Heuristics for Spreading Alarm throughout a Network

Abstract

This paper provides heuristic methods for obtaining a burning number, which is a graph parameter measuring the speed of the spread of alarm, information, or contagion. For discrete time steps, the heuristics determine which nodes (centers, hubs, vertices, users) should be alarmed (in other words, burned) and in which order, when afterwards each alarmed node alarms its neighbors in the network at the next time step. The goal is to minimize the number of discrete time steps (i.e., time) it takes for the alarm to reach the entire network, so that all the nodes in the networks are alarmed. The burning number is the minimum number of time steps (i.e., number of centers in a time sequence alarmed “from outside”) the process must take. Since the problem is NP complete, its solution for larger networks or graphs has to use heuristics. The heuristics proposed here were tested on a wide range of networks. The complexity of the heuristics ranges in correspondence to the quality of their solution, but all the proposed methods provided a significantly better solution than the competing heuristic.

How to Burn a Network or Spread Alarm

“…The problem is, how to define the centre. In [18] an eigenvector index was used, where a vertex with its maximum value was selected as a next centre for the vertex sequence , … , . The simple, greedy like version of the algorithm described above did not work ideally.…”

“…The above-mentioned graph theorists provided algorithms only to determine the upper and lower boundaries of the solution. While the determination of upper boundaries provided also a solution to the problem in a form of sequence of nodes, it was typically rather suboptimal, as shown in further study aimed at heuristic optimisation of burning number [18]. The heuristics studied in [18] used as its part only one way how to determine a current "center" of the network, based on eigenvector.…”

“…While the determination of upper boundaries provided also a solution to the problem in a form of sequence of nodes, it was typically rather suboptimal, as shown in further study aimed at heuristic optimisation of burning number [18]. The heuristics studied in [18] used as its part only one way how to determine a current "center" of the network, based on eigenvector. Moreover, even though the more sophisticated heuristics designed in [18] obtained better solutions by a rather complicated guided search through the search space, they were rather cumbersome and more CPU demanding.…”

“…The heuristics studied in [18] used as its part only one way how to determine a current "center" of the network, based on eigenvector. Moreover, even though the more sophisticated heuristics designed in [18] obtained better solutions by a rather complicated guided search through the search space, they were rather cumbersome and more CPU demanding.…”

“…The computational demands of the existing heuristics were the main reason, why in this paper we are trying to use a range of centrality measures apart from eigenvector values. Here, these centrality indices are combined with the simplest and least demanding heuristic approach from [18], in the hope to obtain "quick and dirty" solution, that can be further speeded up by a parallelization [17]. Such fast heuristic could be used e.g.…”