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On sequential structures in incompressible multidimensional networks

Felipe S. Abrahão, Klaus Wehmuth, Hector Zenil, Artur Ziviani

TL;DR

It is argued that the study of how isomorphic transformations and their respective algorithmic information distortions can characterize sequential interdimensional structures in (multidimensional) networks helps the analysis of network topological properties while being agnostic to the chosen theory, algorithm, computation model, and programming language.

Abstract

In order to deal with multidimensional structure representations of real-world networks, as well as with their worst-case irreducible information content analysis, the demand for new graph abstractions increases. This article investigates incompressible multidimensional networks defined by generalized graph representations. In particular, we mathematically study the lossless incompressibility of snapshot-dynamic networks and multiplex networks in comparison to the lossless incompressibility of more general forms of dynamic networks and multilayer networks, from which snapshot-dynamic networks or multiplex networks are particular cases. Our theoretical investigation first explores fundamental and basic conditions for connecting the sequential growth of information with sequential interdimensional structures such as time in dynamic networks, and secondly it presents open problems demanding future investigation. Although there may be a dissonance between sequential information dynamics and sequential topology in the general case, we demonstrate that incompressibility dissolves it, preventing both the algorithmic dynamics and the interdimensional structure of multidimensional networks from displaying a snapshot-like behavior (as characterized by any arbitrary mathematical theory). Thus, beyond methods based on statistics or probability as traditionally seen in random graphs and complex networks models, representational incompressibility implies a necessary underlying constraint in the multidimensional network topology. We argue that the study of how isomorphic transformations and their respective algorithmic information distortions can characterize sequential interdimensional structures in (multidimensional) networks helps the analysis of network topological properties while being agnostic to the chosen theory, algorithm, computation model, and programming language.

On sequential structures in incompressible multidimensional networks

TL;DR

It is argued that the study of how isomorphic transformations and their respective algorithmic information distortions can characterize sequential interdimensional structures in (multidimensional) networks helps the analysis of network topological properties while being agnostic to the chosen theory, algorithm, computation model, and programming language.

Abstract

In order to deal with multidimensional structure representations of real-world networks, as well as with their worst-case irreducible information content analysis, the demand for new graph abstractions increases. This article investigates incompressible multidimensional networks defined by generalized graph representations. In particular, we mathematically study the lossless incompressibility of snapshot-dynamic networks and multiplex networks in comparison to the lossless incompressibility of more general forms of dynamic networks and multilayer networks, from which snapshot-dynamic networks or multiplex networks are particular cases. Our theoretical investigation first explores fundamental and basic conditions for connecting the sequential growth of information with sequential interdimensional structures such as time in dynamic networks, and secondly it presents open problems demanding future investigation. Although there may be a dissonance between sequential information dynamics and sequential topology in the general case, we demonstrate that incompressibility dissolves it, preventing both the algorithmic dynamics and the interdimensional structure of multidimensional networks from displaying a snapshot-like behavior (as characterized by any arbitrary mathematical theory). Thus, beyond methods based on statistics or probability as traditionally seen in random graphs and complex networks models, representational incompressibility implies a necessary underlying constraint in the multidimensional network topology. We argue that the study of how isomorphic transformations and their respective algorithmic information distortions can characterize sequential interdimensional structures in (multidimensional) networks helps the analysis of network topological properties while being agnostic to the chosen theory, algorithm, computation model, and programming language.

Paper Structure

This paper contains 17 sections, 15 theorems, 78 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Snapshot-like multidimensional networks
  3. Snapshot-dynamic networks
  4. Multiplex networks
  5. Algorithmic-informational characterization of sequential network structures
  6. Conclusions
  7. Acknowledgments
  8. Preliminary definitions and notation
  9. Multiaspect graphs
  10. Algorithmic information theory
  11. Algorithmically random multiaspect graphs
  12. Background results
  13. Algorithmic information theory
  14. Multiaspect graphs and graphs isomorphism
  15. Algorithmically random multiaspect graphs
  16. ...and 2 more sections

Key Result

Theorem 2.1

Let $\mathrm{G'_t} = (\mathrm{V},\mathscr{E},\mathrm{T})$ be a simple spatial TVG that belongs to a recursively labeled infinite family $F_{ G'_t }$ of simple TVGs. Then, there is a binary string $y \in \{ 0 , 1 \}^*$ that is an algorithmically characteristic string of $\mathrm{G'_t}$ such that and hold, where $x$ is the characteristic string of $\mathrm{G'_t}$.

Figures (1)

  • Figure 1: The adjancency matrix of the isomorphic graph to the TVG, which represents a sequentially coupled node-aligned dynamic network.

Theorems & Definitions (35)

  • Definition 2.1
  • Theorem 2.1
  • Corollary 2.2
  • Definition 2.2
  • Theorem 2.3
  • proof
  • Definition A.1
  • Definition A.2
  • Definition A.3
  • Definition A.4
  • ...and 25 more