On Dynamic Monopolies of Graphs With Probabilistic Thresholds

Abstract

Let G be a graph and τ be an assignment of nonnegative thresholds to the vertices of G. A subset of vertices, D, is an irreversible dynamic monopoly of (G, τ) if the vertices of G can be partitioned into subsets D 0 , D 1 , . . . , D k such that D 0 = D and, for all i with 0 ≤ i ≤ k − 1, each vertex v in D i+1 has at least τ(v) neighbours in the union of D 0 , D 1 , . . . , 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, τ) is denoted by dyn τ (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 ≤ X v ≤ deg G (v) + 1. We obtain sharp bounds on the expectation and the concentration of dyn τ (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.2010 Mathematics subject classification: primary 05C69; secondary 91D30.

On the Largest Dynamic Monopolies of Graphs with a Given Average Threshold

“…We first show that the values of dyn τ (G) over all threshold assignments τ with the same average threshold are continuous. We need the following lemma from [11]. Lemma 1.…”

“…Lemma 1. [11] Let G be a graph and τ and τ ′ be two threshold assignments to the vertices of G such that τ (u) = τ ′ (u) for all vertices u of G except for exactly one vertex, say v. Then…”

“…We call such a vertex v self-opinioned (from another interpretation it can be called vaccinated vertex). Irreversible dynamic monopolies and the equivalent concepts target set selection and conversion sets have been the subject of active research in recent years by many authors [3,4,6,7,8,10,11,12,13].…”