arXiv subjects

Stefano Boccaletti

Publications and source records attributed to Stefano Boccaletti.

At least 19 recordsLinked to original sources

arXiv2026

Inertial synchronization of networked oscillators in arbitrary dimensions

The Kuramoto model provides a paradigmatic framework for studying synchronization of interacting oscillators, and has been generalized to arbitrary dimensions to describe swarms, flocks and multi-dimensional opinion dynamics. Yet, existing formulations neglect inertia, a key mechanism known to enhance information propagation and collective responsiveness. Here, we introduce and analyze an inertial Kuramoto model in arbitrary dimensions. We show that inertia fundamentally alters the nature of the synchronization transition, inducing a crossover from continuous to discontinuous behavior, with the onset of hysteresis depending explicitly on both inertia and dimensionality parity. Our analytical theory is supported by extensive numerical simulations. These results establish inertia as a crucial ingredient of high-dimensional collective dynamics and reveal a novel universal structure in synchronization phenomena.

nlin.AO↗

arXiv2026

Topological measures in weighted hypergraphs

Higher-order interactions introduce an additional structural dimension to complex networks, requiring consistent generalizations of classical topological measures. In hypergraphs, the definition of distance between nodes is not unique: beyond the conventional measure derived from clique projection, an alternative formulation that explicitly incorporates the sizes of hyperedges, those of their intersection and their weights has been recently proposed. Here, we generalize three distance-based topological measures, namely closeness centrality, betweenness centrality and node eccentricity, using this new hypergraph distance. Trough tractable illustrative examples, we demonstrate that the differences between results obtained with the two distances are systematic and arise from structurally meaningful features of the higher-order networks. Also, analyzing a series of real-world datasets, we show that hypergraphs can be divided into three distinct classes, corresponding to the possible dominance of specific orders of interaction over their general metric structure. This provides practical guidance on the possibility of limiting the analysis to only some specific interaction orders, reducing its complexity while maintaining the full information of the system.

physics.soc-ph↗

arXiv2026

Paths to synchronization in the Kuramoto model with inertia

Synchronization is ubiquitous across natural and synthetic systems, yet most prior studies focus on the inertia-free Kuramoto model and do so at the macroscopic level. In this study, we instead investigate the inertial Kuramoto model and analyze the kinetics of individual synchronized clusters that emerge in the underdamped dynamics, driven by the interactions among multiple synchronized clusters with different frequencies. Specifically, we explore two forms of intrinsic frequency distribution -- unimodal Gaussian and multimodal uniform -- and show that they give rise to qualitatively different synchronized clusters: a hierarchical organization for the Gaussian distribution and a homogeneous organization for the uniform distribution. This contrast leads to qualitatively different behaviors of the order parameter: for the Gaussian distribution, it increases smoothly with increasing coupling strength, while for the uniform distribution, it grows through a series of discrete jumps that trace out the size of the Devil's staircase (DS). By resolving the kinetics at the cluster level, we further find that the route to synchronization also depends on the distribution type: with a Gaussian distribution, a single dominant cluster forms and gradually entrains the remaining oscillators, whereas with a uniform distribution, synchronization proceeds via successive cluster mergers initiated from peripheral seeds associated with the high-frequency periphery. Taken together, these findings provide a new perspective on collective synchronization dynamics in inertial complex systems.

nlin.AO↗

arXiv2026

Observing the state of networks with directed higher-order interactions

We consider the problem of reconstructing the state of a network of nonlinear dynamical systems in the presence of directed higher-order interactions. Grounded on analytical convergence results, we propose an algorithmic observer design procedure that simultaneously selects the nodes to be measured and the observer gains. We complement the theoretical analysis with an exhaustive numerical investigation campaign that showcases the performance and robustness of the designed observer. Finally, the algorithmic procedure is used to fully reconstruct the opinions of a group of agents.

eess.SY↗

arXiv2026

Offer of a reward does not always promote trust in spatial games

Trust is one of the cornerstones of human society. One of the evolutionary pressure mechanisms that may have led to its emergence is the presence of incentives for trustworthy behavior. However, this type of reward has received relatively little attention in the context of spatial trust games, which are often used to build models in evolutionary game theory. To fill this gap, we introduce an inter-role reward mechanism in the spatial trust game, so that an investing trustor can choose to pay an extra cost to reward a trustworthy trustee. With extensive numerical simulations, we find that this type of reward does not always promote trust. Rather, while moderate rewards break the dominance of mistrust, thereby favoring investment, excessive rewards eventually stimulate a nonreturn strategy, ultimately suppressing the evolution of trust. Additionally, lower reward costs do not necessarily promote trust. Instead, more costly, but not excessive, rewards enhance the advantage of the original investment, consolidating the clusters of rewarders and improving trust. Our model thus provides evidence about the counterintuitive nature of the relationship between trust and rewards in a complex society.

cond-mat.stat-mech↗

arXiv2025

Equitability and explosive synchronisation in multiplex and higher-order networks

Cluster synchronisation is a key phenomenon observed in networks of coupled dynamical units. Its presence has been linked to symmetry and, more generally, to equability of the underlying pattern of interactions between dynamical units. However, it is not known under which conditions equitability-induced synchronisation is the only cluster synchronisation that can occur on a particular system. Here, we reveal a natural linear independent condition such that equitability becomes necessary, and sufficient, for the existence of cluster synchronised solutions on a very general dynamical system which allows multiplex or higher-order, arbitrarily weighted interactions. Our results explain the ubiquity of explosive synchronisation, as opposed to cluster synchronisation, in multiplex and higher-order networks: equitability imposes additional constraints that must be simultaneously satisfied on the same set of nodes. Our results have significant implications for the design of complex dynamical systems of coupled dynamical units with arbitrary cluster synchronisation patterns and coupling functions.

nlin.AO↗

arXiv2025

Solitary states in spiking oscillators with higher-order interactions

We study a system of globally coupled FitzHugh-Nagumo oscillators, showing that the presence of higher-order interactions affects the character of the transition between synchronous and asynchronous states. In particular, we demonstrate that, around the synchronization transition, solitary states emerge due to the presence of second-order interactions. In difference to the phenomenology observed in systems of phase oscillators, we show that, at low coupling strengths, solitary states appear for both transition directions, whereas for higher couplings they only occur in the forward direction, with the backwards one characterized by explosive desynchronization.

nlin.AO↗

arXiv2025

Reconstructing simplicial complexes from evolutionary games

In distributed systems, knowledge of the network structure of the connections among the unitary components is often a requirement for an accurate prediction of the emerging collective dynamics. However, in many real-world situations, one has, at best, access to partial connectivity data, and therefore the entire graph structure needs to be reconstructed from a limited number of observations of the dynamical processes that take place on it. While existing studies predominantly focused on reconstructing traditional pairwise networks, higher-order interactions remain largely unexplored. Here, we introduce three methods to reconstruct a simplicial complex structure of connection from observations of evolutionary games that take place on it, and demonstrate their high accuracy and excellent overall performance in synthetic and empirical complexes. The methods have different requirements and different complexity, thereby constituting a series of approaches from which one can pick the most appropriate one given the specific circumstances of the application under study.

physics.soc-ph↗

arXiv2024

A double explosive Kuramoto transition in hypergraphs

This study aims to develop a generalised concept that will enable double explosive transitions in the forward and backward directions or a combination thereof. We found two essential factors for generating such phase transitions: the use of higher-order (triadic) interactions and the partial adaptation of a global order parameter acting on the triadic coupling. A compromise between the two factors may result in a double explosive transition. To reinforce numerical observations, we employed the Ott--Antonsen ansatz. We observed that for a wide class of hypergraphs, combining two elements can result in a double explosive transition.

nlin.AO↗

arXiv2024

Periodic systems have new classes of synchronization stability

The Master Stability Function is a robust and useful tool for determining the conditions of synchronization stability in a network of coupled systems. While a comprehensive classification exists in the case in which the nodes are chaotic dynamical systems, its application to periodic systems has been less explored. By studying several well-known periodic systems, we establish a comprehensive framework to understand and classify their properties of synchronizability. This allows us to define five distinct classes of synchronization stability, including some that are unique to periodic systems. Specifically, in periodic systems, the Master Stability Function vanishes at the origin, and it can therefore display behavioral classes that are not achievable in chaotic systems, where it starts, instead, at a strictly positive value. Moreover, our results challenge the widely-held belief that periodic systems are easily put in a stable synchronous state, showing, instead, the common occurrence of a lower threshold for synchronization stability.

cond-mat.stat-mech↗

arXiv2024

Taming Cluster Synchronization

Synchronization is a widespread phenomenon observed across natural and artificial networked systems. It often manifests itself by clusters of units exhibiting coincident dynamics. These clusters are a direct consequence of the organization of the Laplacian matrix eigenvalues into spectral localized blocks. We show how the concept of spectral blocks can be leveraged to design straightforward yet powerful controllers able to fully manipulate cluster synchronization of a generic network, thus shaping at will its parallel functioning. Specifically, we demonstrate how to induce the formation of spectral blocks in networks where such structures would not exist, and how to achieve precise mastering over the synchronizability of individual clusters by dictating the sequence in which each of them enters or exits the synchronization stability region as the coupling strength varies. Our results underscore the pivotal role of cluster synchronization control in shaping the parallel operation of networked systems, thereby enhancing their efficiency and adaptability across diverse applications.

nlin.CD↗

arXiv2024

The transition to synchronization of networked systems

We study the synchronization properties of a generic networked dynamical system, and show that, under a suitable approximation, the transition to synchronization can be predicted with the only help of eigenvalues and eigenvectors of the graph Laplacian matrix. The transition comes out to be made of a well defined sequence of events, each of which corresponds to a specific clustered state. The network's nodes involved in each of the clusters can be identified, and the value of the coupling strength at which the events are taking place can be approximately ascertained. Finally, we present large-scale simulations which show the accuracy of the approximation made, and of our predictions in describing the synchronization transition of both synthetic and real-world large size networks, and we even report that the observed sequence of clusters is preserved in heterogeneous networks made of slightly non-identical systems.

physics.soc-ph↗

arXiv2024

Social norms and cooperation in higher-order networks

Recent research has focused on understanding how cooperation is fostered through various mechanisms in cognitive settings, particularly through pairwise interactions. However, real-world interactions often extend beyond simple dyads, including multiple cliques with both pairwise and higher-order interactions. These complex interactions influence how individuals perceive and adapt their strategies based on social norms. We here introduce a model that explores the evolution of collective strategies and social norms within a heterogeneous environment, encompassing both dyadic and three-body interactions. We find that social norms play a crucial role in promoting cooperation in comparison to simply imitating the most successful neighbor. We also show that the rise of prosocial norms leads to increased cooperation across various social dilemmas, often resulting in shifts from defective to cooperative behavior. Additionally, we observe that a moderate level of information privacy helps sustaining prosocial norms and curtails antisocial tendencies, even in situations where mutual defection might seem advantageous. Our research thus offers insights into the evolution of cooperation through the lens of social norm diffusion in higher-order networks.

physics.soc-ph↗

arXiv2023

Cluster States and $π$-Transition in the Kuramoto Model with Higher Order Interactions

We have examined the synchronization and de-synchronization transitions observable in the Kuramoto model with a standard pair-wise first harmonic interaction plus a higher order (triadic) symmetric interaction for unimodal and bimodal Gaussian distributions of the natural frequencies $\{ ω_i \}$. These transitions have been accurately characterized thanks to a self-consistent mean-field approach joined with extensive numerical simulations. The higher-order interactions favour the formation of two cluster states, which emerge from the incoherent regime via continuous (discontinouos) transitions for unimodal (bimodal) distributions. Fully synchronized initial states give rise to two symmetric equally populated bimodal clusters, each characterized by either positive or negative natural frequencies. These bimodal clusters are formed at an angular distance $γ$, which increases for decreasing pair-wise couplings until it reaches $γ=π$ (corresponding to an anti-phase configuration), where the cluster state destabilizes via an abrupt transition: the $π$-transition. The uniform clusters that reform immediately after (with a smaller angle $γ$) are composed by oscillators with positive and negative $\{ ω_i \}$. For bimodal distributions we have obtained detailed phase diagrams involving all the possible dynamical states in terms of standard and novel order parameters. In particular, the clustering order parameter, here introduced, appears quite suitable to characterize the two cluster regime. As a general aspect, hysteretic (non hysteretic) synchronization transitions, mostly mediated by the emergence of standing waves, are observable for attractive (repulsive) higher-order interactions.

nlin.AO↗

arXiv2023

Why are there six degrees of separation in a social network?

A wealth of evidence shows that real world networks are endowed with the small-world property i.e., that the maximal distance between any two of their nodes scales logarithmically rather than linearly with their size. In addition, most social networks are organized so that no individual is more than six connections apart from any other, an empirical regularity known as the six degrees of separation. Why social networks have this ultra-small world organization, whereby the graph's diameter is independent of the network size over several orders of magnitude, is still unknown. We show that the 'six degrees of separation' are the property featured by the equilibrium state of any network where individuals weigh between their aspiration to improve their centrality and the costs incurred in forming and maintaining connections. We show, moreover, that the emergence of such a regularity is compatible with all other features, such as clustering and scale-freeness, that normally characterize the structure of social networks. Thus, our results show how simple evolutionary rules of the kind traditionally associated with human cooperation and altruism can also account for the emergence of one of the most intriguing attributes of social networks.

physics.soc-ph↗

arXiv2023

Emergent stability in complex network dynamics

The stable functionality of networked systems is a hallmark of their natural ability to coordinate between their multiple interacting components. Yet, strikingly, real-world networks seem random and highly irregular, apparently lacking any design for stability. What then are the naturally emerging organizing principles of complex-system stability? Encoded within the system's stability matrix, the Jacobian, the answer is obscured by the scale and diversity of the relevant systems, their broad parameter space, and their nonlinear interaction mechanisms. To make advances, here we uncover emergent patterns in the structure of the Jacobian, rooted in the interplay between the network topology and the system's intrinsic nonlinear dynamics. These patterns help us analytically identify the few relevant control parameters that determine a system's dynamic stability. Complex systems, we find, exhibit discrete stability classes, from asymptotically unstable, where stability is unattainable, to sensitive, in which stability abides within a bounded range of the system's parameters. Most crucially, alongside these two classes, we uncover a third class, asymptotically stable, in which a sufficiently large and heterogeneous network acquires a guaranteed stability, independent of parameters, and therefore insensitive to external perturbation. Hence, two of the most ubiquitous characteristics of real-world networks - scale and heterogeneity - emerge as natural organizing principles to ensure stability in the face of changing environmental conditions.

nlin.AO↗

arXiv2023

Smallworldness in Hypergraphs

Most real-world networks are endowed with the small-world property, by means of which the maximal distance between any two of their nodes scales logarithmically rather than linearly with their size. The evidence sparkled a wealth of studies trying to reveal possible mechanisms through which the pairwise interactions amongst the units of a network are structured in a way to determine such observed regularity. Here we show that smallworldness occurs also when interactions are of higher order. Namely, by considering Q-uniform hypergraphs and a process through which connections can be randomly rewired with given probability p, we find that such systems may exhibit prominent clustering properties in connection with small average path lengths for a wide range of p values, in analogy to the case of dyadic interactions. The nature of small-world transition remains the same at different orders Q of the interactions, however, the increase in the hyperedge order reduces the range of rewiring probability for which smallworldness emerge.

physics.soc-ph↗

arXiv2022

Epidemic spreading under mutually independent intra- and inter-host pathogen evolution

The dynamics of epidemic spreading is often reduced to the single control parameter $R_0$, whose value, above or below unity, determines the state of the contagion. If, however, the pathogen evolves as it spreads, $R_0$ may change over time, potentially leading to a mutation-driven spread, in which an initially sub-pandemic pathogen undergoes a breakthrough mutation. To predict the boundaries of this pandemic phase, we introduce here a modeling framework to couple the network spreading patterns with the intra-host evolutionary dynamics. For many pathogens these two processes, intra- and inter-host, are driven by different selection forces. And yet here we show that even in the extreme case when these two forces are mutually independent, mutations can still fundamentally alter the pandemic phase-diagram, whose transitions are now shaped, not just by $R_0$, but also by the balance between the epidemic and the evolutionary timescales. If mutations are too slow, the pathogen prevalence decays prior to the appearance of a critical mutation. On the other hand, if mutations are too rapid, the pathogen evolution becomes volatile and, once again, it fails to spread. Between these two extremes, however, we identify a broad range of conditions in which an initially sub-pandemic pathogen can break through to gain widespread prevalence.

q-bio.PE↗