Phase transitions in Pareto optimal complex networks

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 with different arrangements of the connections, but the need of drastic topological changes does not determine the presence or 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.

Multiobjective Optimization and Phase Transitions

“…Phase transitions in Pareto optimal systems. a In a design space of complex networks a set of weighted target functions defines a global energy (equation 1) and renders a potential landscape (explored in [18], [13]). b Those same targets map the design space into the target space.…”

“…Some human made structures, such as transportation networks [9], would also benefit from optimal design. In [13] we consider this problem to a greater extent. We take the cost ρ(γ) of network γ as a function of its edges (number or length) and its efficiency is accounted for by the average path length l (γ), a naive proxy for how fast messages can be relayed across the network.…”

“…To give a good account of these selective forces, a Pareto or Multi Objective Optimization (MOO) approach can be useful. Let us introduce this theory through a recent relevant example [13]. (Technical definitions follow below.)…”

Design Principles of Biological Oscillators through Optimization: Forward and Reverse Analysis

“…While some of the evolutionary aims for circadian clocks are independent, other properties were found to be in a trade-off, as it is the case for entrainability to synchronize with external stimuli and regularity to oscillate with a precise period [1]. Multicriteria (Pareto) optimality concepts are being increasingly used to analyze/design complex systems in different contexts [25–27], including RNA design [28], bacterial adaptability [29, 30], metabolic networks [31–35], gene regulation [23, 36] and biosystems engineering [37]. …”

Origem e evolução da desigualdade material hereditária: uma abordagem dos Sistemas Adaptativos Complexos

“…Desta maneira, entendemos que a transição entre as formas de organização igualitárias e aquelas desiguais se daria como uma transição de fase (HENKEL, 2015;LEE;YANG, 1952;SEOANE;SOLÉ, 2015). A igualdade e a desigualdade material, neste sentido, equivaleriam a estados possíveis de organização do sistema social humano, e a sua transição se daria através de um processo de auto-organização (HELBING, 2010), que passaria de uma fase a outra dadas certas condições externas e processos internos ao sistema, sem que fosse necessária uma coordenação central.…”