Parongama Sen

Modelling aging characteristics in citation networks

Growing network models with preferential attachment dependent on both age and degree are proposed to simulate certain features of citation network noted in [Redner, arXiv: physics/0407137 (2004)]. In this directed network, a new node gets attached to an older node with the probability ∼K(k)f(t) where the degree and age of the older node are k and t, respectively. Several functional forms of K(k) and f(t) have been considered. The desirable features of the citation network can be reproduced with K(k)∼k-β and f(t)∼exp(αt) with β=2.0 and α=-0.2 and with simple modifications in the growth scheme.

Disorder induced phase transition in kinetic models of opinion dynamics

We propose a model of continuous opinion dynamics, where mutual interactions can be both positive and negative. Different types of distributions for the interactions, all characterized by a single parameter p denoting the fraction of negative interactions, are considered. Results from exact calculation of a discrete version and numerical simulations of the continuous version of the model indicate the existence of a universal continuous phase transition at p=pc below which a consensus is reached. Although the order–disorder transition is analogous to a ferromagnetic–paramagnetic phase transition with comparable critical exponents, the model is characterized by some distinctive features relevant to a social system.

Phase transitions in a network with a range-dependent connection probability.

We consider a one-dimensional network in which the nodes at Euclidean distance l can have long range connections with a probability P(l) approximately l(-delta) in addition to nearest neighbor connections. This system has been shown to exhibit small-world behavior for delta<2, above which its behavior is like a regular lattice. From the study of the clustering coefficients, we show that there is a transition to a random network at delta=1. The finite size scaling analysis of the clustering coefficients obtained from numerical simulations indicates that a continuous phase transition occurs at this point. Using these results, we find that the two transitions occurring in this network can be detected in any dimension by the behavior of a single quantity, the average bond length. The phase transitions in all dimensions are nontrivial in nature.

Small-world phenomena and the statistics of linear polymers

A regular lattice in which the sites can have long-range connections at a distance l with a probabilty P(l) ~ l −δ, in addition to the short-range nearest neighbour connections, shows small-world behaviour for 0 ≤ δ < δc. In the most appropriate physical example of such a system, namely, the linear polymer network, the exponent δ is related to the exponents of the corresponding n-vector model in the n → 0 limit, and its value is less than δc. Still, the polymer networks do not show small-world behaviour. Here, we show that this is due to a (small value) constraint on the number, q, of long-range connections per monomer in the network. In the general δ-q space, we obtain a phase boundary separating regions with and without small-world behaviour, and show that the polymer network falls marginally in the regular lattice region.

Accelerated growth in outgoing links in evolving networks: deterministic versus stochastic picture.

In several real-world networks such as the Internet, World Wide Web, etc., the number of links grow in time in a nonlinear fashion. We consider growing networks in which the number of outgoing links is a nonlinear function of time but new links between older nodes are forbidden. The attachments are made using a preferential attachment scheme. In the deterministic picture, the number of outgoing links m (t) at any time t is taken as N (t)(theta) where N (t) is the number of nodes present at that time. The continuum theory predicts a power-law decay of the degree distribution: P (k) proportional to k-(1-2/ (1-theta ) ), while the degree of the node introduced at time t(i) is given by k(t(i),t)=t(theta)(i) [t/t(i) ]((1+theta)/2) when the network is evolved till time t. Numerical results show a growth in the degree distribution for small k values at any nonzero theta. In the stochastic picture, m (t) is a random variable. As long as is independent of time, the network shows a behavior similar to the Barabási-Albert (BA) model. Different results are obtained when is time dependent, e.g., when m (t) follows a distribution P (m) proportional to m(-lambda). The behavior of P (k) changes significantly as lambda is varied: for lambda>3, the network has a scale-free distribution belonging to the BA class as predicted by the mean field theory; for smaller values of lambda it shows different behavior. Characteristic features of the clustering coefficients in both models have also been discussed.

Phase transitions in a two-parameter model of opinion dynamics with random kinetic exchanges.

Recently, a model of opinion formation with kinetic exchanges has been proposed in which a spontaneous symmetry-breaking transition was reported [M. Lallouache, A. S. Chakrabarti, A. Chakraborti, and B. K. Chakrabarti, Phys. Rev. E 82, 056112 (2010)]. We generalize the model to incorporate two parameters: λ, to represent conviction, and μ, to represent the influencing ability of individuals. A phase boundary given by λ=1-μ/2 is obtained separating the symmetric and symmetry broken phases: The effect of the influencing term enhances the possibility of reaching a consensus in the society. The time scale diverges near the phase boundary in a power-law manner. The order parameter and the condensate also show power-law growth close to the phase boundary albeit with different exponents. The exponents in general change along the phase boundary, indicating a nonuniversality. The relaxation times, however, become constant with increasing system size near the phase boundary, indicating the absence of any diverging length scale. Consistently, the fluctuations remain finite but show strong dependence on the trajectory along which it is estimated.

Model of binary opinion dynamics: Coarsening and effect of disorder.

We propose a model of binary opinion in which the opinion of the individuals changes according to the state of their neighboring domains. If the neighboring domains have opposite opinions then the opinion of the domain with the larger size is followed. Starting from a random configuration, the system evolves to a homogeneous state. The dynamical evolution shows a scaling behavior with the persistence exponent theta approximately 0.235 and dynamic exponent z approximately 1.02 + or - 0.02. Introducing disorder through a parameter called rigidity coefficient rho (probability that people are completely rigid and never change their opinion), the transition to a heterogeneous society at rho=0(+) is obtained. Close to rho=0, the equilibrium values of the dynamic variables show power-law scaling behavior with rho. We also discuss the effect of having both quenched and annealed disorder in the system.

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.

Persistence and dynamics in the ANNNI chain

We investigate both the local and global persistence behaviour in the ANNNI (axial next-nearest-neighbour Ising) model. We find that when the ratio κ of the second neighbour interaction to the first neighbour interaction is less than 1, P(t), the probability of a spin to remain in its original state up to time t shows a stretched exponential decay. For κ > 1, P(t) has an algebraic decay but the exponent is different from that of the nearest-neighbour Ising model. The global persistence behaviour shows similar features. We also conduct some deeper investigations in the dynamics of the ANNNI model and conclude that it has a different dynamical behaviour compared to the nearest-neighbour Ising model.

Phase transitions in an Ising model on a Euclidean network.

A one-dimensional network on which there are long-range bonds at lattice distances l>1 with the probability P(l) proportional to l(-delta) has been taken under consideration. We investigate the critical behavior of the Ising model on such a network where spins interact with these extra neighbors apart from their nearest neighbors for 0<or=delta<2. It is observed that there is a finite temperature phase transition in the entire range. For 0<or=delta<1, finite-size scaling behavior of various quantities are consistent with mean-field exponents while for 1<or=delta<or=2, the exponents depend on delta. The results are discussed in the context of earlier observations on the topology of the underlying network.