Random Graph ModelsNetwork Science Reading Group October 31, 2011
Modeling Complex Networks• Real-world complex networks contain an extremely large number of nodes (n)• Nodes interact in various ways – Capture interactions via a graph – If two nodes interact, there is an edge between them• Question: How should edges be placed in order to model real world complex networks?
Random Graph Models• Look at three graph models that rely on a “random” placement of edges – Different initial conditions and probability distributions lead to different types of graphs• Three common models: – Erdos-Renyi (Exponential) – Watts-Strogatz (Small-World) – Scale-Free/Barabasi-Albert (Power-Law Distribution)
Erdos-Renyi• Erdos-Renyi graph: G(n,p) – n: number of nodes – p: probability of adding an edge between any two nodes• Mechanism: each possible edge in the graph is included with probability p• What happens as n→∞ for various values of p?
Phase Transitions• If p < 1/n, graph contains many small components• At p = 1/n, a giant component starts to form• At p = log(n)/n, the graph is almost surely connected• There is a phase transition at 1/n• Note that expected number of edges at each node is (n-1)p
Characteristics of Erdos-Renyi Graphs• If connected, average distance between two nodes is small (small-world)• Degree distribution is Poisson:• Clustering coefficient: number of edges between neighbors of a node, divided by total number of possible edges between those neighbors – Erdos-Renyi graphs tend to have small clustering coefficients – do not match real world networks (high coefficients) Figure from “Scale-Free Networks” by Barabasi and Bonabeau
Watts-Strogatz (Small World) Model• An effort to generate small-world networks with high clustering coefficients• Start with regular lattice and rewire each edge with a certain probability p• Small-world and high clustering coefficient, but degree distribution does not match real-world networks Figure from “Statistical Mechanics of Complex Networks” by Albert and Barabasi
Scale-Free Networks• Real world networks display degree distributions that have a power-law distribution P( k ) k • These are called power-law or scale-free networks• Previous random graph models do not generate scale free networks
Preferential Attachment• Start with a small group of nodes• At each time-step, a new node comes in and attaches to existing nodes – Key point: prefer to attach to nodes that have a higher degree• Can show that this leads to a network that has a scale-free distribution – Contains hubs that connect to many nodes
Degree Distribution of Scale-Free Networks Figure from “Scale-Free Networks” by Barabasi and Bonabeau