Critical Phase in Complex Networks: a Numerical Study

We compare phase transition and critical phenomena of bond percolation on Euclidean lattices, nonamenable graphs, and complex networks. On a Euclidean lattice, percolation shows a phase transition between the nonpercolating phase and percolating phase at the critical point. The critical point is stretched to a finite region, called the critical phase, on nonamenable graphs. To investigate the critical phase, we introduce a fractal exponent, which characterizes a subextensive order of the system. We perform the Monte Carlo simulations for percolation on two nonamenable graphs - the binary tree and the enhanced binary tree. The former shows the nonpercolating phase and the critical phase, whereas the latter shows all three phases. We also examine the possibility of critical phase in complex networks. Our conjecture is that networks with a growth mechanism have only the critical phase and the percolating phase. We study percolation on a stochastically growing network with and without a preferential attachment mechanism, and a deterministically growing network, called the decorated flower, to show that the critical phase appears in those models. We provide a finite-size scaling by using the fractal exponent, which would be a powerful method for numerical analysis of the phase transition involving the critical phase.

Hierarchical scale-free network is fragile against random failure

We investigate site percolation in a hierarchical scale-free network known as the Dorogovtsev- Goltsev-Mendes network. We use the generating function method to show that the percolation threshold is 1, i.e., the system is not in the percolating phase when the occupation probability is less than 1. The present result is contrasted to bond percolation in the same network of which the percolation threshold is zero. We also show that the percolation threshold of intentional attacks is 1. Our results suggest that this hierarchical scale-free network is very fragile against both random failure and intentional attacks. Such a structural defect is common in many hierarchical network models.

Discontinuous Transition of a Multistage Independent Cascade Model on Networks

We propose a multistage version of the independent cascade model, which we call a multistage independent cascade (MIC) model, on networks. This model is parameterized by two probabilities: the probability $T_1$ that a node adopting a fad increases the awareness of a neighboring susceptible node, and the probability $T_2$ that an adopter directly causes a susceptible node to adopt the fad. We formulate a tree approximation for the MIC model on an uncorrelated network with an arbitrary degree distribution $p_k$. Applied on a random regular network with degree $k=6$, this model exhibits a rich phase diagram, including continuous and discontinuous transition lines for fad percolation, and a continuous transition line for the percolation of susceptible nodes. In particular, the percolation transition of fads is discontinuous (continuous) when $T_1$ is larger (smaller) than a certain value. A similar discontinuous transition is also observed in random graphs and scale-free networks. Furthermore, assigning a finite fraction of initial adopters dramatically changes the phase boundaries.

Criticality governed by the stable renormalization fixed point of the Ising model in the hierarchical small-world network

We study the Ising model in a hierarchical small-world network by renormalization group analysis, and find a phase transition between an ordered phase and a critical phase, which is driven by the coupling strength of the shortcut edges. Unlike ordinary phase transitions, which are related to unstable renormalization fixed points (FPs), the singularity in the ordered phase of the present model is governed by the FP that coincides with the stable FP of the ordered phase. The weak stability of the FP yields peculiar criticalities including logarithmic behavior. On the other hand, the critical phase is related to a nontrivial FP, which depends on the coupling strength and is continuously connected to the ordered FP at the transition point. We show that this continuity indicates the existence of a finite correlation-length-like quantity inside the critical phase, which diverges upon approaching the transition point.

Generalized scaling theory for critical phenomena including essential singularity and infinite dimensionality

We propose a generic scaling theory for critical phenomena that includes power-law and essential singularities in finite and infinite dimensional systems. In addition, we clarify its validity by analyzing the Potts model in a simple hierarchical network, where a saddle-node bifurcation of the renormalization-group fixed point governs the essential singularity.

Robustness of correlated networks against propagating attacks

We investigate robustness of correlated networks against propagating attacks modeled by a susceptible-infected-removed model. By Monte-Carlo simulations, we numerically determine the first critical infection rate, above which a global outbreak of disease occurs, and the second critical infection rate, above which disease disintegrates the network. Our result shows that correlated networks are robust compared to the uncorrelated ones, regardless of whether they are assortative or disassortative, when a fraction of infected nodes in an initial state is not too large. For large initial fraction, disassortative network becomes fragile while assortative network holds robustness. This behavior is related to the layered network structure inevitably generated by a rewiring procedure we adopt to realize correlated networks.

Abnormal diffusion of a single vortex in the two dimensional XY model

We study thermal diffusion dynamics of a single vortex in two dimensional XY model. By numerical simulations we find an abnormal diffusion such that the mobility decreases with time $t$ as $1/ln t$. In addition we construct a one dimensional diffusion-like equation to model the dynamics and confirm that it conserves quantitative property of the abnormal diffusion. By analyzing the reduced model, we find that the radius of the collectively moving region with the vortex core grows as $R(t) propto t^{1/2}$. This suggests that the mobility of the vortex is described by dynamical correlation length as $1/ln R(t)$.

Ferromagnetic Ising spin systems on the growing random tree

We analyze the ferromagnetic Ising model on a scale-free tree; the growing random network model with the linear attachment kernel $A_k=k+alpha$ introduced by [Krapivsky et al.: Phys. Rev. Lett. {bf 85} (2000) 4629-4632]. We derive an estimate of the divergent temperature $T_s$ below which the zero-field susceptibility of the system diverges. Our result shows that $T_s$ is related to $alpha$ as $tanh(J/T_s)=alpha/[2(alpha+1)]$, where $J$ is the ferromagnetic interaction. An analysis of exactly solvable limit for the model and numerical calculation support the validity of this estimate.

Spatio-temporal scaling for out-of-equilibrium relaxation dynamics of an elastic manifold in random media: crossover between the Larkin regime and thermally activated regime

We study relaxation dynamics of a three dimensional elastic manifold in random potential from a uniform initial condition by numerically solving the Langevin equation.We observe growth of roughness of the system up to larger wavelengths with time.We analyze structure factor in detail and find a compact scaling ansatz describing two distinct time regimes and crossover between them. We find short time regime corresponding to length scale smaller than the Larkin length $L_c$ is well described by the Larkin model which predicts a power law growth of domain size $L(t)$. Longer time behavior exhibits the random manifold regime with slower growth of $L(t)$.