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 $N_{I}=N_{E}$.

Modeling interacting dynamic networks: II. Systematic study of the statistical properties of cross-links between two networks with preferred degrees

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.

Modeling interacting dynamic networks: I. Preferred degree networks and their characteristics

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 ErdH{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.

Extraordinary variability and sharp transitions in a maximally frustrated dynamic network

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 $N_{I}<N_{E}$. At the transition ($N_{I}=N_{E}$), the fraction $X/mathcal{N}$ wanders across a substantial part of $[0,1]$, much like a pure random walk. Mapping this system to an Ising model with spin-flip dynamics and unusual long-range interactions, we note that such fluctuations are far greater than those displayed in either first or second order transitions of the latter. Thus, we refer to the case here as an `extraordinary transition. Thanks to the restoration of detailed balance and the existence of a `Hamiltonian, several qualitative aspects of these remarkable phenomena can be understood analytically.