The linear assignment problem is a fundamental problem in combinatorial optimization with a wide range of applications, from operational research to data science. It consists of assigning “agents” to “tasks” on a one-to-one basis, while minimizing the total cost associated with the assignment. While many exact algorithms have been developed to identify such an optimal assignment, most of these methods are computationally prohibitive for large size problems. In this paper, we propose an alternative approach to solving the assignment problem using techniques adapted from statistical physics. Our first contribution is to fully describe this formalism, including all the proofs of its main claims. In particular we derive a strongly concave effective free-energy function that captures the constraints of the assignment problem at a finite temperature. We prove that this free energy decreases monotonically as a function of $\beta$, the inverse of temperature, to the optimal assignment cost, providing a robust framework for temperature annealing. We prove also that for large enough $\beta$ values the exact solution to the generic assignment problem can be derived using simple roundoff to the nearest integer of the elements of the computed assignment matrix. Our second contribution is to derive a provably convergent method to handle degenerate assignment problems, with a characterization of those problems. We describe computer implementations of our framework that are optimized for parallel architectures, one based on CPU, the other based on GPU. We show that the latter enables solving large assignment problems (of the orders of a few 10 000s) in computing clock times of the orders of minutes.

We study the stochastically driven conserved Kardar-Parisi-Zhang (CKPZ) equation with quenched disorders. Short-ranged quenched disorders are found to be a relevant perturbation on the pure CKPZ equation at one dimension and, as a result, a different universality class different from pure CKPZ equation appears to emerge. At higher dimensions, quenched disorder turns out to be ineffective to influence the universal scaling. This results in the asymptotic long wavelength scaling to be given by the linear theory, a scenario identical with the pure CKPZ equation. For sufficiently long-ranged quenched disorders, the universal scaling is impacted by the quenched disorder even at higher dimensions.

For the three-dimensional hard-sphere model we investigate the inhomogeneous two-body correlations predicted by Rosenfeld's fundamental measure theory. For the special cases in which the density has either planar or spherical symmetry we provide analytic formulas for the Hankel and Legendre transforms, respectively, of the inhomogeneous two-body direct correlation function as explicit functionals of the density. When combined with the Ornstein-Zernike relation our analytical results allow for rapid calculation of inhomogeneous hard-sphere density correlations in real space. These not only provide information about the packing structures of the hard-sphere system but also form an essential building-block for constructing perturbation theories of more realistic models.

The theory of soliton gas had been previously developed for unidirectional integrable dispersive hydrodynamics in which the soliton gas properties are determined by the overtaking elastic pairwise interactions between solitons. In this paper, we extend this theory to soliton gases in bidirectional integrable Eulerian systems where both head-on and overtaking collisions of solitons take place. We distinguish between two qualitatively different types of bidirectional soliton gases: isotropic gases, in which the position shifts accompanying the head-on and overtaking soliton collisions have the same sign, and anisotropic gases, in which the position shifts for head-on and overtaking collisions have opposite signs. We construct kinetic equations for both types of bidirectional soliton gases and solve the respective shock-tube problems for the collision of two “monochromatic” soliton beams consisting of solitons of approximately the same amplitude and velocity. The corresponding weak solutions of the kinetic equations consisting of differing uniform states separated by contact discontinuities for the mean flow are constructed. Concrete examples of bidirectional Eulerian soliton gases for the defocusing nonlinear Schrödinger (NLS) equation and the resonant NLS equation are considered. The kinetic equation of the resonant NLS soliton gas is shown to be equivalent to that of the shallow-water bidirectional soliton gas described by the Kaup-Boussinesq equations. The analytical results for shock-tube Riemann problems for bidirectional soliton gases are shown to be in excellent agreement with direct numerical simulations.

We investigate the statistics of articulation points and bredges (bridge edges) in complex networks in which bonds are randomly removed in a percolation process. Because of the heterogeneous structure of a complex network, the probability of a node to be an articulation point or the probability of an edge to be a bredge will not be homogeneous across the network. We therefore analyze full distributions of articulation point probabilities as well as bredge probabilities, using a message-passing or cavity approach to the problem. Our methods allow us to obtain these distributions both for large single instances of networks and for ensembles of networks in the configuration model class in the thermodynamic limit, through a single unified approach. We also evaluate deconvolutions of these distributions according to degrees of the node or the degrees of both adjacent nodes in the case of bredges. We obtain closed form expressions for the large mean degree limit of Erdős-Rényi networks. Moreover, we reveal and are able to rationalize a significant amount of structure in the evolution of articulation point and bredge probabilities in response to random bond removal. We find that full distributions of articulation point and bredge probabilities in real networks and in their randomized counterparts may exhibit significant differences even where average articulation point and bredge probabilities do not. We argue that our results could be exploited in a variety of applications, including approaches to network dismantling or to vaccination and islanding strategies to prevent the spread of epidemics or of blackouts in process networks.

The pore-size distributions play a critical role in the determination of the properties of nanoporous cellular materials like aerogels. In this paper, we propose a micromechanical model, and by further designing artificial normal pore-size distributions, we inspect their effect on the macroscopic stress-strain curves. We show that the location of the mean pore size as well as the broadness of the distribution strongly affects the overall macroscopic behavior. Moreover, we also show that by using different damage criteria within the proposed model, the elastic, inelastic, and brittle nature of the macroscopic material can be captured. The damage criteria are based on the different modes of deformation in the pore walls, namely, elastic buckling, irreversible bending and brittle collapse under compression, and combined bending and stretching under tension. The proposed model approach serves as a reverse engineering tool to develop cellular solids with desired mechanical properties.

Efficient feature selection from high-dimensional datasets is a very important challenge in many data-driven fields of science and engineering. We introduce a statistical mechanics inspired strategy that addresses the problem of sparse feature selection in the context of binary classification by leveraging a computational scheme known as expectation propagation (EP). The algorithm is used in order to train a continuous-weights perceptron learning a classification rule from a set of (possibly partly mislabeled) examples provided by a teacher perceptron with diluted continuous weights. We test the method in the Bayes optimal setting under a variety of conditions and compare it to other state-of-the-art algorithms based on message passing and on expectation maximization approximate inference schemes. Overall, our simulations show that EP is a robust and competitive algorithm in terms of variable selection properties, estimation accuracy, and computational complexity, especially when the student perceptron is trained from correlated patterns that prevent other iterative methods from converging. Furthermore, our numerical tests demonstrate that the algorithm is capable of learning online the unknown values of prior parameters, such as the dilution level of the weights of the teacher perceptron and the fraction of mislabeled examples, quite accurately. This is achieved by means of a simple maximum likelihood strategy that consists in minimizing the free energy associated with the EP algorithm.

We study numerically a system of athermal, overdamped, frictionless spheres, as in a non-Brownian suspension, in two and three dimensions. Compressing the system isotropically at a fixed rate $\dot{\epsilon }$, we investigate the critical behavior at the jamming transition. The finite compression rate introduces a control timescale, which allows one to probe the critical timescale associated with jamming. As was found previously for steady-state shear-driven jamming, we find for compression-driven jamming that pressure obeys a critical scaling relation as a function of packing fraction $\varphi$ and compression rate $\dot{\epsilon }$, and that the bulk viscosity $p/\dot{\epsilon }$ diverges upon jamming. A scaling analysis determines the critical exponents associated with the compression-driven jamming transition. Our results suggest that stress-isotropic, compression-driven jamming may be in the same universality class as stress-anisotropic, shear-driven jamming.

Complex networks are characterized by latent geometries induced by their topology or by the dynamics on them. In the latter case, different network-driven processes induce distinct geometric features that can be captured by adequate metrics. Random walks, a proxy for a broad spectrum of processes, from simple contagion to metastable synchronization and consensus, have been recently used, Domenico [Phys. Rev. Lett. 118, 168301 (2017)] to define the class of diffusion geometries and pinpoint the functional mesoscale organization of complex networks from a genuine geometric perspective. Here we first extend this class to families of distinct random walk dynamics—including local and nonlocal information—on multilayer networks—a paradigm for biological, neural, social, transportation, and financial systems—overcoming limitations such as the presence of isolated nodes and disconnected components, typical of real-world networks. We then characterize the multilayer diffusion geometry of synthetic and empirical systems, highlighting the role played by different random search dynamics in shaping the geometric features of the corresponding diffusion manifolds.

Living cells sense their environment through the binding of extracellular molecular ligands to cell surface receptors. Puzzlingly, vast numbers of signaling pathways exhibit a high degree of cross talk between different signals whereby different ligands act through the same receptor or shared components downstream. It remains unclear how a cell can accurately process information from the environment in such cross-wired pathways. We show that a feature which commonly accompanies cross talk—signaling pleiotropy (the ability of a receptor to produce multiple outputs)—offers a solution to the cross-talk problem. In a minimal model we show that a single pleiotropic receptor can simultaneously identify and accurately sense the concentrations of arbitrary unknown ligands present individually or in a mixture. We calculate the fundamental limits of the signaling specificity and accuracy of such signaling schemes. The model serves as an elementary “building block” toward understanding more complex cross-wired receptor-ligand signaling networks.

By employing the exact enumeration technique, we study consequences of different apex angles of a wedge-shaped channel on the mean first passage time and free-energy profile of a linear polymer chain translocating from the cis- to the trans-side through an interacting pore. We investigate effects of asymmetry arising in the free-energy profile due to the change in apex angles and its dependence on the first passage time. We report the combined effect of entropy (arising due to apex angles) and pore interaction on the nonmonotonic behavior of the translocation time. The effect of different solvent quality across the channel has also been explored. We show that the increase in monomer-monomer interaction leads to the formation of globules near the pore, which drives the process faster.

Several experiments and models have highlighted the importance of neuronal heterogeneity in brain dynamics and function. However, how such a cell-to-cell diversity can affect cortical computation, synchronization, and neuronal communication is still under debate. Previous studies have focused on the effect of neuronal heterogeneity in one neuronal population. Here we are specifically interested in the effect of neuronal variability on the phase relations between two populations, which can be related to different cortical communication hypotheses. It has been recently shown that two spiking neuron populations unidirectionally connected in a sender-receiver configuration can exhibit anticipated synchronization (AS), which is characterized by a negative phase lag. This phenomenon has been reported in electrophysiological data of nonhuman primates and human EEG during a visual discrimination cognitive task. In experiments, the unidirectional coupling could be accessed by Granger causality and can be accompanied by either positive or negative phase difference between cortical areas. Here we propose a model of two coupled populations in which the neuronal heterogeneity can determine the dynamical relation between the sender and the receiver and can reproduce phase relations reported in experiments. Depending on the distribution of parameters characterizing the neuronal firing patterns, the system can exhibit both AS and the usual delayed synchronization regime (DS, with positive phase) as well as a zero-lag synchronization regime and phase bistability between AS and DS. Furthermore, we show that our network can present diversity in their phase relations maintaining the excitation-inhibition balance.

Using the Poisson-bracket method, we derive continuum equations for a fluid of deformable particles in two dimensions. Particle shape is quantified in terms of two continuum fields: an anisotropy density field that captures the deformations of individual particles from regular shapes and a shape tensor density field that quantifies both particle elongation and nematic alignment of elongated shapes. We explicitly consider the example of a dense biological tissue as described by the Vertex model energy, where cell shape has been proposed as a structural order parameter for a liquid-solid transition. The hydrodynamic model of biological tissue proposed here captures the coupling of cell shape to flow and provides a starting point for modeling the rheology of dense tissue.

We consider a mathematical model that describes the flow of a nematic liquid crystal (NLC) film placed on a flat substrate, across which a spatially varying electric potential is applied. Due to their polar nature, NLC molecules interact with the (nonuniform) electric field generated, leading to instability of a flat film. Implementation of the long wave scaling leads to a partial differential equation that predicts the subsequent time evolution of the thin film. This equation is coupled to a boundary value problem that describes the interaction between the local molecular orientation of the NLC (the director field) and the electric potential. We investigate numerically the behavior of an initially flat film for a range of film heights and surface anchoring conditions.

Voter models are well known in the interdisciplinary community, yet they have not been studied from the perspective of anomalous diffusion. In this paper, we show that the original voter model exhibits a ballistic regime. Nonlinear transformations of the observation variable and time scale allow us to observe other regimes of anomalous diffusion as well as normal diffusion. We show that numerical simulation results coincide with derived analytical approximations describing the temporal evolution of the raw moments.

Virus outbreaks have the potential to be a source of severe sanitarian and economic crisis. We propose a new methodology to study the influence of several parameter combinations on the dynamical behavior of simple epidemiological compartmental models. Using this methodology, we analyze the behavior of a simple vaccination model. We find that for susceptible-infected-recovered (SIR) models with seasonality and natural death rate, a new vaccination can reduce the chaoticity of epidemic trajectories, even with nonvaccinated adults. This strategy has little effect on the first infection wave, but it can stop subsequent waves.

In this paper, we analyze the dynamics of the Coulomb glass lattice model in three dimensions near a local equilibrium state by using mean-field approximations. We specifically focus on understanding the role of localization length ($\xi$) and the temperature ($T$) in the regime where the system is not far from equilibrium. We use the eigenvalue distribution of the dynamical matrix to characterize relaxation laws as a function of localization length at low temperatures. The variation of the minimum eigenvalue of the dynamical matrix with temperature and localization length is discussed numerically and analytically. Our results demonstrate the dominant role played by the localization length on the relaxation laws. For very small localization lengths, we find a crossover from exponential relaxation at long times to a logarithmic decay at intermediate times. No logarithmic decay at the intermediate times is observed for large localization lengths.

Lévy walk process is one of the most effective models to describe superdiffusion, which underlies some important movement patterns and has been widely observed in micro- and macrodynamics. From the perspective of random walk theory, here we investigate the dynamics of Lévy walks under the influences of the constant force field and the one combined with harmonic potential. Utilizing Hermite polynomial approximation to deal with the spatiotemporally coupled analysis challenges, some striking features are detected, including non-Gaussian stationary distribution, faster diffusion, still strongly anomalous diffusion, etc.

Singularities of dynamical large-deviation functions are often interpreted as the signal of a dynamical phase transition and the coexistence of distinct dynamical phases, by analogy with the correspondence between singularities of free energies and equilibrium phase behavior. Here we study models of driven random walkers on a lattice. These models display large-deviation singularities in the limit of large lattice size, but the extent to which each model's phenomenology resembles a phase transition depends on the details of the driving. We also compare the behavior of ergodic and nonergodic models that present large-deviation singularities. We argue that dynamical large-deviation singularities indicate the divergence of a model timescale, but not necessarily one associated with cooperative behavior or the existence of distinct phases.

Although lattice gases composed of particles preventing up to their $k\mathrm{th}$ nearest neighbors from being occupied (the $k\mathrm{NN}$ models) have been widely investigated in the literature, the location and the universality class of the fluid-columnar transition in the 2NN model on the square lattice are still a topic of debate. Here, we present grand-canonical solutions of this model on Husimi lattices built with diagonal square lattices, with $2L(L+1)$ sites, for $L⩽7$. The systematic sequence of mean-field solutions confirms the existence of a continuous transition in this system, and extrapolations of the critical chemical potential ${\mu }_{2,c}\left(L\right)$ and particle density ${\rho }_{2,c}\left(L\right)$ to $L\to \infty$ yield estimates of these quantities in close agreement with previous results for the 2NN model on the square lattice. To confirm the reliability of this approach, we employ it also for the 1NN model, where very accurate estimates for the critical parameters ${\mu }_{1,c}$ and ${\rho }_{1,c}$—for the fluid-solid transition in this model on the square lattice—are found from extrapolations of data for $L⩽6$. The nonclassical critical exponents for these transitions are investigated through the coherent anomaly method (CAM), which in the 1NN case yields $\beta$ and $\nu$ differing by at most 6% from the expected Ising exponents. For the 2NN model, the CAM analysis is somewhat inconclusive, because the exponents sensibly depend on the value of ${\mu }_{2,c}$ used to calculate them. Notwithstanding, our results suggest that $\beta$ and $\nu$ are considerably larger than the Ashkin-Teller exponents reported in numerical studies of the 2NN system.