Phase transitions for modified Erdős–Rényi processes

A fundamental and very well studied region of the Erdős–Rényi process is the phase transition at m∼n/2 edges in which a giant component suddenly appears. We examine the process beginning with an initial graph. We further examine the Bohman–Frieze process in which edges between isolated vertices are more likely. While the positions of the phase transitions vary, the three processes belong, roughly speaking, to the same universality class. In particular, the growth of the giant component in the barely supercritical region is linear in all cases.     

Investigation of the adsorption behavior of glycine peptides on 12% cross-linked agarose gel media

The highly cross-linked 12% agarose gel Superose 12 10/300 GL causes retardation of glycine peptides when mobile phases containing varying concentrations of acetonitrile in water are used. An investigation has been made into the retention mechanism behind this retardation using the glycine dipeptide (GG) and tripeptide (GGG) as models. The dependence of retention times of analytical-size peaks under different experimental conditions was interpreted such that the adsorption most probably was caused by the formation of hydrogen bonds but that electrostatic interactions cannot be ruled out. Thereafter, a nonlinear adsorption study was undertaken at different acetonitrile content in the eluent, using the elution by characteristic points (ECPs) method on strongly overloaded GG and GGG peaks. With a new evaluation tool, the adsorption energy distribution (AED) could be calculated prior to the model selection. These calculations revealed that when the acetonitrile content in the eluent was varied from 0% to 20% the interactions turned from (i) being homogenous (GG) or mildly heterogeneous (GGG), (ii) via a more or less stronger degree of heterogeneity around one site to (iii) finally a typical bimodal energy interaction comprising of two sites (GG at 20% and GGG at 10% and 20%). The Langmuir, Tóth and bi-Langmuir models described these interesting adsorption trends excellently. Thus, the retardation observed for these glycine peptides is interpreted as being of mixed-mode character composed of electrostatic bonds and hydrogen bonds.     

Force generation by polymerizing microtubules

Polymerization ratchets formed by the assembly of actin filaments and microtubules are possibly the simplest realizations of biological thermal ratchets. A variety of experimental evidence exists that significant forces are generated by these processes, but quantitative studies lag far behind similar studies for molecular motors such as kinesin and myosin. Here we present a discussion of the theory of polymerization ratchets as well as experimental techniques used in our laboratory for the study of forces generated by single growing microtubules. Data obtained with these techniques provide us with valuable information that may eventually allow us to distinguish between different models for the growth of microtubules. Received: 15 January 2002 / Accepted: 11 February 2002 / Published online: 22 April 2002     

The phase transition in inhomogeneous random graphs

The “classical” random graph models, in particular G(n,p), are “homogeneous,” in the sense that the degrees (for example) tend to be concentrated around a typical value. Many graphs arising in the real world do not have this property, having, for example, power‐law degree distributions. Thus there has been a lot of recent interest in defining and studying “inhomogeneous” random graph models. One of the most studied properties of these new models is their “robustness”, or, equivalently, the “phase transition” as an edge density parameter is varied. For G(n,p), p = c/n, the phase transition at c = 1 has been a central topic in the study of random graphs for well over 40 years. Many of the new inhomogeneous models are rather complicated; although there are exceptions, in most cases precise questions such as determining exactly the critical point of the phase transition are approachable only when there is independence between the edges. Fortunately, some models studied have this property already, and others can be approximated by models with independence. Here we introduce a very general model of an inhomogeneous random graph with (conditional) independence between the edges, which scales so that the number of edges is linear in the number of vertices. This scaling corresponds to the p = c/n scaling for G(n,p) used to study the phase transition; also, it seems to be a property of many large real‐world graphs. Our model includes as special cases many models previously studied. We show that, under one very weak assumption (that the expected number of edges is “what it should be”), many properties of the model can be determined, in particular the critical point of the phase transition, and the size of the giant component above the transition. We do this by relating our random graphs to branching processes, which are much easier to analyze. We also consider other properties of the model, showing, for example, that when there is a giant component, it is “stable”: for a typical random graph, no matter how we add or delete o(n) edges, the size of the giant component does not change by more than o(n). © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 31, 3–122, 2007     

Multicyclic components in a random graph process

A limit theorem is obtained for the number of components with more than one cycle that are created during the evolution of a random graph process. In particular, it is shown that the probability that the process never contains more than one such component is about 0.87 © 1993 John Wiley & Sons, Inc.     

Synchronization of a large number of continuous one-dimensional stochastic elements with time-delayed mean-field coupling

We study synchronization as a means of control of collective behavior of an ensemble of coupled stochastic units in which oscillations are induced merely by external noise. For a large number of one-dimensional continuous stochastic elements coupled non-homogeneously through the mean field with delay we developed an approach to find a boundary of synchronization domain and the frequency of the mean-field oscillations on it. Namely, the exact location of the synchronization threshold is shown to be a solution of the boundary value problem (BVP) which was derived from the linearized Fokker-Planck equation. Here the synchronization threshold is found by solving this BVP numerically. Approximate analytics is obtained by expanding the solution of the linearized Fokker-Planck equation into a series of eigenfunctions of the stationary Fokker-Planck operator. Bistable systems with a polynomial and piece-wise linear potential are considered as examples. Multistability and hysteresis in the mean-field behavior are observed in the stochastic network at finite noise intensities. In the limit of small noise intensities the critical coupling strength is shown to remain finite, provided that the delay in the coupling function is not infinitely small. Delay in the coupling term can be used as a control parameter that manipulates the location of the synchronization threshold.