On the Robustness of Complex Networks by Using the Algebraic Connectivity

The second smallest eigenvalue of the Laplacian matrix, also known as the algebraic connectivity, plays a special role for the robustness of networks since it measures the extent to which it is difficult to cut the network into independent components. In this paper we study the behavior of the algebraic connectivity in a well-known complex network model, the Erdős-Rényi random graph. 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 [18]. Simulations emphasize the accuracy of the analytical estimation, also for small graph sizes. Furthermore, 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. These two measures are called the node and the link connectivity. Extensive simulations show that the node and the link connectivity converge to a distribution identical to that of the minimal nodal degree, already at small graph sizes. This makes the minimal nodal degree a valuable estimate of the number of nodes or links whose deletion results into disconnected random graph. Moreover, the algebraic connectivity increases with the increasing node and link connectivity, justifies the correctness of our definition that the algebraic connectivity is a measure of the robustness in complex networks.