  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 complementary 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
Erdos-Renyi structures, and then applies optimization so as to increase an
index of the overall complexity of the networks. An optimization approach has
been considered in terms of gradient descent. The complexity index expresses
the dispersion of the several considered measurements. Several interesting
results are reported and discussed, including the fact that the HC networks
define, as the optimization proceeds, a trajectory in the principal component
space of the measurements that tends to depart from the considered theoretical
models (Erdos-Renyi, Barabasi-Albert, Waxman, Random Geometric Graph and
Watts-Strogatz), heading to a previously empty 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.

Physics and Society