On incompressible multidimensional networks

This work presents a theoretical investigation of incompressible multidimensional networks defined by a generalized graph representation. In particular, we study the incompressibility (i.e., algorithmic randomness) of snapshot-dynamic networks and multiplex networks in comparison to the incompressibility of more general forms of multidimensional networks, from which snapshot-dynamic networks or multiplex networks are particular cases. In addition, we study some of their network topological properties and discuss how these may be related to real-world complex networks. First, we show that incompressible snapshot-dynamic (or multiplex) networks carry an amount of algorithmic information that is linearly dominated by the size of the set of time instants (or layers). This contrasts with the algorithmic information carried by an incompressible general dynamic (or multilayer) network that is of the quadratic order of the size of the set of time instants (or layers). Furthermore, incompressible general multidimensional networks inherit most of the topological properties from their respective isomorphic graph. Hence, we show that these networks have very short diameter, high k-connectivity, and degrees of the order of half of the network size within a strong-asymptotically dominated standard deviation. Particularly, we show that incompressible general multidimensional networks have transtemporal (or crosslayer) edges. Thus, this property may not correspond to the underlying structure of many real-world networks that can be properly modeled by snapshot-like multidimensional networks.