arXiv subjects

Diego Garlaschelli

Publications and source records attributed to Diego Garlaschelli.

At least 55 records · Page 3Linked to original sources

arXiv2023

Largest Eigenvalue of the Configuration Model and Breaking of Ensemble Equivalence

We analyse the largest eigenvalue of the adjacency matrix of the configuration model with large degrees, where the latter are treated as hard constraints. In particular, we compute the expectation of the largest eigenvalue for degrees that diverge as the number of vertices $n$ tends to infinity, uniformly on a scale between $1$ and $\sqrt{n}$, and show that a weak law of large numbers holds. We compare with what was derived in our earlier paper "Central limit theorem for the principal eigenvalue and eigenvector of Chung-Lu random graphs" for the Chung-Lu model, which in the regime considered represents the corresponding configuration model with soft constraints, and show that the expectation is shifted down by $1$ asymptotically. This shift is a signature of breaking of ensemble equivalence between the hard and soft (also known as micro-canonical and canonical) versions of the configuration model. The latter result generalizes a previous finding in "A spectral signature of breaking of ensemble equivalence for constrained random graphs" obtained in the case when all degrees are equal.

math.PR↗

arXiv2023

Temporal networks with node-specific memory: unbiased inference of transition probabilities, relaxation times and structural breaks

One of the main challenges in the study of time-varying networks is the interplay of memory effects with structural heterogeneity. In particular, different nodes and dyads can have very different statistical properties in terms of both link formation and link persistence, leading to a superposition of typical timescales, sub-optimal parametrizations and substantial estimation biases. Here we develop an unbiased maximum-entropy framework to study empirical network trajectories by controlling for the observed structural heterogeneity and local link persistence simultaneously. An exact mapping to a heterogeneous version of the one-dimensional Ising model leads to an analytic solution that rigorously disentangles the hidden variables that jointly determine both static and temporal properties. Additionally, model selection via likelihood maximization identifies the most parsimonious structural level (either global, node-specific or dyadic) accounting for memory effects. As we illustrate on a real-world social network, this method enables an improved estimation of dyadic transition probabilities, relaxation times and structural breaks between dynamical regimes. In the resulting picture, the graph follows a generalized configuration model with given degrees and given time-persisting degrees, undergoing transitions between empirically identifiable stationary regimes.

physics.soc-ph↗

arXiv2023

Deterministic, quenched and annealed parameter estimation for heterogeneous network models

At least two, different approaches to define and solve statistical models for the analysis of economic systems exist: the typical, econometric one, interpreting the Gravity Model specification as the expected link weight of an arbitrary probability distribution, and the one rooted into statistical physics, constructing maximum-entropy distributions constrained to satisfy certain network properties. In a couple of recent, companion papers they have been successfully integrated within the framework induced by the constrained minimisation of the Kullback-Leibler divergence: specifically, two, broad classes of models have been devised, i.e. the integrated and the conditional ones, defined by different, probabilistic rules to place links, load them with weights and turn them into proper, econometric prescriptions. Still, the recipes adopted by the two approaches to estimate the parameters entering into the definition of each model differ. In econometrics, a likelihood that decouples the binary and weighted parts of a model, treating a network as deterministic, is typically maximised; to restore its random character, two alternatives exist: either solving the likelihood maximisation on each configuration of the ensemble and taking the average of the parameters afterwards or taking the average of the likelihood function and maximising the latter one. The difference between these approaches lies in the order in which the operations of averaging and maximisation are taken - a difference that is reminiscent of the quenched and annealed ways of averaging out the disorder in spin glasses. The results of the present contribution, devoted to comparing these recipes in the case of continuous, conditional network models, indicate that the annealed estimation recipe represents the best alternative to the deterministic one.

physics.soc-ph↗

arXiv2023

Social media battle for attention: opinion dynamics on competing networks

In the age of information abundance, attention is a coveted resource. Social media platforms vigorously compete for users' engagement, influencing the evolution of their opinions on a variety of topics. With recommendation algorithms often accused of creating "filter bubbles", where like-minded individuals interact predominantly with one another, it's crucial to understand the consequences of this unregulated attention market. To address this, we present a model of opinion dynamics on a multiplex network. Each layer of the network represents a distinct social media platform, each with its unique characteristics. Users, as nodes in this network, share their opinions across platforms and decide how much time to allocate in each platform depending on its perceived quality. Our model reveals two key findings. i) When examining two platforms - one with a neutral recommendation algorithm and another with a homophily-based algorithm - we uncover that even if users spend the majority of their time on the neutral platform, opinion polarization can persist. ii) By allowing users to dynamically allocate their social energy across platforms in accordance to their homophilic preferences, a further segregation of individuals emerges. While network fragmentation is usually associated with "echo chambers", the emergent multi-platform segregation leads to an increase in users' satisfaction without the undesired increase in polarization. These results underscore the significance of acknowledging how individuals gather information from a multitude of sources. Furthermore, they emphasize that policy interventions on a single social media platform may yield limited impact.

physics.soc-ph↗

arXiv2023

On nonlinear compression costs: when Shannon meets Rényi

Shannon entropy is the shortest average codeword length a lossless compressor can achieve by encoding i.i.d. symbols. However, there are cases in which the objective is to minimize the \textit{exponential} average codeword length, i.e. when the cost of encoding/decoding scales exponentially with the length of codewords. The optimum is reached by all strategies that map each symbol $x_i$ generated with probability $p_i$ into a codeword of length $\ell^{(q)}_D(i)=-\log_D\frac{p_i^q}{\sum_{j=1}^Np_j^q}$. This leads to the minimum exponential average codeword length, which equals the Rényi, rather than Shannon, entropy of the source distribution. We generalize the established Arithmetic Coding (AC) compressor to this framework. We analytically show that our generalized algorithm provides an exponential average length which is arbitrarily close to the Rényi entropy, if the symbols to encode are i.i.d.. We then apply our algorithm to both simulated (i.i.d. generated) and real (a piece of Wikipedia text) datasets. While, as expected, we find that the application to i.i.d. data confirms our analytical results, we also find that, when applied to the real dataset (composed by highly correlated symbols), our algorithm is still able to significantly reduce the exponential average codeword length with respect to the classical `Shannonian' one. Moreover, we provide another justification of the use of the exponential average: namely, we show that by minimizing the exponential average length it is possible to minimize the probability that codewords exceed a certain threshold length. This relation relies on the connection between the exponential average and the cumulant generating function of the source distribution, which is in turn related to the probability of large deviations. We test and confirm our results again on both simulated and real datasets.

cs.IT↗

arXiv2023

Pattern detection in bipartite networks: a review of terminology, applications and methods

Two dimensional matrices with binary (0/1) entries are a common data structure in many research fields. Examples include ecology, economics, mathematics, physics, psychometrics and others. Because the columns and rows of these matrices represent distinct entities, they can equivalently be expressed as a pair of bipartite networks that are linked by projection. A variety of diversity statistics and network metrics can then be used to quantify patterns in these matrices and networks. But what should these patterns be compared to? In all of these disciplines, researchers have recognized the necessity of comparing an empirical matrix to a benchmark set of "null" matrices created by randomizing certain elements of the original data. This common need has nevertheless promoted the independent development of methodologies by researchers who come from different backgrounds and use different terminology. Here, we provide a multidisciplinary review of randomization techniques for matrices representing binary, bipartite networks. We aim to translate the concepts from different technical domains into a common language that is accessible to a broad scientific audience. Specifically, after briefly reviewing examples of binary matrix structures across different fields, we introduce the major approaches and common strategies for randomizing these matrices. We then explore the details of and performance of specific techniques, and discuss their limitations and computational challenges. In particular, we focus on the conceptual importance and implementation of structural constraints on the randomization, such as preserving row or columns sums of the original matrix in each of the randomized matrices. Our review serves both as a guide for empiricists in different disciplines, as well as a reference point for researchers working on theoretical and methodological developments in matrix randomization methods.

physics.data-an↗

arXiv2023

Reconstructing supply networks

Network reconstruction is a well-developed sub-field of network science, but it has only recently been applied to production networks, where nodes are firms and edges represent customer-supplier relationships. We review the literature that has flourished to infer the topology of these networks by partial, aggregate, or indirect observation of the data. We discuss why this is an important endeavour, what needs to be reconstructed, what makes it different from other network reconstruction problems, and how different researchers have approached the problem. We conclude with a research agenda.

econ.GN↗

arXiv2023

Multiscale network renormalization: scale-invariance without geometry

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 heterogeneous coarse-grainings, thereby allowing for the definition of 'block-nodes' across multiple scales. This approach identifies a class of scale-invariant networks characterized by a necessary and specific dependence on additive hidden variables attached to nodes, plus optional dyadic factors. If the hidden variables are annealed, they lead to realistic scale-free networks with assortativity 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 arbitrary geographic partitions. These results highlight a deep conceptual distinction between scale-free and scale-invariant networks, and provide a geometry-free route to renormalization.

physics.soc-ph↗

arXiv2023

Critical density for network reconstruction

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.

physics.soc-ph↗

arXiv2023

Learn your entropy from informative data: an axiom ensuring the consistent identification of generalized entropies

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 Rényi 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.

physics.data-an↗

arXiv2022

Reconciling econometrics with continuous maximum-entropy network models

In the study of economic networks, econometric approaches interpret the traditional Gravity Model specification as the expected link weight coming from a probability distribution whose functional form can be chosen arbitrarily, while statistical-physics approaches construct maximum-entropy distributions of weighted graphs, constrained to satisfy a given set of measurable network properties. In a recent, companion paper, we integrated the two approaches and applied them to the World Trade Web, i.e. the network of international trade among world countries. While the companion paper dealt only with discrete-valued link weights, the present paper extends the theoretical framework to continuous-valued link weights. In particular, we construct two broad classes of maximum-entropy models, namely the integrated and the conditional ones, defined by different criteria to derive and combine the probabilistic rules for placing links and loading them with weights. In the integrated models, both rules follow from a single, constrained optimization of the continuous Kullback-Leibler divergence; in the conditional models, the two rules are disentangled and the functional form of the weight distribution follows from a conditional, optimization procedure. After deriving the general functional form of the two classes, we turn each of them into a proper family of econometric models via a suitable identification of the econometric function relating the corresponding, expected link weights to macroeconomic factors. After testing the two classes of models on World Trade Web data, we discuss their strengths and weaknesses.

physics.soc-ph↗

arXiv2022

Central limit theorem for the principal eigenvalue and eigenvector of Chung-Lu random graphs

A Chung-Lu random graph is an inhomogeneous Erdős-Rényi random graph in which vertices are assigned average degrees, and pairs of vertices are connected by an edge with a probability that is proportional to the product of their average degrees, independently for different edges. We derive a central limit theorem for the principal eigenvalue and the components of the principal eigenvector of the adjacency matrix of a Chung-Lu random graph. Our derivation requires certain assumptions on the average degrees that guarantee connectivity, sparsity and bounded inhomogeneity of the graph.

math.PR↗

arXiv2022

Gravity models of networks: integrating maximum-entropy and econometric approaches

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 model 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.

physics.soc-ph↗

arXiv2022

Fluctuating ecological networks: a synthesis of maximum-entropy approaches for pattern detection and process inference

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.

q-bio.QM↗

arXiv2021

Mesoscopic Structure of the Stock Market and Portfolio Optimization

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 co-movements 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

q-fin.PM↗

arXiv2021

Reconstructing firm-level interactions: the Dutch input-output network

Recent crises have shown that the knowledge of the structure of input-output networks at the firm level is crucial when studying economic resilience from the microscopic point of view of firms that rewire their connections under supply and demand shocks. Unfortunately, empirical inter-firm network data are rarely accessible and protected by confidentiality. The available methods of network reconstruction from partial information, which have been devised for financial exposures, are inadequate for inter-firm relationships because they treat all pairs of nodes as potentially interacting, thereby overestimating the rewiring capabilities of the system. Here we use two big data sets of transactions in the Netherlands to represent a large portion of the Dutch inter-firm network and document the properties of one of the few analysed networks of this kind. We, then, introduce a generalized maximum-entropy reconstruction method that preserves the production function of each firm in the data, i.e. the input and output flows of each node for each product type. We confirm that the new method becomes increasingly more reliable as a finer product resolution is considered and can therefore be used as a generative model of inter-firm networks with fine production constraints. The likelihood of the model, being related to the entropy, proxies the rewiring capability of the system for a fixed input-output configuration.

physics.soc-ph↗

arXiv2021

A spectral signature of breaking of ensemble equivalence for constrained random graphs

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.

math.PR↗

arXiv2021

Ensemble nonequivalence and Bose-Einstein condensation in weighted 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.

cond-mat.stat-mech↗