Graph Concatenations to Derive Weighted Fractal Networks

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