Q factor: A measure of competition between the topper and the average in percolation and in self-organized criticality.

We define the Q factor in the percolation problem as the quotient of the size of the largest cluster and the average size of all clusters. As the occupation probability p is increased, the Q factor for the system size L grows systematically to its maximum value Q_{max}(L) at a specific value p_{max}(L) and then gradually decays. Our numerical study of site percolation problems on the square, triangular, and simple cubic lattices exhibits that the asymptotic values of p_{max}, though close, are distinct from the corresponding percolation thresholds of these lattices. We also show, using scaling analysis, that at p_{max} the value of Q_{max}(L) diverges as L^{d} (d denoting the dimension of the lattice) as the system size approaches its asymptotic limit. We further extend this idea to nonequilibrium systems such as the sandpile model of self-organized criticality. Here the Q(ρ,L) factor is the quotient of the size of the largest avalanche and the cumulative average of the sizes of all the avalanches, with ρ the drop density of the driving mechanism. This study was prompted by some observations in sociophysics.

Non-Abelian sandpile automata with height restrictions.

We have studied the properties of a sandpile automata under the constraint of height restriction of sand columns. In this sandpile, an active site transfers a grain to a neighboring site if and only if the height of the sand column at the destination site is less than a preassigned value n_{c}. This sandpile was studied by Dickman et al. [Phys. Rev. E 66, 016111 (2002)1063-651X10.1103/PhysRevE.66.016111] in a conserved system with a fixed number of sand grains. In contrast, we have studied the avalanche dynamics of the driven sandpile under the open boundary conditions. The deterministic dynamics of the Bak, Tang, and Wiesenfeld (BTW) sandpile under the height restriction is found to be non-Abelian. Using numerical results, we argue that the steady states of the sandpile are exactly the recurrent states of the BTW sandpile, but occur with nonuniform probabilities. A detailed analysis of the cluster size distributions indicates that the associated exponent values are likely to be different from those of the BTW sandpile. The other differences include that the drop number distribution decays as a power law, and the largest avalanche size grows as the fourth power of the system size.

Nonstationary but quasisteady states in self-organized criticality.

The notion of self-organized criticality (SOC) was conceived to interpret the spontaneous emergence of long-range correlations in nature. Since then many different models have been introduced to study SOC. All of them have a few common features: externally driven dynamical systems self-organize themselves to nonequilibrium stationary states exhibiting fluctuations of all length scales as the signatures of criticality. In contrast, we have studied here in the framework of the sandpile model a system that has mass inflow but no outflow. There is no boundary, and particles cannot escape from the system by any means. Therefore, there is no current balance, and consequently it is not expected that the system would arrive at a stationary state. In spite of that, it is observed that the bulk of the system self-organizes to a quasisteady state where the grain density is maintained at a nearly constant value. Power law distributed fluctuations of all lengths and time scales have been observed, which are the signatures of criticality. Our detailed computer simulation study gives the set of critical exponents whose values are very close to their counterparts in the original sandpile model. This study indicates that (i) a physical boundary and (ii) the stationary state, though sufficient, may not be the necessary criteria for achieving SOC.

Bond percolation between k separated points on a square lattice.

We consider a percolation process in which k points separated by a distance proportional the system size L simultaneously connect together (k>1), or a single point at the center of a system connects to the boundary (k=1), through adjacent connected points of a single cluster. These processes yield new thresholds p[over ¯]_{ck} defined as the average value of p at which the desired connections first occur. These thresholds not sharp, as the distribution of values of p_{ck} for individual samples remains broad in the limit of L→∞. We study p[over ¯]_{ck} for bond percolation on the square lattice and find that p[over ¯]_{ck} are above the normal percolation threshold p_{c}=1/2 and represent specific supercritical states. The p[over ¯]_{ck} can be related to integrals over powers of the function P_{∞}(p) equal to the probability a point is connected to the infinite cluster; we find numerically from both direct simulations and from measurements of P_{∞}(p) on L×L systems that for L→∞, p[over ¯]_{c1}=0.51755(5), p[over ¯]_{c2}=0.53219(5), p[over ¯]_{c3}=0.54456(5), and p[over ¯]_{c4}=0.55527(5). The percolation thresholds p[over ¯]_{ck} remain the same, even when the k points are randomly selected within the lattice. We show that the finite-size corrections scale as L^{-1/ν_{k}} where ν_{k}=ν/(kß+1), with ß=5/36 and ν=4/3 being the ordinary percolation critical exponents, so that ν_{1}=48/41, ν_{2}=24/23, ν_{3}=16/17, ν_{4}=6/7, etc. We also study three-point correlations in the system and show how for p>p_{c}, the correlation ratio goes to 1 (no net correlation) as L→∞, while at p_{c} it reaches the known value of 1.022.

Branching process in a stochastic extremal model.

We considered a stochastic version of the Bak-Sneppen model (SBSM) of ecological evolution where the number M of sites mutated in a mutation event is restricted to only two. Here the mutation zone consists of only one site and this site is randomly selected from the neighboring sites at every mutation event in an annealed fashion. The critical behavior of the SBSM is found to be the same as the BS model in dimensions d=1 and 2. However on the scale-free graphs the critical fitness value is nonzero even in the thermodynamic limit but the critical behavior is mean-field like. Finally M has been made even smaller than two by probabilistically updating the mutation zone, which also shows the original BS model behavior. We conjecture that a SBSM on any arbitrary graph with any small branching factor greater than unity will lead to a self-organized critical state.

Brittle to quasibrittle transition in a compound fiber bundle.

The brittle to quasibrittle transition has been studied for a compound of two different kinds of fibrous materials, having distinct difference in their breaking strengths under the framework of the fiber bundle model. A random fiber bundle model has been devised with a bimodal distribution of the breaking strengths of the individual fibers. The bimodal distribution is assumed to consist of two symmetrically placed rectangular probability distributions of strengths p and 1-p, each of width d, and separated by a gap 2s. Different properties of the transition have been studied varying these three parameters and using the well-known equal load-sharing dynamics. Our study exhibits a brittle to quasibrittle transition at the critical width d_{c}(s,p)=p(1/2-s)/(1+p) confirmed by our numerical results.

Double transition in a model of oscillating percolation.

Two distinct transition points have been observed in a problem of lattice percolation studied using a system of pulsating disks. Sites on a regular lattice are occupied by circular disks whose radii vary sinusoidally within [0,R_{0}] starting from a random distribution of phase angles. A lattice bond is said to be connected when its two end disks overlap with each other. Depending on the difference of the phase angles of these disks, a bond may be termed as dead or live. While a dead bond can never be connected, a live bond is connected at least once in a complete time period. Two different time scales can be associated with such a system, leading to two transition points. Namely, a percolation transition occurs at R_{0c}=0.908(5) when a spanning cluster of connected bonds emerges in the system. Here, information propagates across the system instantly, i.e., with infinite speed. Secondly, there exists another transition point R_{0}^{*}=0.5907(3) where the giant cluster of live bonds spans the lattice. In this case the information takes finite time to propagate across the system through the dynamical evolution of finite-size clusters. This passage time diverges as R_{0}→R_{0}^{*} from above. Both the transitions exhibit the critical behavior of ordinary percolation transition. The entire scenario is robust with respect to the distribution of frequencies of the individual disks. This study may be relevant in the context of wireless sensor networks.

Colored percolation.

A model called "colored percolation" has been introduced with its infinite number of versions in two dimensions. The sites of a regular lattice are randomly occupied with probability p and are then colored by one of the n distinct colors using uniform probability q=1/n. Denoting different colors by the letters of the Roman alphabet, we have studied different versions of the model like AB,ABC,ABCD,ABCDE,... etc. Here, only those lattice bonds having two different colored atoms at the ends are defined as connected. The percolation threshold p_{c}(n) asymptotically converges to its limiting value of p_{c} as 1/n. The model has been generalized by introducing a preference towards a subset of colors when m out of n colors are selected with probability q/m each and the rest of the colors are selected with probability (1-q)/(n-m). It has been observed that p_{c}(q,m) depends nontrivially on q and has a minimum at q_{min}=m/n. In another generalization the fractions of bonds between similarly and dissimilarly colored atoms have been treated as independent parameters. Phase diagrams in this parameter space have been drawn exhibiting percolating and nonpercolating phases.

Weighted scale-free networks in Euclidean space using local selection rule.

A spatial scale-free network is introduced and studied, whose motivation originated in the growing Internet as well as airport networks. We argue that in these real-world networks a new node necessarily selects one of its neighboring local nodes for connection and is not controlled by preferential attachment as in the Barabási-Albert (BA) model. This observation is mimicked in our model where the nodes pop up at randomly located positions in the Euclidean space and are connected to one end of the nearest link. In spite of this crucial difference it is observed that the leading behavior of our network is like that of the BA model. Defining the link weight as an algebraic power of its Euclidean length, the weight distribution and the nonlinear dependence of the nodal strength on the degree are analytically calculated. It is claimed that a power law decay of the link weights with time ensures such nonlinear behavior. Switching off the Euclidean space from the same model yields a much simpler definition of the BA model where numerical effort grows linearly with N.

Percolation model with an additional source of disorder.

The ranges of transmission of the mobiles in a mobile ad hoc network are not uniform in reality. They are affected by the temperature fluctuation in air, obstruction due to the solid objects, even the humidity difference in the environment, etc. How the varying range of transmission of the individual active elements affects the global connectivity in the network may be an important practical question to ask. Here a model of percolation phenomena, with an additional source of disorder, is introduced for a theoretical understanding of this problem. As in ordinary percolation, sites of a square lattice are occupied randomly with probability p. Each occupied site is then assigned a circular disk of random value R for its radius. A bond is defined to be occupied if and only if the radii R_{1} and R_{2} of the disks centered at the ends satisfy a certain predefined condition. In a very general formulation, one divides the R_{1}-R_{2} plane into two regions by an arbitrary closed curve. One defines a point within one region as representing an occupied bond; otherwise it is a vacant bond. The study of three different rules under this general formulation indicates that the percolation threshold always varies continuously. This threshold has two limiting values, one is p_{c}(sq), the percolation threshold for the ordinary site percolation on the square lattice, and the other is unity. The approach of the percolation threshold to its limiting values are characterized by two exponents. In a special case, all lattice sites are occupied by disks of random radii R∈{0,R_{0}} and a percolation transition is observed with R_{0} as the control variable, similar to the site occupation probability.

Brittle-to-quasibrittle transition in bundles of nonlinear elastic fibers.

Properties of the fiber bundle model have been studied using equal load-sharing dynamics where each fiber obeys a nonlinear stress (s)-strain (x) characteristic function s=G(x) till its breaking threshold. In particular, four different functional forms have been studied: G(x)=e^{αx}, 1+x^{α}, x^{α}, and xe^{αx} where α is a continuously tunable parameter of the model in all cases. Analytical studies, supported by extensive numerical calculations of this model, exhibit a brittle to quasibrittle phase transition at a critical value of α_{c} only in the first two cases. This transition is characterized by the weak power law modulated logarithmic (brittle) and logarithmic (quasibrittle) dependence of the relaxation time on the two sides of the critical point. Moreover, the critical load σ_{c}(α) for the global failure of the bundle depends explicitly on α in all cases. In addition, four more cases have also been studied, where either the nonlinear functional form or the probability distribution of breaking thresholds has been suitably modified. In all these cases similar brittle to quasibrittle transitions have been observed.

Phase transition in a directed traffic flow network.

The generic feature of traffic in a network of flowing electronic data packets is a phase transition from a stationary free-flow phase to a continuously growing congested nonstationary phase. In the most simple network of directed oriented square lattice we have been able to observe all crucial features of such flow systems having nontrivial critical behavior near the critical point of transition. The network here is in the shape of a square lattice and data packets are randomly posted with a rate rho at one side of the lattice. Each packet executes a directed diffusive motion toward the opposite boundary where it is delivered. Packets accumulated at a particular node form a queue and a maximum of m such packets randomly jump out of this node at every time step to its neighbors on a first-in-first-out basis. The phase transition occurs at rho(c) = m. The distribution of travel times through the system is found to have a log-normal behavior and the power spectrum of the load time series shows 1/f-like noise similar to the scenario of Internet traffic.

Sandpile model on a quenched substrate generated by kinetic self-avoiding trails.

Kinetic self-avoiding trails are introduced and used to generate a substrate of randomly quenched flow vectors. A sandpile model is studied on such a substrate with asymmetric toppling matrices where the precise balance between the net outflow of grains from a toppling site, and the total inflow of grains to the same site when all its neighbors topple once, is maintained at all sites. Within numerical accuracy this model behaves in the same way as the multiscaling Bak, Tang, and Wiesenfeld model.

Fiber bundle model with highly disordered breaking thresholds.

We present a study of the fiber bundle model using equal load-sharing dynamics where the breaking thresholds of the fibers are drawn randomly from a power-law distribution of the form p(b)∼b-1 in the range 10-ß to 10ß. Tuning the value of ß continuously over a wide range, the critical behavior of the fiber bundle has been studied both analytically as well as numerically. Our results are: (i) The critical load σc(ß,N) for the bundle of size N approaches its asymptotic value σc(ß) as σc(ß,N)=σc(ß)+AN-1/ν(ß), where σc(ß) has been obtained analytically as σc(ß)=10ß/(2ßeln10) for ß≥ßu=1/(2ln10), and for ß<ßu the weakest fiber failure leads to the catastrophic breakdown of the entire fiber bundle, similar to brittle materials, leading to σ_{c}(ß)=10-ß; (ii) the fraction of broken fibers right before the complete breakdown of the bundle has the form 1-1/(2ßln10); (iii) the distribution D(Δ) of the avalanches of size Δ follows a power-law D(Δ)∼Δ-ξ with ξ=5/2 for Δ≫Δc(ß) and ξ=3/2 for Δ≪Δc(ß), where the crossover avalanche size Δc(ß)=2/(1-e10-2ß)2.

Topological-distance-dependent transition in flocks with binary interactions.

We have studied a flocking model with binary interactions (binary flock), where the velocity of an agent depends on the velocity of only another agent and its own velocity, topped by the angular noise. The other agent is selected as the nth topological neighbor; the specific value of n being a fixed parameter of the problem. On the basis of extensive numerical simulation results, we argue that for n = 1, the phase transition from the ordered to the disordered phase of the flock is a special kind of discontinuous transition. Here, the order parameter does not flip-flop between multiple metastable states. It continues its initial disordered state for a period t(c), then switches over to the ordered state and remains in this state ever after. For n = 2, it is the usual discontinuous transition between two metastable states. Beyond this range, the continuous transitions are observed for n≥3. Such a system of binary flocks has been further studied using the hydrodynamic equations of motion. Linear stability analysis of the homogeneous polarized state shows that such a state is unstable close to the critical point and above some critical speed, which increases as we increase n. The critical noise strengths, which depend on the average correlation between a pair of topological neighbors, are estimated for five different values of n, which match well with their simulated values.

Diffusion-limited friendship network: a model for six degrees of separation.

A dynamic model of a society is studied where each person is an uncorrelated and noninteracting random walker. A dynamical random graph represents the acquaintance network of the society whose nodes are the individuals and links are the pairs of mutual friendships. This network exhibits a different percolationlike phase transition in all dimensions. On introducing simultaneous death and birth rates in the population, we show that the friendship network shows the six degrees of separation for ever after where the precise value of the network diameter depends on the death/birth rate. A susceptible-infected-susceptible-type model of disease spreading shows that this society always remains healthy if the population density is less than certain threshold value.

Scale-free networks from a Hamiltonian dynamics.

Contrary to many recent models of growing networks, we present a model with fixed number of nodes and links, where a dynamics favoring the formation of links between nodes with degree of connectivity as different as possible is introduced. By applying a local rewiring move, the network reaches equilibrium states assuming broad degree distributions, which have a power-law form in an intermediate range of the parameters used. Interestingly, in the same range we find nontrivial hierarchical clustering.

Clustering properties of a generalized critical Euclidean network.

Many real-world networks exhibit a scale-free feature, have a small diameter, and a high clustering tendency. We study the properties of a growing network, which has all these features, in which an incoming node is connected to its ith predecessor of degree k(i) with a link of length l using a probability proportional to k(beta)(i)l(alpha). For alpha>-0.5, the network is scale-free at beta=1 with the degree distribution P(k) proportional to k(-gamma) and gamma=3.0 as in the Barabási-Albert model (alpha=0,beta=1). We find a phase boundary in the alpha-beta plane along which the network is scale-free. Interestingly, we find a scale-free behavior even for beta>1 for alpha<-0.5, where the existence of a different universality class is indicated from the behavior of the degree distribution and the clustering coefficients. The network has a small diameter in the entire scale-free region. The clustering coefficients emulate the behavior of most real networks for increasing negative values of alpha on the phase boundary.

Directed fixed energy sandpile model.

We numerically study the directed version of the fixed energy sandpile. On a closed square lattice, the dynamical evolution of a fixed density of sand grains is studied. The activity of the system shows a continuous phase transition around a critical density. While the deterministic version has the set of nontrivial exponents, the stochastic model is characterized by mean field like exponents.

Modulated scale-free network in Euclidean space.

A random network is grown by introducing at unit rate randomly selected nodes on the Euclidean space. A node is randomly connected to its ith predecessor of degree k(i) with a directed link of length l using a probability proportional to k(i)l(alpha). Our numerical study indicates that the network is scale free for all values of alpha>alpha(c) and the degree distribution decays stretched exponentially for the other values of alpha. The link length distribution follows a power law: D(l) approximately l(delta), where delta is calculated exactly for the whole range of values of alpha.