Explain the nonmonotonic community probability in the Barabási–Albert model

Determine the mechanism responsible for the observed nonmonotonic dependence of the probability of detecting at least two communities, P_C, on network size in the Barabási–Albert model BA(n,m), including its increase to approximately 0.5 near n≈1000 followed by a decline as n increases.

Background

The paper studies whether network communities emerge from local network-evolution rules and compares local models with the nonlocal Barabási–Albert preferential-attachment model. For BA(n,m=2), numerical simulations show that the probability P_C of obtaining at least two communities is nearly zero for small networks, rises to about 0.5 around n≈1000, and then decreases with increasing network size.

The authors explicitly state that they have no explanation for this behavior. They suggest that it might be related to the age structure of preferential attachment, in which older nodes tend to have higher degree, but leave the mechanism unresolved. Understanding this phenomenon would clarify why the Barabási–Albert model fails to exhibit the emergent communities property with almost certainty despite developing some structural organization.