On the Robustness of a Complex Network by Using the Algebraic Connectivity

The second smallest eigenvalue of the Laplacian matrix, also known as the algebraic connectivity, is a remarkable measure to unfold the robustness of complex networks. In this paper we study the asymptotic behavior of the algebraic connectivity in the Erdős-Rényi random graph, the simplest model to describe a complex network. We estimate analytically the mean and the variance of the algebraic connectivity by approximating it with the minimum nodal degree. The resulting estimate improves a known expression for the asymptotic behavior of the algebraic connectivity [19]. Simulations emphasize the accuracy of the analytical estimation. Further, we study the algebraic connectivity in relation to the graph’s robustness to node and link failures, i.e. the number of nodes and links that have to be removed in order to disconnect a graph, called the node and the link connectivity. Extensive simulations show that the node, the edge connectivity and the minimal nodal degree converge to an identical distribution already at small graph sizes.