Statistical physics of complex systems exploits network theory not only to model, but also to effectively extract information from many dynamical real-world systems. A pivotal case of study is given by financial systems: market prediction represents an unsolved scientific challenge yet with crucial implications for society, as financial crises have devastating effects on real economies. Thus, nowadays the quest for a robust estimator of market efficiency is both a scientific and institutional priority. In this work we study the visibility graphs built from the time series of several trade market indices. We propose a validation procedure for each link of these graphs against a null hypothesis derived from ARCH-type modeling of such series. Building on this framework, we devise a market indicator that turns out to be highly correlated and even predictive of financial instability periods.

Although not as wide, and popular, as that of quantum mechanics, the investigation of fundamental aspects of statistical mechanics constitutes an important research field in the building of modern physics. Besides the interest for itself, both for physicists and philosophers, and the obvious pedagogical motivations, there is a further, compelling reason for a thorough understanding of the subject. The fast development of models and methods at the edge of the established domain of the field requires indeed a deep reflection on the essential aspects of the theory, which are at the basis of its success. These elements should never be disregarded when trying to expand the domain of statistical mechanics to systems with novel, little known features. It is thus important to (re)consider in a careful way the main ingredients involved in the foundations of statistical mechanics. Among those, a primary role is covered by the dynamical aspects (e.g. presence of chaos), the emergence of collective features for large systems, and the use of probability in the building of a consistent statistical description of physical systems. With this goal in mind, in the present review we aim at providing a consistent picture of the state of the art of the subject, both in the classical and in the quantum realm. In particular, we will highlight the similarities of the key technical and conceptual steps with emphasis on the relevance of the many degrees of freedom, to justify the use of statistical ensembles in the two domains.

In this letter we study a reference model in theoretical ecology, the disordered Lotka-Volterra model for ecological communities, in the presence of finite demographic noise. Our theoretical analysis, which takes advantage of a mapping to an equilibrium disordered system, proves that for sufficiently heterogeneous interactions and low demographic noise the system displays a multiple equilibria phase, which we fully characterize. In particular, we show that in this phase the number of stable equilibria is exponential in the number of species. Upon further decreasing the demographic noise, we unveil a "Gardner" transition to a marginally stable phase, similar to that observed in jamming of amorphous materials. We confirm and complement our analytical results by numerical simulations. Furthermore, we extend their relevance by showing that they hold for others interacting random dynamical systems, such as the Random Replicant Model. Finally, we discuss their extension to the case of asymmetric couplings.

Motivated by the collective behaviour of biological swarms, we study the critical dynamics of field theories with coupling between order parameter and conjugate momentum in the presence of dissipation. By performing a dynamical renormalization group calculation at one loop, we show that the violation of momentum conservation generates a crossover between a conservative yet IR-unstable fixed point, characterized by a dynamic critical exponent $z=d/2$, and a dissipative IR-stable fixed point with $z=2$. Interestingly, the two fixed points have different upper critical dimensions. The interplay between these two fixed points gives rise to a crossover in the critical dynamics of the system, characterized by a crossover exponent $\kappa=4/d$. Such crossover is regulated by a conservation length scale, $\mathcal R_0$, which is larger the smaller the dissipation: beyond $\mathcal R_0$ the dissipative fixed point dominates, while at shorter distances dynamics is ruled by the conservative fixed point and critical exponent, a behaviour which is all the more relevant in finite-size systems with weak dissipation. We run numerical simulations in three dimensions and find a crossover between the exponents $z=3/2$ and $z=2$ in the critical slowing down of the system, confirming the renormalization group results. From the biophysical point of view, our calculation indicates that in finite-size biological groups mode-coupling terms in the equation of motion can significantly change the dynamical critical exponents even in the presence of dissipation, a step towards reconciling theory with experiments in natural swarms. Moreover, our result provides the scale within which fully conservative Bose-Einstein condensation is a good approximation in systems with weak symmetry-breaking terms violating number conservation, as quantum magnets or photon gases.

We review generalized Fluctuation-Dissipation Relations which are valid under general conditions even in ``non-standard systems'', e.g. out of equilibrium and/or without a Hamiltonian structure. The response functions can be expressed in terms of suitable correlation functions computed in the unperperturbed dynamics. In these relations, typically one has nontrivial contributions due to the form of the stationary probability distribution; such terms take into account the interaction among the relevant degrees of freedom in the system. We illustrate the general formalism with some examples in non-standard cases, including driven granular media, systems with a multiscale structure, active matter and systems showing anomalous diffusion.

We use a relationship between response and correlation function in nonequilibrium systems to establish a connection between the heat production and the deviations from the equilibrium fluctuation-dissipation theorem. This scheme extends the Harada-Sasa formulation [Phys. Rev. Lett. 95, 130602 (2005)], obtained for Langevin equations in steady states, as it also holds for transient regimes and for discrete jump processes involving small entropic changes. Moreover, a general formulation includes two times and the new concepts of two-time work, kinetic energy, and of a two-time heat exchange that can be related to a nonequilibrium "effective temperature". Numerical simulations of a chain of anharmonic oscillators and of a model for a molecular motor driven by ATP hydrolysis illustrate these points.

Collective behaviour is a widespread phenomenon in biology, cutting through a huge span of scales, from cell colonies up to bird flocks and fish schools. The most prominent trait of collective behaviour is the emergence of global order: individuals synchronize their states, giving the stunning impression that the group behaves as one. In many biological systems, though, it is unclear whether global order is present. A paradigmatic case is that of insect swarms, whose erratic movements seem to suggest that group formation is a mere epiphenomenon of the independent interaction of each individual with an external landmark. In these cases, whether or not the group behaves truly collectively is debated. Here, we experimentally study swarms of midges in the field and measure how much the change of direction of one midge affects that of other individuals. We discover that, despite the lack of collective order, swarms display very strong correlations, totally incompatible with models of noninteracting particles. We find that correlation increases sharply with the swarm's density, indicating that the interaction between midges is based on a metric perception mechanism. By means of numerical simulations we demonstrate that such growing correlation is typical of a system close to an ordering transition. Our findings suggest that correlation, rather than order, is the true hallmark of collective behaviour in biological systems.

The impacts of Cosmic Rays on the detectors are a key problem for space-based missions. We are studying the effects of such interactions on arrays of Kinetic Inductance Detectors (KID), in order to adapt this technology for use on board of satellites. Before proposing a new technology such as the Kinetic Inductance Detectors for a space-based mission, the problem of the Cosmic Rays that hit the detectors during in-flight operation has to be studied in detail. We present here several tests carried out with KID exposed to radioactive sources, which we use to reproduce the physical interactions induced by primary Cosmic Rays, and we report the results obtained adopting different solutions in terms of substrate materials and array geometries. We conclude by outlining the main guidelines to follow for fabricating KID for space-based applications.

Drift chambers operated with helium-based gas mixtures represent a common solution for tracking charged particles keeping the material budget in the sensitive volume to a minimum. The drawback of this solution is the worsening of the spatial resolution due to primary ionisation fluctuations, which is a limiting factor for high granularity drift chambers like the MEG II tracker. We report on the measurements performed on three different prototypes of the MEG II drift chamber aimed at determining the achievable single-hit resolution. The prototypes were operated with helium/isobutane gas mixtures and exposed to cosmic rays, electron beams and radioactive sources. Direct measurements of the single hit resolution performed with an external tracker returned a value of 110 $\mu$m, consistent with the values obtained with indirect measurements performed with the other prototypes.

We study the Einstein relation between diffusion and response to an external field in systems showing superdiffusion. In particular, we investigate a continuous time Levy walk where the velocity remains constant for a time \tau, with distribution P(\tau) \tau^{-g}. At varying g the diffusion can be standard or anomalous; in spite of this, if in the unperturbed system a current is absent, the Einstein relation holds. In the case where a current is present the scenario is more complicated and the usual Einstein relation fails. This suggests that the main ingredient for the breaking of the Einstein relation is not the anomalous diffusion but the presence of a mean drift (current).

By employing a path integral formulation, we obtain the entropy production rate for a system of active Ornstein-Uhlenbeck particles (AOUP) both in the presence and in the absence of thermal noise. The present treatment clarifies some contraddictions concerning the definition of the entropy production rate in the AOUP model, recently appeared in the literature. We derive explicit formulas for three different cases: overdamped Brownian particle, AOUP with and without thermal noise. In addition, we show that it is not necessary to introduce additional hypotheses concerning the parity of auxiliary variables under time reversal transformation. Our results agree with those based on a previous mesoscopic approach.

The MEG II experiment at Paul Scherrer Institute (PSI) in Switzerland aims to achieve a sensitivity of $6\times10^{-14}$ on the charged lepton flavor violating decay $\mu^+\to e^+\gamma$. The current upper limit on this decay is $4.2\times10^{-13}$ at 90% Confidence Level (CL), set by the first phase of MEG. This result was achieved using the PSI muon beam at a reduced intensity, $3\times10^7~\mu^+/$s, to keep the background at a manageable level. The upgraded detectors in MEG~II can cope with a higher intensity, thus the experiment is expected to run at a $7\times10^7~\mu^+/$s intensity. The new low mass, single volume, high granularity tracker, together with a new highly segmented timing counter, guarantees better resolutions for the positron detection. Moreover, the replacement of the old PhotoMultiplier Tubes (PMTs) with Multi-Pixel Photon Counters (MPPCs) in the inner face of the liquid xenon photon detector improved its performance. The details of the upgraded detectors and their present status will be discussed, together with the latest results from last year's pre-engineering run and the perspective for the 2021 run, the first with all the detectors and electronics installed.

Macroevolutionary dynamics often display sudden, explosive surges, where systems remain relatively stable for extended periods before experiencing dramatic acceleration that frequently exceeds traditional exponential growth. This pattern is evident in biological evolution, cultural shifts, and technological progress and is often referred to as the emergence of singularities. Despite their widespread occurrence, these explosions arise from distinct underlying mechanisms in different domains. In this context, we present a unified framework that captures these dynamics through a theory of combinatorial innovation. Building on the Theory of the Adjacent Possible, we model macroevolutionary change as a process driven by recombining pre-existing elements within a system. By formalising these qualitative insights, we provide a mathematical structure that explains the emergence of these explosive phenomena, facilitates comparisons across different systems, and enables predictive insights into future evolutionary trajectories. Moreover, by comparing discrete and continuous formalisations of the theory, we emphasise that the occurrence and observation of these presumed singularities should be carefully considered, as they arise from the continuous limit of inherently discrete models.

Scale invariance profoundly influences the dynamics and structure of complex systems, spanning from critical phenomena to network architecture. Here, we propose a precise definition of scale-invariant networks by leveraging the concept of a constant entropy-loss rate across scales in a renormalization-group coarse-graining setting. This framework enables us to differentiate between scale-free and scale-invariant networks, revealing distinct characteristics within each class. Furthermore, we offer a comprehensive inventory of genuinely scale-invariant networks, both natural and artificially constructed, demonstrating, e.g., that the human connectome exhibits notable features of scale invariance. Our findings open new avenues for exploring the scale-invariant structural properties crucial in biological and socio-technological systems.

Diffractive optical elements that divide an input beam into a set of replicas are used in many optical applications ranging from image processing to communications. Their design requires time-consuming optimization processes, which, for a given number of generated beams, are to be separately treated for one-dimensional and two-dimensional cases because the corresponding optimal efficiencies may be different. After generalizing their Fourier treatment, we prove that, once a particular divider has been designed, its transmission function can be used to generate numberless other dividers through affine transforms that preserve the efficiency of the original element without requiring any further optimization.

Non-reciprocal interactions are a defining feature of many complex systems, biological, ecological, and technological, often pushing them far from equilibrium and enabling rich dynamical responses. These asymmetries can arise at multiple levels: locally, in the dynamics of individual units, and globally, in the topology of their interactions. In this work, we investigate how these two forms of non-reciprocity interact in networks of neuronal populations. At the local level, each population is modeled by a non-reciprocally coupled set of excitatory and inhibitory neural populations exhibiting transient amplification and reactivity. At the network level, these populations are coupled via directed, asymmetric connections that introduce structural non-normality. Since non-reciprocal interactions generically lead to non-normal linear operators, we frame both local and global asymmetries in terms of non-normal dynamics. Using a modified Wilson-Cowan framework, we analyze how the interplay between these two types of non-normality shapes the system's behavior. We show that their combination leads to emergent collective dynamics, including fluctuation-driven transitions, dimensionality reduction, and novel nonequilibrium steady states. Our results provide a minimal yet flexible framework to understand how multi-scale non-reciprocities govern complex dynamics in neural and other interconnected systems.

Bootstrap, or $k$-core, percolation displays on the Bethe lattice a mixed first/second order phase transition with both a discontinuous order parameter and diverging critical fluctuations. I apply the recently introduced $M$-layer technique to study corrections to mean-field theory showing that at all orders in the loop expansion the problem is equivalent to a spinodal with quenched disorder. This implies that the mean-field hybrid transition does not survive in physical dimension. Nevertheless, its critical properties as an avoided transition, make it a proxy of the avoided Mode-Coupling-Theory critical point of supercooled liquids.

In this chapter we discuss how the results developed within the theory of fractals and Self-Organized Criticality (SOC) can be fruitfully exploited as ingredients of adaptive network models. In order to maintain the presentation self-contained, we first review the basic ideas behind fractal theory and SOC. We then briefly review some results in the field of complex networks, and some of the models that have been proposed. Finally, we present a self-organized model recently proposed by Garlaschelli et al. [Nat. Phys. 3, 813 (2007)] that couples the fitness network model defined by Caldarelli et al. [Phys. Rev. Lett. 89, 258702 (2002)] with the evolution model proposed by Bak and Sneppen [Phys. Rev. Lett. 71, 4083 (1993)] as a prototype of SOC. Remarkably, we show that the results obtained for the two models separately change dramatically when they are coupled together. This indicates that self-organized networks may represent an entirely novel class of complex systems, whose properties cannot be straightforwardly understood in terms of what we have learnt so far.

The dynamical critical exponent $z$ of natural swarms of insects is calculated using the renormalization group to order $\epsilon = 4-d$. A novel fixed point emerges, where both activity and inertia are relevant. In three dimensions the critical exponent at the new fixed point is $z = 1.35$, in agreement with both experiments ($1.37 \pm 0.11$) and numerical simulations ($1.35 \pm 0.04$).

Assessing the stability of economic systems is a fundamental research focus in economics, that has become increasingly interdisciplinary in the currently troubled economic situation. In particular, much attention has been devoted to the interbank lending market as an important diffusion channel for financial distress during the recent crisis. In this work we study the stability of the interbank market to exogenous shocks using an agent-based network framework. Our model encompasses several ingredients that have been recognized in the literature as pro-cyclical triggers of financial distress in the banking system: credit and liquidity shocks through bilateral exposures, liquidity hoarding due to counterparty creditworthiness deterioration, target leveraging policies and fire-sales spillovers. But we exclude the possibility of central authorities intervention. We implement this framework on a dataset of 183 European banks that were publicly traded between 2004 and 2013. We document the extreme fragility of the interbank lending market up to 2008, when a systemic crisis leads to total depletion of market equity with an increasing speed of market collapse. After the crisis instead the system is more resilient to systemic events in terms of residual market equity. However, the speed at which the crisis breaks out reaches a new maximum in 2011, and never goes back to values observed before 2007. Our analysis points to the key role of the crisis outbreak speed, which sets the maximum delay for central authorities intervention to be effective.