The average distances in random graphs with given expected degrees

摘要

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 k is proportional to 1/k~β 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 < ? < 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, which we call the core, having n~(c/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.