Interplay between function and structure in complex networks

We show that abrupt structural transitions can arise in functionally optimal networks, driven by small changes in the level of transport congestion. Our results offer an explanation as to why so many diverse species of network structure arise in Nature (e.g. fungal systems) under essentially the same environmental conditions. Our findings are based on an exactly solvable model system which mimics a variety of biological and social networks. We then extend our analysis by introducing a novel renormalization scheme involving cost motifs, to describe analytically the average shortest path across multiple-ring-and-hub networks. As a consequence, we uncover a 'skin effect' whereby the structure of the inner multi-ring core can cease to play any role in terms of determining the average shortest path across the network.Comment: Expanded version of physics/0508228 with additional new result

Clustered marginalization of minorities during social transitions induced by co-evolution of behaviour and network structure

Large-scale transitions in societies are associated with both individual behavioural change and restructuring of the social network. These two factors have often been considered independently, yet recent advances in social network research challenge this view. Here we show that common features of societal marginalization and clustering emerge naturally during transitions in a co-evolutionary adaptive network model. This is achieved by explicitly considering the interplay between individual interaction and a dynamic network structure in behavioural selection. We exemplify this mechanism by simulating how smoking behaviour and the network structure get reconfigured by changing social norms. Our results are consistent with empirical findings: The prevalence of smoking was reduced, remaining smokers were preferentially connected among each other and formed increasingly marginalised clusters. We propose that self-amplifying feedbacks between individual behaviour and dynamic restructuring of the network are main drivers of the transition. This generative mechanism for co-evolution of individual behaviour and social network structure may apply to a wide range of examples beyond smoking.Comment: 16 pages, 5 figure

Phase transitions in Pareto optimal complex networks

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.Comment: 14 pages, 7 figure

Coevolution of Cooperation and Partner Rewiring Range in Spatial Social Networks

In recent years, there has been growing interest in the study of coevolutionary games on networks. Despite much progress, little attention has been paid to spatially embedded networks, where the underlying geographic distance, rather than the graph distance, is an important and relevant aspect of the partner rewiring process. It thus remains largely unclear how individual partner rewiring range preference, local vs. global, emerges and affects cooperation. Here we explicitly address this issue using a coevolutionary model of cooperation and partner rewiring range preference in spatially embedded social networks. In contrast to local rewiring, global rewiring has no distance restriction but incurs a one-time cost upon establishing any long range link. We find that under a wide range of model parameters, global partner switching preference can coevolve with cooperation. Moreover, the resulting partner network is highly degree-heterogeneous with small average shortest path length while maintaining high clustering, thereby possessing small-world properties. We also discover an optimum availability of reputation information for the emergence of global cooperators, who form distant partnerships at a cost to themselves. From the coevolutionary perspective, our work may help explain the ubiquity of small-world topologies arising alongside cooperation in the real world

Polyhedral computational geometry for averaging metric phylogenetic trees

This paper investigates the computational geometry relevant to calculations of the Frechet mean and variance for probability distributions on the phylogenetic tree space of Billera, Holmes and Vogtmann, using the theory of probability measures on spaces of nonpositive curvature developed by Sturm. We show that the combinatorics of geodesics with a specified fixed endpoint in tree space are determined by the location of the varying endpoint in a certain polyhedral subdivision of tree space. The variance function associated to a finite subset of tree space has a fixed $C^\infty$ algebraic formula within each cell of the corresponding subdivision, and is continuously differentiable in the interior of each orthant of tree space. We use this subdivision to establish two iterative methods for producing sequences that converge to the Frechet mean: one based on Sturm's Law of Large Numbers, and another based on descent algorithms for finding optima of smooth functions on convex polyhedra. We present properties and biological applications of Frechet means and extend our main results to more general globally nonpositively curved spaces composed of Euclidean orthants.Comment: 43 pages, 6 figures; v2: fixed typos, shortened Sections 1 and 5, added counter example for polyhedrality of vistal subdivision in general CAT(0) cubical complexes; v1: 43 pages, 5 figure