Irreversible <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si38.gif" display="inline" overflow="scroll"><mml:mi>k</mml:mi></mml:math>-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion

Dreyer, Paul; Roberts, Fred S.

doi:10.1016/j.dam.2008.09.012

Cited by 163 publications

(15 citation statements)

References 11 publications

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“…The following theorem is from the literature [21,15] and tells us that the MIN-SEED problem is NP-complete.…”

Section: Definition 1 (Threshold Function)mentioning

confidence: 99%

“…In [34], the authors look at deterministic tipping where each node is activated upon a percentage of neighbors being activated. Dryer and Roberts [15] introduce the MIN-SEED problem, study its complexity, and describe several of its properties w.r.t. certain special cases of graphs/networks.…”

Section: Related Workmentioning

confidence: 99%

“…Based on this data, the desired output is the smallest possible set of individuals (seed set) such that, if initially activated, the entire population will become activated (adopting the new property). This problem is NP-Complete [21,15] so approximation algorithms must be used. Though some such algorithms have been proposed, [24,12,3,13] none seem to scale to very large data sets.…”

Section: Introductionmentioning

confidence: 99%

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A scalable heuristic for viral marketing under the tipping model

Shakarian

Eyre

Paulo

2013

Soc. Netw. Anal. Min.

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In a "tipping" model, each node in a social network, representing an individual, adopts a property or behavior if a certain number of his incoming neighbors currently exhibit the same. In viral marketing, a key problem is to select an initial "seed" set from the network such that the entire network adopts any behavior given to the seed. Here we introduce a method for quickly finding seed sets that scales to very large networks. Our approach finds a set of nodes that guarantees spreading to the entire network under the tipping model. After experimentally evaluating 31 real-world networks, we found that our approach often finds seed sets that are several orders of magnitude smaller than the population size and outperform nodal centrality measures in most cases. In addition, our approach scales well -on a Friendster social network consisting of 5.6 million nodes and 28 million edges we found a seed set in under

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“…The following theorem is from the literature [21,15] and tells us that the MIN-SEED problem is NP-complete.…”

Section: Definition 1 (Threshold Function)mentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A scalable heuristic for viral marketing under the tipping model

Shakarian

Eyre

Paulo

2013

Soc. Netw. Anal. Min.

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“…The Linear Threshold Model [5,6,[24][25][26] is based on the association of a threshold θ(v) to each node v and of a weight w(u, v) to each arc (u, v): node u becomes informed if and only if ∑ v∈N(u)∩I w(u, v) ≥ θ (u), where N(u) is the set of neighbors of u and I is the set of the already informed nodes. In other words, the Linear Threshold Model assumes that any unaware node becomes informed if the weight of its informed neighbors is above a certain threshold (i.e., the node is subject to a large enough amount of "social pressure").…”

Section: Introductionmentioning

confidence: 99%

“…In other words, the Linear Threshold Model assumes that any unaware node becomes informed if the weight of its informed neighbors is above a certain threshold (i.e., the node is subject to a large enough amount of "social pressure"). In [25,27] a special case of the Linear Threshold Model is considered, in which all arc weights are 1 and all nodes have the same threshold θ; in this case, the threshold value just stands for the number of neighbors that have to be informed in order to induce a node to become informed. In that paper, the authors prove that deciding if a graph has a target set of size d is an NP-complete problem for θ ≥ 3.…”

Section: Introductionmentioning

confidence: 99%

Latency-Bounded Target Set Selection in Signed Networks

Ianni

Varricchio

2020

Algorithms

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It is well-documented that social networks play a considerable role in information spreading. The dynamic processes governing the diffusion of information have been studied in many fields, including epidemiology, sociology, economics, and computer science. A widely studied problem in the area of viral marketing is the target set selection: in order to market a new product, hoping it will be adopted by a large fraction of individuals in the network, which set of individuals should we “target” (for instance, by offering them free samples of the product)? In this paper, we introduce a diffusion model in which some of the neighbors of a node have a negative influence on that node, namely, they induce the node to reject the feature that is supposed to be spread. We study the target set selection problem within this model, first proving a strong inapproximability result holding also when the diffusion process is required to reach all the nodes in a couple of rounds. Then, we consider a set of restrictions under which the problem is approximable to some extent.

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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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Irreversible k-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion

Cited by 163 publications

References 11 publications

A scalable heuristic for viral marketing under the tipping model

A scalable heuristic for viral marketing under the tipping model

Latency-Bounded Target Set Selection in Signed Networks

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

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