1. 851-0585-04L – Modeling and Simulating Social Systems with MATLAB
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 |
© ETH Zürich |

2. Schedule of the course Introduction to MATLAB
Schedule of the course
Introduction to MATLAB
20.02.
27.02.
05.03.
12.03.
19.03.
26.03.
02.04.
23.04.
30.04.
07.05.
14.05.
21.05.
28.05.
Introduction to social-science modeling and simulations
Working on projects (seminar thesis)
Handing in seminar thesis and giving a presentation

3. Schedule of the course Different ways of Introduction to MATLAB
Schedule of the course
Different ways of
Representing space
Introduction to MATLAB
20.02.
27.02.
05.03.
12.03.
19.03.
26.03.
02.04.
23.04.
30.04.
07.05.
14.05.
21.05.
28.05.
Dynamical Systems (no-space)
Cellular Automata (grid)
Networks (graphs)
Working on projects (seminar thesis)
Continuous Space (…)
Handing in seminar thesis and giving a presentation

4. Goals of Lecture 7: students will
Consolidate knowledge acquired during lecture 6, through brief repetition of the main points
Translate a research question into a simple model of simulation on networks
Get a firm grasp in algorithms to generate different network topologies
Review a custom implementation of algorithms to efficiently:
Generate different networks topologies
Compute statistical properties
Run a simple simulation of the emergence of giant component in a random network

5. Repetition: Definition of Graph
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

6. Repetition: Networks Topologies
Random:
Small average path length
Small clustering coefficient (in general)
Small World:
High clustering coefficient
Scale Free (Power Law):
Slightly smaller average path length
Highest clustering coefficient

7. Repetition: Networks Topologies
Random:
Small average path length
Small clustering coefficient (in general)
Small World:
High clustering coefficient
Scale Free:
Slightly smaller average path length
Highest clustering coefficient
Homegeneous
(Exponential)
Degree Distr.
Power Law
Degree Distr.

8. Examples of different network topologies
Examples of different network topologies
Differences between an exponential network—a U.S. roadmap and a scale-free network—an air- line routing map. On the roadmap, the nodes are cities that are connected by highways. This is a fairly uniform network: each major city has at least one link to the highway system, and there are no cities served by hundreds of highways. The airline routing map dif- fers drastically from the roadmap. The nodes of this network are airports connected by direct flights among them. There are a few hubs on the airline routing map, including Chicago, Dallas, Denver, Atlanta, and New York, from which flights depart to almost all other U.S. airports. The vast majority of airports are tiny, appearing as nodes with one or a few links connecting them to one or sev- eral hubs.
Source:
Wang (2003)

9. From a problem to a scientific answer
Observation:
The role of social networks in shaping the public opinion is becoming increasingly more important

10. From a problem to a scientific answer
Question:
How easy is to be manipulated in a social network?
Vs.

11. From a problem to a scientific answer
Literature Review:
Was the problem already treated?

12. From a problem to a scientific answer
Literature Review:
How existing models can be improved? ?

13. From a problem to a scientific answer
Creating a new model
Let us assume ‘informed’ agents
Let us consider networks

14. How to generate random, realistic graphs?
Generators
How to generate random, realistic graphs?
Probabilistic generators
Degree-based generators
Process-based generators
Recursive/self-similar generators

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

16. Probabilistic Generators: Erdos-Renyi
random graph – 100 nodes, avg degree = 2
Fascinating properties (phase transition)
But: unrealistic (Poisson degree distribution != power law)

17. E-R model & PercolationPc
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 1
N->infty K K0
K = avg(k)
Pc = Prob( there is a giant connected component)

18. Graphs: Laws and patterns
(C) 2010, C. Faloutsos
Graphs: Laws and patterns
Are real graphs random?
If we look at the data the answer most of the time is: NO!!
count
count k
degree
degree k

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

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…

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

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…

23. Get the lecture files from GitHub
Get the lecture files from GitHub
Lecture files are also available on a GitHub repository:

24. Get the lecture files from GitHub
Get the lecture files from GitHub
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:
git clone
git pull

25. Lecture Files Generate Compute: Export to file: Random Networks
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

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.

27. R and Matlab integration to plot nice graphs
SOCNET Mailing List
R and Matlab integration to plot nice graphs

28. Social Network Analysis Mailing List:
SOCNET Mailing List
Social Network Analysis Mailing List:

29.
References
Jure Leskovec, Deepayan Chakrabarti, Jon M. Kleinberg, Christos Faloutsos: Realistic, Mathematically Tractable Graph Generation and Evolution, Using Kronecker Multiplication. PKDD 2005:
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 pp (2010)
Duncan J. Watts, Steven H. Strogatz, Collective dynamics of 'small- world' networks. Nature 393, (4 June 1998)
Laszlo Barabasi, Scale-Free Networks: A Decade and Beyond. cience 24 July 2009: Vol. 325 no. 5939
Laszlo Barabasi web site:
takedowns-and-meltdowns-but-very-little-truth/?xid=newsletter- europe-weekly

30.
Projects
Today, there are no exercises. Instead, you can work on your projects and we will supervise you.