On hypercomplex networks

The concept of ‘complexity’ plays a central role in complex network science. Traditionally, this term has been taken to express heterogeneity of the node degrees of a therefore complex network. However, given that the degree distribution is not enough to provide an invertible representation of a given network, additional measurements of its topology are required in order to complement its characterization. In the present work, we aim at obtaining a new model of complex networks, called hypercomplex networks — HC, that are characterized by heterogeneity not only of the degree distribution, but also of a relatively complete set of complementary topological measurements including node degree, shortest path length, clustering coefficient, betweenness centrality, matching index, Laplacian eigenvalue and hierarchical node degree. The proposed model starts with uniformly random networks, namely Erdős–Rényi structures, and then applies optimization to change the network structure so as to increase a complexity index relatively to a set of reference networks. Gradient descent has been adopted for implementing this optimization. We also propose a complexity index that expresses the dispersion of the several considered measurements. Several interesting results are reported and discussed, including the fact that the HC network undergoes, as the optimization proceeds, a trajectory in the principal component space of the measurements that tends to depart from the considered theoretical models (Erdős–Rényi, Barabási–Albert, Waxman, Random Geometric Graph and Watts–Strogatz), heading to an empty portion of the feature space (low density of cases). We observed that, after a considerably large number of optimization steps, peripheral branching tends to appear that further enhances the complexity of these networks.