The asymptotic equivalence of canonical and microcanonical ensembles is a central concept in statistical physics, with important consequences for both theoretical research and practical applications. However, this property breaks down under certain c
ircumstances. The most studied violation of ensemble equivalence requires phase transitions, in which case it has a `restricted (i.e. confined to a certain region in parameter space) but `strong (i.e. characterized by a difference between the entropies of the two ensembles that is of the same order as the entropies themselves) form. However, recent research on networks has shown that the presence of an extensive number of local constraints can lead to ensemble nonequivalence even in the absence of phase transitions. This occurs in a `weak (i.e. leading to a subleading entropy difference) but remarkably `unrestricted (i.e. valid in the entire parameter space) form. Here we look for more general manifestations of ensemble nonequivalence in arbitrary ensembles of matrices with given margins. These models have widespread applications in the study of spatially heterogeneous and/or temporally nonstationary systems, with consequences for the analysis of multivariate financial and neural time-series, multi-platform social activity, gene expression profiles and other Big Data. We confirm that ensemble nonequivalence appears in `unrestricted form throughout the entire parameter space due to the extensivity of local constraints. Surprisingly, at the same time it can also exhibit the `strong form. This novel, simultaneously `strong and unrestricted form of nonequivalence is very robust and imposes a principled choice of the ensemble. We calculate the proper mathematical quantities to be used in real-world applications.
The World Trade Web (WTW) is the network of international trade relationships among world countries. Characterizing both the local link weights (observed trade volumes) and the global network structure (large-scale topology) of the WTW via a single m
odel is still an open issue. While the traditional Gravity Model (GM) successfully replicates the observed trade volumes by employing macroeconomic properties such as GDP and geographic distance, it, unfortunately, predicts a fully connected network, thus returning a completely unrealistic topology of the WTW. To overcome this problem, two different classes of models have been introduced in econometrics and statistical physics. Econometric approaches interpret the traditional GM as the expected value of a probability distribution that can be chosen arbitrarily and tested against alternative distributions. Statistical physics approaches construct maximum-entropy probability distributions of (weighted) graphs from a chosen set of measurable structural constraints and test distributions resulting from different constraints. Here we compare and integrate the two approaches by considering a class of maximum-entropy models that can incorporate macroeconomic properties used in standard econometric models. We find that the integrated approach achieves a better performance than the purely econometric one. These results suggest that the maximum-entropy construction can serve as a viable econometric framework wherein extensive and intensive margins can be separately controlled for, by combining topological constraints and dyadic macroeconomic variables.
Ecological networks such as plant-pollinator systems vary systematically in space and time. This variability includes fluctuations in global network properties such as total number and intensity of interactions in the network, but also in the local p
roperties of individual nodes, such as the number and intensity of species-level interactions. Fluctuations of local properties can significantly affect higher-order network features, e.g. robustness and nestedness. These fluctuations should therefore be controlled for in applications that rely on null models, including pattern detection, perturbation experiments and network reconstruction from limited observations. By contrast, most randomization methods used by ecologists treat node-level local properties as hard constraints that cannot fluctuate. Here we synthesise a set of methods based on the statistical mechanics of networks, which we illustrate with some practical examples. We illustrate how this approach can be used by experimental ecologists to study the statistical significance of network patterns and the rewiring of networks under simulated perturbations. Modelling species heterogeneity, while allowing for local fluctuations around a theoretically grounded notion of structural equilibrium, will offer a new generation of models and experiments to understand the assembly and resilience of ecological networks.
The asymptotic (non)equivalence of canonical and microcanonical ensembles, describing systems with soft and hard constraints respectively, is a central concept in statistical physics. Traditionally, the breakdown of ensemble equivalence (EE) has been
associated with nonvanishing relative canonical fluctuations of the constraints in the thermodynamic limit. Recently, it has been reformulated in terms of a nonvanishing relative entropy density between microcanonical and canonical probabilities. The earliest observations of EE violation required phase transitions or long-range interactions. More recent research on binary networks found that an extensive number of local constraints can also break EE, even in absence of phase transitions. Here we study for the first time ensemble nonequivalence in weighted networks with local constraints. Unlike their binary counterparts, these networks can undergo a form of Bose-Einstein condensation (BEC) producing a core-periphery structure where a finite fraction of the link weights concentrates in the core. This phenomenon creates a unique setting where local constraints coexist with a phase transition. We find surviving relative fluctuations only in the condensed phase, as in more traditional BEC settings. However, we also find a non-vanishing relative entropy density for all temperatures, signalling a breakdown of EE due to the presence of an extensive number of constraints, irrespective of BEC. Therefore, in presence of extensively many local constraints, vanishing relative fluctuations no longer guarantee EE.
Systems with lattice geometry can be renormalized exploiting their embedding in metric space, which naturally defines the coarse-grained nodes. By contrast, complex networks defy the usual techniques because of their small-world character and lack of
explicit metric embedding. Current network renormalization approaches require strong assumptions (e.g. community structure, hyperbolicity, scale-free topology), thus remaining incompatible with generic graphs and ordinary lattices. Here we introduce a graph renormalization scheme valid for any hierarchy of coarse-grainings, thereby allowing for the definition of `block nodes across multiple scales. This approach reveals a necessary and specific dependence of network topology on an additive hidden variable attached to nodes, plus optional dyadic factors. Renormalizable networks turn out to be consistent with a unique specification of the fitness model, while they are incompatible with preferential attachment, the configuration model or the stochastic blockmodel. These results highlight a deep conceptual distinction between scale-free and scale-invariant networks, and provide a geometry-free route to renormalization. If the hidden variables are annealed, the model spontaneously leads to realistic scale-free networks with cut-off. If they are quenched, the model can be used to renormalize real-world networks with node attributes and distance-dependence or communities. As an example we derive an accurate multiscale model of the International Trade Network applicable across hierarchical geographic partitions.
For random systems subject to a constraint, the microcanonical ensemble requires the constraint to be met by every realisation (hard constraint), while the canonical ensemble requires the constraint to be met only on average (soft constraint). It is
known that for random graphs subject to topological constraints breaking of ensemble equivalence may occur when the size of the graph tends to infinity, signalled by a non-vanishing specific relative entropy of the two ensembles. We investigate to what extent breaking of ensemble equivalence is manifested through the largest eigenvalue of the adjacency matrix of the graph. We consider two examples of constraints in the dense regime: (1) fix the degrees of the vertices (= the degree sequence); (2) fix the sum of the degrees of the vertices (= twice the number of edges). Example (1) imposes an extensive number of local constraints and is known to lead to breaking of ensemble equivalence. Example (2) imposes a single global constraint and is known to lead to ensemble equivalence. Our working hypothesis is that breaking of ensemble equivalence corresponds to a non-vanishing difference of the expected values of the largest eigenvalue under the two ensembles. We verify that, in the limit as the size of the graph tends to infinity, the difference between the expected values of the largest eigenvalue in the two ensembles does not vanish for (1) and vanishes for (2). A key tool in our analysis is a transfer method that uses relative entropy to determine whether probabilistic estimates can be carried over from the canonical ensemble to the microcanonical ensemble, and illustrates how breaking of ensemble equivalence may prevent this from being possible.
Crowded trades by similarly trading peers influence the dynamics of asset prices, possibly creating systemic risk. We propose a market clustering measure using granular trading data. For each stock the clustering measure captures the degree of tradin
g overlap among any two investors in that stock. We investigate the effect of crowded trades on stock price stability and show that market clustering has a causal effect on the properties of the tails of the stock return distribution, particularly the positive tail, even after controlling for commonly considered risk drivers. Reduced investor pool diversity could thus negatively affect stock price stability.
Many real networks feature the property of nestedness, i.e. the neighbours of nodes with a few connections are hierarchically nested within the neighbours of nodes with more connections. Despite the abstract simplicity of this notion, different mathe
matical definitions of nestedness have been proposed, sometimes giving contrasting results. Moreover, there is an ongoing debate on the statistical significance of nestedness, since even random networks where the number of connections (degree) of each node is fixed to its empirical value are typically as nested as real-world ones. Here we propose a clarification that exploits the recent finding that random networks where the degrees are enforced as hard constraints (microcanonical ensembles) are thermodynamically different from random networks where the degrees are enforced as soft constraints (canonical ensembles). We show that if the real network is perfectly nested, then the two ensembles are trivially equivalent and the observed nestedness, independently of its definition, is indeed an unavoidable consequence of the empirical degrees. On the other hand, if the real network is not perfectly nested, then the two ensembles are not equivalent and alternative definitions of nestedness can be even positively correlated in the canonical ensemble and negatively correlated in the microcanonical one. This result disentangles distinct notions of nestedness captured by different metrics and highlights the importance of making a principled choice between hard and soft constraints in null models of ecological networks.
Multilayer networks represent multiple types of connections between the same set of nodes. Clearly, a multilayer description of a system adds value only if the multiplex does not merely consist of independent layers, i.e. if the inter-layer overlap i
s nontrivial. On real-world multiplexes, it is expected that the observed overlap may partly result from spurious correlations arising from the heterogeneity of nodes and partly from true interdependencies. However, no rigorous way to disentangle these two effects has been developed. In this paper we introduce an unbiased maximum-entropy model of multiplexes with controllable node degrees and controllable overlap. The model can be mapped to a generalized Ising model where the combination of node heterogeneity and inter-layer coupling leads to the possibility of local phase transitions. In particular, we find that an increased heterogeneity in the network results in different critical points for different pairs of nodes, which in turn leads to local phase transitions that may ultimately increase the overlap. The model allows us to quantify how the overlap can be increased by either increasing the heterogeneity of the network (spurious correlation) or the strength of the inter-layer coupling (true correlation), thereby disentangling the two effects. As an application, we show that the empirical overlap in the International Trade Multiplex is not merely a spurious result of the correlation between node degrees across different layers, but requires a non-zero inter-layer coupling in its modeling.
In statistical physics, the challenging combinatorial enumeration of the configurations of a system subject to hard constraints (microcanonical ensemble) is mapped to a mathematically easier calculation where the constraints are softened (canonical e
nsemble). However, the mapping is exact only when the size of the system is infinite and if the property of ensemble equivalence (EE), i.e. the asymptotic identity of canonical and microcanonical large deviations, holds. For finite systems, or when EE breaks down, statistical physics is currently believed to provide no answer to the combinatorial problem. In contrast with this expectation, here we establish exact relationships connecting conjugate ensembles in full generality, even for finite system size and when EE does not hold. We also show that in the thermodynamic limit the ensembles are directly related through the matrix of canonical (co)variances of the constraints, plus a correction term that survives only if this matrix has an infinite number of finite eigenvalues. These new relationships restore the possibility of enumerating microcanonical configurations via canonical probabilities, thus reconnecting statistical physics and combinatorics in realms where they were believed to be no longer in mutual correspondence.
