A solution for the time- and age-dependent connectivity distribution of a growing random network is presented. The built by adding sites that link to earlier with probability A(k) which depends on number preexisting links k site. For homogeneous connection kernels, approximately k(gamma), different behaviors arise gamma<1, gamma>1, gamma = 1. links, N(k), varies as stretched exponential. single site connects nearly all other sites. In borderline case k, power law N(k) k(-nu) found, where...

The organizational development of growing random networks is investigated. These are built by adding nodes successively and linking each to an earlier node degree k with attachment probability A_k. When A_k grows slower than linearly k, the number links, N_k(t), decays faster a power-law in while for single emerges which connects nearly all other nodes. asymptotically linear, N_k(t) tk^{-nu}, nu dependent on details probability, but range 2<nu<infty. combined age distribution shows that old...

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

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 sense a link an imbalanced triad---a triangular loop one or three links---is reversed to make triad balanced. With this dynamics, infinite undergoes dynamic phase transition from steady state ``paradise''---all are friendly---as propensity $p$ for update event passes through $1∕2$. A finite always falls into socially...

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

We study the kinetics of an irreversible monomer-monomer model heterogeneous catalysis. In this model, two reactive species, A and B, adsorb stick to single sites a catalytic substrate. Surface reactions are assumed occur only between dissimilar species that nearest neighbors on The process studied in reaction-controlled limit. map catalysis onto kinetic Ising find dynamics is superposition zero-temperature spin-flip infinite-temperature spin-exchange dynamics. solve analytically determine...

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

We study a simple aggregation model that mimics the clustering of traffic on one-lane roadway. In this model, each ``car'' moves ballistically at its initial velocity until it overtakes preceding car or cluster. After encounter, incident assumes cluster which has just joined. The properties distribution velocities in small-velocity limit control long-time process. For an with power-law tail small velocities, ${\mathit{P}}_{0}$(v)\ensuremath{\sim}${\mathit{v}}^{\mathrm{\ensuremath{\mu}}}$ as...

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

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

We apply macroscopic fluctuation theory to study the diffusion of a tracer in one-dimensional interacting particle system with excluded mutual passage, known as single-file diffusion. In case Brownian point particles hard-core repulsion, we derive cumulant generating function position and its large deviation function. general arbitrary inter-particle interactions, express variance terms collective transport properties, viz. coefficient mobility. Our analysis applies both for fluctuating...

Saturn's rings consist of a huge number water ice particles, with tiny addition rocky material. They form flat disk, as the result an interplay angular momentum conservation and steady loss energy in dissipative inter-particle collisions. For particles size range from few centimeters to meters, power-law distribution radii, $\sim r^{-q}$ $q \approx 3$, has been inferred; for larger sizes, steep cutoff. It suggested that this may arise balance between aggregation fragmentation ring yet...

Viral production from infected cells can occur continuously or in a burst that generally kills the cell. For HIV infection, both modes of have been suggested. Standard viral dynamic models formulated as sets ordinary differential equations not distinguish between these two production, predicted dynamics is identical long produce same total number virions over their lifespan. Here we show stochastic infection yield different early term dynamics. Further, analytically determine probability...

We study relaxation properties of two-body collisions on the mean-field level. show that this process exhibits multiscaling asymptotic behavior as underlying distribution is characterized by an infinite set nontrivial exponents. These nonequilibrium characteristics are found to be closely related steady state system.

The voter model is a simple for coarsening with nonconserved scalar order parameter. We investigate and persistence in the by introducing quantity ${\mathit{P}}_{\mathit{n}}$(t), defined as fraction of voters who changed their opinion n times up to time t. show that ${\mathit{P}}_{\mathit{n}}$(t) exhibits scaling behavior strongly depends on dimension well initial concentrations. Exact results are obtained average number changes, 〈n〉, autocorrelation function,...

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

We study opinion formation in a population of leftists, centrists and rightist. In an interaction between neighbouring agents, centrist leftist can become both or leftists (and similarly for rightist), while rightists do not affect each other. The evolution is controlled by the initial density ρ0. For any spatial dimension system reaches consensus with probability ρ0, 1 − ρ0 final state either extremist consensus, frozen rightists. one dimension, we determine mapping onto spin-1 Ising model...

We investigate the final state of zero-temperature Ising ferromagnets that are endowed with single-spin-flip Glauber dynamics. Surprisingly, ground is generally not reached for zero initial magnetization. In two dimensions, system reaches either a frozen stripe probability approximately 1/3 or 2/3. greater than reaching rapidly vanishes as size increases; instead wanders forever in an isoenergy set metastable states. An external magnetic field changes situation drastically-in dimensions...

We investigate the relaxation of homogeneous Ising ferromagnets on finite lattices with zero-temperature spin-flip dynamics. On square lattice, a frozen two-stripe state is apparently reached approximately 3/10 time, while ground otherwise. The asymptotic characterized by two distinct time scales longer stemming from influence long-lived diagonal stripe defect. In greater than dimensions, probability to reach rapidly vanishes as size increases and system typically ends up wandering forever...

We study the evolution of a system $N$ interacting species which mimics dynamics cyclic food chain. On one-dimensional lattice with $N<5$ species, spatial inhomogeneities develop spontaneously in initially homogeneous systems. The arising patterns form mosaic single-species domains algebraically growing average size, $〈\ensuremath{\ell}(t)〉\ensuremath{\sim}{t}^{\ensuremath{\alpha}}$, where $\ensuremath{\alpha}=\frac{3}{4}(\frac{1}{2}) \mathrm{and} \frac{1}{3}$ for $N=3$ sequential (parallel)...

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

We study the survival of a prey that is hunted by N predators. The predators perform independent random walks on square lattice with V sites and start direct chase whenever appears within their sighting range. caught when predator jumps to site occupied prey. analyze efficacy lazy, minimal-effort evasion strategy according which tries avoid encounters making hop only any its range; otherwise stays still. show if range such lazy equal 1 spacing, at least 3 are needed in order catch lattice....