On algebraic connectivity of directed scale-free networks

In this paper, we study the algebraic connectivity of directed complex networks with scale-free property. Algebraic connectivity of a directed graph is the eigenvalue of its Laplacian matrix whose real part is the second smallest. This is known as an important measure for the diffusion speed of many diffusion processes over networks (e.g. consensus, information spreading, epidemics). We propose an algorithm, extending that of Barabasi and Albert, to generate directed scale-free networks, and show by simulations the relations between algebraic connectivity and network size, exponents of in/out-degree distributions, and minimum in/out degrees. The results are moreover compared to directed small-world networks, and demonstrated on a specific diffusion process, reaching consensus.