Scaling Properties of Complex Networks and Spanning Trees

We present a relation between three properties of networks: the fractal properties of the percolation cluster at criticality, the optimal path between vertices in the network under strong disorder (i.e., a broad distribution of edge weights) and the minimum spanning tree. Based on properties of the percolation cluster we show that the distance between vertices under strong disorder and on the minimum spanning tree behaves as N for the N vertex complete graph and for Erdős– Rényi random graphs, as well as for scale free networks with exponent γ > 4. For scale free networks with 3 < γ < 4 the distance behaves as N (γ−3)/(γ−1). For 2 < γ < 3, our numerical results indicate that the distance scales as lnγ−1 N . We also discuss a fractal property of some real world networks. These networks present self similarity and a finite fractal dimension when measured using the box covering method.