The average distance in a random graph with given expected degrees

Abstract

Random graph theory is used to examine the "small-world phenomenon"- any two strangers are connected through a short chain of mutual acquaintances. We will show that for certain families of random graphs with given expected degrees, the average distance is almost surely of order log n /log d where d is the weighted average of the sum of squares of the expected degrees. Of particular interest are power law random graphs in which the number of vertices of degree κ is proportional to 1/κβ for some fixed exponent β. For the case of β > 3, we prove that the average distance of the power law graphs is almost surely of order log n / log d. However, many Internet, social, and citation networks are power law graphs with exponents in the range 2 < β < 3 for which the power law random graphs have average distance almost surely of order log log n, but have diameter of order log n (provided having some mild constraints for the average distance and maximum degree). In particular, these graphs contain a dense subgraph, that we call the core, having nc/log log n vertices. Almost all vertices are within distance log log n of the core although there are vertices at distance log n from the core. © A K Peters, Ltd.