The task of community detection, which aims to partition a network into clusters of nodes to summarize its large-scale structure, has spawned the development of many competing algorithms with varying objectives. Some community detection methods are inferential, explicitly deriving the clustering objective through a probabilistic generative model, while other methods are descriptive, dividing a network according to an objective motivated by a particular application, making it challenging to compare these methods on the same scale. Here we present a solution to this problem that associates any community detection objective, inferential or descriptive, with its corresponding implicit network generative model. This allows us to compute the description length of a network and its partition under arbitrary objectives, providing a principled measure to compare the performance of different algorithms without the need for “ground-truth” labels. Our approach also gives access to instances of the community detection problem that are optimal to any given algorithm and in this way reveals intrinsic biases in popular descriptive methods, explaining their tendency to overfit. Using our framework, we compare a number of community detection methods on artificial networks and on a corpus of over 500 structurally diverse empirical networks. We find that more expressive community detection methods exhibit consistently superior compression performance on structured data instances, without having degraded performance on a minority of situations where more specialized algorithms perform optimally. Our results undermine the implications of the “no free lunch” theorem for community detection, both conceptually and in practice, since it is confined to unstructured data instances, unlike relevant community detection problems which are structured by requirement.

We study the binary and continuous negative-margin perceptrons as simple nonconvex neural network models learning random rules and associations. We analyze the geometry of the landscape of solutions in both models and find important similarities and differences. Both models exhibit subdominant minimizers which are extremely flat and wide. These minimizers coexist with a background of dominant solutions which are composed by an exponential number of algorithmically inaccessible small clusters for the binary case (the frozen 1-RSB phase) or a hierarchical structure of clusters of different sizes for the spherical case (the full RSB phase). In both cases, when a certain threshold in constraint density is crossed, the local entropy of the wide flat minima becomes nonmonotonic, indicating a breakup of the space of robust solutions into disconnected components. This has a strong impact on the behavior of algorithms in binary models, which cannot access the remaining isolated clusters. For the spherical case the behavior is different, since even beyond the disappearance of the wide flat minima the remaining solutions are shown to always be surrounded by a large number of other solutions at any distance, up to capacity. Indeed, we exhibit numerical evidence that algorithms seem to find solutions up to the SAT/UNSAT transition, that we compute here using an 1RSB approximation. For both models, the generalization performance as a learning device is shown to be greatly improved by the existence of wide flat minimizers even when trained in the highly underconstrained regime of very negative margins.

Since the seminal work of Powell, the relationships between the population growth rate, the probability distributions of generation time, and the distribution of cell age have been known for the bacterial population in a steady state of exponential growth. Here we generalize these relationships to include an unsteady (transient) state for both the batch culture and the mother machine experiment. In particular, we derive a time-dependent Euler-Lotka equation (relating the generation-time distributions to the population growth rate) and a generalization of the inequality between the mean generation time and the population doubling time. To do this, we use a model proposed by Lebowitz and Rubinow, in which each cell is described by its age and generation time. We show that our results remain valid for a class of more complex models that use other state variables in addition to cell age and generation time, as long as the integration of these additional variables reduces the model to Lebowitz-Rubinow form. As an application of this formalism, we calculate the fitness landscapes for phenotypic traits (cell age, generation time) in a population that is not growing exponentially. We clarify that the known fitness landscape formula for the cell age as a phenotypic trait is an approximation to the exact time-dependent formula.

We do extensive simulations of a simple model of shear-driven jamming in two dimensions to determine and analyze the velocity distribution at different densities $\varphi$ around the jamming density ${\varphi }_J$ and at different low shear strain rates, $\dot{\gamma }$. We then find that the velocity distribution is made up of two parts which are related to two different physical processes which we call the slow process and the fast process as they are dominated by the slower and the faster particles, respectively. Earlier scaling analyses have shown that the shear viscosity $\eta$, which diverges as the jamming density is approached from below, consists of two different terms, and we present strong evidence that these terms are related to the two different processes: the leading divergence is due to the fast process, whereas the correction-to-scaling term is due to the slow process. The analysis of the slow process is possible thanks to the observation that the velocity distribution for different $\dot{\gamma }$ and $\varphi$ at and around the shear-driven jamming transition has a peak at low velocities and that the distribution has a constant shape up to and slightly above this peak. We then find that it is possible to express the contribution to the shear viscosity due to the slow process in terms of height and position of the peak in the velocity distribution and find that this contribution matches the correction-to-scaling term, determined through a standard critical scaling analysis. A further observation is that the collective particle motion is dominated by the slow process. In contrast to the usual picture in critical phenomena with a direct link between the diverging correlation length and a diverging order parameter, we find that correlations and shear viscosity decouple since they are controlled by different sets of particles and that shear-driven jamming is thus an unusual kind of critical phenomenon.

Particles anomalously diffusing in contact with a thermal bath are initially released from an asymptotically flat potential well. For temperatures that are sufficiently low compared to the potential depth, the dynamical and thermodynamical observables of the system remain almost constant for long times. We show how these stagnated states are characterized as non-normalizable quasiequilibrium (NNQE) states. We use the fractional-time Fokker-Planck equation (FTFPE) and continuous-time random walk approaches to calculate ensemble averages. We obtain analytical estimates of the durations of NNQE states, depending on the fractional order, from approximate theoretical solutions of the FTFPE. We study and compare two types of observables, the mean square displacement typically used to characterize diffusion, and the thermodynamic energy. We show that the typical timescales for transient stagnation depend exponentially on the value of the depth of the potential well, in units of temperature, multiplied by a function of the fractional exponent.

Transitions between metastable equilibria in the low-temperature phase of dynamical Ising game with activity spillover are studied in the infinite time limit. It is shown that exponential enhancement due to activity spillover, which takes place in finite-time transitions, is absent in the infinite time limit. In order to demonstrate that, the analytical description for infinite time trajectory is developed. An analytical approach to estimate the probability of transition between metastable equilibria in the infinite time limit is introduced and its results are compared with those of kinetic Monte Carlo simulation. Our study sheds light on the dynamics of the Ising game and has implications for the understanding of transitions between metastable states in complex systems.

We study kinetic transport through one-dimensional modular networks consisting of alternating domains using both analytical and numerical methods. We demonstrate that the mean velocity is insensitive to the local structure of the network, and it depends only on global, structural-averaged properties. However, by examining high-order cumulants characterizing the kinetics, we reveal information on the degree of inhomogeneity of blocks and the size of repeating units in the network. Specifically, in unbiased diffusion, the kurtosis is the first transport coefficient that exposes structural information, whereas in biased chains, the diffusion coefficient already reveals structural motifs. Nevertheless, this latter dependence is weak, and it disappears at both low and high biasing. Our study demonstrates that high-order moments of the population distribution over sites provide information about the network structure that is not captured by the first moment (mean velocity) alone. These results are useful towards deciphering mechanisms and determining architectures underlying long-range charge transport in biomolecules and biological and chemical reaction networks.

We explore a complete analogy between the classic susceptible-infected-recovered epidemiological model with natural birth and death rates, and class-B laser equations. As a result, recently derived asymptotic formulas in the former context can be used to describe the switch-on intensity pulse of a laser suddenly brought well above the lasing threshold, as in active $Q$-switching. Conversely, the well-studied laser relaxation oscillations find a companion behavior in epidemiology, emphasizing nontrivial timescales. Finally, we discuss the possible correspondence between multistrain outbreaks and multimode lasing.

We study the effects of noise cross correlations on the steady states of driven, nonequilibrium systems, which are described by two stochastically driven dynamical variables, in one dimension. We use a well-known stochastically driven coupled model with two dynamical variables, where one of the variables is autonomous, being independent of the other, whereas the second one depends explicitly on the former. Introducing cross correlations of the two noises in the two dynamical equations, we show that depending upon the details of the nonlinear coupling between the dynamical fields, such cross correlations can induce instabilities in the models that are otherwise stable in the absence of any cross correlations. We argue that this is reminiscent of the roughening transition found in the Kardar-Parisi-Zhang equation in dimensions greater than two. Phenomenological implications of our results are discussed.

A nonconservative system with nonreciprocal interaction has been found to reveal exotic features where sudden phase transitions can occur. In this paper, the emanation of a chimera in a network of Stuart-Landau oscillators with nonreciprocal interaction is reported. Note that the spins follow the random discrete distribution. In other words, we pick a random oscillator to rotate clockwise or anticlockwise. At the transition points, we find the spectral singularities in the eigenplane, where eigenvalues coalesce, commonly known as exceptional points. We find that the counterrotational symmetry breaking induced by exceptional points antecedents the occurrence of the chimera. We numerically attest to the findings for two cases of initial conditions, namely, bipartite and random. We also extend our study to a two-dimensional array of nonreciprocally interacting, distributed spins. The findings could have pragmatic implications in the areas of active matter, networks, and photonics.

The lifting Hele-Shaw cell setup is a popular modification of the classic, fixed-gap, radial viscous fingering problem. In the lifting cell configuration, the upper cell plate is lifted such that a more viscous inner fluid is invaded by an inward-moving outer fluid. As the fluid-fluid interface contracts, one observes the rising of distinctive patterns in which penetrating fingers having rounded tips compete among themselves, reaching different lengths. Despite the scholarly and practical relevance of this confined lifting flow problem, the impact of interfacial rheology effects on its pattern-forming dynamics has been overlooked. Authors of recent studies on the traditional injection-induced radial Hele-Shaw flow and its centrifugally driven variant have shown that, if the fluid-fluid interface is structured (i.e., laden with surfactants, particles, proteins, or other surface-active entities), surface rheological stresses start to act, influencing the development of the viscous fingering patterns. In this paper, we investigate how interfacial rheology affects the stability as well as the shape of the emerging fingered structures in lifting Hele-Shaw flows, at linear and early nonlinear dynamic stages. We tackle the problem by utilizing the Boussinesq-Scriven model to describe the interface and by employing a perturbative mode-coupling scheme. Our linear stability results show that interfacial rheology effects destabilize the interface. Furthermore, our second-order findings indicate that interfacial rheology significantly alters intrinsically nonlinear morphological features of the shrinking interface, inducing the formation of narrow sharp-tip penetrating fingers and favoring enhanced competition among them.

We introduce a basis set consisting of three-dimensional Deslauriers-Dubuc wavelets and solve numerically the Schrödinger equations of H and He atoms and molecules ${\mathrm{H}}_2, {\mathrm{H}}_2^{+}$, and $\mathrm{LiH}$ with Hartree-Fock and density functional theory (DFT) methods. We also compute the $2\mathit{s}$ and $2\mathit{p}$ excited states of hydrogen. The Coulomb singularity at the nucleus is handled by using a pseudopotential. The eigenvalue problem is solved with Arnoldi and Lanczos methods, Poisson equation with generalized minimal residual method and conjugate gradient on the normal equations methods, and matrix elements are computed using the biorthogonality relations of the interpolating wavelets. Performance is compared with those of CCCBDB and bigdft.

Experimental measurements of the surface tension of colloidal interfaces have long been in conflict with computer simulations. In this Letter we show that the surface tension of colloids as measured by surface fluctuations picks up a gravity-dependent contribution which removes the discrepancy. The presence of this term puts a strong constraint on the structure of the interface which allows one to identify corrections to the fundamental equation of equilibrium capillarity and deduce bottom up the microscopic origin of a growth model with close relation to the Kardar-Parisi-Zhang equation.

Percolation theory and the associated conductance networks have provided deep insights into the flow and transport properties of a vast number of heterogeneous materials and media. In practically all cases, however, the conductance of the networks' bonds remains constant throughout the entire process. There are, however, many important problems in which the conductance of the bonds evolves over time and does not remain constant. Examples include clogging, dissolution and precipitation, and catalytic processes in porous materials, as well as the deformation of a porous medium by applying an external pressure or stress to it that reduces the size of its pores. We introduce two percolation models to study the evolution of the conductivity of such networks. The two models are related to natural and industrial processes involving clogging, precipitation, and dissolution processes in porous media and materials. The effective conductivity of the models is shown to follow known power laws near the percolation threshold, despite radically different behavior both away from and even close to the percolation threshold. The behavior of the networks close to the percolation threshold is described by critical exponents, yielding bounds for traditional percolation exponents. We show that one of the two models belongs to the traditional universality class of percolation conductivity, while the second model yields nonuniversal scaling exponents.

The numerical, analytical, and experimental analyses are presented for synchronizing two rotors under the Yukawa interaction. We report that the rotors exhibit in-phase and mixed-phase measure synchronizations for a pair of coupled rotors. Here, the analytical condition for synchronization is derived, tested numerically, and confirmed experimentally using coupled camphor infused rotors as a test bed. Moreover, the concept of measure synchronization is discussed. We report that, in conservative systems, not only the critical coupling parameter but initial conditions also play an essential role for estimating the measure synchronization region.

Mechanistic network models specify the mechanisms by which networks grow and change, allowing researchers to investigate complex systems using both simulation and analytical techniques. Unfortunately, it is difficult to write likelihoods for instances of graphs generated with mechanistic models, and thus it is near impossible to estimate the parameters using maximum likelihood estimation. In this paper, we propose treating the node sequence in a growing network model as an additional parameter, or as a missing random variable, and maximizing over the resulting likelihood. We develop this framework in the context of a simple mechanistic network model, used to study gene duplication and divergence, and test a variety of algorithms for maximizing the likelihood in simulated graphs. We also run the best-performing algorithm on one human protein-protein interaction network and four nonhuman protein-protein interaction networks. Although we focus on a specific mechanistic network model, the proposed framework is more generally applicable to reversible models.

During embryonic development, structures with complex geometry can emerge from planar epithelial monolayers; studying these shape transitions is of key importance for revealing the biophysical laws involved in the morphogenesis of biological systems. Here, using the example of normal proliferative monkey kidney (COS) cell monolayers, we investigate global and local topological characteristics of this model system in dependence on its shape. The obtained distributions of cells by their valence demonstrate a difference between the spherical and planar monolayers. In addition, in both spherical and planar monolayers, the probability of observing a pair of neighboring cells with certain valences depends on the topological charge of the pair. The zero topological charge of the cell pair can increase the probability for the cells to be the nearest neighbors. We then test and confirm that analogous relationships take place in a more ordered spherical system with a larger fraction of 6-valent cells, namely, in the nonproliferative epithelium (follicular system) of ascidian species oocytes. However, unlike spherical COS cell monolayers, ascidian monolayers are prone to nonrandom agglomeration of 6-valent cells and have linear topological defects called scars and pleats. The reasons for this difference in morphology are discussed. The morphological peculiarities found are compared with predictions of the widely used vertex model of epithelium.

Via computer simulations we study evolution dynamics in systems of continuously moving active Brownian particles. The obtained results are discussed against those from the passive 2D Ising case. Following sudden quenches of random configurations to state points lying within the miscibility gaps and to the critical points, we investigate the far-from-steady-state dynamics by calculating quantities associated with structure and characteristic length scales. We also study aging for quenches into the miscibility gap and provide a quantitative picture for the scaling behavior of the two-time order-parameter correlation function. The overall structure and dynamics are consistent with expectations from the Ising model. This remains true for certain active lattice models as well, for which we present results for quenches to the critical points.

We study the director configurations of nematic liquid crystal (NLC) droplets with homeotropic anchoring in a magnetic field and report observation of a magnetic-field-driven transition from a deformed radial to an axial-with-defect configuration. Magnetic-field-induced transitions in NLC droplets differ fundamentally from the traditional planar Freedericksz transition due to the spherical droplet geometry and resulting topological defect. This transition has been studied theoretically, but the director configurations and mechanism of defect evolution in an applied magnetic field have yet to be observed experimentally. To this end, we combine polarized optical microscopy with a variable electromagnet ($\le 1$ T) for continuous observation of droplet director fields, and we employ Landau–de Gennes numerical simulations to elucidate the director configurations and first-order nature of the transition. We report a configuration transition from point defect to ring defect at a critical field, which varies inversely with droplet radius and is relatively independent of surfactant type and concentration. We also estimate anchoring strengths of commonly used surfactants at the NLC-aqueous interface.

The Enskog kinetic theory is applied to compute the mean square displacement of impurities or intruders (modeled as smooth inelastic hard spheres) immersed in a granular gas of smooth inelastic hard spheres (grains). Both species (intruders and grains) are surrounded by an interstitial molecular gas (background) that plays the role of a thermal bath. The influence of the latter on the motion of intruders and grains is modeled via a standard viscous drag force supplemented by a stochastic Langevin-like force proportional to the background temperature. We solve the corresponding Enskog-Lorentz kinetic equation by means of the Chapman-Enskog expansion truncated to first order in the gradient of the intruder number density. The integral equation for the diffusion coefficient is solved by considering the first two Sonine approximations. To test these results, we also compute the diffusion coefficient from the numerical solution of the inelastic Enskog equation by means of the direct simulation Monte Carlo method. We find that the first Sonine approximation generally agrees well with the simulation results, although significant discrepancies arise when the intruders become lighter than the grains. Such discrepancies are largely mitigated by the use of the second Sonine approximation, in excellent agreement with computer simulations even for moderately strong inelasticities and/or dissimilar mass and diameter ratios. We invoke a random walk picture of the intruders' motion to shed light on the physics underlying the intricate dependence of the diffusion coefficient on the main system parameters. This approach, recently employed to study the case of an intruder immersed in a granular gas, also proves useful in the present case of a granular suspension. Finally, we discuss the applicability of our model to real systems in the self-diffusion case. We conclude that collisional effects may strongly impact the diffusion coefficient of the grains.