arXiv subjects

R. K. P. Zia

Publications and source records attributed to R. K. P. Zia.

At least 19 recordsLinked to original sources

arXiv2025

Pattern formation in a coupled driven diffusive system

We investigate pattern formation in a driven mixture of two repulsive particles by introducing a Field-based Lattice Model (FLM), a hybrid model that combines aspects of the driven Widom-Rowlison lattice gas (DWRLG) and its statistical field theory. We find that the FLM effectively captures the bulk behavior of the DWRLG in both low- and high-density phases, suggesting that phase transitions in these models may share a universality class. Under the effect of the drive, the FLM additionally reveals an intermediate regime, not reported in the previous DWRLG studies, characterized by "irregular stripes" with widely fluctuating widths, contrasting with the "regular", well-ordered stripes found at higher densities. In this intermediate phase, the system exhibits long-range order, predominantly perpendicular to the drive direction. To construct a continuum description, we derive two coupled partial differential equations via a gradient expansion of the FLM mean mass-transfer equations, supplemented with additive noise. Designing a numerical solver using the pseudospectral method with dealiasing and stochastic time differencing, we reproduce the low-density microemulsion phase (characterized by a non-zero characteristic wavenumber q*) and perpendicular stripes at high density. We identify the non-zero difference in the characteristic velocities of the fields as a necessary condition for perpendicular stripe formation in the high-density phase. The continuum model also uncovers novel behaviors not previously observed in the FLM, such as stripes aligned parallel to the drive and chaotic patterns. This work highlights how the interplay of external drive, particle interactions, and noise can lead to a rich phenomenology in strongly driven binary mixtures.

cond-mat.stat-mech↗

arXiv2024

Extreme Value Statistics of Community Detection in Complex Networks with Reduced Network Extremal Ensemble Learning (RenEEL)

Arguably, the most fundamental problem in Network Science is finding structure within a complex network. One approach is to partition the nodes into communities that are more densely connected than one expects in a random network. "The" community structure then corresponds to the partition that maximizes Modularity, an objective function that quantifies this idea. Finding the maximizing partition, however, is a computationally difficult, NP-Complete problem. We explore using a recently introduced machine-learning algorithmic scheme to find the structure of benchmark networks. The scheme, known as RenEEL, creates an ensemble of $K$ partitions and updates the ensemble by replacing its worst member with the best of $L$ partitions found by analyzing a simplified network. The updating continues until consensus is achieved within the ensemble. We perform an empirical study of three real-world networks to explore how the Modularity of the consensus partition depends on the values of $K$ and $L$ and relate the results to the extreme value statistics of record-breaking. We find that increasing $K$ is generally more effective than increasing $L$ for finding the best partition.

physics.comp-ph↗

arXiv2024

Signals of Detailed Balance Violation in Nonequilibrium Stationary States: Subtle, Manifest, and Extraordinary

The evolution of physical systems are often modeled by simple Markovian processes. When settled into stationary states, the probability distributions of such systems are time independent, by definition. However, they do not necessarily fall within the framework of equilibrium statistical mechanics. Instead, they may be non-equilibrium steady states (NESS). One distinguished feature of NESS is the presence of time reversal asymmetry (TRA) and persistent probability current loops. These loops lead naturally to the notion of probability angular momenta, which play a role on the same footing as the noise covariance in stochastic processes. Illustrating with simulations of simple models and physical data, we present ways to detect these signals of TRA, from the subtle to the prominent.

cond-mat.stat-mech↗

arXiv2022

Effects of homophily and heterophily on preferred-degree networks: mean-field analysis and overwhelming transition

We investigate the long-time properties of a dynamic, out-of-equilibrium network of individuals holding one of two opinions in a population consisting of two communities of different sizes. Here, while the agents' opinions are fixed, they have a preferred degree which leads them to endlessly create and delete links. Our evolving network is shaped by homophily/heterophily, which is a form of social interaction by which individuals tend to establish links with others having similar/dissimilar opinions. Using Monte Carlo simulations and a detailed mean-field analysis, we study in detail how the sizes of the communities and the degree of homophily/heterophily affects the network structure. In particular, we show that when the network is subject to enough heterophily, an "overwhelming transition" occurs: individuals of the smaller community are overwhelmed by links from agents of the larger group, and their mean degree greatly exceeds the preferred degree. This and related phenomena are characterized by obtaining the network's total and joint degree distributions, as well as the fraction of links across both communities and that of agents having fewer edges than the preferred degree. We use our mean-field theory to discuss the network's polarization when the group sizes and level of homophily vary.

physics.soc-ph↗

arXiv2021

Microemulsions in the driven Widom-Rowlinson lattice gas

An investigation of the two-dimensional Widom-Rowlinson lattice gas under an applied drive uncovered a remarkable non-equilibrium steady state in which uniform stripes (reminiscent of an equilibrium lamellar phase) form perpendicular to the drive direction [R. Dickman and R. K. P. Zia, Phys. Rev. E 97, 062126 (2018)]. Here we study this model at low particle densities in two and three dimensions, where we find a disordered phase with a characteristic length scale (a "microemulsion") along the drive direction. We develop a continuum theory of this disordered phase to derive a coarse-grained field-theoretic action for the non-equilibrium dynamics. The action has the form of two coupled driven diffusive systems with different characteristic velocities, generated by an interplay between the particle repulsion and the drive. We then show how fluctuation corrections in the field theory may generate the characteristic features of the microemulsion phase, including a peak in the static structure factor corresponding to the characteristic length scale. This work lays the foundation for understanding the stripe phenomenon more generally.

cond-mat.stat-mech↗

arXiv2021

How does homophily shape the topology of a dynamic network?

We consider a dynamic network of individuals that may hold one of two different opinions in a two-party society. As a dynamical model, agents can endlessly create and delete links to satisfy a preferred degree, and the network is shaped by \emph{homophily}, a form of social interaction. Characterized by the parameter $J \in [-1,1]$, the latter plays a role similar to Ising spins: agents create links to others of the same opinion with probability $(1+J)/2$, and delete them with probability $(1-J)/2$. Using Monte Carlo simulations and mean-field theory, we focus on the network structure in the steady state. We study the effects of $J$ on degree distributions and the fraction of cross-party links. While the extreme cases of homophily or heterophily ($J= \pm 1$) are easily understood to result in complete polarization or anti-polarization, intermediate values of $J$ lead to interesting features of the network. Our model exhibits the intriguing feature of an "overwhelming transition" occurring when communities of different sizes are subject to sufficient heterophily: agents of the minority group are oversubscribed and their average degree greatly exceeds that of the majority group. In addition, we introduce an original measure of polarization which displays distinct advantages over the commonly used average edge homogeneity.

cond-mat.stat-mech↗

arXiv2019

Fluctuations and Correlations in a Model of Extreme Introverts and Extroverts

Unlike typical phase transitions of first and second order, a system displaying the Thouless effect exhibits characteristics of both at the critical point (jumps in the order parameter and anomalously large fluctuations). An $extreme$ Thousless effect was observed in a recently introduced model of social networks consisting of `introverts and extroverts' ($XIE$). We study the fluctuations and correlations of this system using both Monte Carlo simulations and analytic methods based on a self-consistent mean field theory. Due to the symmetries in the model, we derive identities between all independent two point correlations and fluctuations in three quantities (degrees of individuals and the total number of links between the two subgroups) in the stationary state. As simulations confirm these identities, we study only the fluctuations in detail. Though qualitatively similar to those in the 2D Ising model, there are several unusual aspects, due to the extreme Thouless effect. All these anomalous fluctuations can be quantitatively understood with our theory, despite the mean-field aspects in the approximations. In our theory, we frequently encounter the `finite Poisson distribution' (i.e., $x^{n}/n!$ for $n\in \left[ 0,N\right] $ and zero otherwise). Since its properties appear to be quite obscure, we include an Appendix on the details and the related `finite exponential series' $\sum_{0}^{N}x^{n}/n!$. Some simulation studies of joint degree distributions, which provide a different perspective on correlations, have also been carried out.

physics.soc-ph↗

arXiv2019

Nonequilibrium oscillations, probability angular momentum, and the climate system

Though the Boltzmann-Gibbs framework of equilibrium statistical mechanics has been successful in many arenas, it is clearly inadequate for describing many interesting natural phenomena driven far from equilibrium. The simplest step towards that goal is a better understanding of nonequilibrium steady-states (NESS). Here we focus on one of the distinctive features of NESS, persistent probability currents, and their manifestations in our climate system. We consider the natural variability of the steady-state climate system, which can be approximated as a NESS. These currents must form closed loops, which are odd under time reversal, providing the crucial difference between systems in thermal equilibrium and NESS. Seeking manifestations of such current loops leads us naturally to the notion of probability angular momentum and oscillations in the space of observables. Specifically, we will relate this concept to the asymmetric part of certain time-dependent correlation functions. Applying this approach, we propose that these current loops give rise to preferred spatio-temporal patterns of natural climate variability that take the form of climate oscillations such as the El-Niño Southern Oscillation (ENSO) and the Madden-Julien Oscillation (MJO). In the space of climate indices, we observe persistent currents and define a new diagnostic for these currents: the probability angular momentum. Using the observed climatic time series of ENSO and MJO, we compute both the averages and the distributions of the probability angular momentum. These results are in good agreement with the analysis from a linear Gaussian model. We propose that, in addition to being a new quantification of climate oscillations across models and observations, the probability angular momentum provides a meaningful characterization for all statistical systems in NESS.

physics.ao-ph↗

arXiv2018

Exact results for the extreme Thouless effect in a model of network dynamics

If a system undergoing phase transitions exhibits some characteristics of both first and second order, it is said to be of 'mixed order' or to display the Thouless effect. Such a transition is present in a simple model of a dynamic social network, in which $N_{I/E}$ extreme introverts/extroverts always cut/add random links. In particular, simulations showed that $\left\langle f\right\rangle $, the average fraction of cross-links between the two groups (which serves as an 'order parameter' here), jumps dramatically when $Δ\equiv N_{I}-N_{E}$ crosses the 'critical point' $Δ_{c}=0$, as in typical first order transitions. Yet, at criticality, there is no phase co-existence, but the fluctuations of $f$ are much larger than in typical second order transitions. Indeed, it was conjectured that, in the thermodynamic limit, both the jump and the fluctuations become maximal, so that the system is said to display an 'extreme Thouless effect.' While earlier theories are partially successful, we provide a mean-field like approach that accounts for all known simulation data and validates the conjecture. Moreover, for the critical system $N_{I}=N_{E}=L$, an analytic expression for the mesa-like stationary distribution, $P\left( f\right) $, shows that it is essentially flat in a range $\left[ f_{0},1-f_{0}\right] $, with $f_0 \ll 1$. Numerical evaluations of $f_{0}$ provides excellent agreement with simulation data for $L\lesssim 2000$. For large $L$, we find $f_{0}\rightarrow \sqrt{\left( \ln L^2 \right) /L}$ , though this behavior begins to set in only for $L>10^{100}$. For accessible values of $L$, we provide a transcendental equation for an approximate $f_{0}$ which is better than $\sim$1% down to $L=100$. We conjecture how this approach might be used to attack other systems displaying an extreme Thouless effect.

physics.soc-ph↗

arXiv2018

Co-evolution of nodes and links: diversity driven coexistence in cyclic competition of three species

When three species compete cyclically in a well-mixed, stochastic system of $N$ individuals, extinction is known to typically occur at times scaling as the system size $N$. This happens, for example, in rock-paper-scissors games or conserved Lotka-Volterra models in which every pair of individuals can interact on a complete graph. Here we show that if the competing individuals also have a "social temperament" to be either introverted or extroverted, leading them to cut or add links respectively, then long-living state in which all species coexist can occur when both introverts and extroverts are present. These states are non-equilibrium quasi-steady states, maintained by a subtle balance between species competition and network dynamcis. Remarkably, much of the phenomena is embodied in a mean-field description. However, an intuitive understanding of why diversity stabilizes the co-evolving node and link dynamics remains an open issue.

physics.soc-ph↗

arXiv2018

Driven Widom-Rowlinson lattice gas

In the Widom-Rowlinson lattice gas, two particle species (A, B) diffuse freely via particle-hole exchange, subject to both on-site exclusion and prohibition of A-B nearest-neighbor pairs. As an athermal system, the overall densities are the only control parameters. As the densities increase, an entropically driven phase transition occurs, leading to ordered states with A- and B-rich domains separated by hole-rich interfaces. Using Monte Carlo simulations, we analyze the effect of imposing a drive on this system, biasing particle moves along one direction. Our study parallels that for a driven Ising lattice gas -- the Katz-Lebowitz-Spohn (KLS) model, which displays atypical collective behavior, e.g., structure factors with discontinuity singularities and ordered states with domains only parallel to the drive. Here, other novel features emerge, including structure factors with kink singularities (best fitted to |q|), maxima at non-vanishing wavevector values, oscillating correlation functions, and ordering into multiple striped domains perpendicular to the drive, with a preferred wavelength depending on density and drive intensity. Moreover, the (hole-rich) interfaces between the domains are statistically rough (whether driven or not), in sharp contrast with those in the KLS model, in which the drive suppresses interfacial roughness. Defining a novel order parameter (to account for the emergence of multistripe states), we map out the phase diagram in the density-drive plane and present preliminary evidence for a critical phase in this driven lattice gas.

cond-mat.stat-mech↗

arXiv2017

Emergence of a spectral gap in a class of random matrices associated with split graphs

Motivated by the intriguing behavior displayed in a dynamic network that models a population of extreme introverts and extroverts (XIE), we consider the spectral properties of ensembles of random split graph adjacency matrices. We discover that, in general, a gap emerges in the bulk spectrum between -1 and 0 that contains a single eigenvalue. An analytic expression for the bulk distribution is derived and verified with numerical analysis. We also examine their relation to chiral ensembles, which are associated with bipartite graphs.

physics.soc-ph↗

arXiv2017

A Heterogeneous Out-of-Equilibrium Nonlinear $q$-Voter Model with Zealotry

We study the dynamics of the out-of-equilibrium nonlinear q-voter model with two types of susceptible voters and zealots, introduced in [EPL 113, 48001 (2016)]. In this model, each individual supports one of two parties and is either a susceptible voter of type $q_1$ or $q_2$, or is an inflexible zealot. At each time step, a $q_i$-susceptible voter ($i = 1,2$) consults a group of $q_i$ neighbors and adopts their opinion if all group members agree, while zealots are inflexible and never change their opinion. This model violates detailed balance whenever $q_1 \neq q_2$ and is characterized by two distinct regimes of low and high density of zealotry. Here, by combining analytical and numerical methods, we investigate the non-equilibrium stationary state of the system in terms of its probability distribution, non-vanishing currents and unequal-time two-point correlation functions. We also study the switching times properties of the model by exploiting an approximate mapping onto the model of [Phys. Rev. E 92, 012803 (2015)] that satisfies the detailed balance, and also outline some properties of the model near criticality.

physics.soc-ph↗

arXiv2016

The Many-agent limit of the Extreme Introvert-Extrovert model

We consider a toy model of interacting extrovert and introvert agents introduced earlier by Liu et al [Europhys. Lett. {\bf 100} (2012) 66007]. The number of extroverts, and introverts is $N$ each. At each time step, we select an agent at random, and allow her to modify her state. If an extrovert is selected, she adds a link at random to an unconnected introvert. If an introvert is selected, she removes one of her links. The set of $N^2$ links evolves in time, and may be considered as a set of Ising spins on an $N \times N$ square-grid with single-spin-flip dynamics. This dynamics satisfies detailed balance condition, and the probability of different spin configurations in the steady state can be determined exactly. The effective hamiltonian has long-range multi-spin couplings that depend on the row and column sums of spins. If the relative bias of choosing an extrovert over an introvert is varied, this system undergoes a phase transition from a state with very few links to one in which most links are occupied. We show that the behavior of the system can be determined exactly in the limit of large $N$. The behavior of large fluctuations in the total numer of links near the phase transition is determined. We also discuss two variations, called egalitarian and elitist agents, when the agents preferentially add or delete links to their least/ most-connected neighbor. These shows interesting cooperative behavior.

physics.soc-ph↗

arXiv2016

Characterization of the Nonequilibrium Steady State of a Heterogeneous Nonlinear $q$-Voter Model with Zealotry

We introduce an heterogeneous nonlinear $q$-voter model with zealots and two types of susceptible voters, and study its non-equilibrium properties when the population is finite and well mixed. In this two-opinion model, each individual supports one of two parties and is either a zealot or a susceptible voter of type $q_1$ or $q_2$. While here zealots never change their opinion, a $q_i$-susceptible voter ($i=1,2$) consults a group of $q_i$ neighbors at each time step, and adopts their opinion if all group members agree. We show that this model violates the detailed balance whenever $q_1 \neq q_2$ and has surprisingly rich properties. Here, we focus on the characterization of the model's non-equilibrium stationary state (NESS) in terms of its probability distribution and currents in the distinct regimes of low and high density of zealotry. We unveil the NESS properties in each of these phases by computing the opinion distribution and the circulation of probability currents, as well as the two-point correlation functions at unequal times (formally related to a "probability angular momentum"). Our analytical calculations obtained in the realm of a linear Gaussian approximation are compared with numerical results.

physics.soc-ph↗

arXiv2015

Spatial structures in a simple model of population dynamics for parasite-host interactions

Spatial patterning can be crucially important for understanding the behavior of interacting populations. Here we investigate a simple model of parasite and host populations in which parasites are random walkers that must come into contact with a host in order to reproduce. We focus on the spatial arrangement of parasites around a single host, and we derive using analytics and numerical simulations the necessary conditions placed on the parasite fecundity and lifetime for the populations long-term survival. We also show that the parasite population can be pushed to extinction by a large drift velocity, but, counterintuitively, a small drift velocity generally increases the parasite population.

q-bio.PE↗

arXiv2015

Networks with preferred degree: A mini-review and some new results

Since their inception about a decade ago, dynamic networks which adapt to the state of the nodes have attracted much attention. One simple case of such an adaptive dynamics is a model of social networks in which individuals are typically comfortable with a certain number of contacts, i.e., preferred degrees. This paper is partly a review of earlier work of single homogeneous systems and ones with two interacting networks, and partly a presentation of some new results. In general, the dynamics does not obey detailed balance and the stationary distributions are not known analytically. A particular limit of the latter is a system of extreme introverts and extroverts - the XIE model. Remarkably, in this case, the detailed balance condition is satisfied, the exact distribution and an effective Hamiltonian can be found explicitly. Further, the model exhibits a phase transition in which the total number of links in the system - a macroscopically interesting quantity, displays an extreme Thouless effect. We show that in the limit of large populations and away from the transition, the model reduces to one with non-interacting agents of the majority subgroup. We determine the nature of fluctuations near the transition. We also introduce variants of the model where the agents show preferential attachment or detachment. There are significant changes to the degree distributions in the steady state, some of which can be understood by theoretical arguments and some remain to be explored. Many intriguing questions are posed, providing some food for thought and avenues for future research.

cond-mat.stat-mech↗

arXiv2015

Extreme Thouless effect in a minimal model of dynamic social networks

In common descriptions of phase transitions, first order transitions are characterized by discontinuous jumps in the order parameter and normal fluctuations, while second order transitions are associated with no jumps and anomalous fluctuations. Outside this paradigm are systems exhibiting `mixed order transitions' displaying a mixture of these characteristics. When the jump is maximal and the fluctuations range over the entire range of allowed values, the behavior has been coined an `extreme Thouless effect'. Here, we report findings of such a phenomenon, in the context of dynamic, social networks. Defined by minimal rules of evolution, it describes a population of extreme introverts and extroverts, who prefer to have contacts with, respectively, no one or everyone. From the dynamics, we derive an exact distribution of microstates in the stationary state. With only two control parameters, $N_{I,E}$ (the number of each subgroup), we study collective variables of interest, e.g., $X$, the total number of $I$-$E $ links and the degree distributions. Using simulations and mean-field theory, we provide evidence that this system displays an extreme Thouless effect. Specifically, the fraction $X/\left( N_{I}N_{E}\right) $ jumps from $0$ to $1$ (in the thermodynamic limit) when $N_{I}$ crosses $N_{E}$, while all values appear with equal probability at $N_{I}=N_{E}$.

cond-mat.stat-mech↗