Shells structure in uncorrelated scale-free networks

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 gamma. To overcome the analytical difficulties when 2 < gamma < 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 gamma, 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. (C) 2019 Elsevier B.V. All rights reserved.