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

Irreversible dynamic monopolies arise from the formulation of the irreversible spread of influence such as disease, opinion, adaptation of a new product, etc., in social networks. In some applications, the influence in the underlying network is unilateral or one-sided. In order to study the latter models we need to introduce the concept of dynamic monopolies in directed graphs. Let GG be a directed graph such that the in-degree of any vertex of GG is at least one. Let also τ:V(G)→Nτ:V(G)→N be an assignment of thresholds to the vertices of GG. A subset MM of vertices of GG is called a dynamic monopoly for (G,τ)(G,τ) if the vertex set of GG can be partitioned into D0∪⋯∪DtD0∪⋯∪Dt such that D0=MD0=M and for any i≥1i≥1 and any v∈Div∈Di, the number of edges from D0∪⋯∪Di−1D0∪⋯∪Di−1 to vv is at least τ(v)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 vv, τ(v)=⌈(deg−(v)+1)/2⌉, where deg−(v) stands for the in-degree of vv. 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 results concerning the smallest size of dynamic monopolies in directed graphs. We prove that any strongly connected directed graph GG admits a strict majority dynamic monopoly with at most ⌈|G|/2⌉⌈|G|/2⌉ vertices. Next we show that any simple directed graph on nn vertices and with positive minimum in-degree admits a strict majority dynamic monopoly with at most n/2n/2 vertices, where by a simple directed graph we mean any directed graph G=(V,E)G=(V,E) such that (u,v)∈E(u,v)∈E implies (v,u)∉E(v,u)∉E for all uu, v∈Vv∈V. We show that this bound is achieved by a polynomial time algorithm. This upper bound improves greatly the previous best known result. The final note of the paper deals with the possibility of the improvement of the latter n/2n/2 bound.