterized by: Degree k: Number of connections. Importance: Degree, Betweenness centrality, Closeness, Eigenvector centrality (e.g. PageRank). (More measures later on in this lecture and in the course) A graph can be characterized by: Degree distribution P(k): Fraction of nodes with k connections. 13

14 Degree Distribution Graphs can be classified by their topology, by measuring the degree-distribution function P(k), of the number of connections k per node: Random graph: P(k) = binomial distribution Scale-free graph: P(k) = k -γ (power law) Source: 14

15 Examples of different network topologies Source: Wang (2003) K. Donnay & S. Balietti / kdonnay@ethz.ch sbalietti@ethz.ch 15

16 Paths Path of length n = ordered collection of n+1 nodes. Eg: A,C,D,E in G =(N,L) n links. Eg: (A,C), (C,D),(D,E) in G =(N,L) Circuit = closed path (last node = first node) 16

17 Paths and connectedness A graph G=(N,L)is connected if and only if there exists a path connecting any two nodes in G is not connected Connected (Tree) Not Connected (Forest) Connected with loops 17

18 Giant Component The giant component connects the vast majority of the nodes of a Graph. 18

19 Shortest paths The shortest path between i and j is minimum number of traversed edges I D B A X Distance l(i,j) = shortest path between i and j Diameter D of the graph = max(l(i,j)) Over time D is shrinking/constant J H I D B A X J H 19

20 Shortest paths: Average Path Length Average path length is the average number of steps along the shortest paths for all possible pairs of network nodes. It is a measure of the efficiency of information or mass transport on a network, e.g. how quick an epidemics can spread.

21 Centrality Measures: Betweeness Centrality Express the number of shortest paths passing through a node v. Namely, v Example of a node v with high betweeness centrality 21

22 Cliques Clique is a complete subgraph. If a clique can not be contained by any larger clique, it is called the maximal clique Lipari-2010

23 Cliques Clique is a complete subgraph. If a clique can not be contained by any larger clique, it is called the maximal clique Lipari-2010

24 Cliques Clique is a complete subgraph. If a clique can not be contained by any larger clique, it is called the maximal clique Lipari-2010

25 Cliques Clique is a complete subgraph. If a clique can not be contained by any larger clique, it is called the maximal clique. {0,1,2}, {0,1,3}, {1,2,3} {2,3,4}, {0,1,2,3} are cliques; {0,1,2,3} and {2,3,4} are the maximal cliques Lipari-2010

26 Clustering Coefficient Clustering coefficient is the average fraction of pairs of neighbors of a node that are also neighbors of each other. It measures how clickish a network is. Question: What is the local clustering coefficient for the node i? Source Costa (2008)

27 The Small World Phenomenon Graphs are useful for modeling social networks, disease spreading, transportation, and so on One of the most famous graph studies is the Small World Experiment (S. Milgram), which shows that the minimum distance between any two persons in the world is almost never longer than through 5 friends. 27

28 Small World Example: Oracle of Bacon There is a web page finding the path from any actor at any time to the Hollywood actor Kevin Bacon. It can also be used to find the shortest path between any two actors. 28

29 Small World Network Properties High clustered networks, like regular lattices, and small average path lengths, like random graphs. A small-world network is defined to be a network where the typical distance L between two randomly chosen nodes grows logarithmically with n. 29

30 MATLAB Implementation A graph can be implemented in MATLAB via its adjacency matrix, i.e. an N x N matrix, defining how N nodes are connected to the other N-1 nodes: N = 10; A = zeros(n, N); A(1,2) = 1; A(10,4) = 1; 30

31 Graphs If the nodes are cities and the links define connections and travel times for the SBB network it looks like this: Basel 4 Geneva 3 Bern 2 1 Zurich 31

32 Graphs If the nodes are cities and the links define connections and travel times for the SBB 4 Geneva network it looks like this: 3 Bern Basel 2 1 Zurich A = A = [ ; ; ; ];

33 Graphs If the nodes are cities and the links define connections and travel times for the SBB 4 Geneva network it looks like this: 1:41 0:55 3 Bern Basel 2 0:57 0:54 1 Zurich 33

34 Graphs If the nodes are cities and the links define connections and travel times for the SBB 4 Geneva network it looks like this: 1:41 0:55 3 Bern Basel 2 0:57 0:54 1 Zurich A =

35 Alternatives Ways to Store Network Data Edge/Arc lists can easily stored to a file and loaded when needed 4 Geneva Basel 2 3 Bern 1 Zurich

36 Alternatives Ways to Store Network Data Cell arrays can contain vectors of different size 4 Geneva Basel 2 3 Bern 1 Zurich >> A = [2 3]; >> B = [1 3]; >> C = [1 2 4]; >> D = [3]; >> Net = {A;B;C;D}; >> Net{1}(1) >> ans = 2 36

37 Alternatives Ways to Store Network Data Cell arrays grants more freedom in representing data structures, in spite of loosing the simplicity and clarity of the matrix notation >> A = [2,54; 3,57]; >> B = [1,54; 3,55]; >> C = [1,57; 2,55; 4,101]; >> D = [3,101]; >> Net = {A;B;C;D}; 37

38 Alternatives Ways to Store Network Data Cell arrays grants more freedom in representing data structures, in spite of loosing the simplicity and clarity of the matrix notation >> A = [2,54; 3,57]; >> B = [1,54; 3,55]; >> C = [1,57; 2,55, 4,101]; >> D = [3,101]; Warning: you must validate your own data structure! >> Net = {A;B;C;D}; 38

39 Software Packages for Graph Visualization The following programs are valuable tools for representing and and visualizing networks: Pajek ( -> Easy to use NWB ( -> Good for Analysis Gephi ( -> New Visone ( -> made in Konstanz JUNG ( -> library Net Draw ( Pegasus ( -> for huge data Use them!! 39

40 Exporting and visualizing a graph in Gephi csvwrite ( filename,matrix)writes a matrix as a list of comma seperated values but works only with adjacency matrixes. Often. we need an edge list (cell array). Download two files from the web site: cell2csv.m export.m 40

41 Exporting and visualizing a graph in Gephi Download Gephi (maybe should do this first) Open the.csv edge list that you just exported Visualize the network Compute the modularity score: 41

42 Modularity Measures the strength of division of a network into modules (groups, clusters or communities). Many methods to compute it (compare edges towards a given groups vs randomly distributed) C A B Modularity scores are used for community detection algorithm. 42

43 Live demo which should get this as a final result

44 Projects Today, there are no exercises. Instead, you can work on your projects and we will supervise you. 44

45 References Handbook of graphs and networks: from the Genome to the Internet, edited by S. Bornholdt, H. G. Schuster. John Wiley and Sons, Jure Leskovec, Jon Kleinberg and Christos Faloutsos Graphs over Time: Densification Laws, Shrinking Diameters and Possible Explanations, KDD 2005 (Best Research paper award). Kleinberg, Jon (1999). "Authoritative sources in a hyperlinked environment" (PDF). Journal of the ACM 46 (5): Xiao Fan Wang and Guanrong Chen Complex Networks: Small- World, Scale-Free and Beyond GEPHI: