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

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

Representation of Networks

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

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

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

Measuring Topological Properties of Networks

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

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

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

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

Several Well-Known Network Models

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

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)

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

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)

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

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

Dynamics on Complex Networks

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

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?

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?

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