A polyhedral study of dynamic monopolies

Soltani, Hossein; Moazzez, Babak

doi:10.1007/s10479-018-3107-5

Cited by 6 publications

(3 citation statements)

References 11 publications

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“…Finally, the integer linear programming formulation of the problem has been studied by prior work, cf. [1,40,14].…”

Section: Related Workmentioning

confidence: 99%

Minimum Target Sets in Non-Progressive Threshold Models: When Timing Matters

Soltani,

Zehmakan,

Hushmandi

2023

Frontiers in Artificial Intelligence and Applications

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Let G be a graph, which represents a social network, and suppose each node v has a threshold value τ(v). Consider an initial configuration, where each node is either positive or negative. In each discrete time step, a node v becomes/remains positive if at least τ(v) of its neighbors are positive and negative otherwise. A node set S is a Target Set (TS) whenever the following holds: if S is fully positive initially, all nodes in the graph become positive eventually. We focus on a generalization of TS, called Timed TS (TTS), where it is permitted to assign a positive state to a node at any step of the process, rather than just at the beginning. We provide graph structures for which the minimum TTS is significantly smaller than the minimum TS, indicating that timing is an essential aspect of successful target selection strategies. Furthermore, we prove tight bounds on the minimum size of a TTS in terms of the number of nodes and maximum degree when the thresholds are assigned based on the majority rule. We show that the problem of determining the minimum size of a TTS is NP-hard and provide an Integer Linear Programming formulation and a greedy algorithm. We evaluate the performance of our algorithm by conducting experiments on various synthetic and real-world networks. We also present a linear-time exact algorithm for trees.

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“…Finally, the integer linear programming formulation of the problem has been studied by prior work, cf. [1,40,14].…”

Section: Related Workmentioning

confidence: 99%

Minimum Target Sets in Non-Progressive Threshold Models: When Timing Matters

Soltani,

Zehmakan,

Hushmandi

2023

Frontiers in Artificial Intelligence and Applications

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“…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.…”

Section: Preliminariesmentioning

confidence: 99%

Target set selection with maximum activation time

Keiler¹,

Lima²,

Maia³

et al. 2020

Preprint

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A target set selection model is a graph G with a threshold function τ : V → N upper-bounded by the vertex degree. For a given model, a set S0 ⊆ V (G) is a target set if V (G) can be partitioned into non-empty subsets S0, S1, . . . , St such that, for i ∈ {1, . . . , t}, Si contains exactly every vertex v having at least τ (v) neighbors in S0 ∪ • • • ∪ Si−1. We say that t is the activation time tτ (S0) of the target set S0. The problem of, given such a model, finding a target set of minimum size has been extensively studied in the literature. In this article, we investigate its variant, which we call TSS-time, in which the goal is to find a target set S0 that maximizes tτ (S0). That is, given a graph G, a threshold function τ in G, and an integer k, the objective of the TSS-time problem is to decide whether G contains a target set S0 such that tτ (S0) ≥ k. Let τ * = max v∈V (G) τ (v). Our main result is the following dichotomy about the complexity of TSS-time when G belongs to a minor-closed graph class C: if C has bounded local treewidth, the problem is FPT parameterized by k and τ ; otherwise, it is NP-complete even for fixed k = 4 and τ = 2. We also prove that, with τ * = 2, the problem is NP-hard in bipartite graphs for fixed k = 5, and from previous results we observe that TSS-time is NP-hard in planar graphs and W[1]-hard parameterized by treewidth. Finally, we present a linear-time algorithm to find a target set S0 in a given tree maximizing tτ (S0). ACM Subject ClassificationMathematics of computing → Graph algorithms.

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“…The minimum subset of nodes required under dynamic activation is also referred to as the dynamic monopoly, where nodes become activated if at least half of the neighbors are activated in the previous time period. Hence, an extra index of time is incorporated into the integer programming formulation to describe step-by-step activation of nodes, see [14,15,16]. We do not consider time dynamics in our problem despite the fact that the influence propagation occurs in discrete steps.…”

Section: Introductionmentioning

confidence: 99%

A Polyhedral Approach to Least Cost Influence Maximization in Social Networks

Chen¹,

Pasiliao²,

Boginski³

2022

Preprint

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The least cost influence maximization problem aims to determine minimum cost of partial (e.g., monetary) incentives initially given to the influential spreaders on a social network, so that these early adopters exert influence toward their neighbors and prompt influence propagation to reach a desired penetration rate by the end of cascading processes. We first conduct polyhedral analysis on a substructure that describes influence propagation assuming influence weights are unequal, linear and additively separable. Two classes of facet-defining inequalities based on a mixed 0-1 knapsack set contained in this substructure are proposed. We characterize another exponential class of valid and facet-defining inequalities utilizing the concept of minimum influencing subset. We show that these inequalities can be separated in polynomial time efficiently. Furthermore, a polynomial-time dynamic programming recursion is presented to solve this problem on a simple cycle graph. For arbitrary graphs, we propose a new exponential class of valid inequalities that dominates the cycle elimination constraints and an efficient separation algorithm for them. A compact convex hull description for a special case is presented. We illustrate the effectiveness of these inequalities via a delayed cut generation algorithm in the computational experiments.

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A polyhedral study of dynamic monopolies

Cited by 6 publications

References 11 publications

Minimum Target Sets in Non-Progressive Threshold Models: When Timing Matters

Minimum Target Sets in Non-Progressive Threshold Models: When Timing Matters

Target set selection with maximum activation time

A Polyhedral Approach to Least Cost Influence Maximization in Social Networks

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