Complex Networks and Symmetry I: A Review

In this review we establish various connections between complex networks and symmetry. While special types of symmetries (e.g., automorphisms) are studied in detail within discrete mathematics for particular classes of deterministic graphs, the analysis of more general symmetries in real complex networks is far less developed. We argue that real networks, as any entity characterized by imperfections or errors, necessarily require a stochastic notion of invariance. We therefore propose a definition of stochastic symmetry based on graph ensembles and use it to review the main results of network theory from an unusual perspective. The results discussed here and in a companion paper show that stochastic symmetry highlights the most informative topological properties of real networks, even in noisy situations unaccessible to exact techniques.Comment: Final accepted versio

Detecting spatial homogeneity in the world trade web with Detrended Fluctuation Analysis

In a spatially embedded network, that is a network where nodes can be uniquely determined in a system of coordinates, links' weights might be affected by metric distances coupling every pair of nodes (dyads). In order to assess to what extent metric distances affect relationships (link's weights) in a spatially embedded network, we propose a methodology based on DFA (Detrended Fluctuation Analysis). DFA is a well developed methodology to evaluate autocorrelations and estimate long-range behaviour in time series. We argue it can be further extended to spatially ordered series in order to assess autocorrelations in values. A scaling exponent of 0.5 (uncorrelated data) would thereby signal a perfect homogeneous space embedding the network. We apply the proposed methodology to the World Trade Web (WTW) during the years 1949-2000 and we find, in some contrast with predictions of gravity models, a declining influence of distances on trading relationships.Comment: 15 pages, 7 figure

The Prism of Elasticity in Rebound Effect Modelling:An Insight from the Freight Transport Sector

If the rebound effect is to be considered a major obstacle to sustainable freight transport, then action and timely policy must be made in advance. This, however, requires a theoretical understanding of the nature of the rebound effect and an empirical grasp of its underlying mechanism. Elasticity is the centrepiece of current models on the rebound effect (or Jevons paradox). Although elasticity is a metric of indisputable usefulness for empirical purposes, it may be misleading when applied to the complex rebound effect. Drawing on the parallel case of the ‘distance puzzle’ in international economics, it will be shown how elasticity can be misinterpreted or how it can misdirect an investigation of the phenomenon by following a predetermined mindset. This particular bias is shown to widen in the long term and evolving systems in which the elasticity metric continues to output a constant number, eliciting a persistent effect. Drawing on previous research, an alternative approach to studying the rebound effect based on complex network theory and statistical mechanics of networks will be described. It will be shown how the interplay between spatial and non-spatial effects in freight transport networks can inform us about the evolution of the effect of distances on trade relationships, upon which a new metric for the rebound effect can be built

Multiplexity and multireciprocity in directed multiplexes

Real-world multi-layer networks feature nontrivial dependencies among links of different layers. Here we argue that, if links are directed, dependencies are twofold. Besides the ordinary tendency of links of different layers to align as the result of `multiplexity', there is also a tendency to anti-align as the result of what we call `multireciprocity', i.e. the fact that links in one layer can be reciprocated by \emph{opposite} links in a different layer. Multireciprocity generalizes the scalar definition of single-layer reciprocity to that of a square matrix involving all pairs of layers. We introduce multiplexity and multireciprocity matrices for both binary and weighted multiplexes and validate their statistical significance against maximum-entropy null models that filter out the effects of node heterogeneity. We then perform a detailed empirical analysis of the World Trade Multiplex (WTM), representing the import-export relationships between world countries in different commodities. We show that the WTM exhibits strong multiplexity and multireciprocity, an effect which is however largely encoded into the degree or strength sequences of individual layers. The residual effects are still significant and allow to classify pairs of commodities according to their tendency to be traded together in the same direction and/or in opposite ones. We also find that the multireciprocity of the WTM is significantly lower than the usual reciprocity measured on the aggregate network. Moreover, layers with low (high) internal reciprocity are embedded within sets of layers with comparably low (high) mutual multireciprocity. This suggests that, in the WTM, reciprocity is inherent to groups of related commodities rather than to individual commodities. We discuss the implications for international trade research focusing on product taxonomies, the product space, and fitness/complexity metrics.Comment: 20 pages, 8 figure

Interactions between financial and environmental networks in OECD countries

We analyse a multiplex of networks between OECD countries during the decade 2002-2010, which consists of five financial layers, given by foreign direct investment, equity securities, short-term, long-term and total debt securities, and five environmental layers, given by emissions of N O x, P M 10 SO 2, CO 2 equivalent and the water footprint associated with international trade. We present a new measure of cross-layer correlations between flows in different layers based on reciprocity. For the assessment of results, we implement a null model for this measure based on the exponential random graph theory. We find that short-term financial flows are more correlated with environmental flows than long-term investments. Moreover, the correlations between reverse financial and environmental flows (i.e. flows of different layers going in opposite directions) are generally stronger than correlations between synergic flows (flows going in the same direction). This suggests a trade-off between financial and environmental layers, where, more financialised countries display higher correlations between outgoing financial flows and incoming environmental flows from lower financialised countries, which could have important policy implications. Five countries are identified as hubs in this finance-environment multiplex: The United States, France, Germany, Belgium-Luxembourg and the United Kingdom.Comment: Supplementary Information provide

The role of distances in the World Trade Web

In the economic literature, geographic distances are considered fundamental factors to be included in any theoretical model whose aim is the quantification of the trade between countries. Quantitatively, distances enter into the so-called gravity models that successfully predict the weight of non-zero trade flows. However, it has been recently shown that gravity models fail to reproduce the binary topology of the World Trade Web. In this paper a different approach is presented: the formalism of exponential random graphs is used and the distances are treated as constraints, to be imposed on a previously chosen ensemble of graphs. Then, the information encoded in the geographical distances is used to explain the binary structure of the World Trade Web, by testing it on the degree-degree correlations and the reciprocity structure. This leads to the definition of a novel null model that combines spatial and non-spatial effects. The effectiveness of spatial constraints is compared to that of nonspatial ones by means of the Akaike Information Criterion and the Bayesian Information Criterion. Even if it is commonly believed that the World Trade Web is strongly dependent on the distances, what emerges from our analysis is that distances do not play a crucial role in shaping the World Trade Web binary structure and that the information encoded into the reciprocity is far more useful in explaining the observed patterns.Comment: Preprint, accepted for SITIS 2012 (http://www.sitis-conf.org/). Final version to be published by IEEE Computer Society as conference proceeding

On the relationship between energy efficiency and complexity: Insight on the causality chain

The relationship between the energy efficiency, energy density and complexity level of the system is here addressed from both thermodynamic and evolutionary perspectives. A case study from economic systems is presented to show that, contrary to widespread opinion, energy efficiency is responsible for energy growth and the complexity leap. This article further examines to what extent complexity, on a historical time scale, may evolve to counterbalance conservative effects brought about by energy efficiency. We analyze structural complexity growth by four different paradigms. An evolutionary pattern is then proposed that may encompass the broad dynamics underlying complexity growth. This evolutionary pattern rests on the hypothesis that thermodynamic evolutionary systems are featured from an ever growing influx of energy driven into the system by self-catalytic processes, which must find its way through the constrains of the system. The system initially disposes of the energy by expanding, in extent and in number of components, up to saturation due to inner or outer constraints. The two counteractive forces, constraints and growing energy flux, expose the systems to new gradients. Every new gradient upon the system represents a symmetry rupture in components' space. By exploring a new gradient, the system imposes further restrictions on its components and increases its overall degree of freedom. © 2008 WIT Press