Evangelism in social networks: Algorithms and complexity

Abstract

We consider a population of interconnected individuals that, with respect to a piece of information, at each time instant can be subdivided into three (time-dependent) categories: agnostics, influenced, and evangelists. A dynamical process of information diffusion evolves among the individuals of the population according to the following rules. Initially, all individuals are agnostic. Then, a set of people is chosen from the outside and convinced to start evangelizing, i.e., to start spreading the information. When a number of evangelists, greater than a given threshold, communicate with a node v, the node v becomes influenced, whereas, as soon as the individual v is contacted by a sufficiently much larger number of evangelists, it is itself converted into an evangelist and consequently it starts spreading the information. The question is: How to choose a bounded cardinality initial set of evangelists so as to maximize the final number of influenced individuals? We prove that the problem is hard to solve, even in an approximate sense. On the positive side, we present exact polynomial time algorithms for trees and complete graphs. For general graphs, we derive exact parameterized algorithms. We also investigate the problem when the objective is to select a minimum number of evangelists capable of influencing the whole network. Our motivations to study these problems come from the areas of Viral Marketing and the analysis of quantitative models of spreading of influence in social networks.

Iterated Type Partitions

“…We start from two recently introduced parameters: modular-width [22] and neighborhood diversity [31]. Both parameters received much attention [1,2,5,7,12,17,18,21,24,25,29] also due to their property of being computable in polynomial time [22,31].…”

Time-Bounded Influence Diffusion with Incentives

“…He proved a strong inapproximability result that makes unlikely the existence of an algorithm with approximation factor better than O(2 log 1−ǫ |V | ). Chen's result stimulated a series of papers including [1,2,3,5,6,10,11,12,13,14,15,18,19,21,32,33,38,43,46,48,50,51,52] that isolated many interesting scenarios in which the problem (and variants thereof) become tractable. Ben-Zwi et al [3] generalized Chen's result on trees to show that target set selection can be solved in n O(w) time where w is the treewidth of the input graph.…”

Parameterized Complexity of Immunization in the Threshold Model

“…Threshold models are widely adopted by sociologists to describe collective behaviours [24] and their use to study of the propagation of innovations through a network was first considered in [27]. The linear threshold model has then been widely used in the literature to study the problem of influence maximization, which aims at identifying a small subset of nodes that can maximize the influence diffusion [4,6,7,9,13,27].…”

“…We consider several parameters associated to the input: the bounds k and , the number ζ related to initially contaminated nodes, and some parameters of the underlying network: The maximum degree ∆, the treewidth tw [35], and the neighborhood diversity nd [31]. The two last parameters, formally defined in Sections 3.4 and 3.5 respectively, are two incomparable parameters of a graph that can be viewed as representing sparse and dense graphs respectively [31]; they received much attention in the literature [1,3,4,7,8,10,13,18,23,20,21,30].…”