High Degree Vertices, Eigenvalues and Diameter of Random Apollonian Networks

ABSTRACT. Upon the discovery of power laws [8, 16, 30], a large body of work in complex network analysis has focused on developing generative models of graphs which mimick real-world network properties such as skewed degree distributions [30], small diameter [2] and large clustering coefficients [38, 45]. Most of these models belong either to the stochastic, e.g., [8, 13, 20, 40], or the strategic e.g., [5, 6, 14, 29], family of network formation models. Despite the fact that planar graphs arise in numerous real-world settings, e.g., in road and railway maps, in printed circuits, in chemical molecules, in river networks [9, 41], comparably less attention has been devoted to the study of planar graph generators. In this work we analyze basic properties of Random Apollonian Networks [46, 47], a popular stochastic model which generates planar graphs with power law properties. Specifically, let k be a constant and ∆1 ≥ ∆2 ≥ .. ≥ ∆k be the degrees of the k highest degree verti...

• Type: Journal
• Year: 2011
• Where: CORR
• Authors: Alan M. Frieze, Charalampos E. Tsourakakis
• Keywords: CORR 2011, Education, Planar Graphs, Power Law, Time T