Sync and Swarm: Solvable Model of Nonidentical Swarmalators.

We study a model of nonidentical swarmalators, generalizations of phase oscillators that both sync in time and swarm in space. The model produces four collective states: asynchrony, sync clusters, vortexlike phase waves, and a mixed state. These states occur in many real-world swarmalator systems such as biological microswimmers, chemical nanomotors, and groups of drones. A generalized Ott-Antonsen ansatz provides the first analytic description of these states and conditions for their existence. We show how this approach may be used in studies of active matter and related disciplines.

A data-driven model for COVID-19 pandemic - Evolution of the attack rate and prognosis for Brazil.

We introduce a compartmental model SEIAHRV (Susceptible, Exposed, Infected, Asymptomatic, Hospitalized, Recovered, Vaccinated) with age structure for the spread of the SARAS-CoV virus. In order to model current different vaccines we use compartments for individuals vaccinated with one and two doses without vaccine failure and a compartment for vaccinated individual with vaccine failure. The model allows to consider any number of different vaccines with different efficacies and delays between doses. Contacts among age groups are modeled by a contact matrix and the contagion matrix is obtained from a probability of contagion p c per contact. The model uses known epidemiological parameters and the time dependent probability p c is obtained by fitting the model output to the series of deaths in each locality, and reflects non-pharmaceutical interventions. As a benchmark the output of the model is compared to two good quality serological surveys, and applied to study the evolution of the COVID-19 pandemic in the main Brazilian cities with a total population of more than one million. We also discuss with some detail the case of the city of Manaus which raised special attention due to a previous report of We also estimate the attack rate, the total proportion of cases (symptomatic and asymptomatic) with respect to the total population, for all Brazilian states since the beginning of the COVID-19 pandemic. We argue that the model present here is relevant to assessing present policies not only in Brazil but also in any place where good serological surveys are not available.

Filtering Statistics on Networks.

Compression, filtering, and cryptography, as well as the sampling of complex systems, can be seen as processing information. A large initial configuration or input space is nontrivially mapped to a smaller set of output or final states. We explored the statistics of filtering of simple patterns on a number of deterministic and random graphs as a tractable example of such information processing in complex systems. In this problem, multiple inputs map to the same output, and the statistics of filtering is represented by the distribution of this degeneracy. For a few simple filter patterns on a ring, we obtained an exact solution of the problem and numerically described more difficult filter setups. For each of the filter patterns and networks, we found three key numbers that essentially describe the statistics of filtering and compared them for different networks. Our results for networks with diverse architectures are essentially determined by two factors: whether the graphs structure is deterministic or random and the vertex degree. We find that filtering in random graphs produces much richer statistics than in deterministic graphs, reflecting the greater complexity of such graphs. Increasing the graph's degree reduces this statistical richness, while being at its maximum at the smallest degree not equal to two. A filter pattern with a strong dependence on the neighbourhood of a node is much more sensitive to these effects.

Finding the Optimal Nets for Self-Folding Kirigami.

Three-dimensional shells can be synthesized from the spontaneous self-folding of two-dimensional templates of interconnected panels, called nets. However, some nets are more likely to self-fold into the desired shell under random movements. The optimal nets are the ones that maximize the number of vertex connections, i.e., vertices that have only two of its faces cut away from each other in the net. Previous methods for finding such nets are based on random search, and thus, they do not guarantee the optimal solution. Here, we propose a deterministic procedure. We map the connectivity of the shell into a shell graph, where the nodes and links of the graph represent the vertices and edges of the shell, respectively. Identifying the nets that maximize the number of vertex connections corresponds to finding the set of maximum leaf spanning trees of the shell graph. This method allows us not only to design the self-assembly of much larger shell structures but also to apply additional design criteria, as a complete catalog of the maximum leaf spanning trees is obtained.

Mapping the Structure of Directed Networks: Beyond the Bow-Tie Diagram.

We reveal a hierarchical, multilayer organization of finite components-i.e., tendrils and tubes-around the giant connected components in directed networks and propose efficient algorithms allowing one to uncover the entire organization of key real-world directed networks, such as the World Wide Web, the neural network of Caenorhabditis elegans, and others. With increasing damage, the giant components decrease in size while the number and size of tendril layers increase, enhancing the susceptibility of the networks to damage.

Localization of nonbacktracking centrality on dense subgraphs of sparse networks.

The nonbacktracking matrix and the related nonbacktracking centrality (NBC) play a crucial role in models of percolation-type processes on networks, such as nonrecurrent epidemics. Here we study the localization of NBC in infinite sparse networks that contain an arbitrary finite subgraph. Assuming the local tree likeness of the enclosing network, and that branches emanating from the finite subgraph do not intersect at finite distances, we show that the largest eigenvalue of the nonbacktracking matrix of the composite network is equal to the highest of the two largest eigenvalues: that of the finite subgraph and of the enclosing network. In the localized state, when the largest eigenvalue of the subgraph is the highest of the two, we derive explicit expressions for the NBCs of nodes in the subgraph and other nodes in the network. In this state, nonbacktracking centrality is concentrated on the subgraph and its immediate neighborhood in the enclosing network. We obtain simple, exact formulas in the case where the enclosing network is uncorrelated. We find that the mean NBC decays exponentially around the finite subgraph, at a rate which is independent of the structure of the enclosing network, contrary to what was found for the localization of the principal eigenvector of the adjacency matrix. Numerical simulations confirm that our results provide good approximations even in moderately sized, loopy, real-world networks.

Localization and spreading of diseases in complex networks.

Using the susceptible-infected-susceptible model on unweighted and weighted networks, we consider the disease localization phenomenon. In contrast to the well-recognized point of view that diseases infect a finite fraction of vertices right above the epidemic threshold, we show that diseases can be localized on a finite number of vertices, where hubs and edges with large weights are centers of localization. Our results follow from the analysis of standard models of networks and empirical data for real-world networks.

Avalanche collapse of interdependent networks.

We reveal the nature of the avalanche collapse of the giant viable component in multiplex networks under perturbations such as random damage. Specifically, we identify latent critical clusters associated with the avalanches of random damage. Divergence of their mean size signals the approach to the hybrid phase transition from one side, while there are no critical precursors on the other side. We find that this discontinuous transition occurs in scale-free multiplex networks whenever the mean degree of at least one of the interdependent networks does not diverge.

Hidden transition in multiplex networks.

Weak multiplex percolation generalizes percolation to multi-layer networks, represented as networks with a common set of nodes linked by multiple types (colors) of edges. We report a novel discontinuous phase transition in this problem. This anomalous transition occurs in networks of three or more layers without unconnected nodes, [Formula: see text]. Above a critical value of a control parameter, the removal of a tiny fraction [Formula: see text] of nodes or edges triggers a failure cascade which ends either with the total collapse of the network, or a return to stability with the system essentially intact. The discontinuity is not accompanied by any singularity of the giant component, in contrast to the discontinuous hybrid transition which usually appears in such problems. The control parameter is the fraction of nodes in each layer with a single connection, [Formula: see text]. We obtain asymptotic expressions for the collapse time and relaxation time, above and below the critical point [Formula: see text], respectively. In the limit [Formula: see text] the total collapse for [Formula: see text] takes a time [Formula: see text], while there is an exponential relaxation below [Formula: see text] with a relaxation time [Formula: see text].

Enhanced robustness of single-layer networks with redundant dependencies.

Dependency links in single-layer networks offer a convenient way of modeling nonlocal percolation effects in networked systems where certain pairs of nodes are only able to function together. We study the percolation properties of the weak variant of this model: Nodes with dependency neighbors may continue to function if at least one of their dependency neighbors is active. We show that this relaxation of the dependency rule allows for more robust structures and a rich variety of critical phenomena, as percolation is not determined strictly by finite dependency clusters. We study Erdos-Rényi and random scale-free networks with an underlying Erdos-Rényi network of dependency links. We identify a special "cusp" point above which the system is always stable, irrespective of the density of dependency links. We find continuous and discontinuous hybrid percolation transitions, separated by a tricritical point for Erdos-Rényi networks. For scale-free networks with a finite degree cutoff we observe the appearance of a critical point and corresponding double transitions in a certain range of the degree distribution exponent. We show that at a special point in the parameter space, where the critical point emerges, the giant viable cluster has the unusual critical singularity S-S_{c}∝(p-p_{c})^{1/4}. We study the robustness of networks where connectivity degrees and dependency degrees are correlated and find that scale-free networks are able to retain their high resilience for strong enough positive correlation, i.e., when hubs are protected by greater redundancy.

Impact of field heterogeneity on the dynamics of the forced Kuramoto model.

We studied the impact of field heterogeneity on entrainment in a system of uniformly interacting phase oscillators. Field heterogeneity is shown to induce dynamical heterogeneity in the system. In effect, the heterogeneous field partitions the system into interacting groups of oscillators that feel the same local field strength and phase. Based on numerical and analytical analysis of the explicit dynamical equations derived from the periodically forced Kuramoto model, we found that the heterogeneous field can disrupt entrainment at different field frequencies when compared to the homogeneous field. This transition occurs when the phase- and frequency-locked synchronization between groups of oscillators is broken at a critical field frequency, causing each group to enter a new dynamical state (disrupted state). Strikingly, it is shown that disrupted dynamics can differ between groups.

Approximating nonbacktracking centrality and localization phenomena in large networks.

Message-passing theories have proved to be invaluable tools in studying percolation, nonrecurrent epidemics, and similar dynamical processes on real-world networks. At the heart of the message-passing method is the nonbacktracking matrix, whose largest eigenvalue, the corresponding eigenvector, and the closely related nonbacktracking centrality play a central role in determining how the given dynamical model behaves. Here we propose a degree-class-based method to approximate these quantities using a smaller matrix related to the joint degree-degree distribution of neighboring nodes. Our findings suggest that in most networks, degree-degree correlations beyond nearest neighbor are actually not strong, and our first-order description already results in accurate estimates, particularly when message-passing itself is a good approximation to the original model in question, that is, when the number of short cycles in the network is sufficiently low. We show that localization of the nonbacktracking centrality is also captured well by our scheme, particularly in large networks. Our method provides an alternative to working with the full nonbacktracking matrix in very large networks where this may not be possible due to memory limitations.

Explosive percolation transition is actually continuous.

Recently a discontinuous percolation transition was reported in a new "explosive percolation" problem for irreversible systems [D. Achlioptas, R. M. D'Souza, and J. Spencer, Science 323, 1453 (2009)] in striking contrast to ordinary percolation. We consider a representative model which shows that the explosive percolation transition is actually a continuous, second order phase transition though with a uniquely small critical exponent of the percolation cluster size. We describe the unusual scaling properties of this transition and find its critical exponents and dimensions.

Choosing among alternative histories of a tree.

The structure of an evolving network contains information about its past. Extracting this information efficiently, however, is, in general, a difficult challenge. We formulate a fast and efficient method to estimate the most likely history of growing trees, based on exact results on root finding. We show that our linear-time algorithm produces the exact stepwise most probable history in a broad class of tree growth models. Our formulation is able to treat very large trees and therefore allows us to make reliable numerical observations regarding the possibility of root inference and history reconstruction in growing trees. We obtain the general formula 〈lnN〉≅NlnN-cN for the size dependence of the mean logarithmic number of possible histories of a given tree, a quantity that largely determines the reconstructability of tree histories. We also reveal an uncertainty principle: a relationship between the inferability of the root and that of the complete history, indicating that there is a tradeoff between the two tasks; the root and the complete history cannot both be inferred with high accuracy at the same time.

Exotic critical behavior of weak multiplex percolation.

We describe the critical behavior of weak multiplex percolation, a generalization of percolation to multiplex or interdependent networks. A node can determine its active or inactive status simply by referencing neighboring nodes. This is not the case for the more commonly studied generalization of percolation to multiplex networks, the mutually connected clusters, which requires an interconnecting path within each layer between any two vertices in the giant mutually connected component. We study the emergence of a giant connected component of active nodes under the weak percolation rule, finding several nontypical phenomena. In two layers, the giant component emerges with a continuous phase transition, but with quadratic growth above the critical threshold. In three or more layers, a discontinuous hybrid transition occurs, similar to that found in the giant mutually connected component. In networks with asymptotically powerlaw degree distributions, defined by the decay exponent γ, the discontinuity vanishes but at γ=1.5 in three layers, more generally at γ=1+1/(M-1) in M layers.

Communication and correlation among communities.

Given a network and a partition in communities, we consider the issues "how communities influence each other" and "when two given communities do communicate." Specifically, we address these questions in the context of small-world networks, where an arbitrary quenched graph is given and long-range connections are randomly added. We prove that, among the communities, a superposition principle applies and gives rise to a natural generalization of the effective field theory already presented by M. Ostilli and J. F. F. Mendes [Phys. Rev. E 78, 031102 (2008)] (n=1), which here (n>1) consists in a sort of effective TAP (Thouless, Anderson, and Palmer) equations in which each community plays the role of a microscopic spin. The relative susceptibilities derived from these equations calculated at finite or zero temperature, where the method provides an effective percolation theory, give us the answers to the above issues. Unlike the case n=1, asymmetries among the communities may lead, via the TAP-like structure of the equations, to many metastable states whose number, in the case of negative shortcuts among the communities, may grow exponentially fast with n. As examples we consider the n Viana-Bray communities model and the n one-dimensional small-world communities model. Despite being the simplest ones, the relevance of these models in network theory, as, e.g., in social networks, is crucial and no analytic solution were known until now. Connections between percolation and the fractal dimension of a network are also discussed. Finally, as an inverse problem, we show how, from the relative susceptibilities, a natural and efficient method to detect the community structure of a generic network arises.

Effective field theory for models defined over small-world networks: first- and second-order phase transitions.

We present an effective field theory to analyze, in a very general way, models defined over small-world networks. Even if the exactness of the method is limited to the paramagnetic regions and to some special limits, it provides, yielding a clear and immediate (also in terms of calculation) physical insight, the exact critical behavior and the exact critical surfaces and percolation thresholds. The underlying structure of the nonrandom part of the model-i.e., the set of spins filling up a given lattice L0 of dimension d_{0} and interacting through a fixed coupling J0 -is exactly taken into account. When J_{0}> or = 0 , the small-world effect gives rise, as is known, to a second-order phase transition that takes place independently of the dimension d_{0} and of the added random connectivity c . When J0<0 , a different and novel scenario emerges in which, besides a spin-glass transition, multiple first- and second-order phase transitions may take place. As immediate analytical applications we analyze the Viana-Bray model (d_{0}=0) , the one-dimensional chain (d_{0}=1) , and the spherical model for arbitrary d_{0} .

Laplacian spectra of, and random walks on, complex networks: are scale-free architectures really important?

We study the Laplacian operator of an uncorrelated random network and, as an application, consider hopping processes (diffusion, random walks, signal propagation, etc.) on networks. We develop a strict approach to these problems. We derive an exact closed set of integral equations, which provide the averages of the Laplacian operator's resolvent. This enables us to describe the propagation of a signal and random walks on the network. We show that the determining parameter in this problem is the minimum degree q(m) of vertices in the network and that the high-degree part of the degree distribution is not that essential. The position of the lower edge of the Laplacian spectrum lambda(c) appears to be the same as in the regular Bethe lattice with the coordination number q(m). Namely, lambda(c)>0 if q(m)>2 , and lambda(c)=0 if q(m)< or =2 . In both of these cases the density of eigenvalues rho(lambda)-->0 as lambda-->lambda(c)+0 , but the limiting behaviors near lambda(c) are very different. In terms of a distance from a starting vertex, the hopping propagator is a steady moving Gaussian, broadening with time. This picture qualitatively coincides with that for a regular Bethe lattice. Our analytical results include the spectral density rho(lambda) near lambda(c) and the long-time asymptotics of the autocorrelator and the propagator.

Simple reaction-diffusion population model on scale-free networks.

We study a simple reaction-diffusion population model [proposed by A. Windus and H. J. Jensen, J. Phys. A: Math. Theor. 40, 2287 (2007)] on scale-free networks. In the case of fully random diffusion, the network topology cannot affect the critical death rate, whereas the heterogeneous connectivity can cause smaller steady population density and critical population density. In the case of modified diffusion, we obtain a larger critical death rate and steady population density, at the meanwhile, lower critical population density, which is good for the survival of species. The results were obtained using a mean-field-like framework and were confirmed by computer simulations.

Percolation on correlated networks.

We reconsider the problem of percolation on an equilibrium random network with degree-degree correlations between nearest-neighboring vertices focusing on critical singularities at a percolation threshold. We obtain criteria for degree-degree correlations to be irrelevant for critical singularities. We present examples of networks in which assortative and disassortative mixing leads to unusual percolation properties and new critical exponents.