Tight Bounds on the Minimum Size of a Dynamic Monopoly

Zehmakan, Ahad N.

doi:10.1007/978-3-030-13435-8_28

Cited by 8 publications

(6 citation statements)

References 23 publications

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“…The rule that we study is the 1-BP rule of bootstrap percolation [2,12,14,21]. For τ ∈ N 0 , the rule τ -BP is defined based on two states, black and white, as follows: If a node is black, then it always remains black; if a node is white, then it turns black if and only if it has at least τ black neighbors.…”

Section: Problem Settingmentioning

confidence: 99%

Embedding Arbitrary Boolean Circuits into Fungal Automata

Modanese

Worsch

2022

Lecture Notes in Computer Science

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Inspired by the works of Goldreich and Ron (J. ACM, 2017) and Nakar and Ron (ICALP, 2021), we initiate the study of property testing in dynamic environments with arbitrary topologies. Our focus is on the simplest non-trivial rule that can be tested, which corresponds to the 1-BP rule of bootstrap percolation and models a simple spreading behavior: Every "infected" node stays infected forever, and each "healthy" node becomes infected if and only if it has at least one infected neighbor. We show various results for both the case where we test a single time step of evolution and where the evolution spans several time steps. In the first, we show that the worst-case query complexity is O(∆/ε) or Õ( √ n/ε) (whichever is smaller), where ∆ and n are the maximum degree of a node and number of vertices, respectively, in the underlying graph, and we also show lower bounds for both one-and two-sided error testers that match our upper bounds up to ∆ = o( √ n) and ∆ = O(n 1/3 ), respectively. In the second setting of testing the environment over T time steps, we show upper bounds of O(∆ T −1 /εT ) and Õ(|E|/εT ), where E is the set of edges of the underlying graph. All of our algorithms are one-sided error, and all of them are also time-conforming and non-adaptive, with the single exception of the more complex Õ( √ n/ε)-query tester for the case T = 2.

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Section: Problem Settingmentioning

confidence: 99%

Embedding Arbitrary Boolean Circuits into Fungal Automata

Modanese

Worsch

2022

Lecture Notes in Computer Science

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“…(This is sometimes also known as dynamic monopoly or percolating set.) For both the majority and threshold model, the minimum size of a target set has been extensively studied on various classes of graphs such as lattice [24], Erdős-Rényi random graph [44], random regular graphs [25], power-law random graphs [1], expander graphs [37], and bipartite graphs [48]. Furthermore, Berger [9] proved that there exist arbitrarily large graphs which have target sets of constant size under the majority model and it was shown in [6] that every n-node graph has a target set of size at most n/2 under the asynchronous variant.…”

Section: Related Workmentioning

confidence: 99%

Majority Opinion Diffusion in Social Networks: An Adversarial Approach

Zehmakan¹

2020

Preprint

Self Cite

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We introduce and study a novel majority-based opinion diffusion model. Consider a graph G, which represents a social network. Assume that initially a subset of nodes, called seed nodes or early adopters, are colored either black or white, which correspond to positive or negative opinion regarding a consumer product or a technological innovation. Then, in each round an uncolored node, which is adjacent to at least one colored node, chooses the most frequent color among its neighbors.Consider a marketing campaign which advertises a product of poor quality and its ultimate goal is that more than half of the population believe in the quality of the product at the end of the opinion diffusion process. We focus on three types of attackers which can select the seed nodes in a deterministic or random fashion and manipulate almost half of them to adopt a positive opinion toward the product (that is, to choose black color). We say that an attacker succeeds if a majority of nodes are black at the end of the process. Our main purpose is to characterize classes of graphs where an attacker cannot succeed. In particular, we prove that if the maximum degree of the underlying graph is not too large or if it has strong expansion properties, then it is fairly resilient to such attacks.Furthermore, we prove tight bounds on the stabilization time of the process (that is, the number of rounds it needs to end) in both settings of choosing the seed nodes deterministically and randomly. We also provide several hardness results for some optimization problems regarding stabilization time and choice of seed nodes.

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“…We should also mention that in addition to the standard bootstrap percolation considered in this paper, there are also several variants. For example, two-way bootstrap percolation, which has been considered in [21,27,28], and the previously mentioned bootstrap percolation on edges, which has been recently considered in [14].…”

Section: Introductionmentioning

confidence: 99%

Deterministic bootstrap percolation on trees

Beeler

Keaton

Norwood

2022

Art Discrete Appl. Math.

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In a graph, k-bootstrap percolation is a process by which an "infection" spreads from an initial set of infected vertices, according to the rule that on each iteration an uninfected vertex with k infected neighbors becomes infected. This process continues until either every vertex is infected or every uninfected vertex has fewer than k infected neighbors. We are particularly interested in the case where every vertex is eventually infected. The cardinality of a smallest set that results in this is the k-bootstrap percolation number of the graph. In this paper, we determine the k-bootstrap percolation number for trees of small diameter, spiders, complete N -ary trees, and caterpillars. For these graph families we also consider the smallest number of iterations needed for any smallest set to spread to the entire graph. Finally, we give an upper bound for the k-bootstrap percolation number for general trees which improves upon previous results.

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Tight Bounds on the Minimum Size of a Dynamic Monopoly

Cited by 8 publications

References 23 publications

Embedding Arbitrary Boolean Circuits into Fungal Automata

Embedding Arbitrary Boolean Circuits into Fungal Automata

Majority Opinion Diffusion in Social Networks: An Adversarial Approach

Deterministic bootstrap percolation on trees

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