Modeling and Simulating Social Systems with MATLAB
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1 Modeling and Simulating Social Systems with MATLAB Lecture 8 Introduction to Graphs/Networks Olivia Woolley, Stefano Balietti, Lloyd Sanders, Dirk Helbing Chair of Sociology, in particular of Modeling and Simulation ETH Zürich
2 Schedule of the course Introduction to MATLAB Flash Talks Working on projects (seminar thesis) Modeling overview Introduction to social-science modeling and simulations Handing in seminar thesis and giving a presentation 2
3 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? 3
4 Seven Bridges of Königsberg (II) In order to approach the problem, Euler represented the important information as a graph: Source: wikipedia.org 4
5 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) 5
6 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.) Links of different types can exist (multiplex networks) A node can also contain information (attributes) 6
7 The social network 7
8 Graphs - examples Internet Map [opte project] Food Web [Martinez 91] Friendship Network [Moody 01] Protein Interactions [genomebiology.com] 8
9 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 9
10 Characterizing networks 1. Path properties 2. Node centrality measures and distributions 3. Local structure e.g. clustering 10
11 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) 11
12 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 12
13 Giant Component The giant component connects the vast majority of the nodes of a Graph. 13
14 Shortest paths The shortest path between i and j is minimum number of traversed edges I B A J I B A J D H D X H X Distance l(i,j) = shortest path between i and j Diameter D of the graph = max(l(i,j)) 14
15 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 transport through a network, e.g. how quick an epidemics can spread.
16 Centrality Measures The importance of a node can be captured by: Degree: number of connections Flux or strength: Sum of strength of all connections Closeness: Average distance (inverse of connection strength) form others. Eigenvector centrality (e.g. PageRank): Centrality score is higher the more high-scoring others a node is connected to. 16
17 Centrality Measures: Betweeness Centrality Idea: Controlling network flows The number of shortest paths passing through a node v. Namely, Example of a node v with high betweeness centrality v σ st = number of shortest paths from s to t σ st (v) = number of shortest paths from s to t passing through v 17
18 Statistical description of network topology: Degree Distribution Probability distribution function P(k) of the degree k of nodes Random graph: P(k) = binomial distribution Scale-free graph: P(k) = k -γ (power law) Source: 18
19 Examples of different network topologies Source: Wang (2003) 19
20 Local structure: Clustering Coefficient Local clustering coefficient C(i): fraction of pairs of neighbors of a node that are also neighbors of each other. Global clustering coefficient: network average It measures how clickish a network is. Source Costa (2008) Question: What is the local clustering coefficient for the node i?
21 Small Worlds: Clustering & small diameter 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. 21
22 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. 22
23 Small World Network Properties High clustered networks, like regular lattices, and small 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 total number of nodes 23
24 Small World model Source: Watts, D. J., & Strogatz, S. H. (1998) 24
25 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; 25
26 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 26
27 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 = [ ; ; ; ];
28 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 28
29 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 =
30 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
31 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 31
32 Alternatives Ways to Store Network Data Cell arrays grants more freedom in representing data structures, in spite of losing 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}; 32
33 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}; 33
34 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!! 34
35 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 35
36 Exporting and visualizing a graph in Gephi Download Gephi Open the.csv edge list that you just exported Visualize the network Compute the modularity score: 36
37 Live demo which should get this as a final result
38 References Handbook of graphs and networks: from the Genome to the Internet, edited by S. Bornholdt, H. G. Schuster. John Wiley and Sons, Watts,D.J.,& Strogatz, S.H. (1998).Collective dynamics of smallworld networks. nature, 393(6684), Newman, M.E. (2003).The structure and function of complex networks. SIAM review, 45(2), Newman, M. E. (2009). Networks: an introduction. Oxford University Press. Easley,D., &Kleinberg,J. (2010). Networks, crowds, and markets. Cambridge: Cambridge University Press. Xiao Fan Wang and Guanrong Chen Complex Networks: Small- World, Scale-Free and Beyond GEPHI:
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