Phase transition in the Ising model on a small-world network with distance-dependent interactions

Quantifying the connectivity of a network: the network correlation function method

Networks are useful for describing systems of interacting objects, where the nodes represent the objects and the edges represent the interactions between them. The applications include chemical and metabolic systems, food webs as well as social networks. Lately, it was found that many of these networks display some common topological features, such as high clustering, small average path length (small-world networks), and a power-law degree distribution (scale-free networks). The topological features of a network are commonly related to the network's functionality. However, the topology alone does not account for the nature of the interactions in the network and their strength. Here, we present a method for evaluating the correlations between pairs of nodes in the network. These correlations depend both on the topology and on the functionality of the network. A network with high connectivity displays strong correlations between its interacting nodes and thus features small-world functionality. We quantify the correlations between all pairs of nodes in the network, and express them as matrix elements in the correlation matrix. From this information, one can plot the correlation function for the network and to extract the correlation length. The connectivity of a network is then defined as the ratio between this correlation length and the average path length of the network. Using this method, we distinguish between a topological small world and a functional small world, where the latter is characterized by long-range correlations and high connectivity. Clearly, networks that share the same topology may have different connectivities, based on the nature and strength of their interactions. The method is demonstrated on metabolic networks, but can be readily generalized to other types of networks.

Phase transition of a one-dimensional Ising model with distance-dependent connections

The critical behavior of the Ising model on a one-dimensional network, which has long-range connections at distances l>1 with the probability theta(l) approximately l(-m), is studied by using Monte Carlo simulations. Through analyzing the Ising model on networks with different m values, this paper discusses the impact of the global correlation, which decays with the increase of m, on the phase transition of the Ising model. Adding the analysis of the finite-size scaling of the order parameter [M], it is observed that in the whole range of 0<m<2, a finite-temperature transition exists, and the critical exponents show consistence with mean-field values, which indicates a mean-field nature of the phase transition.

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.

Inverted Berezinskii-Kosterlitz-Thouless singularity and high-temperature algebraic order in an Ising model on a scale-free hierarchical-lattice small-world network

We have obtained exact results for the Ising model on a hierarchical lattice incorporating three key features characterizing many real-world networks--a scale-free degree distribution, a high clustering coefficient, and the small-world effect. By varying the probability p of long-range bonds, the entire spectrum from an unclustered, non-small-world network to a highly clustered, small-world system is studied. Using the self-similar structure of the network, we obtain analytic expressions for the degree distribution P(k) and clustering coefficient C for all p, as well as the average path length l for p = 0 and 1. The ferromagnetic Ising model on this network is studied through an exact renormalization-group transformation of the quenched bond probability distribution, using up to 562,500 renormalized probability bins to represent the distribution. For p < 0.494, we find power-law critical behavior of the magnetization and susceptibility, with critical exponents continuously varying with p, and exponential decay of correlations away from Tc. For p > or = 0.494, in fact where the network exhibits small-world character, the critical behavior radically changes: We find a highly unusual phase transition, namely an inverted Berezinskii-Kosterlitz-Thouless singularity, between a low-temperature phase with nonzero magnetization and finite correlation length and a high-temperature phase with zero magnetization and infinite correlation length, with power-law decay of correlations throughout the phase. Approaching Tc from below, the magnetization and the susceptibility, respectively, exhibit the singularities of exp(-C/square root of Tc - T) and exp(D/square root of Tc - T), with C and D positive constants. With long-range bond strengths decaying with distance, we see a phase transition with power-law critical singularities for all p, and evaluate an unusually narrow critical region and important corrections to power-law behavior that depend on the exponent characterizing the decay of long-range interactions.

Synchronization landscapes in small-world-connected computer networks

Motivated by a synchronization problem in distributed computing we studied a simple growth model on regular and small-world networks, embedded in one and two dimensions. We find that the synchronization landscape (corresponding to the progress of the individual processors) exhibits Kardar-Parisi-Zhang-like kinetic roughening on regular networks with short-range communication links. Although the processors, on average, progress at a nonzero rate, their spread (the width of the synchronization landscape) diverges with the number of nodes (desynchronized state) hindering efficient data management. When random communication links are added on top of the one and two-dimensional regular networks (resulting in a small-world network), large fluctuations in the synchronization landscape are suppressed and the width approaches a finite value in the large system-size limit (synchronized state). In the resulting synchronization scheme, the processors make close-to-uniform progress with a nonzero rate without global intervention. We obtain our results by "simulating the simulations," based on the exact algorithmic rules, supported by coarse-grained arguments.

Performance of networks of artificial neurons: the role of clustering

The performance of the Hopfield neural network model is numerically studied on various complex networks, such as the Watts-Strogatz network, the Barabási-Albert network, and the neuronal network of Caenorhabditis elegans. Through the use of a systematic way of controlling the clustering coefficient, with the degree of each neuron kept unchanged, we find that the networks with the lower clustering exhibit much better performance. The results are discussed in the practical viewpoint of application, and the biological implications are also suggested.