Dynamic monopolies in directed graphs: the spread of unilateral influence in social networks

Irreversible dynamic monopolies were already defined and widely studied in the literature for undirected graphs. They are arising from formulation of the irreversible spread of influence such as disease, opinion, adaptation of a new product, etc., in social networks, where the influence between any two individuals is assumed to be bilateral or reciprocal. But in many phenomena, the influence in the underlying network is unilateral or one-sided. In order to study the latter models we need to introduce and study the concept of dynamic monopolies in directed graphs. Let G be a directed graph such that the in-degree of any vertex G is at least one. Let also τ : V (G) → N be an assignment of thresholds to the vertices of G. A subset M of vertices of G is called a dynamic monopoly for (G, τ) if the vertex set of G can be partitioned into D0∪ . . .∪Dt such that D0 = M and for any i ≥ 1 and any v ∈ Di, the number of edges from D0 ∪ . . .∪Di−1 to v is at least τ(v). One of the most applicable and widely studied threshold assignments in directed graphs is strict majority threshold assignment in which for any vertex v, τ(v) = ⌈(deg(v) + 1)/2⌉, where deg(v) stands for the in-degree of v. By a strict majority dynamic monopoly of a graph G we mean any dynamic monopoly of G with strict majority threshold assignment for the vertices of G. In this paper we first discuss some basic upper and lower bounds for the size of dynamic monopolies with general threshold assignments and then obtain some hardness complexity results concerning the smallest size of dynamic monopolies in directed graphs. Next we show that any directed graph on n vertices and with positive minimum in-degree admits a strict majority dynamic monopoly with n/2 vertices. ∗Corresponding author: [email protected]