arXiv subjects

Pierpaolo Vivo

Publications and source records attributed to Pierpaolo Vivo.

At least 37 records · Page 2Linked to original sources

arXiv2024

CompLex: legal systems through the lens of complexity science

While "complexity science" has achieved significant successes in several interdisciplinary fields such as economics and biology, it is only a very recent observation that legal systems -- from the way legal texts are drafted and connected to the rest of the corpus, up to the level of how judges and courts reach decisions under a variety of conflicting inputs -- share several features with standard Complex Adaptive Systems. This review is meant as a gentle introduction to the use of quantitative tools and techniques of complexity science to describe, analyse, and tame the complex web of human interactions that the Law is supposed to regulate. We offer an overview of the main directions of research undertaken so far as well as an outlook for future research, and we argue that statistical physicists and complexity scientists should not ignore the opportunities offered by the cross-fertilisation between legal scholarship and complex-systems modelling.

physics.soc-ph↗

arXiv2024

Price of information in games of chance: a statistical physics approach

Information in the form of data, which can be stored and transferred between users, can be viewed as an intangible commodity, which can be traded in exchange for money. Determining the fair price at which a string of data should be traded is an important and open problem in many settings. In this work we develop a statistical physics framework that allows to determine analytically the fair price of information exchanged between players in a game of chance. For definiteness, we consider a game where $N$ players bet on the binary outcome of a stochastic process and share the entry fees pot if successful. We assume that one player holds information about past outcomes of the game, which they may either use exclusively to improve their betting strategy or offer to sell to another player. We find a sharp transition as the number of players $N$ is tuned across a critical value, between a phase where the transaction is always profitable for the seller and one where it may not be. In both phases, different regimes are possible, depending on the "quality" of information being put up for sale: we observe symbiotic regimes, where both parties collude effectively to rig the game in their favor, competitive regimes, where the transaction is unappealing to the data holder as it overly favors a competitor for scarce resources, and even prey-predator regimes, where an exploitative data holder could be giving away bad-quality data to undercut a competitor. Our analytical framework can be generalized to more complex settings and constitutes a flexible tool to address the rich and timely problem of pricing information in games of chance.

cond-mat.stat-mech↗

arXiv2024

Cost of excursions until first crossing of the origin for random walks and Lévy flights: an exact general formula

We consider a discrete-time random walk on a line starting at $x_0\geq 0$ where a cost is incurred at each jump. We obtain an exact analytical formula for the distribution of the total cost of a trajectory until the process crosses the origin for the first time. The formula is valid for arbitrary jump distribution and cost function (heavy- and light-tailed alike), provided they are symmetric and continuous. We analyze the formula in different scaling regimes, and find a high degree of universality with respect to the details of the jump distribution and the cost function. Applications are given to the motion of an active run-and-tumble particle in one dimension and extensions to multiple cost variables are considered. The analytical results are in perfect agreement with numerical simulations.

cond-mat.stat-mech↗

arXiv2024

Random Linear Systems with Quadratic Constraints: from Random Matrix Theory to replicas and back

I present here a pedagogical introduction to the works by Rashel Tublin and Yan V. Fyodorov on random linear systems with quadratic constraints, using tools from Random Matrix Theory and replicas. These notes illustrate and complement the material presented at the Summer School organised within the Puglia Summer Trimester 2023 in Bari (Italy). Consider a system of $M$ linear equations in $N$ unknowns, $\sum_{j=1}^N A_{kj}x_j=b_k$ for $k=1,\ldots,M$, subject to the constraint that the solutions live on the $N$-sphere, $x_1^2+\ldots + x_N^2=N$. Assume that both the coefficients $A_{ij}$ and the parameters $b_i$ be independent Gaussian random variables with zero mean. Using two different approaches -- based on Random Matrix Theory and on a replica calculation -- it is possible to compute whether a large linear system subject to a quadratic constraint is typically solvable or not, as a function of the ratio $α=M/N$ and the variance $σ^2$ of the $b_i$'s. This is done by defining a quadratic loss function $H({\bf x})=\frac{1}{2}\sum_{k=1}^M\left[\sum_{j=1}^NA_{kj} x_j-b_k\right]^2$ and computing the statistics of its minimal value on the sphere, $E_{min}=\min_{||\bf x||^2=N}H({\bf x})$, which is zero if the system is compatible, and larger than zero if it is incompatible. One finds that there exists a compatibility threshold $0<α_c<1$, such that systems with $α>α_c$ are typically incompatible. This means that even weakly under-complete linear systems could become typically incompatible if forced to additionally obey a quadratic constraint.

cond-mat.stat-mech↗

arXiv2024

Distribution of centrality measures on undirected random networks via cavity method

The Katz centrality of a node in a complex network is a measure of the node's importance as far as the flow of information across the network is concerned. For ensembles of locally tree-like and undirected random graphs, this observable is a random variable. Its full probability distribution is of interest but difficult to handle analytically because of its "global" character and its definition in terms of a matrix inverse. Leveraging a fast Gaussian Belief Propagation-cavity algorithm to solve linear systems on a tree-like structure, we show that (i) the Katz centrality of a single instance can be computed recursively in a very fast way, and (ii) the probability $P(K)$ that a random node in the ensemble of undirected random graphs has centrality $K$ satisfies a set of recursive distributional equations, which can be analytically characterized and efficiently solved using a population dynamics algorithm. We test our solution on ensembles of Erdős-Rényi and scale-free networks in the locally tree-like regime, with excellent agreement. The distributions display a crossover between multimodality and unimodality as the mean degree increases, where distinct peaks correspond to the contribution to the centrality coming from nodes of different degrees. We also provide an approximate formula based on a rank-$1$ projection that works well if the network is not too sparse, and we argue that an extension of our method could be efficiently extended to tackle analytical distributions of other centrality measures such as PageRank for directed networks in a transparent and user-friendly way.

physics.soc-ph↗

arXiv2024

Upstreamness and downstreamness in input-output analysis from local and aggregate information

Ranking sectors and countries within global value chains is of paramount importance to estimate risks and forecast growth in large economies. However, this task is often non-trivial due to the lack of complete and accurate information on the flows of money and goods between sectors and countries, which are encoded in Input-Output (I-O) tables. In this work, we show that an accurate estimation of the role played by sectors and countries in supply chain networks can be achieved without full knowledge of the I-O tables, but only relying on local and aggregate information, e.g., the total intermediate demand per sector. Our method, based on a rank-$1$ approximation to the I-O table, shows consistently good performance in reconstructing rankings (i.e., upstreamness and downstreamness measures for countries and sectors) when tested on empirical data from the World Input-Output Database. Moreover, we connect the accuracy of our approximate framework with the spectral properties of the I-O tables, which ordinarily exhibit relatively large spectral gaps. Our approach provides a fast and analytical tractable framework to rank constituents of a complex economy without the need of matrix inversions and the knowledge of finer intersectorial details.

physics.soc-ph↗

arXiv2024

Universal transient behavior in large dynamical systems on networks

We analyze how the transient dynamics of large dynamical systems in the vicinity of a stationary point, modeled by a set of randomly coupled linear differential equations, depends on the network topology. We characterize the transient response of a system through the evolution in time of the squared norm of the state vector, which is averaged over different realizations of the initial perturbation. We develop a mathematical formalism that computes this quantity for graphs that are locally tree-like. We show that for unidirectional networks the theory simplifies and general analytical results can be derived. For example, we derive analytical expressions for the average squared norm for random directed graphs with a prescribed degree distribution. These analytical results reveal that unidirectional systems exhibit a high degree of universality in the sense that the average squared norm only depends on a single parameter encoding the average interaction strength between the individual constituents. In addition, we derive analytical expressions for the average squared norm for unidirectional systems with fixed diagonal disorder and with bimodal diagonal disorder. We illustrate these results with numerical experiments on large random graphs and on real-world networks.

nlin.AO↗

arXiv2023

Classification of Heavy-tailed Features in High Dimensions: a Superstatistical Approach

We characterise the learning of a mixture of two clouds of data points with generic centroids via empirical risk minimisation in the high dimensional regime, under the assumptions of generic convex loss and convex regularisation. Each cloud of data points is obtained via a double-stochastic process, where the sample is obtained from a Gaussian distribution whose variance is itself a random parameter sampled from a scalar distribution $\varrho$. As a result, our analysis covers a large family of data distributions, including the case of power-law-tailed distributions with no covariance, and allows us to test recent "Gaussian universality" claims. We study the generalisation performance of the obtained estimator, we analyse the role of regularisation, and we analytically characterise the separability transition.

stat.ML↗

arXiv2023

Nonlinear-Cost Random Walk: exact statistics of the distance covered for fixed budget

We consider the Nonlinear-Cost Random Walk model in discrete time introduced in [Phys. Rev. Lett. 130, 237102 (2023)], where a fee is charged for each jump of the walker. The nonlinear cost function is such that slow/short jumps incur a flat fee, while for fast/long jumps the cost is proportional to the distance covered. In this paper we compute analytically the average and variance of the distance covered in $n$ steps when the total budget $C$ is fixed, as well as the statistics of the number of long/short jumps in a trajectory of length $n$, for the exponential jump distribution. These observables exhibit a very rich and non-monotonic scaling behavior as a function of the variable $C/n$, which is traced back to the makeup of a typical trajectory in terms of long/short jumps, and the resulting "entropy" thereof. As a byproduct, we compute the asymptotic behavior of ratios of Kummer hypergeometric functions when both the first and last arguments are large. All our analytical results are corroborated by numerical simulations.

cond-mat.stat-mech↗

arXiv2023

Cost of diffusion: nonlinearity and giant fluctuations

We introduce a simple model of diffusive jump process where a fee is charged for each jump. The nonlinear cost function is such that slow jumps incur a flat fee, while for fast jumps the cost is proportional to the velocity of the jump. The model -- inspired by the way taxi meters work -- exhibits a very rich behavior. The cost for trajectories of equal length and equal duration exhibits giant fluctuations at a critical value of the scaled distance travelled. Furthermore, the full distribution of the cost until the target is reached exhibits an interesting ``freezing'' transition in the large-deviation regime. All the analytical results are corroborated by numerical simulations. Our results also apply to elastic systems near the depinning transition, when driven by a random force.

cond-mat.stat-mech↗

arXiv2022

Ranking influential nodes in networks from partial information

Many complex systems exhibit a natural hierarchy in which elements can be ranked according to a notion of "influence". While the complete and accurate knowledge of the interactions between constituents is ordinarily required for the computation of nodes' influence, using a low-rank approximation we show that in a variety of contexts local information about the neighborhoods of nodes is enough to reliably estimate how influential they are, without the need to infer or reconstruct the whole map of interactions. Our framework is successful in approximating with high accuracy different incarnations of influence in systems as diverse as the WWW PageRank, trophic levels of ecosystems, upstreamness of industrial sectors in complex economies, and centrality measures of social networks, as long as the underlying network is not exceedingly sparse. We also discuss the implications of this "emerging locality" on the approximate calculation of non-linear network observables.

physics.soc-ph↗

arXiv2022

Graphie: A network-based visual interface for UK's Primary Legislation

We present Graphie, a novel navigational interface to visualize Acts and Bills included in the UK's legislation digital repository [legislation.gov.uk]. Graphie provides a network representation of the hierarchical structure of an Act of Parliament, which is typically organized in a tree-like fashion according to the content and information contained in each sub-branch. Nodes in Graphie represent sections of an Act (or individual provisions), while links embody the hierarchical connections between them. The legal map provided by Graphie is easily navigable by hovering on nodes, which are also color-coded and numbered to provide easily accessible information about the underlying content. The full textual content of each node is also available on a dedicated hyperlinked canvas. The building block of Graphie is Sofia, an offline data pipeline designed to support different data visualizations by parsing and modelling data provided by [legislation.gov.uk] in open access form. While we focus on the Housing Act 2004 for illustrative purposes, our platform is scalable, versatile, and provides users with a unified toolbox to visualize and explore the UK legal corpus in a fast and user-friendly way.

cs.CY↗

arXiv2022

Rationalizing systematic discrepancies between election outcomes and opinion polls

The Bradley effect concerns the discrepancy between opinion polls and actual election outcomes that emerges when candidates do not exhibit ideological, sexual or racial mainstream features. This effect was first observed during the 1982 election for the Governor of California that resulted in a significant loss for the black Democratic candidate, Tom Bradley, despite him being ahead in polls. It has been argued that poll respondents may tend to mask their true political preference in favour of what is generally considered more socially acceptable. We propose an exactly solvable statistical mechanical model, which allows for a quantitative exploration of this phenomenon. The model includes three main ingredients: (i) the tendency of individuals to align their real preference to the declared (public) opinions of others, (ii) a term accounting for an individual integrity factor, which induces voters to be consistent between their public and private opinions, and (iii) a self-monitoring component, which tunes the strength by which an individual is affected by and wishes to publicly conform to social norms. We observe the emergence of order in a wide range of parameters of the model and discuss the effect of noise on the correlation between hidden and public preference. Finally, we present an application of this model to the prediction of election outcomes.

physics.soc-ph↗

arXiv2022

Delocalization-localization dynamical phase transition of random walks on graphs

We consider random walks evolving on two models of connected and undirected graphs and study the exact large deviations of a local dynamical observable. We prove, in the thermodynamic limit, that this observable undergoes a first-order dynamical phase transition (DPT). This is interpreted as a `co-existence' of paths in the fluctuations that visit the highly connected bulk of the graph (delocalization) and paths that visit the boundary (localization). The methods we used also allow us to characterize analytically the scaling function that describes the finite size crossover between the localized and delocalized regimes. Remarkably, we also show that the DPT is robust with respect to a change in the graph topology, which only plays a role in the crossover regime. All results support the view that a first-order DPT may also appear in random walks on infinite-size random graphs.

cond-mat.stat-mech↗

arXiv2022

Graph-combinatorial approach for large deviations of Markov chains

We consider discrete-time Markov chains and study large deviations of the pair empirical occupation measure, which is useful to compute fluctuations of pure-additive and jump-type observables. We provide an exact expression for the finite-time moment generating function, which is split in cycles and paths contributions, and scaled cumulant generating function of the pair empirical occupation measure via a graph-combinatorial approach. The expression obtained allows us to give a physical interpretation of interaction and entropic terms, and of the Lagrange multipliers, and may serve as a starting point for sub-leading asymptotics. We illustrate the use of the method for a simple two-state Markov chain.

cond-mat.stat-mech↗

arXiv2022

Information retrieval and structural complexity of legal trees

We introduce a model for the retrieval of information hidden in legal texts. These are typically organised in a hierarchical (tree) structure, which a reader interested in a given provision needs to explore down to the "deepest" level (articles, clauses,...). We assess the structural complexity of legal trees by computing the mean first-passage time a random reader takes to retrieve information planted in the leaves. The reader is assumed to skim through the content of a legal text based on their interests/keywords, and be drawn towards the sought information based on keywords affinity, i.e. how well the Chapters/Section headers of the hierarchy seem to match the informational content of the leaves. Using randomly generated keyword patterns, we investigate the effect of two main features of the text -- the horizontal and vertical coherence -- on the searching time, and consider ways to validate our results using real legal texts. We obtain numerical and analytical results, the latter based on a mean-field approximation on the level of patterns, which lead to an explicit expression for the complexity of legal trees as a function of the structural parameters of the model. Policy implications of our results are briefly discussed.

physics.soc-ph↗

arXiv2022

Instabilities of complex fluids with partially structured and partially random interactions

We develop a theory for thermodynamic instabilities of complex fluids composed of many interacting chemical species organised in families. This model includes partially structured and partially random interactions and can be solved exactly using tools from random matrix theory. The model exhibits three kinds of fluid instabilities: one in which the species form a condensate with a local density that depends on their family (family condensation); one in which species demix in two phases depending on their family (family demixing); and one in which species demix in a random manner irrespective of their family (random demixing). We determine the critical spinodal density of the three types of instabilities and find that the critical spinodal density is finite for both family condensation and family demixing, while for random demixing the critical spinodal density grows as the square root of the number of species. We use the developed framework to describe phase-separation instability of the cytoplasm induced by a change in pH.

physics.bio-ph↗

arXiv2022

Exact and Approximate Mean First Passage Times on Trees and other Necklace Structures: a Local Equilibrium Approach

In this work we propose a novel method to calculate mean first-passage times (MFPTs) for random walks on graphs, based on a dimensionality reduction technique for Markov State Models, known as local-equilibrium (LE). We show that for a broad class of graphs, which includes trees, LE coarse-graining preserves the MFPTs between certain nodes, upon making a suitable choice of the coarse-grained states (or clusters). We prove that this relation is exact for graphs that can be coarse-grained into a one-dimensional lattice where each cluster connects to the lattice only through a single node of the original graph. A side result of the proof generalises the well-known essential edge lemma (EEL), which is valid for reversible random walks, to irreversible walkers. Such a generalised EEL leads to explicit formulae for the MFPTs between certain nodes in this class of graphs. For graphs that do not fall in this class, the generalised EEL provides useful approximations if the graph allows a one-dimensional coarse-grained representation and the clusters are sparsely interconnected. We first demonstrate our method for the simple random walk on the $c$-ary tree, then we consider other graph structures and more general random walks, including irreversible random walks.

cond-mat.stat-mech↗