Offdiagonal complexity: A computationally quick network complexity measure. Application to protein networks and cell division

TL;DR

The paper introduces Offdiagonal Complexity (OdC), a fast computational measure to quantify network complexity, applied to biological protein networks and cell division structures, revealing insights into their structural organization.

Contribution

It presents a novel, computationally efficient complexity measure based on node-link cross-distribution, applicable to biological networks and cell aggregates, extending beyond traditional metrics.

Findings

OdC effectively distinguishes complex biological networks from random graphs.

Application to protein interaction networks shows meaningful complexity differences.

OdC quantifies spatial complexity in embryonic cell arrangements.

Abstract

Many complex biological, social, and economical networks show topologies drastically differing from random graphs. But, what is a complex network, i.e.\ how can one quantify the complexity of a graph? Here the Offdiagonal Complexity (OdC), a new, and computationally cheap, measure of complexity is defined, based on the node-node link cross-distribution, whose nondiagonal elements characterize the graph structure beyond link distribution, cluster coefficient and average path length. The OdC apporach is applied to the {\sl Helicobacter pylori} protein interaction network and randomly rewired surrogates thereof. In addition, OdC is used to characterize the spatial complexity of cell aggregates. We investigate the earliest embryo development states of Caenorhabditis elegans. The development states of the premorphogenetic phase are represented by symmetric binary-valued cell connection matrices with dimension growing from 4 to 385. These matrices can be interpreted as adjacency matrix of an undirected graph, or network. The OdC approach allows to describe quantitatively the complexity of the cell aggregate geometry.