The Polynomial Volume Law of Complex Networks in the Context of Local and Global Optimization

Franz-Benjamin Mocnik

doi:10.1038/s41598-018-29131-0

Abstract

Many complex networks expose global hub structures: for some nodes, the number of incident edges far exceeds the average, leading to a small average shortest path length. Such ‘small-world properties’ are often guided by a scale-free power-law distribution of the node degrees, and self-organization inside the network has been identified as a reason driving the emergence of this structure. Small-world networks have recently raised lots of interest, because they capture the global topology of the World-Wide Web, metabolic, and social networks. While small-world networks reflect global structures, little attention is paid to the local structure of complex networks. In this article neighbourhoods are demonstrated to share a common local structure in many real complex networks, manifested by a polynomial volume law. This law can, in case of networks that are embedded in space, be explained in terms of the embedding and the properties of Euclidean space. A model of hierarchical spatial networks is introduced to examine the effect of global structures, in particular of hierarchies, on the polynomial volume law. It turns out that the law is robust against the coexistence of such global structures. The local structure of space and global optimization can both be found in transport, brain, and communication networks, which suggests the polynomial volume law, often in combination with hierarchies or other global optimization principles, to be a generic property inherent to many networks.

Highlights

Networks are used to describe and analyse systems that expose relations between objects
Local optimization refers to the fact that the optimization can be executed on smaller neighbourhoods inside the network, while global optimization on the contrary always refers to the entire network
The examination of the Mocnik model has demonstrated that global optimization leads to an underestimation of the volume at lower radii and an overestimation of the dimension when fitting to the polynomial volume law

Summary

Introduction

Networks are used to describe and analyse systems that expose relations between objects. Shortest paths contain only very few nodes in average, when many hubs are present in a network or the network exposes a strong hierarchy; a network is said to expose small-world properties[2,8] in case of such a short average shortest path length. Instead of a short average shortest path length, the average shortest path length of a road network grows much faster for an increasing size of the network; and the node degree is usually limited by 4 or 5 This is despite the fact that road networks are optimized for travel time or travelled distance. Thereby, the influence of the spatial layout of the network on the evaluation of the dimension is discussed It is examined at the introduced hierarchical spatial network model in how far local optimization masks the polynomial volume law and impairs the estimation of the dimension. The results are set into context by the analysis of a number of real-world networks, which expose both local and global optimization

Methods

Results

Discussion

Conclusion