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.
The totally asymmetric simple exclusion process (TASEP) is a well studied example of far-from-equilibrium dynamics. Here, we consider a TASEP with open boundaries but impose a global constraint on the total number of particles. In other words, the bo
undary reservoirs and the system must share a finite supply of particles. Using simulations and analytic arguments, we obtain the average particle density and current of the system, as a function of the boundary rates and the total number of particles. Our findings are relevant to biological transport problems if the availability of molecular motors becomes a rate-limiting factor.
As a solvable and broadly applicable model system, the totally asymmetric exclusion process enjoys iconic status in the theory of non-equilibrium phase transitions. Here, we focus on the time dependence of the total number of particles on a 1-dimensi
onal open lattice, and its power spectrum. Using both Monte Carlo simulations and analytic methods, we explore its behavior in different characteristic regimes. In the maximal current phase and on the coexistence line (between high/low density phases), the power spectrum displays algebraic decay, with exponents -1.62 and -2.00, respectively. Deep within the high/low density phases, we find pronounced emph{oscillations}, which damp into power laws. This behavior can be understood in terms of driven biased diffusion with conserved noise in the bulk.
