Distribution of number of neighbors on semi-directed Barab'asi-Albert networks with many initial neighbors

Abstract

The Barba'asi-Albert network [1, 2] is growing such that the probability of a new site to be connected to one of the already existing sites, is proportional to the number of previous connections to this already existing site: The rich gets richer. In this way, each new site selects exactly m old sites as neighbors. In directed (DBA) and undirected (UBA) Barba'asi-Albert networks, the network itself was built in the standard way, but when agents (spins) were put on the network nodes [2, 3, 4] the neighbor relations were such that if A has B as a neighbor, B in general does not have A as a neighbor for DBA while it does have for UBA. The present work continues the study of two semi-directed BA networks, SDBA1 [5] and SDBA2 [6] for much larger m than before. In these semidirected networks the exponent γ for the power law governing the decay of the number n (k) of nodes having k neighbors, n (k) α 1/kγ, depends continuously on the parameter m, and we want to know its behavior for m