We study a general set of models of social network evolution and dynamics. The models consist of both a dynamics on the network and evolution of the network. Links are formed preferentially between "similar" nodes, where the similarity is defined by the particular process taking place on the network. The interplay between the two processes produces phase transitions and hysteresis, as seen using numerical simulations for three specific processes. We obtain analytic results using mean-field approximations, and for a particular case we derive an exact solution for the network. In common with real-world social networks, we find coexistence of high and low connectivity phases and history dependence.
We introduce the concept of 'discrete-time persistence', which deals with zero-crossings of a continuous stochastic process, X(T ), measured at discrete times, T = n∆T . For a Gaussian Markov process with relaxation rate µ, we show that the persistence (no crossing) probability decays as [ρ(a)] n for large n, where a = exp(−µ∆T ), and we compute ρ(a) to high precision. We also define the concept of 'alternating persistence', which corresponds to a < 0. For a > 1, corresponding to motion in an unstable potential (µ < 0), there is a nonzero probability of having no zero-crossings in infinite time, and we show how to calculate it.PACS numbers: 05.70. Ln, 05.40.+j, 81.10.Aj Persistence of a continuous stochastic process has generated much recent interest in a wide variety of nonequilibrium systems including various models of phase ordering kinetics, diffusion, fluctuating interfaces and reactiondiffusion processes [1]. Persistence has also been recently used in fields as diverse as ecology [2] and seismology [3]. Persistence is simply the probability P (t) that a stochastic process x(t) does not change sign up to time t. In most of the systems mentioned above, P (t) ∼ t −θ for large t, where the persistence exponent θ is nontrivial. Apart from various analytical and numerical results, this exponent has also been measured experimentally in systems such as breath figures [4], liquid crystals [5], soap bubbles [6], and more recently in laser-polarized Xe gas using NMR techniques [7].Persistence has also remained a popular subject among applied mathematicians for many decades [8]. They are most interested in the probability of 'no zero crossing' of a Gaussian stationary process (GSP) between times T 1 and T 2 [9]. It is well known that this probability usually decays as ∼ exp(−θT ) for large T = |T 2 − T 1 | where θ is nontrivial [9,8]. The persistence of some of the nonstationary processes mentioned in the previous paragraph such as the diffusion processes, can be mapped to that of a corresponding GSP [10]. This makes the two sets of problems related to each other and the power law exponent in the former problem becomes the inverse decay rate in the latter. Even though θ is, in general, hard to compute analytically, it is very easy to evaluate numerically in most cases. Given this fact, and the combined interest of both statistical physicists and applied mathematicians, much recent effort has been devoted to computing θ numerically to extremely high precision.This raises a natural question: How accurately can one measure θ? Is there a natural limitation and if so, can it be overcome? This issue arises from the following simple observation. All the stochastic processes mentioned above occur in continuous time. However, when one performs numerical simulations or experiments on persistence, one has to discretize time in some way and sample the data only at these discrete time points to check if the process has retained its sign. Due to this discretization, some information is lost. For example, the process may have cro...
Segregation and mixing; phase separation. PACS. 82.70.Dd -Colloids.Abstract. -We report the results of an experimental investigation of segregation in a binary mixture of dry particles subjected to horizontal oscillatory excitation. The thin layer of particles was driven by the stick-slip frictional interaction with the surface of a horizontal tray. As the packing fraction of the mixture was increased the evolution of distinct phases was observed. We identified them as a binary gas, segregation liquid and segregation crystal and provide both microscopic and macroscopic measures to identify their properties. Finally, we draw some analogies between segregation in our granular system and self-assembly in binary colloidal mixtures.
We consider an arbitrary Gaussian stationary process X(T) with known correlator C(T), sampled at discrete times Tn = nDeltaT. The probability that (n+1) consecutive values of X have the same sign decays as Pn approximately exp(-theta(D)Tn). We calculate the discrete persistence exponent theta(D) as a series expansion in the correlator C(DeltaT) up to fourteenth order, and extrapolate to DeltaT = 0 using constrained Padé approximants to obtain the continuum persistence exponent thetas. For the diffusion equation our results are in exceptionally good agreement with recent numerical estimates.
We consider the persistence probability, the occupation-time distribution and the distribution of the number of zero crossings for discrete or (equivalently) discretely sampled Gaussian Stationary Processes (GSPs) of zero mean. We first consider the Ornstein-Uhlenbeck process, finding expressions for the mean and variance of the number of crossings and the 'partial survival' probability. We then elaborate on the correlator expansion developed in an earlier paper (G. C. M. A. Ehrhardt and A. J. Bray, Phys. Rev. Lett. 88, 070602 (2001)) to calculate discretely sampled persistence exponents of GSPs of known correlator by means of a series expansion in the correlator. We apply this method to the processes d n x/dt n = η(t) with n ≥ 3, incorporating an extrapolation of the series to the limit of continuous sampling. We then extend the correlator method to calculate the occupation-time and crossing-number distributions, as well as their partial-survival distributions and the means and variances of the occupation time and number of crossings. We apply these general methods to the d n x/dt n = η(t) processes for n = 1 (random walk), n = 2 (random acceleration) and larger n, and to simple diffusion from random initial conditions in 1-3 dimensions. The results for discrete sampling are extrapolated to the continuum limit where possible.
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