Efficient methods for computing the reliability polynomials of graphs and complex networks

Various methods have been proposed to evaluate the reliability of a graph, one of the most well known of which is the reliability polynomial, R(G, p). It is assumed that G(V, E) is a simple and unweighted connected graph whose nodes are perfect and edges are operational with an independent probability p. Thus, the edge reliability polynomial is a function of p of the number of network edges. There are various methods for calculating the coefficients of reliability polynomial, all of which are related to their recursive nature, which has led to an increase in their computational complexity. Therefore, if the difference between the number of links and nodes in the network exceeds a certain amount, the exact calculation of the coefficients R(G, p) is practically in the NP-hard complexity class. In this paper, while examining the problems in the previous methods, four new approaches for estimating the coefficients of reliability polynomial are presented. In the first approach, using an iterative method, the coefficients are estimated. This method, on average, has the same accuracy as common methods in the related studies. In addition, the second method as an intelligent scheme for integrating the values of coefficients has been proposed. The values of coefficients for smaller, larger, and finally intermediate indices have been determined with the help of this intelligent approach. Further, as a third proposed method, Benford's law is utilized to combine the coefficients. Finally, in the fourth approach, using the Legendre interpolation method, the coefficients are effectively estimated with an appropriate accuracy. To compare these approaches fairly and accurately with each other, they have been carried out on synthetic and real-world underlying graphs. Then, their efficiency and accuracy have been evaluated, compared, and analyzed according to the experimental results.