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 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 JSTA

Bond percolation of polymers

We study bond percolation of $N$ non-interacting Gaussian polymers of $\ell$ segments on a 2D square lattice of size $L$ with reflecting boundaries. Through simulations, we find the fraction of configurations displaying {\em no} connected cluster which span from one edge to the opposite edge. From this fraction, we define a critical segment density $\rho_{c}^L(\ell)$ and the associated critical fraction of occupied bonds $p_{c}^L(\ell)$, so that they can be identified as the percolation threshold in the $L \to \infty$ limit. Whereas $p_{c}^L(\ell)$ is found to decrease monotonically with $\ell$ for a wide range of polymer lengths, $\rho_{c}^L(\ell)$ is non-monotonic. We give physical arguments for this intriguing behavior in terms of the competing effects of multiple bond occupancies and polymerization.Comment: 4 pages with 6 figure

Universal aspects of vacancy-mediated disordering dynamics: the effect of external fields

We investigate the disordering of an initially phase-segregated binary alloy, due to a highly mobile defect which couples to an electric or gravitational field. Using both mean-field and Monte Carlo methods, we show that the late stages of this process exhibit dynamic scaling, characterized by a set of exponents and scaling functions. A new scaling variable emerges, associated with the field. While the scaling functions carry information about the field and the boundary conditions, the exponents are universal. They can be computed analytically, in excellent agreement with simulation results.Comment: 15 pages, 6 figure

The Dynamics of Supply and Demand in mRNA Translation

We study the elongation stage of mRNA translation in eukaryotes and find that, in contrast to the assumptions of previous models, both the supply and the demand for tRNA resources are important for determining elongation rates. We find that increasing the initiation rate of translation can lead to the depletion of some species of aa-tRNA, which in turn can lead to slow codons and queueing. Particularly striking “competition” effects are observed in simulations of multiple species of mRNA which are reliant on the same pool of tRNA resources. These simulations are based on a recent model of elongation which we use to study the translation of mRNA sequences from the Saccharomyces cerevisiae genome. This model includes the dynamics of the use and recharging of amino acid tRNA complexes, and we show via Monte Carlo simulation that this has a dramatic effect on the protein production behaviour of the system

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, labeledintroverts (I), prefers fewer contacts (a lower degree) than the other, labeled extroverts (E). As a starting point, we consider anextreme 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, NI and NE. 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 NI > NE or near its maximum () if NI NE. At the transition (NI = NE), the fraction 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.This is an author's manuscript of an article from Europhysics Letters 100 (2012): 66007, doi:10.1209/0295-5075/100/66007.</p

Epidemic spreading on preferred degree adaptive networks

We study the standard SIS model of epidemic spreading on networks where individuals have a fluctuating number of connections around a preferred degree . Using very simple rules for forming such preferred degree networks, we find some unusual statistical properties not found in familiar Erdös-Rényi or scale free networks. By letting depend on the fraction of infected individuals, we model the behavioral changes in response to how the extent of the epidemic is perceived. In our models, the behavioral adaptations can be either ‘blind’ or ‘selective’ – depending on whether a node adapts by cutting or adding links to randomly chosen partners or selectively, based on the state of the partner. For a frozen preferred network, we find that the infection threshold follows the heterogeneous mean field result and the phase diagram matches the predictions of the annealed adjacency matrix (AAM) approach. With ‘blind’ adaptations, although the epidemic threshold remains unchanged, the infection level is substantially affected, depending on the details of the adaptation. The ‘selective’ adaptive SIS models are most interesting. Both the threshold and the level of infection changes, controlled not only by how the adaptations are implemented but also how often the nodes cut/add links (compared to the time scales of the epidemic spreading). A simple mean field theory is presented for the selective adaptations which capture the qualitative and some of the quantitative features of the infection phase diagram.This article is from PLoS ONE 7 (2012): e48686, doi:10.1371/journal.pone.0048686.</p

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, κ. Each node with degree k attempts to maintain its κ and will add (cut) a link with probability w(k;κ) (1 − w(k;κ)). As a starting point, we consider a homogeneous population, where each node has the same κ, and examine several forms of w(k;κ), inspired by Fermi–Dirac functions. Using Monte Carlo simulations, we find the degree distribution in the steady state. In contrast to the well known Erdős–Rényi 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.This is an author's manuscript of an article from Journal of Statistical Mechanics: Theory and Experiment (2013): P08001, doi:10.1088/1742-5468/2013/08/P08001.</p