On dynamic monopolies of graphs with probabilistic thresholds

H. Soltani; M. Zaker

On dynamic monopolies of graphs with probabilistic thresholds

Authors

H. Soltani
M. Zaker

Keywords:

graph theory

Abstract

Let

$G$ be a graph and

${\mathcal{\tau}}$ be an assignment of nonnegative thresholds to the vertices of

$G$ . A subset of vertices,

$D$ , is an irreversible dynamic monopoly of

$(G, \tau)$ if the vertices of

$G$ can be partitioned into subsets

$D_0, D_1, \ldots, D_k$ such that

$D_0=D$ and for all

$i\in \{0, \ldots, k-1\}$ , each vertex

$v$ in

$D_{i+1}$ has at least

$\tau(v)$ neighbours in the union of

$D_0\cup \ldots \cup D_i$ . Dynamic monopolies model the spread of influence or propagation of opinion in social networks, where the graph

$G$ represents the underlying network. The smallest cardinality of any dynamic monopoly of

$(G,\tau)$ is denoted by

$dyn_{\tau}(G)$ . In this paper we assume that the threshold of each vertex

$v$ of the network is a random variable

$X_v$ such that

$0\leq X_v \leq deg_G(v)+1$ . We obtain sharp bounds on the expectation and the concentration of

$dyn_{\tau}(G)$ around its mean value. We also obtain some lower bounds for the size of dynamic monopolies in terms of the order of graph and expectation of the thresholds. DOI:- 10.1017/S0004972714000604

Published

2014-09-20

Issue

Vol. 90 No. 3 (2014)

Section

Articles

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