Complexity of Networks (reprise)

TL;DR

This paper introduces a new, computationally efficient network complexity measure based on information content, addressing previous limitations and enabling analysis of network evolution in complex systems.

Contribution

A novel network complexity measure using a new representation language that is both practical and theoretically sound, improving upon previous methods like zcomplexity.

Findings

New representation language encodes links, nodes, and linklist.

Measure is computationally feasible for large networks.

Addresses previous issues with maximal complexity of fully connected and empty networks.

Abstract

Network or graph structures are ubiquitous in the study of complex systems. Often, we are interested in complexity trends of these system as it evolves under some dynamic. An example might be looking at the complexity of a food web as species enter an ecosystem via migration or speciation, and leave via extinction. In a previous paper, a complexity measure of networks was proposed based on the {\em complexity is information content} paradigm. To apply this paradigm to any object, one must fix two things: a representation language, in which strings of symbols from some alphabet describe, or stand for the objects being considered; and a means of determining when two such descriptions refer to the same object. With these two things set, the information content of an object can be computed in principle from the number of equivalent descriptions describing a particular object. The previously proposed representation language had the deficiency that the fully connected and empty networks were the most complex for a given number of nodes. A variation of this measure, called zcomplexity, applied a compression algorithm to the resulting bitstring representation, to solve this problem. Unfortunately, zcomplexity proved too computationally expensive to be practical. In this paper, I propose a new representation language that encodes the number of links along with the number of nodes and a representation of the linklist. This, like zcomplexity, exhibits minimal complexity for fully connected and empty networks, but is as tractable as the original measure. ...