The zero temperature quenching dynamics of the ferromagnetic Ising model on a densely connected s... more The zero temperature quenching dynamics of the ferromagnetic Ising model on a densely connected small world network is studied where long range bonds are added randomly with a finite probability p. We find that in contrast to the sparsely connected networks and random graph, there is no freezing and an initial random configuration of the spins reaches the equilibrium configuration within a very few Monte Carlo time steps in the thermodynamic limit for any p = 0. The residual energy and the number of spins flipped at any time shows an exponential relaxation to equilibrium. The persistence probability is also studied and it shows a saturation within a few time steps, the saturation value being 0.5 in the thermodynamic limit. These results are explained in the light of the topological properties of the network which is highly clustered and has a novel small world behaviour.
We propose a model of continuous opinion dynamics, where mutual interactions can be both positive... more 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 = p c 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.
We consider a one-dimensional network in which the nodes at Euclidean distance l can have long ra... more We consider a one-dimensional network in which the nodes at Euclidean distance l can have long range connections with a probabilty P (l) ∼ l -δ in addition to nearest neighbour connections. This system has been shown to exhibit small world behaviour for δ < 2 above which its behaviour is like a regular lattice. From the study of the clustering coefficients, we show that there is a transition to a random network at δ = 1. The finite size scaling analysis of the clustering coefficients obtained from numerical simulations indicate 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 behaviour of a single quantity, the average bond length. The phase transitions in all dimensions are non-trivial in nature.
We consider the ordering dynamics of the Ising model on a square lattice where an additional fixe... more We consider the ordering dynamics of the Ising model on a square lattice where an additional fixed number of bonds connect any two sites chosen randomly. The total number of shortcuts added is controlled by two parameters p and α. The structural properties of the network are investigated which show that that the small world behaviour is obtained along the line α = ln(N/2p) ln N , which separates regions with ultra small world like behaviour and short ranged lattice like behaviour. We obtain a rich phase diagram in the pα plane showing the existence of different types of active and absorbing states to which Ising model evolves and their boundaries.
We study by Monte Carlo simulations the short-time exponent θ in an antiferromagnetic Ising syste... more We study by Monte Carlo simulations the short-time exponent θ in an antiferromagnetic Ising system for which the magnetisation is conserved but the sublattice magnetisation (which is the order parameter in this case) is not. This system belongs to the dynamic class of model C. We use nearest neighbour Kawasaki dynamics so that the magnetisation is conserved locally. We find that in three dimensions θ is independent of the conserved magnetisation. This is in agreement with the available theoretical studies, but in disagreement with previous simulation studies with global conservation algorithm. However, we agree with both these studies regarding the result θC = θA. We also find that in two dimensions, θC = θA.
We consider a one dimensional Euclidean network which is grown using a preferential attachment. H... more We consider a one dimensional Euclidean network which is grown using a preferential attachment. Here the jth incoming node gets attached to the ith existing node with the probability Πi ∝ kil α ij , where lij is the Euclidean distance between them and ki the degree of the ith node. This network is known to have a static phase transition point at αc ≃ 0.5. On this network, we employ three different searching strategies based on degrees or distances or both, where the possibility of termination of search chains is allowed. A detailed analysis shows that these strategies are significantly affected by the presence of the static critical point. The distributions of the search path lengths and the success rates are also estimated and compared for the different strategies. These distributions appear to be marginally affected by the static phase transition.
We consider a discrete time quantum walker in one dimension, where at each step, the step length ... more We consider a discrete time quantum walker in one dimension, where at each step, the step length ℓ is chosen from a distribution P (ℓ) ∝ ℓ -δ-1 with ℓ ≤ ℓ max . We evaluate the probability f (x, t) that the walker is at position x at time t and its first two moments. As expected, the disorder effectively localizes the walk even for large values of δ. Asymptotically, x 2 ∝ t 3/2 and x ∝ t 1/2 independent of δ and ℓ, both finite. The scaled distribution f (x, t)t 1/2 plotted versus x/t 1/2 shows a data collapse for x/t < α(δ, ℓ max ) ∼ O(1) indicating the existence of a universal scaling function. The scaling function is shown to have a crossover behaviour at δ = δ * ≈ 4.0 beyond which the results are independent of ℓ max . We also calculate the von Neumann entropy of entanglement which gives a larger asymptotic value compared to the quantum walk with unique step length even for large δ, with negligible dependence on the initial condition.
We study the probability distribution Q(n, t) of n(t), the fraction of spins unflipped till time ... more We study the probability distribution Q(n, t) of n(t), the fraction of spins unflipped till time t, in a Ising chain with ferromagnetic interactions. The distribution shows a peak at n = nmax and in general is non-Gaussian and asymmetric in nature. However for n > nmax it shows a Gaussian decay. A data collapse can be obtained when Q(n, t)/L α versus (n -nmax)L β is plotted with α ∼ 0.45 and β ∼ 0.6. Interestingly, nmax(t) shows a different behaviour compared to n(t) = P (t), the persistence probability which follows the well-known behaviour P (t) ∼ t -θ . A quantitative estimate of the asymmetry and non-Gaussian nature of Q(n, t) is made by calculating its skewness and kurtosis.
We explore the effect of interplay of interfacial noise and curvature driven dynamics in a binary... more We explore the effect of interplay of interfacial noise and curvature driven dynamics in a binary spin system. An appropriate model is the generalised two dimensional voter model proposed earlier (J. Phys. A: Math. Gen. 26, 2317), where the flipping probability of a spin depends on the state of its neighbours and is given in terms of two parameters x and y. x = 0.5, y = 1 corresponds to the conventional voter model which is purely interfacial noise driven while x = 1 and y = 1 corresponds to the Ising model, where coarsening is fully curvature driven. The coarsening phenomena for 0.5 < x < 1 keeping y = 1 is studied in detail. The dynamical behaviour of the relevant quantities show characteristic differences from both x = 0.5 and 1. The most remarkable result is the existence of two time scales for x ≥ xc where xc ≈ 0.7. On the other hand, we have studied the exit probability which shows Ising like behaviour with an universal exponent for any value of x > 0.5; the effect of x appears in altering the value of the parameter occurring in the scaling function only.
We consider the critical dynamics of a system with an n-component non-conserved order parameter c... more We consider the critical dynamics of a system with an n-component non-conserved order parameter coupled to a conserved field with long range diffusion. An exponent σ characterizes the long range transport, σ = 2 being the known locally conserved case. With renormalisation group calculations done upto one loop order, several regions are found with different values of the dynamic exponent z in the σn plane. For n < 4, there are three regimes, I: nonuniversal, σ dependent z, II: universal with z depending on n and III': conservation law irrelevant, z being equal to that in the nonconserved case. The known locally conserved case belongs to regions I and II.
The dynamical evolution of a recently introduced one dimensional model in (henceforth referred to... more The dynamical evolution of a recently introduced one dimensional model in (henceforth referred to as model I), has been made stochastic by introducing a parameter β such that β = 0 corresponds to the Ising model and β → ∞ to the original model I. The equilibrium behaviour for any value of β is identical: a homogeneous state. We argue, from the behaviour of the dynamical exponent z, that for any β = 0, the system belongs to the dynamical class of model I indicating a dynamic phase transition at β = 0. On the other hand, the persistence probabilities in a system of L spins saturate at a value Psat(β, L) = (β/L) α f (β), where α remains constant for all β = 0 supporting the existence of the dynamic phase transition at β = 0. The scaling function f (β) shows a crossover behaviour with f (β) = constant for β << 1 and f (β) ∝ β -α for β >> 1.
Philosophical Transactions of the Royal Society A, Apr 11, 2022
We propose two different schemes of realizing a virtual walk corresponding to a kinetic exchange ... more We propose two different schemes of realizing a virtual walk corresponding to a kinetic exchange model of opinion dynamics. The walks are either Markovian or non-Markovian in nature. The opinion dynamics model is characterized by a parameter p which drives an order disorder transition at a critical value pc. The distribution S(X, t) of the displacements X from the origin of the walkers is computed at different times. Below pc, two time scales associated with a crossover behavior in time are detected, which diverge in a power law manner at criticality with different exponent values. S(X, t) also carries the signature of the phase transition as it changes its form at pc. The walks show the features of a biased random walk below pc, and above pc, the walks are like unbiased random walks. The bias vanishes in a power law manner at pc and the width of the resulting Gaussian function shows a discontinuity. Some of the features of the walks are argued to be comparable to the critical quantities associated with the mean field Ising model, to which class the opinion dynamics model belongs. The results for the Markovian and non-Markovian walks are almost identical which is justified by considering the different fluxes. We compare the present results with some earlier similar studies.
We consider a growing network in which an incoming node gets attached to the i th existing node w... more We consider a growing network in which an incoming node gets attached to the i th existing node with the probability Πi ∝ ki β τ α i , where ki is the degree of the i th node and τi its present age. The phase diagram in the αβ plane is obtained. The network shows scale-free behaviour, i.e., the degree distribution P (k) ∼ k -γ with γ = 3 only along a line in this plane. Small world property, on the other hand, exists over a large region in the phase diagram.
We investigate the dynamical fixed points of the zero temperature Glauber dynamics in Ising-like ... more We investigate the dynamical fixed points of the zero temperature Glauber dynamics in Ising-like models. The stability analysis of the fixed points in the mean field calculation shows the existence of an exponent that depends on the coordination number z in the Ising model. For the generalised voter model, a phase diagram is obtained based on this study. Numerical results for the Ising model for both the mean field case and short ranged models on lattices with different values of z are also obtained. A related study is the behaviour of the exit probability E(x0), defined as the probability that a configuration ends up with all spins up starting with x0 fraction of up spins. An interesting result is E(x0) = x0 in the mean field approximation when z = 2, which is consistent with the conserved magnetisation in the system. For larger values of z, E(x0) shows the usual finite size dependent non linear behaviour both in the mean field model and in Ising model with nearest neighbour interaction on different two dimensional lattices. For such a behaviour, a data collapse of E(x0) is obtained using y = (x 0 -xc) xc L 1/ν as the scaling variable and f (y) = 1+tanh(λy) 2 appears as the scaling function. The universality of the exponent and the scaling factor is investigated. We first consider the Ising model in the mean field approximation. The master equation for the variable x(t), the fraction of up spins at time t is set up after calculating the spin flip probabilities. The system evolves
We consider navigation or search schemes on networks which have a degree distribution of the form... more We consider navigation or search schemes on networks which have a degree distribution of the form P (k) ∝ exp(-k γ ). In addition, the linking probability is taken to be dependent on social distances and is governed by a parameter λ. The searches are realistic in the sense that not all search chains can be completed. An estimate of µ = ρ/s d , where ρ is the success rate and s d the dynamic path length, shows that for a network of N nodes, µ ∝ N -δ in general. Dynamic small world effect, i.e., δ ≃ 0 is shown to exist in a restricted region of the λγ plane.
Journal of Statistical Mechanics: Theory and Experiment, Apr 17, 2007
We consider navigation or search schemes on networks which are realistic in the sense that not al... more We consider navigation or search schemes on networks which are realistic in the sense that not all search chains can be completed. We show that the quantity µ = ρ/s d , where s d is the average dynamic shortest distance and ρ the success rate of completion of a search, is a consistent measure for the quality of a search strategy. Taking the example of realistic searches on scale-free networks, we find that µ scales with the system size N as N -δ , where δ decreases as the searching strategy is improved. This measure is also shown to be sensitive to the distintinguishing characteristics of networks. In this new approach, a dynamic small world (DSW) effect is said to exist when δ ≈ 0. We show that such a DSW indeed exists in social networks in which the linking probability is dependent on social distances.
We consider the Susceptible-Infected-Recovered (SIR) epidemic model on a Euclidean network in one... more We consider the Susceptible-Infected-Recovered (SIR) epidemic model on a Euclidean network in one dimension in which nodes at a distance l are connected with probability P (l) ∝ l -δ in addition to nearest neighbors. The topology of the network changes as δ is varied and its effect on the SIR model is studied. R(t), the recovered fraction of population up to time t, and τ , the total duration of the epidemic are calculated for different values of the infection probability q and δ. A threshold behavior is observed for all δ up to δ ≈ 2.0; above the threshold value q = qc, the saturation value Rsat attains a finite value. Both Rsat and τ show scaling behavior in a finite system of size N ; Rsat ∼ N -β/ν g1[(qqc)N 1/ν ] and τ ∼ N µ/ν g2[(qqc)N 1/ν ]. qc is constant for 0 ≤ δ < 1 and increases with δ for 1 < δ 2. Mean field behavior is seen up to δ ≈ 1.3; weak dependence on δ is observed beyond this value of δ. The distribution of the outbreak sizes is also estimated and found to be unimodal for q < qc and bimodal for q > qc. The results are compared to static percolation phenomena and also to mean field results for finite systems. Discussions on the properties of the Euclidean network are made in the light of the present results.
A national voting population, when segmented into groups like, for example, different states, can... more A national voting population, when segmented into groups like, for example, different states, can yield a counter-intuitive scenario where the winner may not necessarily get the most number of total votes. A recent example is the 2016 presidential election in the US. We model the situation by using interacting opinion dynamics models and look at the effect of coarse graining near the critical points where the spatial fluctuations are high. We establish that the sole effect of coarse graining, which mimics the 'winner takes all' electoral college system in the US, can give rise to finite probabilities of such events of minority candidate winning even in the large size limit near the critical point. The overall probabilities of victory of the minority candidate can be predicted from the models which indicates that one may expect more instances of minority candidate winning in the future.
We report exact numerical diagonalization results of the infinite-range Ising spin glass in a tra... more We report exact numerical diagonalization results of the infinite-range Ising spin glass in a transverse field Γ at zero temperature. Eigenvalues and eigenvectors are determined for various strengths of Γ and for system sizes N ≤ 16. We obtain the moments of the distribution of the spin-glass order parameter, the spin-glass susceptibility and the mass gap at different values of Γ. The disorder averaging is done typically over 1000 configurations. Our finite size scaling analysis indicates a spin glass transition at Γ c ≃ 1.5. Our estimates for the exponents at the transition are in agreement with those known from other approaches. For the dynamic exponent, we get z = 2.1 ± 0.1 which is in contradiction with a recent estimate (z = 4). Our cumulant analysis indicates the existence of a replica symmetric spin glass phase for Γ < Γ c .
The zero temperature quenching dynamics of the ferromagnetic Ising model on a densely connected s... more The zero temperature quenching dynamics of the ferromagnetic Ising model on a densely connected small world network is studied where long range bonds are added randomly with a finite probability p. We find that in contrast to the sparsely connected networks and random graph, there is no freezing and an initial random configuration of the spins reaches the equilibrium configuration within a very few Monte Carlo time steps in the thermodynamic limit for any p = 0. The residual energy and the number of spins flipped at any time shows an exponential relaxation to equilibrium. The persistence probability is also studied and it shows a saturation within a few time steps, the saturation value being 0.5 in the thermodynamic limit. These results are explained in the light of the topological properties of the network which is highly clustered and has a novel small world behaviour.
We propose a model of continuous opinion dynamics, where mutual interactions can be both positive... more 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 = p c 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.
We consider a one-dimensional network in which the nodes at Euclidean distance l can have long ra... more We consider a one-dimensional network in which the nodes at Euclidean distance l can have long range connections with a probabilty P (l) ∼ l -δ in addition to nearest neighbour connections. This system has been shown to exhibit small world behaviour for δ < 2 above which its behaviour is like a regular lattice. From the study of the clustering coefficients, we show that there is a transition to a random network at δ = 1. The finite size scaling analysis of the clustering coefficients obtained from numerical simulations indicate 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 behaviour of a single quantity, the average bond length. The phase transitions in all dimensions are non-trivial in nature.
We consider the ordering dynamics of the Ising model on a square lattice where an additional fixe... more We consider the ordering dynamics of the Ising model on a square lattice where an additional fixed number of bonds connect any two sites chosen randomly. The total number of shortcuts added is controlled by two parameters p and α. The structural properties of the network are investigated which show that that the small world behaviour is obtained along the line α = ln(N/2p) ln N , which separates regions with ultra small world like behaviour and short ranged lattice like behaviour. We obtain a rich phase diagram in the pα plane showing the existence of different types of active and absorbing states to which Ising model evolves and their boundaries.
We study by Monte Carlo simulations the short-time exponent θ in an antiferromagnetic Ising syste... more We study by Monte Carlo simulations the short-time exponent θ in an antiferromagnetic Ising system for which the magnetisation is conserved but the sublattice magnetisation (which is the order parameter in this case) is not. This system belongs to the dynamic class of model C. We use nearest neighbour Kawasaki dynamics so that the magnetisation is conserved locally. We find that in three dimensions θ is independent of the conserved magnetisation. This is in agreement with the available theoretical studies, but in disagreement with previous simulation studies with global conservation algorithm. However, we agree with both these studies regarding the result θC = θA. We also find that in two dimensions, θC = θA.
We consider a one dimensional Euclidean network which is grown using a preferential attachment. H... more We consider a one dimensional Euclidean network which is grown using a preferential attachment. Here the jth incoming node gets attached to the ith existing node with the probability Πi ∝ kil α ij , where lij is the Euclidean distance between them and ki the degree of the ith node. This network is known to have a static phase transition point at αc ≃ 0.5. On this network, we employ three different searching strategies based on degrees or distances or both, where the possibility of termination of search chains is allowed. A detailed analysis shows that these strategies are significantly affected by the presence of the static critical point. The distributions of the search path lengths and the success rates are also estimated and compared for the different strategies. These distributions appear to be marginally affected by the static phase transition.
We consider a discrete time quantum walker in one dimension, where at each step, the step length ... more We consider a discrete time quantum walker in one dimension, where at each step, the step length ℓ is chosen from a distribution P (ℓ) ∝ ℓ -δ-1 with ℓ ≤ ℓ max . We evaluate the probability f (x, t) that the walker is at position x at time t and its first two moments. As expected, the disorder effectively localizes the walk even for large values of δ. Asymptotically, x 2 ∝ t 3/2 and x ∝ t 1/2 independent of δ and ℓ, both finite. The scaled distribution f (x, t)t 1/2 plotted versus x/t 1/2 shows a data collapse for x/t < α(δ, ℓ max ) ∼ O(1) indicating the existence of a universal scaling function. The scaling function is shown to have a crossover behaviour at δ = δ * ≈ 4.0 beyond which the results are independent of ℓ max . We also calculate the von Neumann entropy of entanglement which gives a larger asymptotic value compared to the quantum walk with unique step length even for large δ, with negligible dependence on the initial condition.
We study the probability distribution Q(n, t) of n(t), the fraction of spins unflipped till time ... more We study the probability distribution Q(n, t) of n(t), the fraction of spins unflipped till time t, in a Ising chain with ferromagnetic interactions. The distribution shows a peak at n = nmax and in general is non-Gaussian and asymmetric in nature. However for n > nmax it shows a Gaussian decay. A data collapse can be obtained when Q(n, t)/L α versus (n -nmax)L β is plotted with α ∼ 0.45 and β ∼ 0.6. Interestingly, nmax(t) shows a different behaviour compared to n(t) = P (t), the persistence probability which follows the well-known behaviour P (t) ∼ t -θ . A quantitative estimate of the asymmetry and non-Gaussian nature of Q(n, t) is made by calculating its skewness and kurtosis.
We explore the effect of interplay of interfacial noise and curvature driven dynamics in a binary... more We explore the effect of interplay of interfacial noise and curvature driven dynamics in a binary spin system. An appropriate model is the generalised two dimensional voter model proposed earlier (J. Phys. A: Math. Gen. 26, 2317), where the flipping probability of a spin depends on the state of its neighbours and is given in terms of two parameters x and y. x = 0.5, y = 1 corresponds to the conventional voter model which is purely interfacial noise driven while x = 1 and y = 1 corresponds to the Ising model, where coarsening is fully curvature driven. The coarsening phenomena for 0.5 < x < 1 keeping y = 1 is studied in detail. The dynamical behaviour of the relevant quantities show characteristic differences from both x = 0.5 and 1. The most remarkable result is the existence of two time scales for x ≥ xc where xc ≈ 0.7. On the other hand, we have studied the exit probability which shows Ising like behaviour with an universal exponent for any value of x > 0.5; the effect of x appears in altering the value of the parameter occurring in the scaling function only.
We consider the critical dynamics of a system with an n-component non-conserved order parameter c... more We consider the critical dynamics of a system with an n-component non-conserved order parameter coupled to a conserved field with long range diffusion. An exponent σ characterizes the long range transport, σ = 2 being the known locally conserved case. With renormalisation group calculations done upto one loop order, several regions are found with different values of the dynamic exponent z in the σn plane. For n < 4, there are three regimes, I: nonuniversal, σ dependent z, II: universal with z depending on n and III': conservation law irrelevant, z being equal to that in the nonconserved case. The known locally conserved case belongs to regions I and II.
The dynamical evolution of a recently introduced one dimensional model in (henceforth referred to... more The dynamical evolution of a recently introduced one dimensional model in (henceforth referred to as model I), has been made stochastic by introducing a parameter β such that β = 0 corresponds to the Ising model and β → ∞ to the original model I. The equilibrium behaviour for any value of β is identical: a homogeneous state. We argue, from the behaviour of the dynamical exponent z, that for any β = 0, the system belongs to the dynamical class of model I indicating a dynamic phase transition at β = 0. On the other hand, the persistence probabilities in a system of L spins saturate at a value Psat(β, L) = (β/L) α f (β), where α remains constant for all β = 0 supporting the existence of the dynamic phase transition at β = 0. The scaling function f (β) shows a crossover behaviour with f (β) = constant for β << 1 and f (β) ∝ β -α for β >> 1.
Philosophical Transactions of the Royal Society A, Apr 11, 2022
We propose two different schemes of realizing a virtual walk corresponding to a kinetic exchange ... more We propose two different schemes of realizing a virtual walk corresponding to a kinetic exchange model of opinion dynamics. The walks are either Markovian or non-Markovian in nature. The opinion dynamics model is characterized by a parameter p which drives an order disorder transition at a critical value pc. The distribution S(X, t) of the displacements X from the origin of the walkers is computed at different times. Below pc, two time scales associated with a crossover behavior in time are detected, which diverge in a power law manner at criticality with different exponent values. S(X, t) also carries the signature of the phase transition as it changes its form at pc. The walks show the features of a biased random walk below pc, and above pc, the walks are like unbiased random walks. The bias vanishes in a power law manner at pc and the width of the resulting Gaussian function shows a discontinuity. Some of the features of the walks are argued to be comparable to the critical quantities associated with the mean field Ising model, to which class the opinion dynamics model belongs. The results for the Markovian and non-Markovian walks are almost identical which is justified by considering the different fluxes. We compare the present results with some earlier similar studies.
We consider a growing network in which an incoming node gets attached to the i th existing node w... more We consider a growing network in which an incoming node gets attached to the i th existing node with the probability Πi ∝ ki β τ α i , where ki is the degree of the i th node and τi its present age. The phase diagram in the αβ plane is obtained. The network shows scale-free behaviour, i.e., the degree distribution P (k) ∼ k -γ with γ = 3 only along a line in this plane. Small world property, on the other hand, exists over a large region in the phase diagram.
We investigate the dynamical fixed points of the zero temperature Glauber dynamics in Ising-like ... more We investigate the dynamical fixed points of the zero temperature Glauber dynamics in Ising-like models. The stability analysis of the fixed points in the mean field calculation shows the existence of an exponent that depends on the coordination number z in the Ising model. For the generalised voter model, a phase diagram is obtained based on this study. Numerical results for the Ising model for both the mean field case and short ranged models on lattices with different values of z are also obtained. A related study is the behaviour of the exit probability E(x0), defined as the probability that a configuration ends up with all spins up starting with x0 fraction of up spins. An interesting result is E(x0) = x0 in the mean field approximation when z = 2, which is consistent with the conserved magnetisation in the system. For larger values of z, E(x0) shows the usual finite size dependent non linear behaviour both in the mean field model and in Ising model with nearest neighbour interaction on different two dimensional lattices. For such a behaviour, a data collapse of E(x0) is obtained using y = (x 0 -xc) xc L 1/ν as the scaling variable and f (y) = 1+tanh(λy) 2 appears as the scaling function. The universality of the exponent and the scaling factor is investigated. We first consider the Ising model in the mean field approximation. The master equation for the variable x(t), the fraction of up spins at time t is set up after calculating the spin flip probabilities. The system evolves
We consider navigation or search schemes on networks which have a degree distribution of the form... more We consider navigation or search schemes on networks which have a degree distribution of the form P (k) ∝ exp(-k γ ). In addition, the linking probability is taken to be dependent on social distances and is governed by a parameter λ. The searches are realistic in the sense that not all search chains can be completed. An estimate of µ = ρ/s d , where ρ is the success rate and s d the dynamic path length, shows that for a network of N nodes, µ ∝ N -δ in general. Dynamic small world effect, i.e., δ ≃ 0 is shown to exist in a restricted region of the λγ plane.
Journal of Statistical Mechanics: Theory and Experiment, Apr 17, 2007
We consider navigation or search schemes on networks which are realistic in the sense that not al... more We consider navigation or search schemes on networks which are realistic in the sense that not all search chains can be completed. We show that the quantity µ = ρ/s d , where s d is the average dynamic shortest distance and ρ the success rate of completion of a search, is a consistent measure for the quality of a search strategy. Taking the example of realistic searches on scale-free networks, we find that µ scales with the system size N as N -δ , where δ decreases as the searching strategy is improved. This measure is also shown to be sensitive to the distintinguishing characteristics of networks. In this new approach, a dynamic small world (DSW) effect is said to exist when δ ≈ 0. We show that such a DSW indeed exists in social networks in which the linking probability is dependent on social distances.
We consider the Susceptible-Infected-Recovered (SIR) epidemic model on a Euclidean network in one... more We consider the Susceptible-Infected-Recovered (SIR) epidemic model on a Euclidean network in one dimension in which nodes at a distance l are connected with probability P (l) ∝ l -δ in addition to nearest neighbors. The topology of the network changes as δ is varied and its effect on the SIR model is studied. R(t), the recovered fraction of population up to time t, and τ , the total duration of the epidemic are calculated for different values of the infection probability q and δ. A threshold behavior is observed for all δ up to δ ≈ 2.0; above the threshold value q = qc, the saturation value Rsat attains a finite value. Both Rsat and τ show scaling behavior in a finite system of size N ; Rsat ∼ N -β/ν g1[(qqc)N 1/ν ] and τ ∼ N µ/ν g2[(qqc)N 1/ν ]. qc is constant for 0 ≤ δ < 1 and increases with δ for 1 < δ 2. Mean field behavior is seen up to δ ≈ 1.3; weak dependence on δ is observed beyond this value of δ. The distribution of the outbreak sizes is also estimated and found to be unimodal for q < qc and bimodal for q > qc. The results are compared to static percolation phenomena and also to mean field results for finite systems. Discussions on the properties of the Euclidean network are made in the light of the present results.
A national voting population, when segmented into groups like, for example, different states, can... more A national voting population, when segmented into groups like, for example, different states, can yield a counter-intuitive scenario where the winner may not necessarily get the most number of total votes. A recent example is the 2016 presidential election in the US. We model the situation by using interacting opinion dynamics models and look at the effect of coarse graining near the critical points where the spatial fluctuations are high. We establish that the sole effect of coarse graining, which mimics the 'winner takes all' electoral college system in the US, can give rise to finite probabilities of such events of minority candidate winning even in the large size limit near the critical point. The overall probabilities of victory of the minority candidate can be predicted from the models which indicates that one may expect more instances of minority candidate winning in the future.
We report exact numerical diagonalization results of the infinite-range Ising spin glass in a tra... more We report exact numerical diagonalization results of the infinite-range Ising spin glass in a transverse field Γ at zero temperature. Eigenvalues and eigenvectors are determined for various strengths of Γ and for system sizes N ≤ 16. We obtain the moments of the distribution of the spin-glass order parameter, the spin-glass susceptibility and the mass gap at different values of Γ. The disorder averaging is done typically over 1000 configurations. Our finite size scaling analysis indicates a spin glass transition at Γ c ≃ 1.5. Our estimates for the exponents at the transition are in agreement with those known from other approaches. For the dynamic exponent, we get z = 2.1 ± 0.1 which is in contradiction with a recent estimate (z = 4). Our cumulant analysis indicates the existence of a replica symmetric spin glass phase for Γ < Γ c .
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Papers by Parongama Sen