Graph Concatenations to Derive Weighted Fractal Networks

Given an initial weighted graph G0, an integer m>1, and m scaling factors f1,…,fm∈0,1, we define a sequence of weighted graphs Gkk=0∞ iteratively. Provided that Gk−1 is given for k≥1, we let Gk−11,…,Gk−1m be m copies of Gk−1, whose weighted edges have been scaled by f1,…,fm, respectively. Then, Gk is constructed by concatenating G0 with all the m copies. The proposed framework shares several properties with fractal sets, and the similarity dimension dfract has a great impact on the topology of the graphs Gk (e.g., node strength distribution). Moreover, the average geodesic distance of Gk increases logarithmically with the system size; thus, this framework also generates the small-world property