1. Hiroki Sayama sayama@binghamton.edu
NECSI Summer School 2008 Week 2: Complex Systems Modeling and Networks Network Models
Hiroki Sayama

2. Models of complex systems
Cellular automata
Parts are connected in a regular grid
Agent-based models
Parts can move in a continuous space and interact based on physical proximity
Network models
Parts can be connected arbitrarily, not restricted by their spatial locations
Logical/social relationships among parts

3. Representation of Networks

4. Networks Network (graph) = nodes (vertices) & links (edges) 1 3 2 4 5
1<->2, 1<->3, 1<->5,
2<->3, 2<->4, 2<->5,
3<->4, 3<->5, 4<->5
(Nodes may have states; links may have directions and weights) 3 2 4 5

5. Representation of a network
Adjacency matrix:
A matrix with rows and columns labeled by nodes, where element aij shows the number of links going from node i to node j
(becomes symmetric for undirected graph)
Adjacency list:
A list of links whose element i->j shows a link going from node i to node j

6. Exercise Represent the following network in: 1 3 2 4 5
Adjacency matrix
Adjacency list 1 2 3 4 5

7. Measuring Topological Properties of Networks

8. Topological properties of networks
Number of nodes
Number of links
Average degree (# of links per node)
Number of connected components
Connectivity measures
Diameter
Clustering coefficient
Degree distribution

9. Diameter
In mathematics: Maximum of shortest path lengths between pairs of nodes
In recent network theory: Average shortest path lengths
Characterizes how large the world being modeled is
A small diameter implies that the network is well connected globally

10. Clustering coefficient
For each node:
Let n be the number of its neighbor nodes
Let m be the number of links among the k neighbors
Calculate c = m / (n choose 2)
Then C = <c> (the average of c)
C indicates the average probability for two of one’s friends to be friends too
A large C implies that the network is well connected locally to form a cluster

11. Degree distribution P(k) = # of nodes with degree k
Gives a rough profile of how the connectivity is distributed within the network
Sk P(k) = total number of nodes

12. Several Well-Known Network Models

13. Erdös-Rényi random network model
N nodes are provided from the beginning
For each of the N(N-1)/2 pairs of nodes, a link will be created independently with probability p

14. Exercise Create and plot several ER random networks using NetworkX
Measure their properties
Study how the number of connected components and the diameter of random networks change with increasing link probability (for the same number of nodes, e.g. n=100)

15. Watts-Strogatz small-world network model
Nodes are initially arranged in a circle
Each node is connected to k nearest neighbors
Then edges are rewired randomly with probability p

16. Exercise
Create and plot several WS small-world networks using NetworkX
Measure their properties
Study how the diameter and the clustering coefficient of WS networks change with increasing rewiring probability (for the same number of nodes, e.g. n=100)

17. Barabási-Albert scale-free network model
Nodes are sequentially added to the network one by one
When adding a new node, it is connected to m nodes chosen from the existing network
Probability for a node to be chosen is proportional to its degree

18. Exercise
Plot degree distributions of several different networks described here (use large number of nodes, e.g. 10,000)
Compare their properties

19. Dynamics on Complex Networks

20. Dynamics on complex networks
Dynamical state changes considered on complex network topologies
Regulatory dynamics on gene/protein networks
Population dynamics on ecological networks
Disease infection on social networks
Information/culture propagation on organizational/social networks

21. Example: Epidemics on networks
Initially, a small fraction of nodes are infected by a disease
An infected node will recover and become susceptible with probability pr
If a susceptible node has an infected neighbor, it will be infected with probability pi (per infected neighbor)
Does the disease stay in the network?

22. Exercise
Study the effects of infection/ recovery probabilities on the fixation of a disease on a random social network
In what condition will the disease remain within society?
In what condition will it go away?
Is the transition smooth, or sharp?

23. Exercise
Do the same experiments with WS small-world networks and BA scale-free networks
Compare their properties