arXiv subjects

P. L. Krapivsky

Publications and source records attributed to P. L. Krapivsky.

At least 217 records · Page 12Linked to original sources

arXiv2005

Cliques and duplication-divergence network growth

A population of complete subgraphs or cliques in a network evolving via duplication-divergence is considered. We find that a number of cliques of each size scales linearly with the size of the network. We also derive a clique population distribution that is in perfect agreement with both the simulation results and the clique statistic of the protein-protein binding network of the fruit fly. In addition, we show that such features as fat-tail degree distribution, various rates of average degree growth and non-averaging, revealed recently for only the particular case of a completely asymmetric divergence, are present in a general case of arbitrary divergence.

q-bio.MN↗

arXiv2005

Percolation with Multiple Giant Clusters

We study the evolution of percolation with freezing. Specifically, we consider cluster formation via two competing processes: irreversible aggregation and freezing. We find that when the freezing rate exceeds a certain threshold, the percolation transition is suppressed. Below this threshold, the system undergoes a series of percolation transitions with multiple giant clusters ("gels") formed. Giant clusters are not self-averaging as their total number and their sizes fluctuate from realization to realization. The size distribution F_k, of frozen clusters of size k, has a universal tail, F_k ~ k^{-3}. We propose freezing as a practical mechanism for controlling the gel size.

cond-mat.stat-mech↗

arXiv2004

Log-Networks

We introduce a growing network model in which a new node attaches to a randomly-selected node, as well as to all ancestors of the target node. This mechanism produces a sparse, ultra-small network where the average node degree grows logarithmically with network size while the network diameter equals 2. We determine basic geometrical network properties, such as the size dependence of the number of links and the in- and out-degree distributions. We also compare our predictions with real networks where the node degree also grows slowly with time -- the Internet and the citation network of all Physical Review papers.

cond-mat.stat-mech↗

arXiv2004

Statistics of Weighted Networks

We study the statistics of growing networks in which each link carries a weight (k_i k_j)^theta, where k_i and k_j are the node degrees at the endpoints of link ij. Network growth is governed by preferential attachment in which a newly-added node attaches to a node of degree k with rate A_k=k+lambda. For general values of theta and lambda, we compute the total weight of a network as a function of the number of nodes N and the distribution of link weights. Generically, the total weight grows as N for lambda>theta-1, and super-linearly otherwise. The link weight distribution is predicted to have a power law form that is modified by a logarithmic correction for the case lambda=0. We also determine the node strength, defined as the sum of the weights of the links that attach to the node, as function of k. Using known results for degree correlations, we deduce the scaling of the node strength on k and N.

cond-mat.stat-mech↗

arXiv2004

Duplication-divergence model of protein interaction network

We show that the protein-protein interaction networks can be surprisingly well described by a very simple evolution model of duplication and divergence. The model exhibits a remarkably rich behavior depending on a single parameter, the probability to retain a duplicated link during divergence. When this parameter is large, the network growth is not self-averaging and an average vertex degree increases algebraically. The lack of self-averaging results in a great diversity of networks grown out of the same initial condition. For small values of the link retention probability, the growth is self-averaging, the average degree increases very slowly or tends to a constant, and a degree distribution has a power-law tail.

q-bio.MN↗

arXiv2004

Understanding Search Trees via Statistical Physics

We study the random m-ary search tree model (where m stands for the number of branches of a search tree), an important problem for data storage in computer science, using a variety of statistical physics techniques that allow us to obtain exact asymptotic results. In particular, we show that the probability distributions of extreme observables associated with a random search tree such as the height and the balanced height of a tree have a traveling front structure. In addition, the variance of the number of nodes needed to store a data string of a given size N is shown to undergo a striking phase transition at a critical value of the branching ratio m_c=26. We identify the mechanism of this phase transition, show that it is generic and occurs in various other problems as well. New results are obtained when each element of the data string is a D-dimensional vector. We show that this problem also has a phase transition at a critical dimension, D_c= π/\sin^{-1}(1/\sqrt{8})=8.69363...

cond-mat.stat-mech↗

arXiv2004

Kinetic Theory of Random Graphs: from Paths to Cycles

Structural properties of evolving random graphs are investigated. Treating linking as a dynamic aggregation process, rate equations for the distribution of node to node distances (paths) and of cycles are formulated and solved analytically. At the gelation point, the typical length of paths and cycles, l, scales with the component size k as l ~ k^{1/2}. Dynamic and finite-size scaling laws for the behavior at and near the gelation point are obtained. Finite-size scaling laws are verified using numerical simulations.

cond-mat.stat-mech↗

arXiv2004

Weight-driven growing networks

We study growing networks in which each link carries a certain weight (randomly assigned at birth and fixed thereafter). The weight of a node is defined as the sum of the weights of the links attached to the node, and the network grows via the simplest weight-driven rule: A newly-added node is connected to an already existing node with the probability which is proportional to the weight of that node. We show that the node weight distribution n(w) has a universal, that is independent on the link weight distribution, tail: n(w) ~ w^-3 as w->oo. Results are particularly neat for the exponential link weight distribution when n(w) is algebraic over the entire weight range.

cond-mat.dis-nn↗

arXiv2004

Universal Properties of Growing Networks

Networks growing according to the rule that every new node has a probability p_k of being attached to k preexisting nodes, have a universal phase diagram and exhibit power law decays of the distribution of cluster sizes in the non-percolating phase. The percolation transition is continuous but of infinite order and the size of the giant component is infinitely differentiable at the transition (though of course non-analytic). At the transition the average cluster size (of the finite components) is discontinuous.

cond-mat.dis-nn↗

arXiv2004

Dynamical Critical Behaviors of the Ising Spin Chain: Swendsen-Wang and Wolff Algorithms

We study the zero-temperature Ising chain evolving according to the Swendsen-Wang dynamics. We determine analytically the domain length distribution and various ``historical'' characteristics, e.g., the density of unreacted domains is shown to scale with the average domain length as ^{-d} with d=3/2 (for the q-state Potts model, d=1+1/q). We also compute the domain length distribution for the Ising chain endowed with the zero-temperature Wolff dynamics.

cond-mat.stat-mech↗

arXiv2004

A Tractable Complex Network Model based on the Stochastic Mean-field Model of Distance

Much recent research activity has been devoted to empirical study and theoretical models of complex networks (random graphs) with three qualitative features: power-law degree distribution, local clustering of edges, and small diameter. We point out a new (in this context) platform for such models -- the stochastic mean-field model of distance -- and within this platform study a simple two-parameter proportional attachment model. The model is mathematicallly natural, permits a wide variety of explicit calculations, has the desired three qualitative features, and fits the complete range of degree scaling exponents and clustering parameters; in these respects it compares favorably to existing models.

cond-mat.dis-nn↗

arXiv2004

Unicyclic Components in Random Graphs

The distribution of unicyclic components in a random graph is obtained analytically. The number of unicyclic components of a given size approaches a self-similar form in the vicinity of the gelation transition. At the gelation point, this distribution decays algebraically, U_k ~ 1/(4k) for k>>1. As a result, the total number of unicyclic components grows logarithmically with the system size.

cond-mat.stat-mech↗

arXiv2004

Random Geometric Series

Integer sequences where each element is determined by a previous randomly chosen element are investigated analytically. In particular, the random geometric series x_n=2x_p with 0<=p<=n-1 is studied. At large n, the moments grow algebraically, n^beta(s) with beta(s)=2^s-1, while the typical behavior is x_n n^ln 2. The probability distribution is obtained explicitly in terms of the Stirling numbers of the first kind and it approaches a log-normal distribution asymptotically.

cond-mat.stat-mech↗

arXiv2004

Size of Outbreaks Near the Epidemic Threshold

The spread of infectious diseases near the epidemic threshold is investigated. Scaling laws for the size and the duration of outbreaks originating from a single infected individual in a large susceptible population are obtained. The maximal size of an outbreak n_* scales as N^{2/3} with N the population size. This scaling law implies that the average outbreak size scales as N^{1/3}. Moreover, the maximal and the average duration of an outbreak grow as t_* ~ N^{1/3} and ~ ln N, respectively.

q-bio.PE↗

arXiv2004

Dynamics of an Unbounded Interface Between Ordered Phases

We investigate the evolution of a single unbounded interface between ordered phases in two-dimensional Ising ferromagnets that are endowed with single-spin-flip zero-temperature Glauber dynamics. We examine specifically the cases where the interface initially has either one or two corners. In both examples, the interface evolves to a limiting self-similar form. We apply the continuum time-dependent Ginzburg-Landau equation and a microscopic approach to calculate the interface shape. For the single corner system, we also discuss a correspondence between the interface and the Young tableau that represents the partition of the integers.

cond-mat.stat-mech↗

arXiv2003

Finite Size Fluctuations in Interacting Particle Systems

Fluctuations may govern the fate of an interacting particle system even on the mean-field level. This is demonstrated via a three species cyclic trapping reaction with a large, yet finite number of particles, where the final number of particles N_f scales logarithmically with the system size N, N_f ~ ln N. Statistical fluctuations, that become significant as the number of particles diminishes, are responsible for this behavior. This phenomenon underlies a broad range of interacting particle systems including in particular multispecies annihilation processes.

cond-mat.stat-mech↗

arXiv2003

Random Walk with Shrinking Steps

We outline basic properties of a symmetric random walk in one dimension, in which the length of the nth step equals lambda^n, with lambda<1. As the number of steps N-->oo, the probability that the endpoint is at x, P_{lambda}(x;N), approaches a limiting distribution P_{lambda}(x) that has many beautiful features. For lambda<1/2, the support of P_{lambda}(x) is a Cantor set. For 1/2<=lambda<1, there is a countably infinite set of lambda values for which P_{lambda}(x) is singular, while P_{lambda}(x) is smooth for almost all other lambda values. In the most interesting case of lambda=(sqrt{5}-1)/2=g, P_g(x) is riddled with singularities and is strikingly self-similar. The self-similarity is exploited to derive a simple form for the probability measure M(a,b)= int_a^b P_g(x) dx.

physics.ed-ph↗

arXiv2003

Correlations in Ballistic Processes

We investigate a class of reaction processes in which particles move ballistically and react upon colliding. We show that correlations between velocities of colliding particles play a crucial role in the long time behavior. In the reaction-controlled limit when particles undergo mostly elastic collisions and therefore are always near equilibrium, the correlations are accounted analytically. For ballistic aggregation, for instance, the density decays as $n\sim t^{-ξ}$ with $ξ=2d/(d+3)$ in the reaction-controlled limit in $d$ dimensions, in contrast with well-known mean-field prediction $ξ=2d/(d+2)$.

cond-mat.stat-mech↗