In this paper we study a random graph with @math nodes, where node @math has degree @math and @math are i.i.d. with @math . We assume that @math for some @math and some constant @math . This graph model is a variant of the so-called configuration model, and includes heavy tail degrees with finite variance. The minimal number of edges between two arbitrary connected nodes, also known as the graph distance or the hopcount, is investigated when @math . We prove that the graph distance grows like @math , when the base of the logarithm equals @math . This confirms the heuristic argument of Newman, Strogatz and Watts NSW00 . In addition, the random fluctuations around this asymptotic mean @math are characterized and shown to be uniformly bounded. In particular, we show convergence in distribution of the centered graph distance along exponentially growing subsequences.
A second related model can be found in @cite_15 and @cite_42 , where edges between nodes @math and @math are present with probability equal to @math for some expected degree vector' @math . Chung and Lu @cite_15 show that when @math is proportional to @math the average distance between pairs of nodes is @math when @math , and @math when @math . The difference between this model and ours is that the nodes are not exchangeable in @cite_15 , but the observed phenomena are similar. This result can be heuristically understood as follows. Firstly, the actual degree vector in @cite_15 should be close to the expected degree vector. Secondly, for the expected degree vector, we can compute that the number of nodes for which the degree is less than or equal to @math equals @math Thus, one expects that the number of nodes with degree at most @math decreases as @math , similarly as in our model. In @cite_42 , Chung and Lu study the sizes of the connected components in the above model. The advantage of this model is that the edges are independently present, which makes the resulting graph closer to a traditional random graph.
@cite_15 @cite_42
2027377866 2112976607
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 k is proportional to 1 kβ 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, which 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. We consider a family of random graphs with a given expected degree sequence. Each edge is chosen independently with probability proportional to the product of the expected degrees of its endpoints. We examine the distribution of the sizes volumes of the connected components which turns out depending primarily on the average degree d and the second-order average degree d . Here d denotes the weighted average of squares of the expected degrees. For example, we prove that the giant component exists if the expected average degree d is at least 1, and there is no giant component if the expected second-order average degree d is at most 1. Examples are given to illustrate that both bounds are best possible.