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2012-11-05 © ETH Zürich |
851-0585-04L – Modeling and Simulating Social Systems with MATLAB Lecture 7 – Simulations with Networks
© ETH Zürich |
Chair of Sociology, in particular of
Modeling and Simulation
Karsten Donnay and Stefano Balietti
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2012-11-05 K. Donnay & S. Balietti / [email protected] [email protected] 2
Schedule of the course 24.09. 01.10. 08.10. 15.10. 22.10. 29.10. 05.11. 12.11. 19.11. 26.11. 03.12. 10.12. 17.12.
Introduction to MATLAB
Introduction to social-science modeling and simulations
Working on projects (seminar thesis)
Handing in seminar thesis and giving a presentation
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2012-11-05 K. Donnay & S. Balietti / [email protected] [email protected] 3
Schedule of the course 24.09. 01.10. 08.10. 15.10. 22.10. 29.10. 05.11. 12.11. 19.11. 26.11. 03.12. 10.12. 17.12.
Introduction to MATLAB
Working on projects (seminar thesis)
Handing in seminar thesis and giving a presentation
Dynamical Systems (no-space) Cellular Automata (grid)
Networks (graphs)
Continuous Space (…)
Different ways of Representing space
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2012-11-05 K. Donnay & S. Balietti / [email protected] [email protected]
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
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2012-11-05 K. Donnay & S. Balietti / [email protected] [email protected]
Repetition: Definition of Graph A graph consists of two entities:
Nodes (vertices): N
Links: L Edge: undirected link Arc: directed link
The graph is defined as G = (N,L)
Source: Batagelj
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2012-11-05 K. Donnay & S. Balietti / [email protected] [email protected]
Repetition: Networks Topologies
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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
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2012-11-05 K. Donnay & S. Balietti / [email protected] [email protected]
Repetition: Networks Topologies
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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
Homegeneous (Exponential) Degree Distr.
Power Law Degree Distr.
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2012-11-05 K. Donnay & S. Balietti / [email protected] [email protected]
Examples of different network topologies
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Source: Wang (2003)
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2012-11-05 K. Donnay & S. Balietti / [email protected] [email protected]
From a problem to a scientific answer Observation:
The role of social networks in shaping the public opinion is becoming increasingly more important
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2012-11-05 K. Donnay & S. Balietti / [email protected] [email protected]
From a problem to a scientific answer Question:
How easy is to be manipulated in a social network?
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Vs.
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2012-11-05 K. Donnay & S. Balietti / [email protected] [email protected]
From a problem to a scientific answer Literature Review:
Was the problem already treated?
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2012-11-05 K. Donnay & S. Balietti / [email protected] [email protected]
From a problem to a scientific answer Literature Review:
How existing models can be improved?
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?
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2012-11-05 K. Donnay & S. Balietti / [email protected] [email protected]
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?
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https://github.com/msssm/Informed_Agents
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2012-11-05 K. Donnay & S. Balietti / [email protected] [email protected]
From a problem to a scientific answer Results
Let us assume ‘informed’ agents Let us consider networks
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https://github.com/msssm/Informed_Agents
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2012-11-05 K. Donnay & S. Balietti / [email protected] [email protected]
Generators How to generate random, realistic graphs?
1. Probabilistic generators
2. Degree-based generators
3. Process-based generators
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2012-11-05 K. Donnay & S. Balietti / [email protected] [email protected]
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
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2012-11-05 K. Donnay & S. Balietti / [email protected] [email protected]
Probabilistic Generators: Erdos-Renyi
random graph – 100 nodes, avg degree = 2
Fascinating properties (phase transition)
But: unrealistic (Poisson degree distribution != power law)
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2012-11-05 K. Donnay & S. Balietti / [email protected] [email protected]
E-R model & Percolation
K
Pc
0
1 N->infty
K0
K = avg(k) Pc = Prob( there is a giant connected component)
The formation of the Giant Component is not a smooth process.
It emerge all of sudden when
p > 1/n
This phenomenon is called 1st order phase-transition
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2012-11-05 K. Donnay & S. Balietti / [email protected] [email protected]
Graphs: Laws and patterns Are real graphs random? If we look at the data the answer most of the
time is: NO!!
degree degree
count count
k k
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2012-11-05 K. Donnay & S. Balietti / [email protected] [email protected]
Degree-based generators Figure out the degree distribution (eg., ‘Zipf’)
Assign degrees to nodes
Put edges, so that they match the original degree distribution
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2012-11-05 K. Donnay & S. Balietti / [email protected] [email protected]
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…
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2012-11-05 K. Donnay & S. Balietti / [email protected] [email protected]
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
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2012-11-05 K. Donnay & S. Balietti / [email protected] [email protected]
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…
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2012-11-05 K. Donnay & S. Balietti / [email protected] [email protected]
Get the lecture files from GitHub
Lecture files are also available on a GitHub repository:
[email protected]:msssm/lectures_files.git
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2012-11-05 K. Donnay & S. Balietti / [email protected] [email protected]
Get the lecture files from GitHub
git clone [email protected]:msssm/lectures_files.git
git pull
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If you have already uploaded your public key in GitHub (see live presentation),
You can dowload the latest lecture files with the following command in GIT Bash
to stay updated:
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2012-11-05 K. Donnay & S. Balietti / [email protected] [email protected]
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
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2012-11-05 K. Donnay & S. Balietti / [email protected] [email protected]
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.
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2012-11-05 K. Donnay & S. Balietti / [email protected] [email protected]
R igraph R and Matlab integration to plot nice
graphs (igraph library)
https://github.com/tconring/Arabian-Spring
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2012-11-05 K. Donnay & S. Balietti / [email protected] [email protected]
SOCNET Mailing List Social Network Analysis Mailing List:
http://www.insna.org/pubs/socnet.html
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2012-11-05 K. Donnay & S. Balietti / [email protected] [email protected]
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/
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2012-11-05 K. Donnay & S. Balietti / [email protected] [email protected]
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 'small-world' 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/
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2012-11-05 K. Donnay & S. Balietti / [email protected] [email protected]
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=newsletter-europe-weekly
http://www.youtube.com/watch?v=Y4MnpzG5Sqc
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2012-11-05 K. Donnay & S. Balietti / [email protected] [email protected]
Projects Today, there are no exercises. Instead,
you can work on your projects and we will supervise you.
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