Sergio Gómez

We study the finite-size Kuramoto model of all-to-all coupled phase oscillators with heterogeneous natural frequencies and characterize the minimal coupling strength required for the existence of a fully phase-locked equilibrium (in a co-rotating frame). To remove the degeneracy due to uniform phase shifts, we move to a reduced co-rotating frame and assess stability through the Jacobian of the reduced system: a fully phase-locked state is stable when this Jacobian is negative definite. This defines a stability region in the phase space. The Kuramoto vector field maps this region to a convex set in frequency space, so a fully-locked state at coupling K exists exactly when the rescaled frequency vector ω ^ / K lies inside that convex image. The critical coupling K ℓ is defined as the smallest coupling strength for which a fully phase-locked equilibrium exists; geometrically, it corresponds to the first intersection of the ray t ω ^ with the boundary of this convex set. Building on this convex-geometric structure, we construct an explicit polytope from analytically computable boundary points of the stability region, providing a closed-form upper bound K b ⩾ K ℓ . The bound is exact for frequencies aligned with polytope vertices and offers a fully explicit outer approximation for general frequency vectors. While not uniformly sharp in a quantitative sense, this construction exposes the underlying geometry of stable fully phase-locking solutions. These results provide a practical use the convex-geometric structure underlying stable fully-locked states in the Kuramoto model.

Cryptocurrencies and Decentralized Finance (DeFi) currently represent a fast growing trend in finance, which enables financial services on public blockchains. In contrast to traditional financial markets, ruled by well established corporations, DeFi is completely transparent, as it keeps publicly available records of all transactions that occur in the network. This availability of the data represents an opportunity to analyze and understand the market from the point of view of the complexity that emerges from the interactions among actors (users, bots, and companies) operating in the embedded market. In this paper, we focus on Ethereum to show that the underlying transaction network bears further and useful information to forecast the evolution of the market. We aim to separate the non-redundant effects of the blockchain transaction network from technical analysis and social media trends in the future price of the Ethereum native cryptocurrency. To this end, we build two machine learning models to predict the future trend of the price time series. The first model, serving as a base, considers the set of most relevant features according to the current scientific literature—including technical analysis and social media trends. The second model considers the features of the base model, incorporating the network properties computed from the transaction network. We find that the second model outperforms the base model and can anticipate 46% more rises in the price than the base model and 19% more falls. Thus, we conclude that new indicators based on network properties provide valuable information to forecast the future direction of the market that cannot be explained neither by technical analysis nor by social media trends alone. Hence, our results represent an important first step toward the definition of a new family of DeFi market indicators based on the complexity of the underlying transaction network.

Models of network diffusion typically rely on the Laplacian matrix, capturing interactions via direct connections. Beyond direct interactions, information in many systems can also flow via indirect pathways, where influence typically diminishes over distance. In this work, we analyze diffusion dynamics incorporating such indirect connections using the d -path Laplacian framework. We introduce a parameter, the indirect influence, based on the change in the second smallest eigenvalue of the generalized path Laplacian, to quantify the impact of these pathways on diffusion timescales relative to direct-only models. Using perturbation theory and mean-field approximations, we derive analytical expressions for the indirect influence in terms of structural properties of random networks. Theoretical predictions align well with numerical simulations, providing a phase diagram for when indirect influence becomes significant. We also identify a structural phase transition governed by the emergence of d -paths and derive the critical connection probability above which they dramatically alter diffusion. This study provides a quantitative understanding of how indirect pathways shape network dynamics and reveals their collective structural onset.

Mdendro is an R package that provides a comprehensive collection of linkage methods for agglomerative hierarchical clustering on a matrix of proximity data (distances or similarities), returning a multifurcated dendrogram or multidendrogram. Multidendrograms can group more than two clusters at the same time, solving the nonuniqueness problem that arises when there are ties in the data. This problem causes that different binary dendrograms are possible depending both on the order of the input data and on the criterion used to break ties. Weighted and unweighted versions of the most common linkage methods are included in the package, which also implements two parametric linkage methods. In addition, package mdendro provides five descriptive measures to analyze the resulting dendrograms: cophenetic correlation coefficient, space distortion ratio, agglomerative coefficient, chaining coefficient and tree balance.

Accurate estimates of the incidence of infectious diseases are key for the control of epidemics. However, healthcare systems are often unable to test the population exhaustively, especially when asymptomatic and paucisymptomatic cases are widespread; this leads to significant and systematic under-reporting of the real incidence. Here, we propose a machine learning approach to estimate the incidence of a pandemic in real-time, using reported cases and the overall test rate. In particular, we use Bayesian symbolic regression to automatically learn the closed-form mathematical models that most parsimoniously describe incidence. We develop and validate our models using COVID-19 incidence values for nine different countries, confirming their ability to accurately predict daily incidence. Remarkably, despite the differences in epidemic trajectories and dynamics across countries, we find that a single model for all countries offers a more parsimonious description and is more predictive of actual incidence compared to separate models for each country. Our results show the potential to accurately model incidence in real-time using closed-form mathematical models, providing a valuable tool for public health decision-makers.

We consider a system of n coupled oscillators described by the Kuramoto model with the dynamics given by θ˙=ω+Kf(θ). In this system, an equilibrium solution θ∗ is considered stable when ω+Kf(θ∗)=0, and the Jacobian matrix Df(θ∗) has a simple eigenvalue of zero, indicating the presence of a direction in which the oscillators can adjust their phases. Additionally, the remaining eigenvalues of Df(θ∗) are negative, indicating stability in orthogonal directions. A crucial constraint imposed on the equilibrium solution is that |Γ(θ∗)|≤π, where |Γ(θ∗)| represents the length of the shortest arc on the unit circle that contains the equilibrium solution θ∗. We provide a proof that there exists a unique solution satisfying the aforementioned stability criteria. This analysis enhances our understanding of the stability and uniqueness of these solutions, offering valuable insights into the dynamics of coupled oscillators in this system.

The spontaneous emergence of ordered structures, known as Turing patterns, in complex networks is a phenomenon that holds potential applications across diverse scientific fields, including biology, chemistry, and physics. Here, we present a novel delayed fractional-order susceptible-infected-recovered-susceptible (SIRS) reaction-diffusion model functioning on a network, which is typically used to simulate disease transmission but can also model rumor propagation in social contexts. Our theoretical analysis establishes the Turing instability resulting from delay, and we support our conclusions through numerical experiments. We identify the unique impacts of delay, average network degree, and diffusion rate on pattern formation. The primary outcomes of our study are: (i) Delays cause system instability, mainly evidenced by periodic temporal fluctuations; (ii) The average network degree produces periodic oscillatory states in uneven spatial distributions; (iii) The combined influence of diffusion rate and delay results in irregular oscillations in both time and space. However, we also find that fractional-order can suppress the formation of spatiotemporal patterns. These findings are crucial for comprehending the impact of network structure on the dynamics of fractional-order systems.