On connectivity and robustness of random graphs with inhomogeneity

Abstract

AbstractThe study of threshold functions has a long history in random graph theory. It is known that the thresholds for minimum degreek,k-connectivity, as well ask-robustness coincide for a binomial random graph. In this paper we consider an inhomogeneous random graph model, which is obtained by including each possible edge independently with an individual probability. Based on an intuitive concept of neighborhood density, we show two sufficient conditions guaranteeingk-connectivity andk-robustness, respectively, which are asymptotically equivalent. Our framework sheds some light on extending uniform threshold values in homogeneous random graphs to threshold landscapes in inhomogeneous random graphs.