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

Abstract

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 D 0 ∪ . . . ∪ D t such that D 0 = M and for any i ≥ 1 and any v ∈ D i , the number of edges from D 0 ∪ . . . ∪ D i−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 in (v) + 1)/2⌉, where deg in (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: mzaker@iasbs.ac.irWe show that this bound is achieved by a polynomial time algorithm. This upper bound improves greatly the best known result. The final note of the paper deals with the possibility of the improvement of the latter n/2 bound.Mathematics Subject Classification: 05C20, 05C82, 91D30

On Dynamic Monopolies of Graphs With Probabilistic Thresholds

“…Let G be a directed graph whose minimum in-degree is at least one. Also let τ : V(G) → N be an assignment of thresholds to the vertices of G such that [1,6,15]. As proved in [1], Theorem 2.4 is also valid for directed graphs.…”

Target set selection with maximum activation time

“…Convexity. Activation problems appear in the literature under a number of different names, such as r-neighbor bootstrap percolation [2-6, 18, 42, 51, 54], dynamic monopolies [15,19,46,55,58], irreversible conversion [17,43,47,56], or graph convexities, and were studied by researchers of various fields. As mentioned in the introduction, in the particular case in which all thresholds are equal to 2, generalized TSS model is also called 2-neighbor bootstrap percolation or P 3 -convexity, which is studied in the field of graph convexities.…”

On irreversible spread of influence in edge-weighted graphs

“…In simple majority assignment we have τ (v) = deg(v)/2 and in strict majority assignment τ (v) = ⌈(deg(v) + 1)/2⌉. Dynamic monopolies with strict majority thresholds have been studied in [1,5,6,9,10,12,13].…”

“…The other related concepts and quantities are defined similarly for directed graphs. Dynamic monopolies in directed graphs have been investigated in [1,5,13]. Algorithmic problems concerning dynamic monopolies were also widely studied in the literature.…”