The surviving rate of an infected network

Wang, Weifan; Finbow, Stephen; Wang, Ping

doi:10.1016/j.tcs.2010.06.009

Cited by 28 publications

(10 citation statements)

References 10 publications

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“…In a conjecture made in [15], a minimal budget b = 2d − 1 is needed to contain a deterministic contact process starting at a single node on the d-dimensional grid. This conjecture was later proven in [16]. In this section we prove a more general result.…”

Section: D-dimensional Gridmentioning

confidence: 55%

Sequential Vaccination for Containing Epidemics

Tennenholtz

Caramanis

Mannor

2020

Preprint

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The dynamics of infectious diseases spread is crucial in determining their risk and offering ways to contain them. We study sequential vaccination of individuals in networks, where there is a limit on the number of individuals that can be vaccinated every day. Effective allocation of vaccine will play a critical role in preventing the spread and reducing the effects of a future pandemic. We derive methods for calculating upper and lower bounds of the expected number of infected individuals, as well as provide estimates on the number of vaccinations that is needed for containment. We calculate these explicitly on trees, d-dimensional grids, and Erdős Rényi graphs. Finally, we construct a time-dependent budget allocation strategy and demonstrate its superiority over constant budget allocation on real networks following first acquaintance vaccination. Our results provide a principled approach to assess the needed vaccination rate given the social graph topology.

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Section: D-dimensional Gridmentioning

confidence: 55%

Sequential Vaccination for Containing Epidemics

Tennenholtz

Caramanis

Mannor

2020

Preprint

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“…Wang et al proved that

ff ({scriptP}_{9}) = 1

. The ideas of the proof of Theorem can be adapted to prove that

ff ({scriptP}_{8}) = 1

, since it is not hard to show that a worst case scenario in this case is if locally, every face close to the firestart has size 8, and around the boundary of these faces the vertices alternatingly have degree 2 and 4.…”

Section: Resultsmentioning

confidence: 99%

“…Wang et al recently proved that

ρ_{1} (P_{9}) \geq 2 / 35

, and that if a graph G is d ‐degenerate (and has at least two vertices),

ρ_{2 d - 1} (G) \geq 2 / (5 d)

. This implies that the firefighter number of every proper minor‐closed class of graphs is finite.…”

Section: Introductionmentioning

confidence: 99%

Fire Containment in Planar Graphs

Esperet

Heuvel

Maffray

et al. 2012

Journal of Graph Theory

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In a graph G, a fire starts at some vertex. At every time step, firefighters can protect up to k vertices, and then the fire spreads to all unprotected neighbours. The k-surviving rate ρ k (G) of G is the expectation of the proportion of vertices that can be saved from the fire, if the starting vertex of the fire is chosen uniformly at random. For a given class of graphs G we are interested in the minimum value k such that for some constant ǫ > 0 and all G ∈ G, ρ k (G) ≥ ǫ (i.e., such that linearly many vertices are expected to be saved in every graph from G).In this note, we prove that for planar graphs this minimum value is at most 4, and that it is precisely 2 for triangle-free planar graphs.

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“…The study of this concept has become very fruitful in the literature and the evidence is the existence of many works on the subject for different graph structures [9,6,14,16,29,36,41,44,42,43].…”

Section: Introductionmentioning

confidence: 99%

Firefighting as a Strategic Game

Àlvarez¹,

Blesa²,

Molter³

2015

Internet Mathematics

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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.

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The surviving rate of an infected network

Cited by 28 publications

References 10 publications

Sequential Vaccination for Containing Epidemics

Sequential Vaccination for Containing Epidemics

Fire Containment in Planar Graphs

Firefighting as a Strategic Game

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