arXiv subjects

Anna Gallo

Publications and source records attributed to Anna Gallo.

13 recordsLinked to original sources

arXiv2026

Synchronization of higher-dimensional Kuramoto oscillators on networks: from scalar to matrix-weighted couplings

The Kuramoto model is the paradigmatic model to study synchronization in coupled oscillator systems. In its classical formulation, the oscillators move on the unit circle, each characterized by a scalar phase and a natural frequency, by interacting through a sinusoidal coupling. In this work, we propose a d-dimensional generalization in which oscillators are represented as unit vectors on the (d-1)-sphere and interact through a matrix-weighted network (MWN), a recently introduced framework where links are endowed with a matrix weight instead of a scalar one. We derive necessary conditions for global synchronization via a Master Stability Function approach: the existence of a synchronous solution requires identical frequency matrices across nodes and, in the MWN case, a coherence condition on the network structure. Through a suitable change of variables, the stability analysis reduces the full Nd-dimensional problem to a family of d-dimensional eigenvalue problems, each one parametrized by the eigenvalue of a suitable scalar weighted Laplacian, showing that the synchronous solution is locally stable for any positive coupling strength K on any connected network. Analytical results are complemented by numerical simulations.

cond-mat.stat-mech↗

arXiv2026

Turing patterns in Matrix-Weighted Networks

Diffusion-driven instability is a fundamental mechanism underlying pattern formation in spatially extended systems. In almost all existing works, diffusion across the links of the underlying network is modeled through scalar weights, possibly complemented by cross-diffusion terms that are homogeneous across links. In this work, we investigate the emergence of Turing patterns on Matrix Weighted Networks (MWNs), a recently introduced framework in which each edge is associated with a matrix weight. Focusing on the class of coherent MWNs, we provide a novel characterization of coherence in terms of node-dependent orthonormal matrices, showing that link transformations can be written as relative rotations between nodes. This representation allows us to deal with coherent MWNs of any size and to introduce an orthonormal change of variables capable to reduce diffusion on a coherent MWN to diffusion on a standard weighted network with scalar weights. Building on this, we extend the classical Turing instability analysis to MWNs and derive the conditions under which a homogeneous equilibrium of the local dynamics loses stability due to matrix-weighted diffusion. Moving beyond the dimensional constraints of previous approaches, our results show how network topology, scalar weights, and inter-node transformations jointly shape pattern formation, and provide a constructive framework to analyze and design Turing patterns on matrix-weighted and higher-order networked systems.

cond-mat.stat-mech↗

arXiv2026

The hidden structure of innovation networks

Innovation emerges from complex collaboration patterns - among inventors, firms, or institutions. However, not much is known about the overall mesoscopic structure around which inventive activity self-organizes. Here, we tackle this problem by employing patent data to analyze both individual (\textit{co-inventorship}) and organization (\textit{co-ownership}) networks in three strategic domains (\textit{artificial intelligence}, \textit{biotechnology} and \textit{semiconductors}). We characterize the mesoscale structure (in terms of clusters) of each domain by comparing two alternative methods: a standard baseline - modularity maximization - and one based on the minimization of the Bayesian Information Criterion, within the Stochastic Block Model and its degree-corrected variant. We find that, across sectors, inventor networks are denser and more clustered than organization ones - consistently with the presence of small recurrent teams embedded into broader institutional hierarchies - whereas organization networks have a neater role-based structures, with few bridging firms coordinating the most peripheral ones; still, both are characterized by the presence of local core-periphery structures. We also find that the discovered meso-structures are connected to innovation output. In particular, Lorenz curves of forward citations show a pervasive inequality in technological influence: across sectors and methods, both inventor (especially) and organization networks consistently show high levels of concentration of citations in a few of the discovered clusters. Our results demonstrate that the baseline modularity-based method may not be capable of fully capturing the way collaborations drive the spreading of inventive impact across technological domains. This is due to the presence of local hierarchies that call for the more refined tools of Bayesian inference.

econ.GN↗

arXiv2025

Global Synchronization in Matrix-Weighted Networks

Synchronization phenomena in complex systems are fundamental to understanding collective behavior across disciplines. While classical approaches model such systems by using scalar-weighted networks and simple diffusive couplings, many real-world interactions are inherently multidimensional and transformative. To address this limitation, Matrix-Weighted Networks (MWNs) have been introduced as a versatile framework where edges are associated with matrix weights that encode both interaction strength and directional transformation. In this work, we investigate the emergence and stability of global synchronization (GS) in MWNs by studying coupled Stuart-Landau (SL) oscillators, an archetypal model of nonlinear dynamics near a Hopf bifurcation. Besides the SL, we considered a generalization of regular oscillators to higher dimensions and also the Lorenz model as a prototype of chaotic oscillators. We derive a generalized Master Stability Function (MSF) tailored to MWNs and establish necessary and sufficient conditions for GS to occur. Central to our analysis is the concept of coherence, a structural property of MWNs ensuring path-independent transformations. Our results show that coherence is necessary to have global synchronization and provides a theoretical foundation for analyzing multidimensional dynamical processes in complex networked systems.

cond-mat.stat-mech↗

arXiv2025

Statistically validated projection of bipartite signed networks

Bipartite networks provide a major insight into the organisation of many real-world systems. One of the most relevant issues encountered when modelling a bipartite network is that of facing the information shortage concerning intra-layer linkages. In the present contribution, we propose an unsupervised algorithm to obtain statistically validated projections of bipartite signed networks, according to which any two nodes sharing a statistically significant number of concordant (discordant) relationships are connected by a positive (negative) edge. Our algorithm outputs a matrix of link-specific $p-$values, from which a validated projection can be obtained upon running a multiple-hypothesis testing procedure. After testing our method on synthetic configurations output by a fully controllable generative model, we apply it to several real-world configurations: in all cases, non-trivial mesoscopic structures, induced by relationships that cannot be traced back to the constraints defining the employed benchmarks, hence revealing genuine traces of self-organisation, are detected.

physics.soc-ph↗

arXiv2024

Patterns of link reciprocity in directed, signed networks

Most of the analyses concerning signed networks have focused on the balance theory, hence identifying frustration with undirected, triadic motifs having an odd number of negative edges; much less attention has been paid to their directed counterparts. To fill this gap, we focus on signed, directed connections, with the aim of exploring the notion of frustration in such a context. When dealing with signed, directed edges, frustration is a multi-faceted concept, admitting different definitions at different scales: if we limit ourselves to consider cycles of length two, frustration is related to reciprocity, i.e. the tendency of edges to admit the presence of partners pointing in the opposite direction. As the reciprocity of signed networks is still poorly understood, we adopt a principled approach for its study, defining quantities and introducing models to consistently capture empirical patterns of the kind. In order to quantify the tendency of empirical networks to form either mutualistic or antagonistic cycles of length two, we extend the Exponential Random Graphs framework to binary, directed, signed networks with global and local constraints and, then, compare the empirical abundance of the aforementioned patterns with the one expected under each model. We find that the (directed extension of the) balance theory is not capable of providing a consistent explanation of the patterns characterising the directed, signed networks considered in this work. Although part of the ambiguities can be solved by adopting a coarser definition of balance, our results call for a different theory, accounting for the directionality of edges in a coherent manner. In any case, the evidence that the empirical, signed networks can be highly reciprocated leads us to recommend to explicitly account for the role played by bidirectional dyads in determining frustration at higher levels (e.g. the triadic one).

physics.soc-ph↗

arXiv2024

Assessing frustration in real-world signed networks: a statistical theory of balance

According to the so-called strong version of structural balance theory, actors in signed social networks avoid establishing triads with an odd number of negative links. Generalising, the weak version of balance theory allows for nodes to be partitioned into any number of blocks with positive internal links, mutually connected by negative links. If this prescription is interpreted rigidly, i.e. without allowing for statistical noise in the observed link signs, then most real graphs will appear to require a larger number of blocks than the actual one, or even to violate both versions of the theory. This might lead to conclusions invoking even more relaxed notions of balance. Here, after rephrasing structural balance theory in statistically testable terms, we propose an inference scheme to unambiguously assess whether a real-world, signed graph is balanced. We find that the proposed statistical balance theory leads to interpretations that are quite different from those derived from the current, deterministic versions of the theory.

physics.soc-ph↗

arXiv2023

Testing structural balance theories in heterogeneous signed networks

The abundance of data about social relationships allows the human behavior to be analyzed as any other natural phenomenon. Here we focus on balance theory, stating that social actors tend to avoid establishing cycles with an odd number of negative links. This statement, however, can be supported only after a comparison with a benchmark. Since the existing ones disregard actors' heterogeneity, we extend Exponential Random Graphs to signed networks with both global and local constraints and employ them to assess the significance of empirical unbalanced patterns. We find that the nature of balance crucially depends on the null model: while homogeneous benchmarks favor the weak balance theory, according to which only triangles with one negative link should be under-represented, heterogeneous benchmarks favor the strong balance theory, according to which also triangles with all negative links should be under-represented. Biological networks, instead, display strong frustration under any benchmark, confirming that structural balance inherently characterizes social networks.

physics.soc-ph↗

arXiv2022

Ising model on clustered networks: A model for opinion dynamics

We study opinion dynamics on networks with a nontrivial community structure, assuming individuals can update their binary opinion as the result of the interactions with an external influence with strength $h\in [0,1]$ and with other individuals in the network. To model such dynamics, we consider the Ising model with an external magnetic field on a family of finite networks with a clustered structure. Assuming a unit strength for the interactions inside each community, we assume that the strength of interaction across different communities is described by a scalar $\epsilon \in [-1,1]$, which allows a weaker but possibly antagonistic effect between communities. We are interested in the stochastic evolution of this system described by a Glauber-type dynamics parameterized by the inverse temperature $\beta$. We focus on the low-temperature regime $\beta\rightarrow\infty$, in which homogeneous opinion patterns prevail and, as such, it takes the network a long time to fully change opinion. We investigate the different metastable and stable states of this opinion dynamics model and how they depend on the values of the parameters $\epsilon$ and $h$. More precisely, using tools from statistical physics, we derive rigorous estimates in probability, expectation, and law for the first hitting time between metastable (or stable) states and (other) stable states, together with tight bounds on the mixing time and spectral gap of the Markov chain describing the network dynamics. Lastly, we provide a full characterization of the critical configurations for the dynamics, i.e., those which are visited with high probability along the transitions of interest.

math.PR↗

arXiv2022

Metastability of the three-state Potts model with general interactions

We consider the Potts model on a two-dimensional periodic rectangular lattice with general coupling constants $J_{ij}>0$, where $i,j\in\{1,2,3\}$ are the possible spin values (or colors). The resulting energy landscape is thus significantly more complex than in the original Ising or Potts models. The system evolves according to a Glauber-type spin-flipping dynamics. We focus on a region of the parameter space where there are two symmetric metastable states and a stable state, and the height of a direct path between the metastable states is equal to the height of a direct path between any metastable state and the stable state. We study the metastable transition time in probability and in expectation, the mixing time of the dynamics and the spectral gap of the system when the inverse temperature $\beta$ tends to infinity. Then, we identify all the critical configurations that are visited with high probability during the metastable transition.

math.PR↗

arXiv2021

Metastability for the degenerate Potts Model with positive external magnetic field under Glauber dynamics

We consider the ferromagnetic q-state Potts model on a finite grid graph with non-zero external field and periodic boundary conditions. The system evolves according to Glauber-type dynamics described by the Metropolis algorithm, and we focus on the low temperature asymptotic regime. We analyze the case of positive external magnetic field. In this energy landscape there are $1$ stable configuration and $q-1$ metastable states. We study the asymptotic behavior of the first hitting time from any metastable state to the stable configuration as $\beta\to\infty$ in probability, in expectation, and in distribution. We also identify the exponent of the mixing time and find an upper and a lower bound for the spectral gap. We also geometrically identify the union of all minimal gates and the tube of typical trajectories for the transition from any metastable state to the unique stable configuration.

math.PR↗

arXiv2021

Metastability for the degenerate Potts Model with negative external magnetic field under Glauber dynamics

We consider the ferromagnetic q-state Potts model on a finite grid graph with non-zero external field and periodic boundary conditions. The system evolves according to Glauber-type dynamics described by the Metropolis algorithm, and we focus on the low temperature asymptotic regime. We analyze the case of negative external magnetic field. In this scenario there are $q-1$ stable configurations and a unique metastable state. We describe the asymptotic behavior of the first hitting time from the metastable to the set of the stable states as $\beta\to\infty$ in probability, in expectation, and in distribution. We also identify the exponent of the mixing time and find an upper and a lower bound for the spectral gap. We identify the union of all minimal gates for the transition from the metastable state both to the set of the stable states and to a fixed stable state. Furthermore, we identify the tube of typical trajectories for these two transitions. The accurate knowledge of the energy landscape allows us to give precise asymptotics for the expected transition time from the unique metastable state to the set of the stable configurations.

math.PR↗

arXiv2021

Critical configurations and tube of typical trajectories for the Potts and Ising models with zero external field

We consider the ferromagnetic q-state Potts model with zero external field in a finite volume evolving according to Glauber-type dynamics described by the Metropolis algorithm in the low temperature asymptotic limit. Our analysis concerns the multi-spin system that has q stable equilibria. Focusing on grid graphs with periodic boundary conditions, we study the tunneling between two stable states and from one stable state to the set of all other stable states. In both cases we identify the set of gates for the transition and prove that this set has to be crossed with high probability during the transition. Moreover, we identify the tube of typical paths and prove that the probability to deviate from it during the transition is exponentially small.

math.PR↗