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S. Redner

Publications and source records attributed to S. Redner.

At least 73 records · Page 4Linked to original sources
arXiv2015

Optimal strategy to capture a skittish lamb wandering near a precipice

We study the splitting probabilities for a one-dimensional Brownian motion in a cage whose two boundaries move at constant speeds $c_1$ and $c_2$. This configuration corresponds to the capture of a diffusing, but skittish lamb, with an approaching shepherd on the left and a precipice on the right. We derive compact expressions for these splitting probabilities when the cage is expanding. We also obtain the time-dependent first-passage probability to the left boundary, as well as the splitting probability to this boundary, when the cage is either expanding or contracting. The boundary motions have a non-trivial impact on the splitting probabilities, leading to multiple regimes of behavior that depend on the expansion or contraction speed of the cage. In particular, the probability to capture the lamb is maximized when the shepherd moves at a non-zero optimal speed if the initial lamb position and the ratio between the two boundary speeds satisfy certain conditions.

cond-mat.stat-mech↗
arXiv2015

Mortality, Redundancy, and Diversity in Stochastic Search

We investigate a stochastic search process in one dimension under the competing roles of mortality, redundancy, and diversity of the searchers. This picture represents a toy model for the fertilization of an oocyte by sperm. A population of $N$ independent and mortal diffusing searchers all start at $x=L$ and attempt to reach the target at $x=0$. When mortality is irrelevant, the search time scales as $τ_D/\ln N$ for $\ln N\gg 1$, where $τ_D\sim L^2/D$ is the diffusive time scale. Conversely, when the mortality rate $μ$ of the searchers is sufficiently large, the search time scales as $\sqrt{τ_D/μ}$, independent of $N$. When searchers have distinct and high mortalities, a subpopulation with a non-trivial optimal diffusivity are most likely to reach the target. We also discuss the effect of chemotaxis on the search time and its fluctuations.

cond-mat.stat-mech↗
arXiv2015

Safe Leads and Lead Changes in Competitive Team Sports

We investigate the time evolution of lead changes within individual games of competitive team sports. Exploiting ideas from the theory of random walks, the number of lead changes within a single game follows a Gaussian distribution. We show that the probability that the last lead change and the time of the largest lead size are governed by the same arcsine law, a bimodal distribution that diverges at the start and at the end of the game. We also determine the probability that a given lead is "safe" as a function of its size $L$ and game time $t$. Our predictions generally agree with comprehensive data on more than 1.25 million scoring events in roughly 40,000 games across four professional or semi-professional team sports, and are more accurate than popular heuristics currently used in sports analytics.

physics.data-an↗
arXiv2014

Depletion-Controlled Starvation of a Diffusing Forager

We study the starvation of a lattice random walker in which each site initially contains one food unit and the walker can travel $\mathcal{S}$ steps without food before starving. When the walker encounters food, the food is completely eaten, and the walker can again travel $\mathcal{S}$ steps without food before starving. When the walker hits an empty site, the time until the walker starves decreases by 1. In spatial dimension $d=1$, the average lifetime of the walker $<τ>\propto \mathcal{S}$, while for $d > 2$, $<τ>\simeq\exp(\mathcal{S}^ω)$, with $ω\to 1$ as $d\to\infty$. In the marginal case of $d=2$, $<τ>\propto \mathcal{S}^z$, with $z\approx 2$. Long-lived walks explore a highly ramified region so they always remains close to sources of food and the distribution of distinct sites visited does not obey single-parameter scaling.

cond-mat.stat-mech↗
arXiv2014

Emergence of Clustering in an Acquaintance Model without Homophily

We introduce an agent-based acquaintance model in which social links are created by processes in which there is no explicit homophily. In spite of the homogeneous nature of the social interactions, highly-clustered social networks can arise. The crucial feature of our model is that of variable transitive interactions. Namely, when an agent introduces two unconnected friends, the rate at which a connection actually occurs between them depends on the number of their mutual acquaintances. As this transitive interaction rate is varied, the social network undergoes a dramatic clustering transition. Close to the transition, the network consists of a collection of well-defined communities. As a function of time, the network can also undergo an \emph{incomplete} gelation transition, in which the gel, or giant cluster, does not constitute the entire network, even at infinite time. Some of the clustering properties of our model also arise, but in a more gradual manner, in Facebook networks. Finally, we discuss a more realistic variant of our original model in which there is a soft cutoff in the rate of transitive interactions. With this variant, one can construct network realizations that quantitatively match Facebook networks.

physics.soc-ph↗
arXiv2014

Gradual Diffusive Capture: Slow Death by Many Mosquito Bites

We study the dynamics of a single diffusing particle (a "man") with diffusivity $D_M$ that is attacked by another diffusing particle (a "mosquito") with fixed diffusivity $D_m$. Each time the mosquito meets and bites the man, the diffusivity of the man is reduced by a fixed amount, while the diffusivity of the mosquito is unchanged. The mosquito is also displaced by a small distance $\pm a$ with respect to the man after each encounter. The man is defined as dead when $D_M$ reaches zero. At the moment when the man dies, his probability distribution of displacements $x$ is given by a Cauchy form, which asymptotically decays as $x^{-2}$, while the distribution of times $t$ when the man dies asymptotically decays as $t^{-3/2}$, which has the same form as the one-dimensional first-passage probability.

cond-mat.stat-mech↗
arXiv2014

Large fluctuations in diffusion-controlled absorption

Suppose that $N_0$ independently diffusing particles, each with diffusivity $D$, are initially released at $x=\ell>0$ on the semi-infinite interval $0\leq x<\infty$ with an absorber at $x=0$. We determine the probability ${\cal P}(N)$ that $N$ particles survive until time $t=T$. We also employ macroscopic fluctuation theory to find the most likely history of the system, conditional on there being exactly $N$ survivors at time $t=T$. Depending on the basic parameter $\ell/\sqrt{4DT}$, very different histories can contribute to the extreme cases of $N=N_0$ (all particles survive) and $N=0$ (no survivors). For large values of $\ell/\sqrt{4DT}$, the leading contribution to ${\cal P}(N=0)$ comes from an effective point-like quasiparticle that contains all the $N_0$ particles and moves ballistically toward the absorber until absorption occurs.

cond-mat.stat-mech↗
arXiv2014

Choice-Driven Phase Transition in Complex Networks

We investigate choice-driven network growth. In this model, nodes are added one by one according to the following procedure: for each addition event a set of target nodes is selected, each according to linear preferential attachment, and a new node attaches to the target with the highest degree. Depending on precise details of the attachment rule, the resulting networks has three possible outcomes: (i) a non-universal power-law degree distribution; (ii) a single macroscopic hub (a node whose degree is of the order of N, the number of network nodes), while the remainder of the nodes comprises a non-universal power-law degree distribution; (iii) a degree distribution that decays as (k ln k)^{-2} at the transition between cases (i) and (ii). These properties are robust when attachment occurs to the highest-degree node from at least two targets. When attachment is made to a target whose degree is not the highest, the degree distribution has the ultra-narrow double-exponential form exp(-const. x e^k), from which the largest degree grows only as ln(ln N).

cond-mat.stat-mech↗
arXiv2014

Highly Dispersed Networks Generated by Enhanced Redirection

We analyze growing networks that are built by enhanced redirection. Nodes are sequentially added and each incoming node attaches to a randomly chosen 'target' node with probability 1-r, or to the parent of the target node with probability r. When the redirection probability r is an increasing function of the degree of the parent node, with r-->1 as the parent degree diverges, networks grown via this enhanced redirection mechanism exhibit unusual properties, including: (i) multiple macrohubs---nodes with degrees proportional to the number of network nodes N; (ii) non-extensivity of the degree distribution in which the number of nodes of degree k, N_k, scales as N^{nu-1}/k^{nu}, with 1 =4(ln2)-2=0.77258...

cond-mat.stat-mech↗
arXiv2013

Fate of 2D Kinetic Ising Ferromagnets and Critical Percolation Crossing Probabilities

We present evidence for a deep connection between the zero-temperature coarsening of the two-dimensional kinetic Ising model (KIM) and critical continuum percolation. In addition to reaching the ground state, the KIM can also fall into a variety of topologically distinct metastable stripe states. The probability to reach a stripe state that winds a times horizontally and b times vertically on a square lattice with periodic boundary conditions equals the corresponding exactly-solved critical percolation crossing probability P_{a,b} for a spanning path with winding numbers a and b.

cond-mat.stat-mech↗
arXiv2013

Highly Dispersed Networks

We introduce a new class of networks that grow by enhanced redirection. Nodes are introduced sequentially, and each either attaches to a randomly chosen target node with probability 1-r or to the ancestor of the target with probability r, where r an increasing function of the degree of the ancestor. This mechanism leads to highly-dispersed networks with unusual properties: (i) existence of multiple macrohubs---nodes whose degree is a finite fraction of the total number of network nodes N, (ii) lack of self averaging, and (iii) anomalous scaling, in which N_k, the number of nodes of degree k scales as N_k N^{nu-1}/k^{nu}, with 1<nu<2.

cond-mat.stat-mech↗
arXiv2013

Distinct Degrees and Their Distribution in Complex Networks

We investigate a variety of statistical properties associated with the number of distinct degrees that exist in a typical network for various classes of networks. For a single realization of a network with N nodes that is drawn from an ensemble in which the number of nodes of degree k has an algebraic tail, N_k ~ N/k^nu for k>>1, the number of distinct degrees grows as N^{1/nu}. Such an algebraic growth is also observed in scientific citation data. We also determine the N dependence of statistical quantities associated with the sparse, large-k range of the degree distribution, such as the location of the first hole (where N_k=0), the last doublet (two consecutive occupied degrees), triplet, dimer (N_k=2), trimer, etc.

cond-mat.stat-mech↗
arXiv2013

Zero-Temperature Coarsening in the 2d Potts Model

We study the fate of the 2d kinetic q-state Potts model after a sudden quench to zero temperature. Both ground states and complicated static states are reached with non-zero probabilities. These outcomes resemble those found in the quench of the 2d Ising model; however, the variety of static states in the q-state Potts model (with q>=3) is much richer than in the Ising model, where static states are either ground or stripe states. Another possibility is that the system gets trapped on a set of equal-energy blinker states where a subset of spins can flip ad infinitum; these states are similar to those found in the quench of the 3d Ising model. The evolution towards the final energy is also unusual---at long times, sudden and massive energy drops may occur that are accompanied by macroscopic reordering of the domain structure. This indeterminacy in the zero-temperature quench of the kinetic Potts model is at odds with basic predictions from the theory of phase-ordering kinetics. We also propose a continuum description of coarsening with more than two equivalent ground states. The resulting time-dependent Ginzburg-Landau equations reproduce the complex cluster patterns that arise in the quench of the kinetic Potts model.

cond-mat.stat-mech↗
arXiv2013

Sublinear but Never Superlinear Preferential Attachment by Local Network Growth

We investigate a class of network growth rules that are based on a redirection algorithm wherein new nodes are added to a network by linking to a randomly chosen target node with some probability 1-r or linking to the parent node of the target node with probability r. For fixed 0<r<1, the redirection algorithm is equivalent to linear preferential attachment. We show that when r is a decaying function of the degree of the parent of the initial target, the redirection algorithm produces sublinear preferential attachment network growth. We also argue that no local redirection algorithm can produce superlinear preferential attachment.

cond-mat.stat-mech↗
arXiv2012

Survival of the Scarcer

We investigate extinction dynamics in the paradigmatic model of two competing species A and B that reproduce (A-->2A, B-->2B), self-regulate by annihilation (2A-->0, 2B-->0), and compete (A+B-->A, A+B-->B). For a finite system that is in the well-mixed limit, a quasi-stationary state arises which describes coexistence of the two species. Because of discrete noise, both species eventually become extinct in time that is exponentially long in the quasi-stationary population size. For a sizable range of asymmetries in the growth and competition rates, the paradoxical situation arises in which the numerically disadvantaged species according to the deterministic rate equations survives much longer.

cond-mat.stat-mech↗
arXiv2012

Rounding Effects in Record Statistics

We analyze record-breaking events in time series of continuous random variables that are subsequently discretized by rounding down to integer multiples of a discretization scale $Δ>0$. Rounding leads to ties of an existing record, thereby reducing the number of new records. For an infinite number of random variables that are drawn from distributions with a finite upper limit, the number of discrete records is finite, while for distributions with a thinner than exponential upper tail, fewer discrete records arise compared to continuous variables. In the latter case the record sequence becomes highly regular at long times.

physics.data-an↗
arXiv2012

Randomness in Competitions

We study the effects of randomness on competitions based on an elementary random process in which there is a finite probability that a weaker team upsets a stronger team. We apply this model to sports leagues and sports tournaments, and compare the theoretical results with empirical data. Our model shows that single-elimination tournaments are efficient but unfair: the number of games is proportional to the number of teams N, but the probability that the weakest team wins decays only algebraically with N. In contrast, leagues, where every team plays every other team, are fair but inefficient: the top $\sqrt{N}$ of teams remain in contention for the championship, while the probability that the weakest team becomes champion is exponentially small. We also propose a gradual elimination schedule that consists of a preliminary round and a championship round. Initially, teams play a small number of preliminary games, and subsequently, a few teams qualify for the championship round. This algorithm is fair and efficient: the best team wins with a high probability and the number of games scales as $N^{9/5}$, whereas traditional leagues require N^3 games to fairly determine a champion.

physics.soc-ph↗