Glassy behaviour in the ferromagnetic Ising model on a Cayley tree

Dynamics and processing in finite self-similar networks

A common feature of biological networks is the geometrical property of self-similarity. Molecular regulatory networks through to circulatory systems, nervous systems, social systems and ecological trophic networks show self-similar connectivity at multiple scales. We analyse the relationship between topology and signalling in contrasting classes of such topologies. We find that networks differ in their ability to contain or propagate signals between arbitrary nodes in a network depending on whether they possess branching or loop-like features. Networks also differ in how they respond to noise, such that one allows for greater integration at high noise, and this performance is reversed at low noise. Surprisingly, small-world topologies, with diameters logarithmic in system size, have slower dynamical time scales, and may be less integrated (more modular) than networks with longer path lengths. All of these phenomena are essentially mesoscopic, vanishing in the infinite limit but producing strong effects at sizes and time scales relevant to biology.

Ising model on a hyperbolic plane with a boundary

A hyperbolic plane can be modeled by a structure called the enhanced binary tree. We study the ferromagnetic Ising model on top of the enhanced binary tree using the renormalization-group analysis in combination with transfer-matrix calculations. We find a reasonable agreement with Monte Carlo calculations on the transition point, and the resulting critical exponents suggest the mean-field surface critical behavior.

Ferromagnetic Ising spin systems on the growing random tree

We analyze the ferromagnetic Ising model on a scale-free tree; the growing random tree model with the linear attachment kernel A(k) = k + alpha . 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 supports the validity of this estimate.

Phase transition of q-state clock models on heptagonal lattices

We study the q-state clock models on heptagonal lattices assigned on a negatively curved surface. We show that the system exhibits three classes of equilibrium phases; in between ordered and disordered phases, an intermediate phase characterized by a diverging susceptibility with no magnetic order is observed at every q>or=2. The persistence of the third phase for all q is in contrast with the disappearance of the counterpart phase in a planar system for small q, which indicates the significance of nonvanishing surface-volume ratio that is peculiar in the heptagonal lattice. Analytic arguments based on Ginzburg-Landau theory and generalized Cayley trees make clear that the two-stage transition in the present system is attributed to an energy gap of spin-wave excitations and strong boundary-spin contributions. We further demonstrate that boundary effects break the mean-field character in the bulk region, which establishes the consistency with results of clock models on boundary-free hyperbolic lattices.

Ising model on the scale-free network with a Cayley-tree-like structure

We derive an exact expression for the magnetization and the zero-field susceptibility of the Ising model on a random graph with degree distribution P(k) proportional, k-gamma and with a boundary consisting of leaves, that is, vertices whose degree is 1. The system has no magnetization at any finite temperature, and the susceptibility diverges below a certain temperature Ts depending on the exponent gamma. In particular, Ts reaches infinity for gamma<or=4. These results are completely different from those of the case having no boundary, indicating the nontrivial roles of the leaves in the networks.

Dynamic pattern evolution on scale-free networks

A general class of dynamic models on scale-free networks is studied by analytical methods and computer simulations. Each network consists of N vertices and is characterized by its degree distribution, P(k), which represents the probability that a randomly chosen vertex is connected to k nearest neighbors. Each vertex can attain two internal states described by binary variables or Ising-like spins that evolve in time according to local majority rules. Scale-free networks, for which the degree distribution has a power law tail P(k) approximately k(-gamma), are shown to exhibit qualitatively different dynamic behavior for gamma < 5/2 and gamma > 5/2, shedding light on the empirical observation that many real-world networks are scale-free with 2 < gamma < 5/2. For 2 < gamma < 5/2, strongly disordered patterns decay within a finite decay time even in the limit of infinite networks. For gamma > 5/2, on the other hand, this decay time diverges as ln(N) with the network size N. An analogous distinction is found for a variety of more complex models including Hopfield models for associative memory networks. In the latter case, the storage capacity is found, within mean field theory, to be independent of N in the limit of large N for gamma > 5/2 but to grow as N(alpha) with alpha = (5 - 2gamma)/(gamma - 1) for 2 < gamma < 5/2.

Kinetics of phase transformation on a Bethe lattice

A kinetic Ising model is applied to the description of phase transformations on a Bethe lattice. A closed set of kinetic equations for a model with the coordination number q=3 is obtained using a procedure developed in a previous paper. For T close to Tc(T>Tc), where Tc is the phase transition temperature, and zero external field (absence of supersaturation), the rate of phase transformation (RPT) for small deviations from equilibrium is independent of time and tends to zero as (T-Tc). At T=Tc, the RPT depends on time and for large times behaves as t(-1). For T<Tc, we examine the transformation from the initial state with almost all spins "down" to the state with almost all spin "up" after the external field jumped from Bi<0 to Bf>0. The role of different mechanisms responsible for growth (decay), splitting (coagulation), and creation (annihilation) of clusters are examined separately. In all cases there is a critical value Bc of the external field, such that the phase transformation takes place only for Bf>Bc. This result is also obtained from a more simple consideration involving spherical-like clusters on a Bethe lattice. The characteristic time tR at which the polarization becomes larger than zero diverges as (Bf-Bc)(-b) for Bf-->Bc with b=0.47. The RPT has a rapid growth near tR and remains constant for t>tR. The average cluster size (number of spins in a cluster) exhibits a rapid unrestricted growth at a time td approximately tR which indicates the creation of infinite clusters. The only exception to the latter behavior occurs when the kinetics is dominated by cluster growth and decay processes. In this case, the average cluster size remains finite during the transformation process. In contrast to the classical theory, the present approach does not separate the processes of creation of clusters of critical size (nucleation) and of their growth, both being accounted for by the kinetic equations employed.

Freezing in Ising ferromagnets

We investigate the final state of zero-temperature Ising ferromagnets that are endowed with single-spin-flip Glauber dynamics. Surprisingly, the ground state is generally not reached for zero initial magnetization. In two dimensions, the system reaches either a frozen stripe state with probability approximately 1/3 or the ground state with probability approximately 2/3. In greater than two dimensions, the probability of reaching the ground state or a frozen state rapidly vanishes as the system size increases; instead the system wanders forever in an isoenergy set of metastable states. An external magnetic field changes the situation drastically-in two dimensions the favorable ground state is always reached, while in three dimensions the field must exceed a threshold value to reach the ground state. For small but nonzero temperature, relaxation to the final state proceeds first by formation of very long-lived metastable states, as in the zero-temperature case, before equilibrium is reached.

Freezing in random graph ferromagnets

Using T=0 Monte Carlo and simulated annealing simulation, we study the energy relaxation of ferromagnetic Ising and Potts models on random graphs. In addition to the expected exponential decay to a zero energy ground state, a range of connectivities for which there is power law relaxation and freezing to a metastable state is found. For some connectivities this freezing persists even using simulated annealing to find the ground state. The freezing is caused by dynamic frustration in the graphs, and is a feature of the local search nature of the Monte Carlo dynamics used. The implications of the freezing on agent-based complex system models are briefly considered.