Rainbow connectivity and rainbow index of inhomogeneous random graphs

We investigate the rainbow k-connectivity rck and (t,k)-rainbow index rxt,k of the in homogeneous random graph G(n,p), where any two vertices i and j are joined by an edge eij with probability p(eij) independently of all other edges, and p={p(eij)}. We show that the known threshold functions for the monotone properties rck(G(n,p))≤r and rxt,k(G(n,p))≤t for integers k, r and t in the Erdős–Rényi random graph G(n,p) can be extended to ‘threshold landscapes’ in terms of G(n,p). In contrast to the traditional plain thresholds characterized as a watershed, our threshold landscapes have two surfaces that are inherently interwoven with each other. This sheds some light on the network connectivity as appropriate trade-offs are allowed and is potentially applicable in network science where connections are not always equal.