Maximal planar networks with exponential degree distribution

Networks with exponential degree distribution have attracted little attention in comparison with ones obeying power-law degree distribution in current complex network study, besides the fact that the former are observed in many real-world networks. In this paper, we propose a maximal planar graph based approach to establish networks of such type, and generate two growth models G1(t) and G2(t) for instance. Then, we study topological structure of the proposed networks in detail, and obtain that (1) models G1(t) and G2(t) are sparse and display small-world phenomena, (2) models G1(t) and G2(t) obey the same degree distribution, but they exhibit distinct mixing structure, and (3) model G2(t) is more resilient to a random removal of edges than G1(t) as it has a larger spanning tree entropy. Additionally, we also consider the trapping problem on two models with a fixed number of traps and derive the precise solution for average trapping time (ATT). More importantly, model G1(t) turns out to have optimal trapping efficiency because the analytical value for ATT achieves the theoretical lower bound completely. Furthermore, based on the proposed approach, we also provide some guides for generating networks with other richer topological structure, such as random network G1(t).