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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., Comment: 24 pages, 10 figures

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., Comment: The paper includes 12 pages and 9 figures

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.

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 $\Delta \equiv N_{I}-N_{E}$ crosses the 'critical point' $\Delta _{c}=0$, 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., Comment: 6 pages, 4 figures

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., Comment: 8 pages, 4 figures

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., Comment: 17 pages, 6 figures

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., Comment: 17 pages, 12 figures

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., Comment: Minor correction in one equation

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., Comment: 6 pages, 2 figures, supplementary material and movie available at https://dx.doi.org/10.6084/m9.figshare.2060595

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., Comment: 38 pages, 18 figures (some with multiple parts), to appear in JSTAT

In a recent work \cite{LiuJoladSchZia13}, we introduced dynamic networks with preferred degrees and presented simulation and analytic studies of a single, homogeneous system as well as two interacting networks. Here, we extend these studies to a wider range of parameter space, in a more systematic fashion. Though the interaction we introduced seems simple and intuitive, it produced dramatically different behavior in the single- and two-network systems. Specifically, partitioning the single network into two identical sectors, we find the cross-link distribution to be a sharply peaked Gaussian. In stark contrast, we find a very broad and flat plateau in the case of two interacting identical networks. A sound understanding of this phenomenon remains elusive. Exploring more asymmetric interacting networks, we discover a kind of `universal behavior' for systems in which the `introverts' (nodes with smaller preferred degree) are far outnumbered. Remarkably, an approximation scheme for their degree distribution can be formulated, leading to very successful predictions., Comment: Accepted by JSTAT

We study a simple model of dynamic networks, characterized by a set preferred degree, $\kappa$. Each node with degree $k$ attempts to maintain its $\kappa$ and will add (cut) a link with probability $w(k;\kappa)$ ($1-w(k;\kappa)$). As a starting point, we consider a homogeneous population, where each node has the same $\kappa$, and examine several forms of $w(k;\kappa)$, inspired by Fermi-Dirac functions. Using Monte Carlo simulations, we find the degree distribution in steady state. In contrast to the well-known Erd\H{o}s-R\'{e}nyi network, our degree distribution is not a Poisson distribution; yet its behavior can be understood by an approximate theory. Next, we introduce a second preferred degree network and couple it to the first by establishing a controllable fraction of inter-group links. For this model, we find both understandable and puzzling features. Generalizing the prediction for the homogeneous population, we are able to explain the total degree distributions well, but not the intra- or inter-group degree distributions. When monitoring the total number of inter-group links, $X$, we find very surprising behavior. $X$ explores almost the full range between its maximum and minimum allowed values, resulting in a flat steady-state distribution, reminiscent of a simple random walk confined between two walls. Both simulation results and analytic approaches will be discussed., Comment: Accepted by JSTAT

Using Monte Carlo and analytic techniques, we study a minimal dynamic network involving two populations of nodes, characterized by different preferred degrees. Reminiscent of introverts and extroverts in a population, one set of nodes, labeled \textit{introverts} ($I$), prefers fewer contacts (a lower degree) than the other, labeled \textit{extroverts} ($E$). As a starting point, we consider an \textit{extreme} case, in which an $I$ simply cuts one of its links at random when chosen for updating, while an $E$ adds a link to a random unconnected individual (node). The model has only two control parameters, namely, the number of nodes in each group, $N_{I}$ and $N_{E}$). In the steady state, only the number of crosslinks between the two groups fluctuates, with remarkable properties: Its average ($X$) remains very close to 0 for all $N_{I}>N_{E}$ or near its maximum ($\mathcal{N}\equiv N_{I}N_{E}$) if $N_{I}

We study the dynamical properties of a finite dynamical network composed of two interacting populations, namely; extrovert ($a$) and introvert ($b$). In our model, each group is characterized by its size ($N_a$ and $N_b$) and preferred degree ($\kappa_a$ and $\kappa_b\ll\kappa_a$). The network dynamics is governed by the competing microscopic rules of each population that consist of the creation and destruction of links. Starting from an unconnected network, we give a detailed analysis of the mean field approach which is compared to Monte Carlo simulation data. The time evolution of the restricted degrees $\moyenne{k_{bb}}$ and $\moyenne{k_{ab}}$ presents three time regimes and a non monotonic behavior well captured by our theory. Surprisingly, when the population size are equal $N_a=N_b$, the ratio of the restricted degree $\theta_0=\moyenne{k_{ab}}/\moyenne{k_{bb}}$ appears to be an integer in the asymptotic limits of the three time regimes. For early times (defined by $t

We propose an exactly solvable model for the dynamics of voters in a two-party system. The opinion formation process is modeled on a random network of agents. The dynamical nature of interpersonal relations is also reflected in the model, as the connections in the network evolve with the dynamics of the voters. In the infinite time limit, an exact solution predicts the emergence of consensus, for arbitrary initial conditions. However, before consensus is reached, two different metastable states can persist for exponentially long times. One state reflects a perfect balancing of opinions, the other reflects a completely static situation. An estimate of the associated lifetimes suggests that lack of consensus is typical for large systems., Comment: 4 pages, 6 figures, submitted to Phys. Rev. Lett

We present high-precision Monte Carlo data for the phase diagram of a two-species driven diffusive system, reminiscent of traffic across a narrow bridge. Earlier studies reported two phases with broken symmetry; the existence of one of these has been the subject of some debate. We show that the disputed phase disappears for sufficiently large systems and/or sufficiently low bulk mobility., Comment: 8 pages, 3 figures, JPA style

Motivated by an analogy with traffic, we simulate two species of particles (`vehicles'), moving stochastically in opposite directions on a two-lane ring road. Each species prefers one lane over the other, controlled by a parameter $0 \leq b \leq 1$ such that $b=0$ corresponds to random lane choice and $b=1$ to perfect `laning'. We find that the system displays one large cluster (`jam') whose size increases with $b$, contrary to intuition. Even more remarkably, the lane `charge' (a measure for the number of particles in their preferred lane) exhibits a region of negative response: even though vehicles experience a stronger preference for the `right' lane, more of them find themselves in the `wrong' one! For $b$ very close to 1, a sharp transition restores a homogeneous state. Various characteristics of the system are computed analytically, in good agreement with simulation data., Comment: 7 pages, 3 figures; to appear in Europhysics Letters (2005)