12 research outputs found
On the Strength of Connectivity of Inhomogeneous Random K-out Graphs
Random graphs are an important tool for modelling and analyzing the
underlying properties of complex real-world networks. In this paper, we study a
class of random graphs known as the inhomogeneous random K-out graphs which
were recently introduced to analyze heterogeneous sensor networks secured by
the pairwise scheme. In this model, first, each of the nodes is classified
as type-1 (respectively, type-2) with probability (respectively,
independently from each other. Next, each type-1 (respectively,
type-2) node draws 1 arc towards a node (respectively, arcs towards
distinct nodes) selected uniformly at random, and then the orientation of the
arcs is ignored. From the literature on homogeneous K-out graphs wherein all
nodes select neighbors (i.e., ), it is known that when , the graph is -connected asymptotically almost surely (a.a.s.) as
gets large. In the inhomogeneous case (i.e., ), it was recently
established that achieving even 1-connectivity a.a.s. requires .
Here, we provide a comprehensive set of results to complement these existing
results. First, we establish a sharp zero-one law for -connectivity, showing
that for the network to be -connected a.a.s., we need to set for all .
Despite such large scaling of being required for -connectivity, we
show that the trivial condition of for all is sufficient to
ensure that inhomogeneous K-out graph has a connected component of size
whp