Network Decontamination with a Single Agent

Abstract

Faults and viruses often spread in networked environments by propa-10 gating from site to neighboring site. We model this process of network contamina-11 tion by graphs. Consider a graph G = (V, E), whose vertex set is contaminated 12 and our goal is to decontaminate the set V (G) using mobile decontamination 13 agents that traverse along the edge set of G. Temporal immunity τ (G) ≥ 0 is 14 defined as the time that a decontaminated vertex of G can remain continuously 15 exposed to some contaminated neighbor without getting infected itself. The im-16 munity number of G, ι k (G), is the least τ that is required to decontaminate G 17 using k agents. We study immunity number for some classes of graphs corre-18 sponding to network topologies and present upper bounds on ι1(G), in some 19 cases with matching lower bounds. Variations of this problem have been exten-20 sively studied in literature, but proposed algorithms have been restricted to mono-21 tone strategies, where a vertex, once decontaminated, may not be recontaminated. 22 We exploit nonmonotonicity to give bounds which are strictly better than those 23 derived using monotone strategies. 24 arXiv:1307.7307v1 [math.CO] 27 Jul 2013 amount of time after which it becomes contaminated. Actual decontamination is per-41 formed by mobile cleaning agents which which move from host to host over network 42 connections. 43 1.1 Previous Work 44 Graph Search. The decontamination problem considered in this paper is a variation of 45 a problem extensively studied in the literature known as graph search. The graph search 46 problem was first introduced by Breish in [5], where an approach for the problem of 47 finding an explorer that is lost in a complicated system of dark caves is given. Parsons 48 ([20][21]) proposed and studied the pursuit-evasion problem on graphs. Members of 49 a team of searchers traverse the edges of a graph in pursuit of a fugitive, who moves 50 along the edges of the graph with complete knowledge of the locations of the pursuers. 51 The efficiency of a graph search solution is based on the size of the search team. Size of 52 smallest search team that can clear a graph G is called search number, and is denoted in 53 literature by s(G). In [19], Megiddo et al. approached the algorithmic question: Given 54 an arbitrary G, how should one calculate s(G)? They proved that for arbitrary graphs, 55 determining if the search number is less than or equal to an integer k is NP-Hard. They 56 also gave algorithms to compute s(G) where G is a special case of trees. For their 57 results, they use the fact that recontamination of a cleared vertex does not help reduce 58 s(G), which was proved by LaPaugh in [16]. A search plan for G that does not involve 59 recontamination of cleared vertices is referred to as a monotone plan.60 Decontamination. The model for decontamination studied in literature is defined as 61 follows. A team of agents is initially located at the same node, the homebase, and all 62 the other nodes are contaminated. A decontamination strategy co...

Combinatorial trace method for network immunization

“…Monotonicity is assumed in [7,13,12,14] that a node cleaned by the agent will not get affected again. Non-monotonic strategies are given in [10]. Some other problems related to graph immunization include the influence maximization [21], the filter placement [11] and the critical node detection problem (CNDP) [5,20,35].…”

Network Decontamination

“…In the case of strong grids, k searchers are sufficient to clear them when the immunity is ≥ 2(2n−1) k [DFZ10]. Finally, ι 1 (G) has been studied in several classes of graphs [DJS16] such as paths: ι 1 (P n ) = 0 for every n; cycles: ι 1 (C n ) = 2 for every n, and goes to n − 1 if monotonicity is required; complete graphs: ι 1 (K n ) = n − 1 for every n; complete bipartite graphs: ι 1 (K n,m ) = 2m−1 for 3 ≤ m ≤ n; n-node trees: ι 1 (T ) ≤ 30 √ n; p × q grids: p/2 ≤ ι 1 ≤ p for p ≤ q, etc. It can be shown that there are n-node trees T for which ι 1 (T ) = Ω(n 1/3+ ) for some constant > 0 [DJS16] and a challenge would be to close the gap with the upper bound 30 √ n. The question of general planar graphs is also open.…”

Spectral Methods for Immunization of Large Networks

“…once a node is decontaminated then it cannot get contaminated again (Bienstock 1991), (Flocchini 2008), (Flocchini 2007), (Fraigniaud 2008). But non-monotonic strategies are also studied (Daadaa 2016).…”