Tomography and Stability of Complex Networks

We study the structure of generalized random graphs with a given degree distribution P(k), and review studies on their behavior under both random breakdown of nodes and intentional attack on the most highly connected nodes. We focus on scale free networks, where P(k)∝k−\l, for m<k<K. We first examine the “Tomography” of these networks, i.e. the structure of layers around a network node. It is shown that the distance distribution of all nodes from the maximally connected node of the network consists of two regimes. The first is characterized by rapid growth in the number of nodes, and the second decays exponentially. We also show analytically that the nodes degree distribution at each layer is a power law with an exponential cut-off. We then show that scale free networks with \l<3 are robust to random breakdown, but vulnerable to intentional attack. We also describe the behavior of the network near the phase transition and show that the critical exponents are influenced by the scale free nature of the network. We show that the critical exponent for the infinite cluster size behaves as β=1/|\l−3|, and the exponent for the finite clusters size distribution behaves as τ=2\l−3\l−2, for 2<\l<4. For \l>4 the exponents are β=1 and τ=2.5 as in normal infinite dimensional percolation. It is also shown that for all \l>3 the exponent for the correlation length is ν=1 and formulas for the fractal dimensions are obtained. The size of the largest cluster at the transition point, known to scale as N2/3 in regular random graphs, is shown to scale as N(\l−2)/(\l−1) for 3<\l<4 and as N2/3 for \l>4.