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The structure of many financial networks is protected by privacy and has to be inferred from aggregate observables. Here we consider one of the most successful network reconstruction methods, producing random graphs with desired link density and where the observed constraints (related to the market size of each node) are replicated as averages over the graph ensemble, but not in individual realizations. We show that there is a minimum critical link density below which the method exhibits an `unreconstructability' phase where at least one of the constraints, while still reproduced on average, is far from its expected value in typical individual realizations. We establish the scaling of the critical density for various theoretical and empirical distributions of interbank assets and liabilities, showing that the threshold differs from the critical densities for the onset of the giant component and of the unique component in the graph. We also find that, while dense networks are always reconstructable, sparse networks are unreconstructable if their structure is homogeneous, while they can display a crossover to reconstructability if they have an appropriate core-periphery or heterogeneous structure. Since the reconstructability of interbank networks is related to market clearing, our results suggest that central bank interventions aimed at lowering the density of links should take network structure into account to avoid unintentional liquidity crises where the supply and demand of all financial institutions cannot be matched simultaneously.

The abundance of data about social, economic and political relationships allows social theories to be tested against empirical evidence and human behaviour to be analyzed just as any other natural phenomenon. Here we focus on balance theory, stating that actors in signed social networks tend to avoid the formation of `unbalanced', or `frustrated', cycles, i.e. cycles with an odd number of negative links. This statement can be supported statistically only after a comparison with a null model. Since the existing benchmarks do not typically account for the heterogeneity of individual actors, here we first extend the Exponential Random Graphs framework to signed networks with both global (homogeneous) and local (heterogeneous) constraints and then employ them to assess the significance of unbalanced patterns in several real-world networks. We find that the nature and level of balance in social networks crucially depends on the null model employed. In particular, the study of signed triangles and signed communities reveals that homogeneous null models favour the weak version of balance theory, according to which only triangles with one negative link should be under-represented in social networks, while heterogeneous null models favour the strong version of balance theory, according to which also triangles with all negative links should be under-represented. Biological networks, instead, display almost inverted patterns and strong frustration under any benchmark, confirming that structural balance inherently distinguishes social networks from other signed networks., 33 pages, 11 figures, 4 tables

Analysing weighted networks requires modelling the binary and weighted properties simultaneously. We highlight three approaches for estimating the parameters responsible for them: econometric techniques treating topology as deterministic and statistical techniques either ensemble-averaging parameters or maximising an averaged likelihood over the topological randomness. In homogeneous models, equivalence holds; in heterogeneous network models, the local disorder breaks it, in a way reminiscent of the difference between `quenched' and `annealed' averages in the physics of disordered systems., 12 pages, 5 figures

Shannon entropy, a cornerstone of information theory, statistical physics and inference methods, is uniquely identified by the Shannon-Khinchin or Shore-Johnson axioms. Generalizations of Shannon entropy, motivated by the study of non-extensive or non-ergodic systems, relax some of these axioms and lead to entropy families indexed by certain `entropic' parameters. In general, the selection of these parameters requires pre-knowledge of the system or encounters inconsistencies. Here we introduce a simple axiom for any entropy family: namely, that no entropic parameter can be inferred from a completely uninformative (uniform) probability distribution. When applied to the Uffink-Jizba-Korbel and Hanel-Thurner entropies, the axiom selects only R\'enyi entropy as viable. It also extends consistency with the Maximum Likelihood principle, which can then be generalized to estimate the entropic parameter purely from data, as we confirm numerically. Remarkably, in a generalized maximum-entropy framework the axiom implies that the maximized log-likelihood always equals minus Shannon entropy, even if the inferred probability distribution maximizes a generalized entropy and not Shannon's, solving a series of problems encountered in previous approaches.

We consider an inhomogeneous Erd\H{o}s-R\'enyi random graph ensemble with exponentially decaying random disconnection probabilities determined by an i.i.d. field of variables with heavy tails and infinite mean associated to the vertices of the graph. This model was recently investigated in the physics literature in Garuccio et al. (2020) as a scale-invariant random graph within the context of network renormalization. From a mathematical perspective, the model fits in the class of scale-free inhomogeneous random graphs whose asymptotic geometrical features have been recently attracting interest. While for this type of graphs several results are known when the underlying vertex variables have finite mean and variance, here instead we consider the case of one-sided stable variables with necessarily infinite mean. To simplify our analysis, we assume that the variables are sampled from a Pareto distribution with parameter $\alpha\in(0,1)$. We start by characterizing the asymptotic distributions of the typical degrees and some related observables. In particular, we show that the degree of a vertex converges in distribution, after proper scaling, to a mixed Poisson law. We then show that correlations among degrees of different vertices are asymptotically non-vanishing, but at the same time a form of asymptotic tail independence is found when looking at the behavior of the joint Laplace transform around zero. Moreover, we present some findings concerning the asymptotic density of wedges and triangles, show a cross-over for the existence of dust (i.e. disconnected nodes), and identify a strongly connected regime in the ensemble., Comment: Some typos and inconsistencies corrected, a new result on connectedness added

Ecological networks such as plant-pollinator systems and food webs vary in space and time. This variability includes fluctuations in global network properties such as total number and intensity of interactions but also in the local properties of individual nodes such as the number and intensity of species-level interactions. Fluctuations of species properties can significantly affect higher-order network features, e.g. robustness and nestedness. Local fluctuations should therefore be controlled for in applications that rely on null models, especially pattern and perturbation detection. By contrast, most randomization methods for null models used by ecologists treat node-level local properties as hard constraints that cannot fluctuate. Here, we synthesise a set of methods that resolves the limit of hard constraints and is based on statistical mechanics. We illustrate the methods with some practical examples making available open source computer codes. We clarify how this approach can be used by experimental ecologists to detect non-random network patterns with null models that not only rewire but also redistribute interaction strengths by allowing fluctuations in the null model constraints (soft constraints). Null modelling of species heterogeneity through local fluctuations around typical topological and quantitative constraints offers a statistically robust and expanded (e.g. quantitative null models) set of tools to understand the assembly and resilience of ecological networks., Comment: submitted

The idiosyncratic (microscopic) and systemic (macroscopic) components of market structure have been shown to be responsible for the departure of the optimal mean-variance allocation from the heuristic 'equally-weighted' portfolio. In this paper, we exploit clustering techniques derived from Random Matrix Theory (RMT) to study a third, intermediate (mesoscopic) market structure that turns out to be the most stable over time and provides important practical insights from a portfolio management perspective. First, we illustrate the benefits, in terms of predicted and realized risk profiles, of constructing portfolios by filtering out both random and systemic comovements from the correlation matrix. Second, we redefine the portfolio optimization problem in terms of stock clusters that emerge after filtering. Finally, we propose a new wealth allocation scheme that attaches equal importance to stocks belonging to the same community and show that it further increases the reliability of the constructed portfolios. Results are robust across different time spans, cross-sectional dimensions and set of constraints defining the optimization problem

Systems with lattice geometry can be renormalized exploiting their coordinates in metric space, which naturally define the coarse-grained nodes. By contrast, complex networks defy the usual techniques, due to their small-world character and lack of explicit geometric 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 additive hidden variables 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, they lead to realistic scale-free networks with density-dependent cut-off, assortatitivy and finite local clustering, even in the sparse regime and in absence of geometry. If they are quenched, they can guide the renormalization of real-world networks with node attributes and distance-dependence or communities. As an application, we derive an accurate multiscale model of the International Trade Network applicable across hierarchically nested geographic partitions.

We provide an up-to-date view on the knowledge management system ScienceWISE (SW) and address issues related to the automatic assignment of articles to research topics. So far, SW has been proven to be an effective platform for managing large volumes of technical articles by means of ontological concept-based browsing. However, as the publication of research articles accelerates, the expressivity and the richness of the SW ontology turns into a double-edged sword: a more fine-grained characterization of articles is possible, but at the cost of introducing more spurious relations among them. In this context, the challenge of continuously recommending relevant articles to users lies in tackling a network partitioning problem, where nodes represent articles and co-occurring concepts create edges between them. In this paper, we discuss the three research directions we have taken for solving this issue: i) the identification of generic concepts to reinforce inter-article similarities; ii) the adoption of a bipartite network representation to improve scalability; iii) the design of a clustering algorithm to identify concepts for cross-disciplinary articles and obtain fine-grained topics for all articles., 6 pages; 5 figures

In economic and financial networks, the strength (total value of the connections) of a given node has always an important economic meaning, such as the size of supply and demand, import and export, or financial exposure. Constructing null models of networks matching the observed strengths of all nodes is crucial in order to either detect interesting deviations of an empirical network from economically meaningful benchmarks or reconstruct the most likely structure of an economic network when the latter is unknown. However, several studies have proved that real economic networks are topologically very different from networks inferred only from node strengths. Here we provide a detailed analysis for the World Trade Web (WTW) by comparing it to an enhanced null model that simultaneously reproduces the strength and the number of connections of each node. We study several temporal snapshots and different aggregation levels (commodity classes) of the WTW and systematically find that the observed properties are extremely well reproduced by our model. This allows us to introduce the concept of extensive and intensive bias, defined as a measurable tendency of the network to prefer either the formation of new links or the reinforcement of existing ones. We discuss the possible economic interpretation in terms of trade margins.

Network topology plays a key role in many phenomena, from the spreading of diseases to that of financial crises. Whenever the whole structure of a network is unknown, one must resort to reconstruction methods that identify the least biased ensemble of networks consistent with the partial information available. A challenging case is when there is only local (node-specic) information available. For binary networks, the relevant ensemble is one where the degree (number of links) of each node is constrained to its observed value. However, for weighted networks the problem is much more complicated. While the naive approach prescribes to constrain the strengths (total link weights) of all nodes, recent counter-intuitive results suggest that in weighted networks the degrees are often more informative than the strengths, and as 'fundamental' as the latter. This implies that the reconstruction of weighted networks would be signicantly enhanced by the specication of both quantities, a computationally hard and bias-prone procedure. Here we solve this problem by introducing an analytical and unbiased maximum-entropy method that works in the shortest possible time and does not require the explicit generation of reconstructed samples. We consider several real-world applications and show that, while the strengths alone give poor results, the additional knowledge of the degrees yields accurately reconstructed networks. Information-theoretic criteria rigorously conrm that the binary information is irreducible to the weighted one. Our results have strong implications for the analysis of motifs and communities and whenever the reconstructed ensemble is required as a null model to detect higher-order patterns.

In self-organizing networks, topology and dynamics coevolve in a continuous feedback, without exogenous driving. The World Trade Network (WTN) is one of the few empirically well documented examples of self-organizing networks: its topology strongly depends on the GDP of world countries, which in turn depends on the structure of trade. Therefore, understanding which are the key topological properties of the WTN that deviate from randomness provides direct empirical information about the structural effects of self-organization. Here, using an analytical pattern-detection method that we have recently proposed, we study the occurrence of triadic "motifs" (subgraphs of three vertices) in the WTN between 1950 and 2000. We find that, unlike other properties, motifs are not explained by only the in- and out-degree sequences. By contrast, they are completely explained if also the numbers of reciprocal edges are taken into account. This implies that the self-organization process underlying the evolution of the WTN is almost completely encoded into the dyadic structure, which strongly depends on reciprocity., 12 pages, 3 figures; Best Paper Award at the 6th International Conference on Self-Organizing Systems, Delft, The Netherlands, 15-16/03/2012

In this sequel to a companion paper, we complement our analysis of the binary projections of the International Trade Network (ITN) by considering its weighted representations. We show that, unlike the binary case, all possible weighted representations of the ITN (directed/undirected, aggregated/disaggregated) cannot be traced back to local structural properties, which are therefore of limited informativeness. Our results highlight that any topological property representing only partial information (e.g., degree sequences) cannot in general be obtained from the corresponding weighted property (e.g., strength sequences). Therefore the expectation that weighted structural properties offer a more complete description than purely topological ones is misleading. Our analysis of the ITN detects indirect effects that are not captured by traditional macroeconomic analyses focused only on weighted rst-order country-specic properties, and highlights the limitations of models and theories that overemphasize the need to reproduce and explain such properties.