851 0585 04 L Modeling and Simulating Social

851 -0585 -04 L – Modeling and Simulating Social Systems with MATLAB Lecture 7 – Simulations with Networks Karsten Donnay and Stefano Balietti Chair of Sociology, in particular of Modeling and Simulation © ETH Zürich | 2012 -11 -05

Schedule of the course Introduction to MATLAB 24. 09. 01. 10. 08. 10. 15. 10. 22. 10. 29. 10. Working on projects (seminar thesis) 05. 11. 12. 11. 19. 11. Introduction to social-science modeling and simulations 26. 11. 03. 12. 10. 12. 17. 12. 2012 -11 -05 Handing in seminar thesis and giving a presentation K. Donnay & S. Balietti / kdonnay@ethz. ch sbalietti@ethz. ch 2

Schedule of the course Introduction to MATLAB 01. 10. Different ways of Representing space 08. 10. Dynamical Systems (no-space) 15. 10. Cellular Automata (grid) 24. 09. 22. 10. 29. 10. Working on projects (seminar thesis) 05. 11. 12. 11. Networks (graphs) Continuous Space (…) 19. 11. 26. 11. 03. 12. 10. 12. 17. 12. 2012 -11 -05 Handing in seminar thesis and giving a presentation K. Donnay & S. Balietti / kdonnay@ethz. ch sbalietti@ethz. ch 3

Goals of Lecture 7: students will 1. Consolidate knowledge acquired during lecture 6, through brief repetition of the main points 2. Translate a research question into a simple model of simulation on networks� 3. Get a firm grasp in algorithms to generate different network topologies 4. Review a custom implementation of algorithms to efficiently: § Generate different networks topologies § Compute statistical properties 5. Run a simple simulation of the emergence of giant component in a random network 2012 -11 -05 K. Donnay & S. Balietti / kdonnay@ethz. ch sbalietti@ethz. ch 4

Repetition: 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) 2012 -11 -05 K. Donnay & S. Balietti / kdonnay@ethz. ch sbalietti@ethz. ch 5

Repetition: Networks Topologies § Random: Small average path length § Small clustering coefficient (in general) § § Small World: Small average path length § High clustering coefficient § § Scale Free: Slightly smaller average path length § Highest clustering coefficient § 2012 -11 -05 K. Donnay & S. Balietti / kdonnay@ethz. ch sbalietti@ethz. ch 6

Repetition: Networks Topologies § Random: Small average path length § Small clustering coefficient (in general) § § Small World: Homegeneous (Exponential) Degree Distr. Small average path length § High clustering coefficient § § Scale Free: Slightly smaller average path length § Highest clustering coefficient § 2012 -11 -05 K. Donnay & S. Balietti / kdonnay@ethz. ch sbalietti@ethz. ch Power Law Degree Distr. 7

Examples of different network topologies Source: Wang (2003) 2012 -11 -05 K. Donnay & S. Balietti / kdonnay@ethz. ch sbalietti@ethz. ch 8

From a problem to a scientific answer § Observation: § 2012 -11 -05 The role of social networks in shaping the public opinion is becoming increasingly more important K. Donnay & S. Balietti / kdonnay@ethz. ch sbalietti@ethz. ch 9

From a problem to a scientific answer § Question: § How easy is to be manipulated in a social network? Vs. 2012 -11 -05 K. Donnay & S. Balietti / kdonnay@ethz. ch sbalietti@ethz. ch 10

From a problem to a scientific answer § Literature Review: § 2012 -11 -05 Was the problem already treated? K. Donnay & S. Balietti / kdonnay@ethz. ch sbalietti@ethz. ch 11

From a problem to a scientific answer § Literature Review: § How existing models can be improved? ? 2012 -11 -05 K. Donnay & S. Balietti / kdonnay@ethz. ch sbalietti@ethz. ch 12

From a problem to a scientific answer § Creating a new model Let us assume ‘informed’ agents § Let us consider networks § § How easily can the opinion of other persons be drifted in online social netwrks? https: //github. com/msssm/Informed_Agents 2012 -11 -05 K. Donnay & S. Balietti / kdonnay@ethz. ch sbalietti@ethz. ch 13

From a problem to a scientific answer § Results Let us assume ‘informed’ agents § Let us consider networks § https: //github. com/msssm/Informed_Agents 2012 -11 -05 K. Donnay & S. Balietti / kdonnay@ethz. ch sbalietti@ethz. ch 14

Generators § How to generate random, realistic graphs? 1. Probabilistic generators 2. Degree-based generators 3. Process-based generators 2012 -11 -05 K. Donnay & S. Balietti / kdonnay@ethz. ch sbalietti@ethz. ch 15

Probabilistic Generators: Erdos-Renyi § Algorithm: Start with a number of nodes n (fully not connected) § Define probability of connection P § § For all the possible couples of nodes a link is created with probability P § The average number of links is given by: § p*n*(n-1)/2 § The greater P the higher the average degree of the network 2012 -11 -05 K. Donnay & S. Balietti / kdonnay@ethz. ch sbalietti@ethz. ch 16

Probabilistic Generators: Erdos-Renyi § random graph – 100 nodes, avg degree = 2 § Fascinating properties (phase transition) § But: unrealistic (Poisson degree distribution != power law) 2012 -11 -05 K. Donnay & S. Balietti / kdonnay@ethz. ch sbalietti@ethz. ch 17

E-R model & Percolation § The formation of the Giant Component is not a smooth process. Pc 1 N->infty § It emerge all of sudden when 0 p > 1/n § This phenomenon is 1 st called order phase -transition 2012 -11 -05 K K 0 K = avg(k) Pc = Prob( there is a giant connected component) K. Donnay & S. Balietti / kdonnay@ethz. ch sbalietti@ethz. ch 18

Graphs: Laws and patterns § Are real graphs random? § If we look at the data the answer most of the time is: NO!! count k 2012 -11 -05 degree K. Donnay & S. Balietti / kdonnay@ethz. ch sbalietti@ethz. ch k degree 19

Degree-based generators § Figure out the degree distribution (eg. , ‘Zipf’) § Assign degrees to nodes § Put edges, so that they match the original degree distribution 2012 -11 -05 K. Donnay & S. Balietti / kdonnay@ethz. ch sbalietti@ethz. ch 20

Process-based: Preferential attachment § Algorithm: Start with a random connected graph § At each time step create a new node and attach it to the others with probability: § § Ki = degree of node i § That is: if a node has many links, it will get more in the future… 2012 -11 -05 K. Donnay & S. Balietti / kdonnay@ethz. ch sbalietti@ethz. ch 21

Process-based: Preferential attachment § Generates power-law tails (richer-get-richer) § The degree distribution is a power law of the form: P(K) ~ k-3 2012 -11 -05 K. Donnay & S. Balietti / kdonnay@ethz. ch sbalietti@ethz. ch 22

Process-based: Preferential attachment § Generates power-law tails (richer-get-richer) § The degree distribution is a power law of the form: P(K) ~ k-3 But still, it does not reproduce the property of shrinking diameter in real evolving networks… 2012 -11 -05 K. Donnay & S. Balietti / kdonnay@ethz. ch sbalietti@ethz. ch 23

Get the lecture files from Git. Hub § Lecture files are also available on a Git. Hub repository: § git@github. com: msssm/lectures_files. git 2012 -11 -05 K. Donnay & S. Balietti / kdonnay@ethz. ch sbalietti@ethz. ch 24

Get the lecture files from Git. Hub § If you have already uploaded your public key in Git. Hub (see live presentation), § You can dowload the latest lecture files with the following command in GIT Bash git clone git@github. com: msssm/lectures_files. git § to stay updated: git pull 2012 -11 -05 K. Donnay & S. Balietti / kdonnay@ethz. ch sbalietti@ethz. ch 25

Lecture Files § Generate Random Networks § Small World Networks § Scale Free Networks § § Compute: Average Path Length § Clustering Coefficient § § Export to file: Parse an adjacency matrix and create an arc list § Write a cell array as a csv file § 2012 -11 -05 K. Donnay & S. Balietti / kdonnay@ethz. ch sbalietti@ethz. ch 26

Lecture Files § Simulate the emergence of the giant component in a random network (files online in. zip archive). emergence. m § emergence_video. m § emergence_smooth. m § § See live demo. 2012 -11 -05 K. Donnay & S. Balietti / kdonnay@ethz. ch sbalietti@ethz. ch 27

R igraph § R and Matlab integration to plot nice graphs (igraph library) § https: //github. com/tconring/Arabian-Spring 2012 -11 -05 K. Donnay & S. Balietti / kdonnay@ethz. ch sbalietti@ethz. ch 28

SOCNET Mailing List § Social Network Analysis Mailing List: http: //www. insna. org/pubs/socnet. html 2012 -11 -05 K. Donnay & S. Balietti / kdonnay@ethz. ch sbalietti@ethz. ch 29

Network Analysis Literacy Talk Tomorrow § Why Pretty Woman is a good recommendation for somebody who loves Star Wars V. ? § Prof. Katharina Zweig (Technical University Kaiserslautern, Germany) § Tuesday, 13 Nov 2012, 11. 00 -12. 00 ETH Zürich, Main Building, HG E 22 § http: //www. sg. ethz. ch/ 2012 -11 -05 K. Donnay & S. Balietti / kdonnay@ethz. ch sbalietti@ethz. ch 30

References § Jure Leskovec, Deepayan Chakrabarti, Jon M. Kleinberg, Christos § § § Faloutsos: Realistic, Mathematically Tractable Graph Generation and Evolution, Using Kronecker Multiplication. PKDD 2005: 133 -145 Xiao Fan Wang and Guanrong Chen Complex Networks: Small. World, Scale-Free and Beyond Damon Centola, The Spread of Behavior in an Online Social Network Experiment. Science, Vol. 329 no. 5996 pp. 1194 -1197 (2010) Duncan J. Watts, Steven H. Strogatz, Collective dynamics of 'smallworld' networks. Nature 393, 440 -442 (4 June 1998) Laszlo Barabasi, Scale-Free Networks: A Decade and Beyond. cience 24 July 2009: Vol. 325 no. 5939 Laszlo Barabasi web site: http: //nd. edu/~alb/ 2012 -11 -05 K. Donnay & S. Balietti / kdonnay@ethz. ch sbalietti@ethz. ch 31

References § About the KONY phenomenon: § http: //invisiblechildren. com/kony/ § http: //globalspin. blogs. time. com/2012/03/19/kony-2012 -mobs- takedowns-and-meltdowns-but-very-little-truth/? xid=newslettereurope-weekly § http: //www. youtube. com/watch? v=Y 4 Mnpz. G 5 Sqc 2012 -11 -05 K. Donnay & S. Balietti / kdonnay@ethz. ch sbalietti@ethz. ch 32

Projects § Today, there are no exercises. Instead, you can work on your projects and we will supervise you. 2012 -11 -05 K. Donnay & S. Balietti / kdonnay@ethz. ch sbalietti@ethz. ch 33