Properties of random and small-world graphs.

<p>Random, small-world and scale-free networks containing 20 nodes and 73 connections were generated by computer algorithms. The number of nodes and connections were chosen to be small to highlight differences between graphs. A. A structural representation of each graph. The nodes are arranged in a ring and connections are lines drawn between the nodes. The sparser appearance of the small-world graph occurs because most connections are between nearest neighbors along the perimeter of the ring, whereas in the random graph, short and long distance connections are equally probable. The scale-free graph has a hub in the upper right portion of the graph. B. Degree distribution. In random graphs, degrees are roughly normally distributed. In the small-world network, the degree distribution has a smaller standard deviation relative to random graphs because most nodes have a similar connectivity pattern. The scale-free graph has a “power-law” (approximately exponential) degree distribution. C. Adjacency matrices. Both axes show node number, and connections are represented as dots in the matrix. There is a uniform distribution of connections in the random graph. The thick diagonal band in the small-world distribution results from the high frequency of connections between nodes in neighboring positions on the ring. The thick horizontal and vertical bands in the adjacency matrix for the scale-free network correspond to hubs. D. Description of connectivity patterns.