Shells structure in uncorrelated scale-free networks

Abstract We study structure of shells in uncorrelated scale-free networks. Using probabilistic arguments, we obtain explicit expression for the distance distribution (i.e., average number of nodes at the l -th shell) for different ranges of the degree exponent γ . To overcome the analytical difficulties when 2 γ 3 , we show that the heterogeneous network can be approximated by a disassortative ordered network, and the average degree of neighbors of a node must depends on shells. We also deduce the mean distance between nodes, 〈 l 〉 , as the distance at which distance distribution is maximum. Taking number of nodes large, we retrieve the known scaling forms for the different ranges of γ , mainly the small-world and the ultra-small world behaviors. Very good accordance with simulations is also found. The expressions of 〈 l 〉 involve all the network’s parameters, and can be used as good approximations of mean distance in real-world problems.