Mean-Field and Anomalous Behavior on a Small-World Network

Abstract

We use scaling results to identify the crossover to mean-field behavior of equilibrium statistical mechanics models on a variant of the small world network. The results are generalizable to a widerange of equilibrium systems. Anomalous scaling is found in the width of the mean-field region, as well as in the mean-field amplitudes. Finally, we consider non-equilibrium processes.The appropriate description for many complex realworld systems is as a network [1], a general connection of nodes and vertices which need not have the structure of a regular lattice. The small-world network model combines both long-range and short-range aspects, and interpolates between regular lattices and random graphs. This model [2], in which a regular lattice is modified by either randomly "re-wiring" links or else by simply randomly adding long-range links, has become a standard model of real-world networks. It incorporates some notion of locality, as most of the links remain the same as that of the original underlying lattice. Yet it also includes the "small world effect", that the average path length between sites on the network scales only as the logarithm of the network size.Cooperative behavior on networks is a major topic of interest. Studies of equilibrium statistical on these systems have shown rich behavior [3][4][5][6]. Work [7,8] on networks with long tails in the probability distribution of the coordination number has found alternatives to mean-field behavior. We will instead study the behavior on smallworld systems. The technique will be generically applicable to a wide-range of problems. We show the existence of mean-field behavior, albeit with anomalous exponents describing the width of the critical region and various mean-field amplitudes. These anomalous exponents can complicate the interpretation of numerical data. Finally, we will consider a non-equilibrium case, describing the relaxation to a stationary state via a branching process.The presence of quenched randomness makes the small-world model difficult to treat analytically. We instead consider a different, but related model, which lacks quenched randomness, considering a variety of equilibrium and non-equilibrium models. We again start with a regular lattice of V sites in d dimensions. Rather than adding long-range links with probability p, we give each site of the lattice a weak coupling, of order p/V , to every other site in the lattice [9]. We will refer to this as the long-range model. We will find that many results can be obtained on this system by combining mean-field with standard renormalization group techniques. Later, we will compare this model to the standard small world model; we will argue that in many cases the presence of quenched randomness in the small world model is irrelevant, and thus scaling results for the small world model can be obtained from the long-range model. It is interesting to compare to a related mean-field solution of path lengths on a small work network [10]. Finally, we will show the long-range model provides an upper bound ...

Critical behavior of ising models with random long-range (small-world) interactions

“…In [17] it is argued that this behavior will also be seen as the number of SW bonds becomes small, i.e. when z=4+p→4 for our square-lattice model.…”

“…There is also a prediction [17] that in the N→∞ limit the order parameter should scale for T <T c as…”

“…Such graphs have been used, for example, to improve scalability of parallel computer algorithms [6][7][8][9][10]. Furthermore, the critical behavior of models of materials, such as the Ising model, Heisenberg model, and random-walker models have been studied on SW graphs [11][12][13][14][15][16][17][18][19][20][21]. The result of these studies is that models on SW graphs exhibit mean-field scaling, namely they have an effective dimension at or above the upper critical dimension of the model (which for the Ising model without disorder is d=4).…”

The smallest of all worlds: Pollination networks

“…This logarithmic scaling of /lS is often termed the ''small-world effect'' (e.g. Hastings, 2003). In fact, /lS of a network is of the order of the logarithm of its size (Watts and Strogatz, 1998) (// l pollinators SS ¼ 1.7 and log /AS ¼ 1.9; //l plants SS ¼ 1.5 and log /PS ¼ 1.5).…”

Recurrent epidemics in small world networks

“…occurs (Dammer & Hinrichsen 2003). In the presence of random long-range interactions on the complex network the transitions become mean-field with an anomalous asymptotic region (Hastings 2003). In addition, finite size effects broaden the transition region considerably.…”