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The field of complex networks studies a wide variety of interacting systems by representing them as networks. To understand their properties and mutual relations, the randomisation of network connections is a commonly used tool. However, information-theoretic randomisation methods with well-established foundations mostly provide a stationary description of these systems, while stochastic randomisation methods that account for their dynamic nature lack such general foundations and require extensive repetition of the stochastic process to measure statistical properties. In this work, we extend the applicability of information-theoretic methods beyond stationary network models. By using the information-theoretic principle of maximum caliber we construct dynamic network ensemble distributions based on constraints representing statistical properties with known values throughout the evolution. We focus on the particular cases of dynamics constrained by the average number of connections of the whole network and each node, comparing each evolution to simulations of stochastic randomisation that obey the same constraints. We find that ensemble distributions estimated from simulations match those calculated with maximum caliber and that the equilibrium distributions to which they converge agree with known results of maximum entropy given the same constraints. Finally, we discuss further the connections to other maximum entropy approaches to network dynamics and conclude by proposing some possible avenues of future research.

The largest 6,529 international corporations are accountable for almost 30% of global CO2e emissions. A growing awareness of the role of the corporate world in the path toward sustainability has led many shareholders and stakeholders to pursue increasingly stringent and ambitious environmental goals. However, how to assess the corporate environmental performance objectively and efficiently remains an open question. This study reveals underlying dynamics and structures that can be used to construct a unified quantitative picture of the environmental impact of companies. This study shows that the environmental impact (metabolism) of companies CO2e energy used, water withdrawal and waste production, scales with their size according to a simple power law which is often sublinear, and can be used to derive a sector-specific, size-dependent benchmark to asses unambiguously a company's environmental performance. Enforcing such a benchmark would potentially result in a 15% emissions reduction, but a fair and effective environmental policy should consider the size of the corporation and the super or sublinear nature of the scaling relationship, Comment: Another update copy was uploaded

Electricity storage systems (ESS) are hailed by many scholars and practitioners as a key element of the future electricity systems and a key step toward the transition to renewables . Nonetheless, the global speed of ESS implementation is relatively slow, and among possible reasons is the lack of viable business models. We developed an agent-based model to simulate the behavior of ESS within the Dutch electricity market. We adopted an exploratory modeling analysis (EMA) approach to investigate the effects of two specific business models on the value of ESS from the perspective of both investors and the government under uncertainties in the ESS technical and economics characteristics, and uncertainties in market conditions and regulations. Our results show ESS is not profitable in most scenarios, and generally wholesale arbitrage business model leads to more profit than reserve capacity. In addition, ESS economic and technical characteristics play more important roles in the value of ESS than market conditions, and carbon pricing., Comment: 39 Pages, 6 Figures, 2 tables, article under review

Over the last two decades, network theory has shown to be a fruitful paradigm in understanding the organization and functioning of real-world complex systems. One technique helpful to this endeavor is identifying functionally influential subgraphs, shedding light on underlying evolutionary processes. Such overrepresented subgraphs, "motifs", have received much attention in simple networks, where edges are either on or off. However, for weighted networks, motif analysis is still undeveloped. Here, we proposed a novel methodology - based on a random walker taking a fixed maximum number of steps - to study weighted motifs of limited size. We introduce a sink node to balance the network and allow the detection of configurations within an a priori fixed number of steps for the random walker. We applied this approach to different real networks and selected a specific benchmark model based on maximum entropy to test the significance of weighted motifs occurrence. We found that identified similarities enable the classifications of systems according to functioning mechanisms associated with specific configurations: economic networks exhibit close patterns while differentiating from ecological systems without any a priori assumption., Comment: 13 pages, 10 figures

One major hurdle in the road toward a low carbon economy is the present entanglement of developed economies with oil. This tight relationship is mirrored in the correlation between most of economic indicators with oil price. This paper addresses the role of oil compared to the other three main energy commodities -coal, gas and electricity, in shaping the international trading network (ITW or WTW, world trade web) in the light of network theory. It initially surveys briefly the literature on the correlation between oil prices with economic growth and compares the concepts of time correlation with the concept of spatial correlation brought about by network theory. It outlines the conceptual framework underpinning the network measures adopted in the analysis and results are presented. Three measures are taken into account: the ratio of mutual exchanges in the network (reciprocity); the role of distances in determining trades (spatial filling); and the spatial correlation of energy commodities with the whole trade network and with four trading categories: food, capital goods, intermediate goods and consumption goods. The analysis deliver five main results:1) the the energy commodities network was structurally stable amid dramatic growth during the decade considered; 2) that oil is the most correlated energy commodity to the world trade web; 3) that oil is the most pervasive network, though coal is the less affected by distances; 4) that oil has a remarkably high level of internal reciprocity and external overlapping 5) that the reciprocity of the trade network is negative correlated in time with the price of oil.

Our work sheds new light on the role of oil prices in shaping the world economy by investigating flows of goods and services through global value chains between 1960 and 2011, by means of Markov Chain and network analysis. We show that over that time period the international division of labor and trade patterns are tightly linked to the price of oil. We demonstrate that this correlation does not depend on the balance of payments nor on the nominal value of trade or trade agreements; it is instead linked to the way the Global Value Chains (GVCs) shape global trade. Our study suggests that transport played an important structural role in shaping GVCs., Comment: 11 pages, 6 figures, 2 tables

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 provided

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 figures

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 figures

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 proceedings

All types of networks arise as intricate combinations of dyadic building blocks formed by pairs of vertices. In directed networks, the dyadic patterns are entirely determined by reciprocity, i.e. the tendency to form, or to avoid, mutual links. Reciprocity has dramatic effects on every networks dynamical processes and the emergence of structures like motifs and communities. The binary reciprocity has been extensively studied: that of weighted networks is still poorly understood. We introduce a general approach to it, by defining quantities capturing the observed patterns (from dyad-specific to vertex-specific and network-wide) and introducing analytically solved models (Exponential Random Graphs-type). Counter-intuitively, the previous reciprocity measures based on the similarity of the mutual links-weights are uninformative. By contrast, our measures can classify different weighted networks, track the temporal evolution of a networks reciprocity, identify patterns. We show that in some networks the local reciprocity structure can be inferred from the global one., Comment: 21 pages, 7 figures, 1 table

Spatially embedded networks are shaped by a combination of purely topological (space-independent) and space-dependent formation rules. While it is quite easy to artificially generate networks where the relative importance of these two factors can be varied arbitrarily, it is much more difficult to disentangle these two architectural effects in real networks. Here we propose a solution to the problem by introducing global and local measures of spatial effects that, through a comparison with adequate null models, effectively filter out the spurious contribution of non-spatial constraints. Our filtering allows us to consistently compare different embedded networks or different historical snapshots of the same network. As a challenging application we analyse the World Trade Web, whose topology is expected to depend on geographic distances but is also strongly determined by non-spatial constraints (degree sequence or GDP). Remarkably, we are able to detect weak but significant spatial effects both locally and globally in the network, showing that our method succeeds in retrieving spatial information even when non-spatial factors dominate. We finally relate our results to the economic literature on gravity models and trade globalization.

We exploit the symmetry concepts developed in the companion review of this article to introduce a stochastic version of link reversal symmetry, which leads to an improved understanding of the reciprocity of directed networks. We apply our formalism to the international trade network and show that a strong embedding in economic space determines particular symmetries of the network, while the observed evolution of reciprocity is consistent with a symmetry breaking taking place in production space. Our results show that networks can be strongly affected by symmetry-breaking phenomena occurring in embedding spaces, and that stochastic network symmetries can successfully suggest, or rule out, possible underlying mechanisms., Comment: Final accepted version

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 version