The Diameter of a Scale-Free Random Graph

Abstract

We consider a random graph process in which vertices are added to the graph one at a time and joined to a fixed number m of earlier vertices, where each earlier vertex is chosen with probability proportional to its degree. This process was introduced by Barabási and Albert [3], as a simple model of the growth of real-world graphs such as the world-wide web. Computer experiments presented by Barabási, Albert and Jeong [1,5] and heuristic arguments given by Newman, Strogatz and Watts [23] suggest that after n steps the resulting graph should have diameter approximately log n. We show that while this holds for m = 1, for m ≥ 2 the diameter is asymptotically log n/ log log n.

A Phase Transition for the Diameter of the Configuration Model

“…In [5], it was proved that in the PAM and for δ = 0, for which τ = 3, the diameter of the resulting graph is equal to log t log log t (1 + o (1)). Unfortunately, the matching result for the CM has not been proved, so that this does not allow us to verify whether the models have similar distances.…”

“…because, when N → ∞, 5) so that, whp, N (1) = ∅. For some constant C > 0, which will be specified later, and k ≥ 2 we define recursively…”

“…Distances in the CM have attracted considerable attention (see e.g., [15,17,18,19]), but distances in PAM's far less (see [5,20]), which makes it hard to verify this prediction. Together with [20], the current paper takes a first step towards a rigorous verification of this conjecture.…”

Some Typical Properties of the Spatial Preferred Attachment Model

“…Many well-known models for complex networks, including the preferential attachment model by Barabási and Albert [5], have diameters growing at most logarithmically with time. (In fact, in [8] Bollobás and Riordan showed that a.a.s. the diameter of the preferential attachment model is asymptotic to log t/ log log t.) Consider a graph G t produced by the SPA model.…”

Combinatorial Models for Cooperation Networks

“…One rigorous approach is due to [3]. They introduced a random multigraph which is built of random forests which are then formed into multigraphs by partitioning the vertex set and identifying vertices in the same block of the partition.…”