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.
Highlights
Let us imagine that we want to spread a piece of information throughout a terrestrial-based network, where the nodes are collected by cables or Wi-Fi and each node can be reached by a satellite
The first network is a path graph, similar to the one in Figure 2, only with 49 nodes instead of nine nodes. It was used for testing, as it is clear that its optimal burning number is seven
Some of the required functions, like finding eccentricity, radius, eigenvalue centrality, number of components or the largest component, average path length, the vertices of the neighborhood of the given vertex within the given graph distance (EGO), or the weighted aggregated sum product assessment (WASPAS) method are directly available in standard R modules, mostly in the igraph module
Summary
Let us imagine that we want to spread a piece of information (e.g., an alarm) throughout a terrestrial-based network, where the nodes are collected by cables or Wi-Fi and each node can be reached by a satellite. The problem was further studied in papers [3,4], where the bounds for the burning number (i.e., the minimum number of time steps, equal to the minimum size of the set of nodes, which are informed or alarmed “from the outside”, not from their neighbor) are analyzed. Unlike in k center or in the “burning number” problem, parallel distribution of information to all the neighbors of a node in one time step is not considered. The burning number problem is slightly related to the broadcasting of control or emergency packets to all nodes of a network [16,17] Such tasks are needed e.g., for route maintenance or critical alert dissemination. Is described in more detail, followed by the heuristics designed here, which are tested on a range of complex networks
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