Modeling the spread of fault in majority-based network systems: Dynamic monopolies in triangular grids

Adams, Sarah Spence; Booth, Paul; Troxell, Denise Sakai; Zinnen, S. Luke

doi:10.1016/j.dam.2012.02.011

Cited by 11 publications

(5 citation statements)

References 12 publications

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“…By the structure of G u , the events v ∈ H u are all independent for v ∈ N G (u). Let X u be a random subset of N 2 ∪ N 3 that contains each vertex of N 2 ∪ N 3 independently at random with probability p (2) . Note that…”

Section: This Implies Thatmentioning

confidence: 99%

Dynamic monopolies for degree proportional thresholds in connected graphs of girth at least five and trees

Gentner

Rautenbach

2017

Theoretical Computer Science

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Section: This Implies Thatmentioning

confidence: 99%

Dynamic monopolies for degree proportional thresholds in connected graphs of girth at least five and trees

Gentner

Rautenbach

2017

Theoretical Computer Science

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“…Irreversible dynamic monopolies have been widely studied in the literature in [5-7, 12, 14, 16, 19, 20], and also under the equivalent terms 'conversion sets' [2,4,11] and 'target set selection' [1,8,9,18]. Dynamic monopolies have applications in viral marketing [10].…”

Section: Introductionmentioning

confidence: 99%

On Dynamic Monopolies of Graphs With Probabilistic Thresholds

Soltani

Zaker

2014

Bull. Aust. Math. Soc.

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Let G be a graph and τ be an assignment of nonnegative thresholds to the vertices of G. A subset of vertices, D, is an irreversible dynamic monopoly of (G, τ) if the vertices of G can be partitioned into subsets D 0 , D 1 , . . . , D k such that D 0 = D and, for all i with 0 ≤ i ≤ k − 1, each vertex v in D i+1 has at least τ(v) neighbours in the union of D 0 , D 1 , . . . , D i . Dynamic monopolies model the spread of influence or propagation of opinion in social networks, where the graph G represents the underlying network. The smallest cardinality of any dynamic monopoly of (G, τ) is denoted by dyn τ (G). In this paper we assume that the threshold of each vertex v of the network is a random variable X v such that 0 ≤ X v ≤ deg G (v) + 1. We obtain sharp bounds on the expectation and the concentration of dyn τ (G) around its mean value. We also obtain some lower bounds for the size of dynamic monopolies in terms of the order of graph and expectation of the thresholds.2010 Mathematics subject classification: primary 05C69; secondary 91D30.

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“…Graph theoretical models have been used in the past decade to study the spreads of (a) disease through a population, (b) opinion through a social network, and (c) faults in a distributed computer network [2,5,6,8,9,10,11,12]. In these models, each vertex represents a person/object that is either colored black (e.g., infected, holding a certain opinion, corrupted) or white (e.g., uninfected, not holding a certain opinion, not corrupted), and each edge represents a relevant interaction between two people/objects.…”

Section: Introductionmentioning

confidence: 99%

“…In an irreversible majority conversion process, if at least half of the neighbors of a white vertex are black at time t − 1, the vertex will be permanently colored black at time t. Although irreversible majority conversion processes may be used to model the spread of opinion through a social network, these processes are more often used to model the propagation of computer faults [2,3,7,8,11,12,13,14]. For example, in a distributed computer network, nodes communicate with each other to maintain uncorrupted copies of the same variables.…”

Section: Introductionmentioning

confidence: 99%

“…These nodes may determine the state of a variable based on a majority vote of their neighbors' states of the same variable. A set of vertices that, when colored black at time t = 0, will eventually cause the entire graph to be colored black under an irreversible majority conversion process is called a dynamic monopoly, or dynamo [2,3,4,7,8,14]. We are concerned with such sets of minimum number of vertices, called minimum dynamos, and we let min D (G) denote the number of vertices in a minimum dynamo of a graph G. Exact values of min D (G) have been found when G presents certain toroidal square grid structures [8], while upper bounds of min D (G) have been found when G is a planar, cylindrical, or toroidal triangular grid [2].…”

Section: Introductionmentioning

confidence: 99%

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Irreversible k-threshold and majority conversion processes on complete multipartite graphs and graph products

Adams,

Brass,

Stokes

et al. 2011

Preprint

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In graph theoretical models of the spread of disease through populations, the spread of opinion through social networks, and the spread of faults through distributed computer networks, vertices are in two states, either black or white, and these states are dynamically updated at discrete time steps according to the rules of the particular conversion process used in the model. This paper considers the irreversible k-threshold and majority conversion processes. In an irreversible k-threshold (resp., majority) conversion process, a vertex is permanently colored black in a certain time period if at least k (resp., at least half) of its neighbors were black in the previous time period. A k-conversion set (resp., dynamic monopoly) is a set of vertices which, if initially colored black, will result in all vertices eventually being colored black under a k-threshold (resp., majority) conversion process. We answer several open problems by presenting bounds and some exact values of the minimum number of vertices in k-conversion sets and dynamic monopolies of complete multipartite graphs, as well as of Cartesian and tensor products of two graphs.

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Modeling the spread of fault in majority-based network systems: Dynamic monopolies in triangular grids

Cited by 11 publications

References 12 publications

Dynamic monopolies for degree proportional thresholds in connected graphs of girth at least five and trees

Dynamic monopolies for degree proportional thresholds in connected graphs of girth at least five and trees

On Dynamic Monopolies of Graphs With Probabilistic Thresholds

Irreversible k-threshold and majority conversion processes on complete multipartite graphs and graph products

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