arXiv subjects

S. Redner

Publications and source records attributed to S. Redner.

At least 163 records · Page 9Linked to original sources

arXiv2002

Ordering of Random Walks: The Leader and the Laggard

We investigate two complementary problems related to maintaining the relative positions of N random walks on the line: (i) the leader problem, that is, the probability {\cal L}_N(t) that the leftmost particle remains the leftmost as a function of time and (ii) the laggard problem, the probability {\cal R}_N(t) that the rightmost particle never becomes the leftmost. We map these ordering problems onto an equivalent (N-1)-dimensional electrostatic problem. From this construction we obtain a very accurate estimate for {\cal L}_N(t) for N=4, the first case that is not exactly soluble: {\cal L}_4(t) ~ t^{-β_4}, with β_4=0.91342(8). The probability of being the laggard also decays algebraically, {\cal R}_N(t) ~ t^{-γ_N}; we derive γ_2=1/2, γ_3=3/8, and argue that γ_N--> ln N/N$ as N-->oo.

cond-mat.stat-mech↗

arXiv2002

Statistics of Lead Changes in Popularity-Driven Systems

We study statistical properties of the highest degree, or most popular, nodes in growing networks. We show that the number of lead changes increases logarithmically with network size N, independent of the details of the growth mechanism. The probability that the first node retains the lead approaches a finite constant for popularity-driven growth, and decays as N^{-phi}(ln N)^{-1/2}, with phi=0.08607..., for growth with no popularity bias.

cond-mat.stat-mech↗

arXiv2002

Infinite-Order Percolation and Giant Fluctuations in a Protein Interaction Network

We investigate a model protein interaction network whose links represent interactions between individual proteins. This network evolves by the functional duplication of proteins, supplemented by random link addition to account for mutations. When link addition is dominant, an infinite-order percolation transition arises as a function of the addition rate. In the opposite limit of high duplication rate, the network exhibits giant structural fluctuations in different realizations. For biologically-relevant growth rates, the node degree distribution has an algebraic tail with a peculiar rate dependence for the associated exponent.

cond-mat.stat-mech↗

arXiv2002

Finiteness and Fluctuations in Growing Networks

We study the role of finiteness and fluctuations about average quantities for basic structural properties of growing networks. We first determine the exact degree distribution of finite networks by generating function approaches. The resulting distributions exhibit an unusual finite-size scaling behavior and they are also sensitive to the initial conditions. We argue that fluctuations in the number of nodes of degree k become Gaussian for fixed degree as the size of the network diverges. We also characterize the fluctuations between different realizations of the network in terms of higher moments of the degree distribution.

cond-mat.stat-mech↗

arXiv2002

A Statistical Physics Perspective on Web Growth

Approaches from statistical physics are applied to investigate the structure of network models whose growth rules mimic aspects of the evolution of the world-wide web. We first determine the degree distribution of a growing network in which nodes are introduced one at a time and attach to an earlier node of degree k with rate A_ksim k^gamma. Very different behaviors arise for gamma<1, gamma=1, and gamma>1. We also analyze the degree distribution of a heterogeneous network, the joint age-degree distribution, the correlation between degrees of neighboring nodes, as well as global network properties. An extension to directed networks is then presented. By tuning model parameters to reasonable values, we obtain distinct power-law forms for the in-degree and out-degree distributions with exponents that are in good agreement with current data for the web. Finally, a general growth process with independent introduction of nodes and links is investigated. This leads to independently growing sub-networks that may coalesce with other sub-networks. General results for both the size distribution of sub-networks and the degree distribution are obtained.

cs.NI↗

arXiv2002

Synchronization and Coarsening (without SOC) in a Forest-Fire Model

We study the long-time dynamics of a forest-fire model with deterministic tree growth and instantaneous burning of entire forests by stochastic lightning strikes. Asymptotically the system organizes into a coarsening self-similar mosaic of synchronized patches within which trees regrow and burn simultaneously. We show that the average patch length grows linearly with time as t-->oo. The number density of patches of length L, N(L,t), scales as ^{-2}M(L/ ), and within a mean-field rate equation description we find that this scaling function decays as e^{-1/x} for x-->0, and as e^{-x} for x-->oo. In one dimension, we develop an event-driven cluster algorithm to study the asymptotic behavior of large systems. Our numerical results are consistent with mean-field predictions for patch coarsening.

cond-mat.stat-mech↗

arXiv2002

Scaling Theory for Migration-Driven Aggregate Growth

We give a comprehensive rate equation description for the irreversible growth of aggregates by migration from small to large aggregates. For a homogeneous rate K(i;j) at which monomers migrate from aggregates of size i to those of size j, that is, K(ai;aj) ~ a^{lambda} K(i,j), the mean aggregate size grows with time as t^{1/(2-lambda)} for lambda<2. The aggregate size distribution exhibits distinct regimes of behavior which are controlled by the scaling properties of the migration rate from the smallest to the largest aggregates. Our theory applies to diverse phenomena, such as the distribution of city populations, late stage coarsening of non-symmetric binary systems, and models for wealth exchange.

cond-mat.stat-mech↗

arXiv2001

Organization of Growing Random Networks

The organizational development of growing random networks is investigated. These growing networks are built by adding nodes successively and linking each to an earlier node of degree k with attachment probability A_k. When A_k grows slower than linearly with k, the number of nodes with k links, N_k(t), decays faster than a power-law in k, while for A_k growing faster than linearly in k, a single node emerges which connects to nearly all other nodes. When A_k is asymptotically linear, N_k(t) tk^{-nu}, with nu dependent on details of the attachment probability, but in the range 2<nu<infty. The combined age and degree distribution of nodes shows that old nodes typically have a large degree. There is also a significant correlation in the degrees of neighboring nodes, so that nodes of similar degree are more likely to be connected. The size distributions of the in-components and out-components of the network with respect to a given node -- namely, its "descendants" and "ancestors" -- are also determined. The in-component exhibits a robust s^{-2} power-law tail, where s is the component size. The out component has a typical size of order ln t and it provides basic insights about the genealogy of the network.

cond-mat.stat-mech↗

arXiv2001

Degree Distributions of Growing Networks

The in-degree and out-degree distributions of a growing network model are determined. The in-degree is the number of incoming links to a given node (and vice versa for out-degree. The network is built by (i) creation of new nodes which each immediately attach to a pre-existing node, and (ii) creation of new links between pre-existing nodes. This process naturally generates correlated in- and out-degree distributions. When the node and link creation rates are linear functions of node degree, these distributions exhibit distinct power-law forms. By tuning the parameters in these rates to reasonable values, exponents which agree with those of the web graph are obtained.

cond-mat.stat-mech↗

arXiv2001

Dissolution in a field

We study the dissolution of a solid by continuous injection of reactive ``acid'' particles at a single point, with the reactive particles undergoing biased diffusion in the dissolved region. When acid encounters the substrate material, both an acid particle and a unit of the material disappear. We find that the lengths of the dissolved cavity parallel and perpendicular to the bias grow as t^{2/(d+1)} and t^{1/(d+1)}, respectively, in d-dimensions, while the number of reactive particles within the cavity grows as t^{2/(d+1)}. We also obtain the exact density profile of the reactive particles and the relation between this profile and the motion of the dissolution boundary. The extension to variable acid strength is also discussed.

cond-mat.stat-mech↗

arXiv2001

Steady State of an Inhibitory Neural Network

We investigate the dynamics of a neural network where each neuron evolves according to the combined effects of deterministic integrate-and-fire dynamics and purely inhibitory coupling with K randomly-chosen "neighbors". The inhibition reduces the voltage of a given neuron by an amount Delta when one of its neighbors fires. The interplay between the integration and inhibition leads to a steady state which is determined by solving the rate equations for the neuronal voltage distribution. We also study the evolution of a single neuron and find that the mean lifetime between firing events equals 1+K*Delta and that the probability that a neuron has not yet fired decays exponentially with time.

nlin.AO↗

arXiv2001

Freezing in Ising Ferromagnets

We investigate the final state of zero-temperature Ising ferromagnets which are endowed with single-spin flip Glauber dynamics. Surprisingly, the ground state is generally not reached for zero initial magnetization. In two dimensions, the system either reaches a frozen stripe state with probability approximately equal to 1/3 or the ground state with probability approximately equal to 2/3. In greater than two dimensions, the probability to reach the ground state or a frozen state rapidly vanishes as the system size increases and the system wanders forever in an iso-energy set of metastable states. An external magnetic field changes the situation drastically -- in two dimensions the favorable ground state is always reached, while in three dimensions the field must exceed a threshold value to reach the ground state. For small but non-zero temperature, relaxation to the final state first proceeds by the formation of very long-lived metastable states, similar to the zero-temperature case, before equilibrium is reached.

cond-mat.stat-mech↗

arXiv2000

Connectivity of Growing Random Networks

A solution for the time- and age-dependent connectivity distribution of a growing random network is presented. The network is built by adding sites which link to earlier sites with a probability A_k which depends on the number of pre-existing links k to that site. For homogeneous connection kernels, A_k ~ k^gamma, different behaviors arise for gamma<1, gamma>1, and gamma=1. For gamma<1, the number of sites with k links, N_k, varies as stretched exponential. For gamma>1, a single site connects to nearly all other sites. In the borderline case A_k ~ k, the power law N_k ~k^{-nu} is found, where the exponent nu can be tuned to any value in the range 2<nu<infinity.

cond-mat.stat-mech↗

arXiv2000

Infiltration through porous media

We study the kinetics of infiltration in which contaminant particles, which are suspended in a flowing carrier fluid, penetrate a porous medium. The progress of the ``invader'' particles is impeded by their trapping on active ``defender'' sites which are on the surfaces of the medium. As the defenders are used up, the invader penetrates further and ultimately breaks through. We study this process in the regime where the particles are much smaller than the pores so that the permeability change due to trapping is negligible. We develop a family of microscopic models of increasing realism to determine the propagation velocity of the invasion front, as well as the shapes of the invader and defender profiles. The predictions of our model agree qualitatively with experimental results on breakthrough times and the time dependence of the invader concentration at the output. Our results also provide practical guidelines for improving the design of deep bed filters in which infiltration is the primary separation mechanism.

cond-mat.stat-mech↗

arXiv2000

Fate of Zero-Temperature Ising Ferromagnets

We investigate the relaxation of homogeneous Ising ferromagnets on finite lattices with zero-temperature spin-flip dynamics. On the square lattice, a frozen two-stripe state is apparently reached approximately 1/4 of the time, while the ground state is reached otherwise. The asymptotic relaxation is characterized by two distinct time scales, with the longer stemming from the influence of a long-lived diagonal stripe ``defect''. In greater than two dimensions, the probability to reach the ground state rapidly vanishes as the size increases and the system typically ends up wandering forever within an iso-energy set of stochastically ``blinking'' metastable states.

cond-mat.stat-mech↗

arXiv2000

Clogging Time of a Filter

We study the time until a filter becomes clogged due to the trapping of suspended particles as they pass through a porous medium. This trapping progressively impedes and eventually stops the flow of the carrier fluid. We develop a simple description for the pore geometry and the motion of the suspended particles which, together with extreme-value statistics, predicts that the distribution of times until a filter clogs has a power-law long-time tail, with an infinite mean clogging time. These results and its consequences are in accord with simulations on a square lattice porous network.

cond-mat.stat-mech↗

arXiv1999

Fitness versus Longevity in Age-Structured Population Dynamics

We examine the dynamics of an age-structured population model in which the life expectancy of an offspring may be mutated with respect to that of the parent. While the total population of the system always reaches a steady state, the fitness and age characteristics exhibit counter-intuitive behavior as a function of the mutational bias. By analytical and numerical study of the underlying rate equations, we show that if deleterious mutations are favored, the average fitness of the population reaches a steady state, while the average population age is a decreasing function of the overall fitness. When advantageous mutations are favored, the average population fitness grows linearly with time t, while the average age is independent of fitness. For no mutational bias, the average fitness grows as t^{2/3}.

adap-org↗

arXiv1999

Shock-Like Dynamics of Inelastic Gases

We provide a simple physical picture which suggests that the asymptotic dynamics of inelastic gases in one dimension is independent of the degree of inelasticity. Statistical characteristics, including velocity fluctuations and the velocity distribution are identical to those of a perfectly inelastic sticky gas, which in turn is described by the inviscid Burgers equation. Asymptotic predictions of this continuum theory, including the t^{-2/3} temperature decay and the development of discontinuities in the velocity profile, are verified numerically for inelastic gases.

cond-mat.stat-mech↗