[Paper Review] On sequential structures in incompressible multidimensional networks

Felipe S. Abrahão, Klaus Wehmuth+2 more|arXiv (Cornell University)|Dec 4, 2018

Graph theory and applications29 references4 citations

TL;DR

This paper investigates algorithmic incompressibility in multidimensional networks using algorithmic information theory, showing that snapshot-dynamic and multiplex networks carry algorithmic information linearly proportional to the number of time instants or layers, unlike general dynamic or multilayer networks, which exhibit quadratic growth. The results reveal that incompressibility implies structural constraints, such as transtemporal or cross-layer edges, in general multidimensional networks.

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.

Motivation & Objective

• To analyze the worst-case lossless compressibility of multidimensional networks using algorithmic information theory.
• To compare the algorithmic information content of snapshot-dynamic and multiplex networks with that of general dynamic and multilayer networks.
• To identify topological constraints implied by incompressibility in multidimensional network structures.
• To investigate the role of transtemporal and cross-layer edges in incompressible multidimensional networks.
• To establish a theoretical foundation for network summarization and reducibility in complex network analysis.

Proposed method

• Application of algorithmic information theory to analyze the incompressibility of network representations, focusing on the algorithmic complexity of edge sets.
• Use of the invariance theorem to ensure language-independent measures of algorithmic complexity.
• Mathematical analysis of the algorithmic information content of generalized multidimensional networks, particularly in terms of time instants or layers.
• Derivation of bounds on algorithmic information for snapshot-dynamic and multiplex networks, showing linear dependence on the number of time instants or layers.
• Extension of results from multidimensional networks to multilayer and dynamic network abstractions via formal network representation frameworks.
• Identification of topological properties—such as short diameter, high k-connectivity, and degree distributions—using theoretical results on incompressible networks.

Experimental results

Research questions

• RQ1 How does the algorithmic information content scale with the number of time instants or layers in snapshot-dynamic and multiplex networks?
• RQ2 What is the difference in algorithmic information content between snapshot-like multidimensional networks and general multidimensional networks?
• RQ3 What topological properties are necessarily present in incompressible multidimensional networks?
• RQ4 Do transtemporal or cross-layer edges necessarily exist in incompressible multidimensional networks, and what is their significance?
• RQ5 How does algorithmic incompressibility constrain the underlying network topology in real-world or artificial networks?

Key findings

• Incompressible snapshot-dynamic and multiplex networks have algorithmic information content linearly dominated by the number of time instants or layers.
• In contrast, general dynamic or multilayer networks exhibit algorithmic information content of quadratic order in the number of time instants or layers.
• Incompressible general multidimensional networks necessarily contain transtemporal or cross-layer edges, indicating non-sequential connections across time or layers.
• These networks display strong topological constraints, including short diameter, high k-connectivity, and degrees on the order of half the network size with small standard deviation.
• The results imply that algorithmic incompressibility enforces structural constraints, such as the presence of non-sequential edges, in multidimensional network topologies.
• The findings suggest that network summarization and reducibility tools must account for such incompressibility-induced constraints to preserve core network properties.