Networks with Exponentially Decaying Degree Distribution from Entropy-Driven Growth

We introduce a deterministic network growth model where each new vertex connects to an existing vertex so as to maximize the Shannon entropy of the degree distribution. Numerical simulations up to N = 10,000 vertices reveal a novel class of networks with (i) geometric degree distribution P(k) = (1/2)^k, and (ii) logarithmically growing maximum degree (deg_max(10000)=13). These features distinguish the model from Erdős–Rényi random graphs and Barabási–Albert scale-free networks. We prove analytically that the geometric distribution is the unique maximum-entropy distribution for a fixed mean degree <k> = 2.