Estimation of the Shortest Average Distance in Bipartite Networks with Given Density

Abstract

Complex networks have become a very popular research domain among physicists since the very beginning of this century. Despite network diversity, many real world networks are observed to have small average distance, i.e., the small-world phenomenon. In such cases, the average distance l in a network typically scales with the logarithm of the number of nodes n. The issue of the small-world phenomenon is of great importance for network studies, such as the information processing in communication networks, epidemic spreading in social networks, network designing and optimization. Despite the usefulness and universality of the smallworld concept, it is somewhat unclear what average distance values general networks can produce. It is well known that the classical random graphs of Erdős and Renyi almost surely have short average distance. Relatively short average distances have also been found in many variant random graphs as well as some scale-free network models. However, if a typical random graph has short average distance, does this mean that all graphs share this property? The answer is clearly no. Examples are regular lattices and paths. The construction of such examples naturally depends on the ratio of present edges amongst all the possible edges, i.e., the density d. Following these considerations, the authors in ref. 1 provide a lower estimation for the shortest average distance values in connected graphs with given density, among other things. They show the lower estimation as 2 d independent of the network size. A class of networks that has attracted scrutiny is bipartite networks, which is the topic of the current work. The collaboration networks of scientists, company directors, and movie actors are all examples of bipartite networks. Many important social networks are bipartite. In this paper, we present a lower estimation of shortest average distance in connected bipartite networks with given size and density. This is followed by simulations for random bipartite graphs which demonstrate the correctness of our estimation. Let G 1⁄4 ðV1; V2; EÞ be a bipartite graph with vertex parts V1, V2 and edge set E. Suppose jV1j 1⁄4 n and jV2j 1⁄4 m. Therefore, the density d can be defined as d 1⁄4 jEj= jV1jjV2j 1⁄4 jEj=nm. It is clear that 0 d 1. We establish the following estimation. Theorem 1. For any connected bipartite graph G with size n m and density d, the shortest average distance of G is not less than minf3 2d; 2g.