2 research outputs found
Tight Bounds for Connectivity of Random K-out Graphs
Random K-out graphs are used in several applications including modeling by
sensor networks secured by the random pairwise key predistribution scheme, and
payment channel networks. The random K-out graph with nodes is constructed
as follows. Each node draws an edge towards distinct nodes selected
uniformly at random. The orientation of the edges is then ignored, yielding an
undirected graph. An interesting property of random K-out graphs is that they
are connected almost surely in the limit of large for any . This
means that they attain the property of being connected very easily, i.e., with
far fewer edges () as compared to classical random graph models including
Erd\H{o}s-R\'enyi graphs (). This work aims to reveal to what
extent the asymptotic behavior of random K-out graphs being connected easily
extends to cases where the number of nodes is small. We establish upper and
lower bounds on the probability of connectivity when is finite. Our lower
bounds improve significantly upon the existing results, and indicate that
random K-out graphs can attain a given probability of connectivity at much
smaller network sizes than previously known. We also show that the established
upper and lower bounds match order-wise; i.e., further improvement on the order
of in the lower bound is not possible. In particular, we prove that the
probability of connectivity is for all .
Through numerical simulations, we show that our bounds closely mirror the
empirically observed probability of connectivity
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