Biomedical subjects

P L Krapivsky

Publications and source records attributed to P L Krapivsky.

At least 19 recordsLinked to original sources

PubMed2006

Alignment of rods and partition of integers.

We study dynamical ordering of rods. In this process, rod alignment via pairwise interactions competes with diffusive wiggling. Under strong diffusion, the system is disordered, but at weak diffusion, the system is ordered. We present an exact steady-state solution for the nonlinear and nonlocal kinetic theory of this process. We find the Fourier transform as a function of the order parameter, and show that Fourier modes decay exponentially with the wave number. We also obtain the order parameter in terms of the diffusion constant. This solution is obtained using iterated partitions of the integer numbers.

Journal Article↗

PubMed2005

"Burnt-bridge" mechanism of molecular motor motion.

Motivated by a biased diffusion of molecular motors with the bias dependent on the state of the substrate, we investigate a random walk on a one-dimensional lattice that contains weak links (called "bridges") which are affected by the walker. Namely, a bridge is destroyed with probability when p the walker crosses it; the walker is not allowed to cross it again and this leads to a directed motion. The velocity of the walker is determined analytically for equidistant bridges. The special case of p = 1 is more tractable--both the velocity and the diffusion constant are calculated for uncorrelated locations of bridges, including periodic and random distributions.

Journal Article↗

PubMed2005

Dynamics of social balance on networks.

We study the evolution of social networks that contain both friendly and unfriendly pairwise links between individual nodes. The network is endowed with dynamics in which the sense of a link in an imbalanced triad--a triangular loop with one or three unfriendly links--is reversed to make the triad balanced. With this dynamics, an infinite network undergoes a dynamic phase transition from a steady state to "paradise"--all links are friendly--as the propensity p for friendly links in an update event passes through 1/2 . A finite network always falls into a socially balanced absorbing state where no imbalanced triads remain. If the additional constraint that the number of imbalanced triads in the network not increase in an update is imposed, then the network quickly reaches a balanced final state.

Adaptation, Physiological↗

PubMed2005

Duplication-divergence model of protein interaction network.

We investigate a very simple model describing the evolution of protein-protein interaction networks via 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 node degree increases algebraically. The lack of self-averaging results in a great diversity of networks grown out of the same initial condition. When less than a half of links are (on average) preserved after divergence, 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. The predicted degree distributions are in a very good agreement with the distributions observed in real protein networks.

Computer Simulation↗

PubMed2005

Statistics of weighted treelike networks.

We study the statistics of growing networks with a tree topology in which each link carries a weight (k(i) k(j))(theta) , where k(i) and k(j) are the node degrees at the end points 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 superlinearly 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 .

Journal Article↗

PubMed2005

Network growth by copying.

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, ultrasmall 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.

Journal Article↗

PubMed2005

Kinetic theory of random graphs: from paths to cycles.

The 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 approximately 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.

Journal Article↗

PubMed2005

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 tail, that is, it is independent of the link weight distribution: n (w) approximately w(-3) as w-->infinity . Results are particularly neat for the exponential link weight distribution when n (w) is algebraic over the entire weight range.

Adaptation, Physiological↗

PubMed2004

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 [n]scales as N(1/3). Moreover, the maximal and the average duration of an outbreak grow as t(*) approximately N(1/3) and [t] approximately ln N, respectively.

Communicable Diseases↗

PubMed2004

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) approximately 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.

Journal Article↗

PubMed2004

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 diagram that represents the partition of the integers.

Algorithms↗

PubMed2003

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 approximately t(-xi) with xi=2d/(d+3) in the reaction-controlled limit in d dimensions, in contrast with the well-known mean-field prediction xi=2d/(d+2).

Journal Article↗

PubMed2003

Exchange-driven growth.

We study a class of growth processes in which clusters evolve via exchange of particles. We show that depending on the rate of exchange there are three possibilities: (I) Growth-clusters grow indefinitely, (II) gelation-all mass is transformed into an infinite gel in a finite time, and (III) instant gelation. In regimes I and II, the cluster size distribution attains a self-similar form. The large size tail of the scaling distribution is Phi(x) approximately exp(-x(2-nu)), where nu is a homogeneity degree of the rate of exchange. At the borderline case nu=2, the distribution exhibits a generic algebraic tail, Phi(x) approximately x(-5). In regime III, the gel nucleates immediately and consumes the entire system. For finite systems, the gelation time vanishes logarithmically, T approximately [lnN](-(nu-2)), in the large system size limit N--> infinity. The theory is applied to coarsening in the infinite range Ising-Kawasaki model and in electrostatically driven granular layers.

Journal Article↗

PubMed2003

Shattering transitions in collision-induced fragmentation.

We investigate the kinetics of nonlinear collision-induced fragmentation. We obtain the fragment mass distribution analytically by utilizing its traveling wave behavior. The system undergoes a shattering transition in which a finite fraction of the mass is lost to infinitesimal fragments (dust). The nature of the shattering transition depends on the fragmentation process. When the larger of the two colliding fragments splits, the transition is discontinuous and the entire mass is transformed into dust at the transition point. When the smaller fragment splits, the transition is continuous with the dust gaining mass steadily on the account of the fragments. At the transition point, the fragment mass distribution diverges algebraically for small masses, c(m) approximately m(-alpha), with alpha=1.201 91 em leader.

Journal Article↗

PubMed2003

Dynamics of majority rule in two-state interacting spin systems.

We introduce a two-state opinion dynamics model where agents evolve by majority rule. In each update, a group of agents is specified whose members then all adopt the local majority state. In the mean-field limit, where a group consists of randomly selected agents, consensus is reached in a time that scales ln(N, where N is the number of agents. On finite-dimensional lattices, where a group is a contiguous cluster, the consensus time fluctuates strongly between realizations and grows as a dimension-dependent power of N. The upper critical dimension appears to be larger than 4. The final opinion always equals that of the initial majority except in one dimension.

Journal Article↗

PubMed2002

Statistics of changes in lead node in connectivity-driven networks.

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 em leader, for growth with no popularity bias.

Journal Article↗

PubMed2002

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.

Journal Article↗

PubMed2002

Scaling, multiscaling, and nontrivial exponents in inelastic collision processes.

We investigate velocity statistics of homogeneous inelastic gases using the Boltzmann equation. Employing an approximate uniform collision rate, we obtain analytic results valid in arbitrary dimension. In the freely evolving case, the velocity distribution is characterized by an algebraic large-velocity tail, P(v,t) approximately v(-sigma). The exponent sigma(d,epsilon), a nontrivial root of an integral equation, varies continuously with the spatial dimension d and the dissipation coefficient epsilon. Although the velocity distribution follows a scaling form, its moments exhibit multiscaling asymptotic behavior. Furthermore, the velocity autocorrelation function decays algebraically with time, A(t)= approximately t(-alpha), with a nonuniversal dissipation-dependent exponent alpha=1/epsilon. In the forced case, the steady state Fourier transform is obtained via a cumulant expansion. Even in this case, velocity correlations develop and the velocity distribution is non-Maxwellian.

Journal Article↗