Burning a Graph as a Model of Social Contagion

Bonato, Anthony; Janssen, Jeannette; Roshanbin, Elham

doi:10.1007/978-3-319-13123-8_2

Cited by 56 publications

(90 citation statements)

References 15 publications

(15 reference statements)

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Bonato et al [5,6] first introduced the burning process as a way to model spread of contagion in a social network; they characterized the burning number for some graph classes and proved some properties for the burning number. The results of [31,32] extended these results for additional graph families and also studied a variant of burning number in which the burning sequence is selected according to some probabilistic rule.…”

Section: Previous Workmentioning

confidence: 99%

“…Clearly, in order to burn a tree in less than g rounds, the tree should pass the g-site condition; otherwise, any set of at most g activators leaves a vertex outside of combination of the spheres of all the activators and hence, the burning process cannot complete within g rounds. 5 4 2 3 1 6 2 4 3 5 6 6 5 3 4 2 7 3 5 4 6 7 13 12 10 11 9 14 10 12 11 13 14 15 14 12 13 11 15 10 12 11 13 14 14 15 14 15 13 15 12 12 11 13 14 7 8 15 15 15 14 13 15 14 15 15 14 12 13 13 The tree is rooted at s. The first selected vertex is A and the first center is c1. The next unmarked vertex with maximum level is B and c2 is selected as the next center.…”

Section: Approximation Algorithm For Treesmentioning

confidence: 99%

“…In this wya, by the end of round k all marked nodes will be burned. Since the total size of items in S i is at leas m i /C, the number of marked vertices by the time k would be 2m i + 1 ≥ n i ; that is, all vertices will be burned by the end of round k. 5 4 2 3 1 6 2 4 3 5 6 6 5 3 4 2 7 3 5 4 6 7 13 12 10 11 9 14 10 12 11 13 14 15 14 12 13 11 15 10 12 11 13 14 14 15 14 15 13 15 12 12 11 13 14 7 8 15 15 15 14 13 15 14 15 15 14 12 13 13 Lemma 11. Given a burning scheme that completes within k − 1 rounds, it is possible to create a solution for the k-instance of bin covering, in polynomial time, so that at least b bins are covered in the solution.…”

Section: Fptas For Non-constant Number Of Regular Disjoint Pathsmentioning

confidence: 99%

“…... 5 4 2 3 1 6 2 4 3 5 6 6 5 3 4 2 7 3 5 4 6 7 13 12 10 11 9 14 10 12 11 13 14 15 14 12 13 11 15 10 12 11 13 14 14 15 14 15 13 15 12 12 11 13 14 7 8 15 15 15 14 13 15 14 15 15 14 12 13 13 14 15 p1 p2 ...…”

Section: Introductionunclassified

See 3 more Smart Citations

Approximation Algorithms for Graph Burning

Bonato

Kamali

2019

Lecture Notes in Computer Science

Self Cite

View full text Add to dashboard Cite

Numerous approaches study the vulnerability of networks against social contagion. Graph burning studies how fast a contagion, modeled as a set of fires, spreads in a graph. The burning process takes place in synchronous, discrete rounds. In each round, a fire breaks out at a vertex, and the fire spreads to all vertices that are adjacent to a burning vertex. The selection of vertices where fires start defines a schedule that indicates the number of rounds required to burn all vertices. Given a graph, the objective of an algorithm is to find a schedule that minimizes the number of rounds to burn graph. Finding the optimal schedule is known to be NP-hard, and the problem remains NP-hard when the graph is a tree or a set of disjoint paths. The only known algorithm is an approximation algorithm for disjoint paths, which has an approximation ratio of 1.5.We present approximation algorithms for graph burning. For general graphs, we introduce an algorithm with an approximation ratio of 3. When the graph is a tree, we present another algorithm with approximation ratio 2. Moreover, we consider a setting where the graph is a forest of disjoint paths. In this setting, when the number of paths is constant, we provide an optimal algorithm which runs in polynomial time. When the number of paths is more than a constant, we provide two approximation schemes: first, under a regularity condition where paths have asymptotically equal lengths, we show the problem admits an approximation scheme which is fully polynomial. Second, for a general setting where the regularity condition does not necessarily hold, we provide another approximation scheme which runs in time polynomial in the size of the graph.

show abstract

Section: Previous Workmentioning

confidence: 99%

Section: Approximation Algorithm For Treesmentioning

confidence: 99%

Section: Fptas For Non-constant Number Of Regular Disjoint Pathsmentioning

confidence: 99%

Section: Introductionunclassified

See 2 more Smart Citations

Approximation Algorithms for Graph Burning

Bonato

Kamali

2019

Lecture Notes in Computer Science

Self Cite

View full text Add to dashboard Cite

show abstract

“…It indicates how fast a graph can burn down completely if there is a new fire in every round. It can be used as a measure of the speed of the spread of contagion in a graph [5].…”

Section: Introductionmentioning

confidence: 99%

Firefighting as a Strategic Game

Àlvarez¹,

Blesa²,

Molter³

2015

Internet Mathematics

View full text Add to dashboard Cite

The Firefighter Problem was proposed in 1995 [25] as a deterministic discrete-time model for the spread and containment of a fire. The problem is defined on an undirected finite graph G = (V, E), where initially fire breaks out at f nodes. In each subsequent time-step, two actions occur: A certain number b of firefighters are placed on nonburning nodes, permanently protecting them from the fire. Then the fire spreads to all non-defended neighbors of the nodes on fire. Since the graph is finite, at some point each node is either on fire or saved, and thus the fire cannot spread further. One of the objectives for the problem is to place the firefighters in such a way that the number of saved nodes is maximized.The applications of the Firefighter problem reach from real fires to the spreading of diseases and the containment of floods. Furthermore, it can be used to model the spread of computer viruses or viral marketing in communication networks. Most research on the problem considers the case in which the fire starts in a single place (i.e., f = 1), and in which the budget of available firefighters per time-step is one (i.e., b = 1). So does the work in this paper too. This configuration already leads to hard problems and even in this case the problem is known to be NP-hard. * This work was supported by grants TIN2013-46181-C2-1-R and TIN2012-37930-C02-2 of the Spanish Ministry of Science and Innovation, and by the project SGR 2014-1034 of the Generalitat de Catalunya.† A preliminary version of this work was presented in 11th Workshop on Algorithms and Models for the Web Graph (waw'14), Beijing, China, December 2014. 1In this work, we study the problem from a game-theoretical perspective. We introduce a strategic game model for the Firefighter Problem to tackle its complexity from a different angle. We refer to it as the Firefighter Game. Such a game-based context seems very appropriate when applied to large networks, where entities may act and make decisions based on their own interests, without global coordination.At every time-step of the game, a player decides whether to place a new firefighter in a non-burning node of the graph. If so, he must decide where to place it. By placing it, the player is indirectly deciding which nodes to protect at that time-step. We define different utility functions in order to model selfish and non-selfish scenarios, which lead to equivalent games. We show that the Price of Anarchy (PoA) is linear for a particular family of graphs, but it is at most 2 for trees. We also analyze the quality of the equilibria when coalitions among players are allowed. It turns out that it is possible to compute an equilibrium in polynomial time, even for constant size coalitions. This yields to a polynomial time approximation algorithm for the problem and its approximation ratio equals the PoA of the corresponding game. We show that for some specific topologies, the PoA is constant when constant size coalitions are considered.

show abstract