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 inhomogeneous 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)) & LE; r and rxt ,k(G(n , p)) & LE; t for integers k , r and tin the Erdos-Renyi 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 land-scapes 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.& COPY; 2023 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).