Phase transitions in Pareto optimal complex networks

TL;DR

This paper investigates how Pareto optimization influences the topology of complex networks, revealing phase transitions driven by the functions optimized rather than intrinsic properties.

Contribution

It demonstrates that phase transitions in network structures result from the interplay between external constraints and the functions under optimization.

Findings

Different phases correspond to distinct connection arrangements.

Topological changes are not necessary for phase transitions to occur.

The functions being optimized are crucial in determining phase transition types.

Abstract

The organization of interactions in complex systems can be described by networks connecting different units. These graphs are useful representations of the local and global complexity of the underlying systems. The origin of their topological structure can be diverse, resulting from different mechanisms including multiplicative processes and optimization. In spatial networks or in graphs where cost constraints are at work, as it occurs in a plethora of situations from power grids to the wiring of neurons in the brain, optimization plays an important part in shaping their organization. In this paper we study network designs resulting from a Pareto optimization process, where different simultaneous constraints are the targets of selection. We analyze three variations on a problem finding phase transitions of different kinds. Distinct phases are associated to different arrangements of the connections; but the need of drastic topological changes does not determine the presence, nor the nature of the phase transitions encountered. Instead, the functions under optimization do play a determinant role. This reinforces the view that phase transitions do not arise from intrinsic properties of a system alone, but from the interplay of that system with its external constraints.