On incompressible high order networks
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This work presents a theoretical investigation of incompressible high order networks defined by a generalized graph representation. In particular, we study the incompressibility (i.e., algorithmic randomness) of snapshot-like dynamic networks in comparison to the incompressibility of a more general form of dynamic networks. In addition, we study some of their network topological properties and how these may be related to real-world complex networks. First, we show that incompressible snapshot networks carry an amount of topological information that is linearly dominated by the size of the set of time instants. This contrasts with the topological information carried by an incompressible general dynamic network that is on the quadratic order of the size of the set of time instants. Furthermore, incompressible dynamic networks, multilayered networks, or dynamic multilayered 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, degrees on the order of half of the network size within a strong-asymptotically dominated standard deviation, and rigidity in respect to automorphisms. Particularly, we show that incompressible dynamic networks (or dynamic multilayered) networks have transtemporal (or crosslayer) edges and, thus, its underlying structure may not correspond to many real-world dynamic networks, e.g., snapshot-like dynamic networks.
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