Analysis of Biological Networks: Random Network Models

So far (lecture 2), two extreme views of network topology were given and analyzed: structured and random networks. However, many biological, technological and social networks lie somewhere between these two extremes. In order to explain the topology of these real networks, Watts and Strogatz [12] suggested a model, which combines these two views, thus having the advantages of both models when compared to real networks. The model starts from a structured network, in which random edge replacement is done. Interpolating between a structured network and a random network is done without altering the number of vertices and edges in the network. Networks derived from this model gain high clustering from the regular lattices they start with, as well as small characteristic path length from the randomization process (hence the term ”Small World”). To interpolate between structured and random networks, we initiate a random rewiring procedure (Figure 1). Starting from a ring lattice with N vertices and 2k edges per vertex, we rewire each edge at random with probability p. This construction allows us to tune the graph between regularity (p = 0) and disorder (p = 1), thereby probing the intermediate region 0 < p < 1. Note that this can be further generalized to multidimensional lattices, but we will only review the one-dimensional case.