The combination of the compactness of networks, featuring small diameters, and their complex architectures results in a variety of critical effects dramatically different from those in cooperative systems on lattices. In the last few years, important steps have been made toward understanding the qualitatively new critical phenomena in complex networks. The results, concepts, and methods of this rapidly developing field are reviewed. Two closely related classes of these critical phenomena are considered, namely, structural phase transitions in the network architectures and transitions in cooperative models on networks as substrates. Systems where a network and interacting agents on it influence each other are also discussed. A wide range of critical phenomena in equilibrium and growing networks including the birth of the giant connected component, percolation, $k$-core percolation, phenomena near epidemic thresholds, condensation transitions, critical phenomena in spin models placed on networks, synchronization, and self-organized criticality effects in interacting systems on networks are mentioned. Strong finite-size effects in these systems and open problems and perspectives are also discussed.

/pdf/critical-phenomena-in-complex-networks-4vxnv3f75f.pdf

Critical phenomena in complex networks

This is a pedagogical introduction to Brownian motion on the occasion of the 100th anniversary of Einstein's 1905 paper on the subject. After briefly reviewing Einstein's work in a contemporary context, we pursue several lines of further developments and applications to soft condensed matter and biology. Over the last century Brownian motion has been promoted from an odd curiosity of marginal scientific interest to a guiding theme pervading all of the modern life sciences.

Brownian motion: a paradigm of soft matter and biological physics

The kinetics of an irreversible catalytic reaction on substrate of arbitrary dimension is examined. In the limit of infinitesimal reaction rate (reaction-controlled limit), we solve the dimer-dimer surface reaction model (or voter model) exactly in arbitrary dimension $D$. The density of reactive interfaces is found to exhibit a power law decay for $D<2$ and a slow logarithmic decay in two dimensions. We discuss the relevance of these results for the monomer-monomer surface reaction model.

Exact results for kinetics of catalytic reactions.

The Sznajd model, which describes opinion formation and social influence, is treated analytically on a complete graph. We prove the existence of the phase transition in the original formulation of the model, while for the Ochrombel modification we find smooth behaviour without transition. We calculate the average time to reach the stationary state as well as the exponential tail of its probability distribution. An analytical argument for the observed 1/n dependence in the distribution of votes in Brazilian elections is provided.

Analytical results for the Sznajd model of opinion formation

The Sznajd model, which describes opinion formation and social influence, is treated analytically on a complete graph. We prove the existence of the phase transition in the original formulation of the model, while for the Ochrombel modification we find smooth behaviour without transition. We calculate the average time to reach the stationary state as well as the exponential tail of its probability distribution. An analytical argument for the observed $1/n$ dependence in the distribution of votes in Brazilian elections is provided.

The dynamics of a monomer-monomer and a monomer-dimer surface catalytic reaction is investigated. From the mean-field solution, finite systems eventually 'poison' at an exponential rate to a fully occupied, non-reactive state. For the monomer-monomer process, this poisoning is driven by concentration fluctuations of a diffusive nature, leading to poisoning times which vary as a power of the linear system size L. A comparison of the Monte Carlo simulations with the mean-field result suggests that the upper critical dimension for the monomer-monomer model is d, = 2. For the monomer-dimer process, there is an effective potential that needs to be surmounted by fluctuations, leading to poisoning times which grow at least as fast as eL. This gives rise to an apparent reactive steady state. Heterogeneous catalysis is a fundamental kinetic process in which the rate of a chemical reaction is enhanced by the presence of suitable catalyst material ( 13. A typical example is where reactants adsorb on a catalytic surface which then promotes the bonding of reactants. Once a reaction occurs, the reaction product desorbs, thereby allowing for continued operation of the system. This type of catalytic reaction underlies a host of technological processes and a substantial fraction of all chemical production ( 11. It has only been very recently, however, that investigations of microscopic models have begun to identify the general principles underlying the dynamical behaviour of heterogeneous catalysis. Ziff et a1 (2) introduced a simple lattice model which appears to describe various features of the surface catalytic reaction of carbon monoxide (monomers, which adsorb onto a single lattice site) and oxygen (which are deposited as dimers and disassociate upon adsorption). Depending on the relative deposition rate of the monomers and dimers, there may be 'poisoning', where the surface eventually becomes covered by only one species, or there may be an apparent reactive steady state. Novel kinetic phase transitions demarcate these two possibilities. This intriguing behaviour has stimulated further work on this and a simpler monomer-monomer process to be defined below (3-71. Our goal, in this letter, is to show that the mean-field solution, together with numerical simulations of very small systems, provide a rather complete description of the dynamics of idealised surface catalytic reactions. For the monomer-monomer process, we show that the final state of a finite system is always poisoned and that poisoning is approached at an exponential rate in mean-field theory. These facts are actually quite difficult to establish in numerical simulations of large systems (5).

Finite-size 'poisoning' in heterogeneous catalysis

The authors generalize the monomer-dimer model of heterogeneous catalysis, introduced by Ziff et al (1986) to allow for variable reaction and adsorption rates Numerical simulations indicate that a steady state exists for a narrowing range of relative deposition rate of dimers to monomers as the reaction rate decreases This steady-state regime disappears at a finite value of the reaction probability A mean-field analysis suggests, however, that the steady-state regime extends to zero reaction probability

/pdf/a-nonequilibrium-tricritical-point-in-the-monomer-dimer-223zyty5s9.pdf

A nonequilibrium tricritical point in the monomer-dimer catalysis model

Comment on "Noise-induced bistability in a Monte Carlo surface-reaction model"

The authors discuss the kinetics of an irreversible monomer-monomer model of heterogeneous catalysis. In this model, two reactive species, A and B, adsorb irreversibly onto single sites of a catalytic substrate; subsequently nearest-neighbour adsorbed AB pairs can bond to form a reaction product which desorbs from the substrate. The kinetics of this process is investigated in a mean-field approximation, where the catalytic substrate is considered to be an N-site complete graph. Two fundamental limits are identified: (a) the adsorption-controlled limit, where the reaction on the surface occurs quickly, so that the overall process is limited by the adsorption rate, and (b) the reaction-controlled limit, where adsorption occurs readily so that the overall reaction is limited by the conversion of unlike neighbouring monomers to an AB pair. By analysing the master equation for the probability density of coverage, they determine the rate at which the catalyst becomes 'saturated', i.e. completely covered by only one species. they show that the saturation time is proportional to N, and they also derive the probability distribution for the substrate coverage. Numerical simulations are performed on lattice substrates of finite spatial dimensionality d to test the range of validity of the mean-field approach. They find good agreement between simulations and mean-field theory for d=2 and 3, but not for d=1, suggesting that d=2 is a critical dimensionality for the monomer-monomer process.

http://physics.bu.edu/~redner/pubs/pdf/jpa23p4297.pdf

Saturation transition in a monomer-monomer model of heterogeneous catalysis

The authors investigate the repulsion of random walks (RWs) and self-avoiding walks (SAWs) induced by (a) excluding a single lattice point or (b) excluding all points on a half-line from - infinity to the origin. For SAWs, they use exact enumeration and Monte Carlo methods to study the asymptotic behaviour of the displacement away from the excluded set on three-, four- and five-dimensional hypercubic lattices. When the SAW begins one lattice site away from an excluded point along the x direction, the mean displacement after N steps, (xN) approaches a finite limit, at a power-law rate, as N to infinity . However, the distribution of projected displacements exhibits a residual asymmetry in the asymptotic limit, reflective of a long-range influence of the excluded point. This general pattern of behaviour also occurs for a purely random walk. For a SAW which starts one lattice spacing away from the axis of the excluded half-line, enumeration data suggest that the mean displacement along the axis diverges as N nu , with v=3/4 in two dimensions, but as Nz, with z approximately=0.35, in three dimensions. This unexpected behaviour appears to be corroborated by constant-fugacity Monte Carlo data. However, for the corresponding RW model, both the solution to the diffusion equation and a heuristic argument indicate that (xN) diverges as N1/2 in two dimensions, and as N1/2/ln N in three dimensions.

/pdf/repulsion-of-random-and-self-avoiding-walks-from-excluded-4w6zw1bjzy.pdf

Repulsion of random and self-avoiding walks from excluded points and lines

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