Contagious sets in dense graphs

Freund, Daniel; Poloczek, Matthias; Reichman, Daniël

doi:10.1016/j.ejc.2017.07.011

Cited by 18 publications

(30 citation statements)

References 20 publications

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“…In this paper, we are interested in degree-based density conditions that ensure that a graph G will percolate from a small set of initially activated vertices. Let δ G ( ) denote the minimum degree of G. Freund et al [22] showed that for each ≥ r 2, if G has order n and…”

Section: Degree-based Resultsmentioning

confidence: 99%

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Ore and Chvátal‐type degree conditions for bootstrap percolation from small sets

Dairyko

Ferrara

Lidický

et al. 2019

Journal of Graph Theory

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Bootstrap percolation is a deterministic cellular automaton in which vertices of a graph G begin in one of two states, “dormant” or “active.” Given a fixed positive integer r, a dormant vertex becomes active if at any stage it has at least r active neighbors, and it remains active for the duration of the process. Given an initial set of active vertices A, we say that G r‐percolates (from A) if every vertex in G becomes active after some number of steps. Let m(G,r) denote the minimum size of a set A such that G r‐percolates from A. Bootstrap percolation has been studied in a number of settings and has applications to both statistical physics and discrete epidemiology. Here, we are concerned with degree‐based density conditions that ensure m(G,2)=2. In particular, we give an Ore‐type degree sum result that states that if a graph G satisfies σ2(G)≥n−2, then either m(G,2)=2 or G is in one of a small number of classes of exceptional graphs. (Here, σ2(G) is the minimum sum of degrees of two nonadjacent vertices in G.) We also give a Chvátal‐type degree condition: If G is a graph with degree sequence d1≤d2≤⋯≤dn such that di≥i+1 or dn−i≥n−i−1 for all 1≤i

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Section: Degree-based Resultsmentioning

confidence: 99%

“…2 Ore [28] proved that every graph G of order ≥ n 3 that satisfies ≥ σ G n ( ) 2 is hamiltonian. Freund et al [22] also showed that Ore's condition is sufficient to ensure that a graph 2percolates from the smallest possible initially activated set.…”

Section: Degree-based Resultsmentioning

confidence: 99%

Ore and Chvátal‐type degree conditions for bootstrap percolation from small sets

Dairyko

Ferrara

Lidický

et al. 2019

Journal of Graph Theory

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“…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.…”

Section: Related Work and Our Resultsmentioning

confidence: 99%

Time-Bounded Influence Diffusion with Incentives

Cordasco

Gargano

Peters

et al. 2018

Structural Information and Communication Complexity

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A widely studied model of influence diffusion in social networks represents the network as a graph G = (V, E) with an influence threshold t(v) for each node. Initially the members of an initial set S ⊆ V are influenced. During each subsequent round, the set of influenced nodes is augmented by including every node v that has at least t(v) previously influenced neighbours. The general problem is to find a small initial set that influences the whole network. In this paper we extend this model by using incentives to reduce the thresholds of some nodes. The goal is to minimize the total of the incentives required to ensure that the process completes within a given number of rounds. The problem is hard to approximate in general networks. We present polynomial-time algorithms for paths, trees, and complete networks.

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“…A polynomial-time algorithm for trees was given in the same paper. Chen's inapproximability result stimulated a series of papers (see for instance [1,2,3,6,7,11,12,13,14,15,16,24,25,30,36,39,40,42,44,45] and references therein quoted) 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.…”

Section: Related Workmentioning

confidence: 99%

Whom to befriend to influence people

Cordasco

Gargano

Lafond

et al. 2020

Theoretical Computer Science

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Alice wants to join a new social network, and influence its members to adopt a new product or idea. Each person v in the network has a certain threshold t(v) for activation, i.e adoption of the product or idea. If v has at least t(v) activated neighbors, then v will also become activated. If Alice wants to activate the entire social network, whom should she befriend? More generally, we study the problem of finding the minimum number of links that a set of external influencers should form to people in the network, in order to activate the entire social network. This Minimum Links Problem has applications in viral marketing and the study of epidemics. Its solution can be quite different from the related and widely studied Target Set Selection problem. We prove that the Minimum Links problem cannot be approximated to within a ratio of O(2 log 1−ε n ), for any fixed ε > 0, unless NP ⊆ DT IME(n polylog(n) ), where n is the number of nodes in the network. On the positive side, we give linear time algorithms to solve the problem for trees, cycles, and cliques, for any given set of external influencers, and give precise bounds on the number of links needed. For general graphs, we design a polynomial time algorithm to compute size-efficient link sets that can activate the entire graph.1 G = (V, E,t) with V (G) representing individuals in the social network, E(G) denoting the social connections, and t an integer-valued threshold function. Starting with a target set, that is, a subset S ⊆ V of nodes in the graph, that are activated by some external incentive, influence propagates deterministically in discrete time steps, and activates nodes.For any unactivated node v, if the number of its activated neighbors at time step t − 1 is at least t(v), then node v will be activated in step t. A node once activated stays activated. It is easy to see that if S is non-empty, then the process terminates after at most |V | − 1 steps. We call the set of nodes that are activated when the process terminates as the activated set. The problem proposed by Domingo and Richardson [21,43] can now be formulated as follows: Given a social network G = (V, E,t), and an integer k, find a subset S ⊆ V of size k so that the resulting activated set is as large as possible. In the context of viral marketing, the parameter k corresponds to the budget, and S is a target set that maximizes the size of the activated set. One question of interest is to find the cheapest way to activate the entire network, when possible. The optimization problem that results has been called the Target Set Selection Problem, and has been widely studied (see for eg. [8,3,40]): the goal is to find a minimum-sized set S ⊆ V that activates the entire network (if such a set exists). In a certain sense, the elements of this minimum target set S are the most influential people in the network; if they are activated, the entire network will eventually be activated.There are, however, two hidden flaws in the formulation of the target set problem. First, the nodes in the target set are as...

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Contagious sets in dense graphs

Cited by 18 publications

References 20 publications

Ore and Chvátal‐type degree conditions for bootstrap percolation from small sets

Ore and Chvátal‐type degree conditions for bootstrap percolation from small sets

Time-Bounded Influence Diffusion with Incentives

Whom to befriend to influence people

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