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Distance closures on complex networks

Published online by Cambridge University Press:  30 March 2015

TIAGO SIMAS  and
LUIS M. ROCHA
TIAGO SIMAS
Affiliation:
Cognitive Science Program, Indiana University, Bloomington, IN 47406, USA (e-mail: tdesimas@indiana.edu)
LUIS M. ROCHA
Affiliation:
Center for Complex Networks and Systems, School of Informatics & Computing, Indiana University, Bloomington, IN 47406, USA Instituto Gulbenkian de Ciencia, Oeiras, Portugal (e-mail: rocha@indiana.edu)
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Abstract

To expand the toolbox available to network science, we study the isomorphism between distance and Fuzzy (proximity or strength) graphs. Distinct transitive closures in Fuzzy graphs lead to closures of their isomorphic distance graphs with widely different structural properties. For instance, the All Pairs Shortest Paths (APSP) problem, based on the Dijkstra algorithm, is equivalent to a metric closure, which is only one of the possible ways to calculate shortest paths in weighted graphs. We show that different closures lead to different distortions of the original topology of weighted graphs. Therefore, complex network analyses that depend on the calculation of shortest paths on weighted graphs should take into account the closure choice and associated topological distortion. We characterize the isomorphism using the max-min and Dombi disjunction/conjunction pairs. This allows us to: (1) study alternative distance closures, such as those based on diffusion, metric, and ultra-metric distances; (2) identify the operators closest to the metric closure of distance graphs (the APSP), but which are logically consistent; and (3) propose a simple method to compute alternative path length measures and corresponding distance closures using existing algorithms for the APSP. In particular, we show that a specific diffusion distance is promising for community detection in complex networks, and is based on desirable axioms for logical inference or approximate reasoning on networks; it also provides a simple algebraic means to compute diffusion processes on networks. Based on these results, we argue that choosing different distance closures can lead to different conclusions about indirect associations on network data, as well as the structure of complex networks, and are thus important to consider.

Keywords

complex networksalgebraic structuresFuzzy Setsfuzzy graphstransitive closuredistance graphssemi-metric graphsAll Pairs Shortest Pathsdiffusion on networkscommunity detection
Type
Research Article
Information
Network Science , Volume 3 , Issue 2 , June 2015 , pp. 227 - 268
Copyright
Copyright © Cambridge University Press 2015 

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