An Analysis of the Transition Zone Between the Various Scaling Regimes in the Small-World Model

Abstract

We analyse the so-called small-world network model (originally devised by Strogatz and Watts), treating it, among other things, as a case study of non-linear coupled difference or differential equations. We derive a system of evolution equations containing more of the previously neglected (possibly relevant) non-linear terms. As an exact solution of this entangled system of equations is out of question we develop a (as we think, promising) method of enclosing the "exact" solutions for the expected quantities by upper and lower bounds, which represent solutions of a slightly simpler system of differential equation. Furthermore we discuss the relation between difference and differential equations and scrutinize the limits of the spreading idea for random graphs. We then show that there exists in fact a "broad" (with respect to scaling exponents) crossover zone, smoothly interpolating between linear and logarithmic scaling of the diameter or average distance. We are able to corroborate earlier findings in certain regions of phase or parameter space (as e.g. the finite size scaling ansatz) but find also deviations for other choices of the parameters. Our analysis is supplemented by a variety of numerical calculations, which, among other things, quantify the effect of various approximations being made. With the help of our analytical results we manage to calculate another important network characteristic, the (fractal) dimension, and provide numerical values for the case of the small-world network.

Quantum Theory as emergent from an undulatory translocal Sub-Quantum Level

“…It is an interesting coincidence that roughly at the same time such translocal network architectures (usually of a static type, i.e., with a fixed wiring) have been studied in network theory, called small world networks (cf. [16] and [17], see also the literature cited in [35]).…”

“…Thus we exactly get a network type of the above structure which is also called a small-world network (cf. [35]). Furthermore, our networks are of the structurally dynamic type, i.e., also the wiring is (clock-)time dependent.…”

“…We called such a structure a wormhole space (cf. [21], [35]). The above Kuramoto model, which inherits its near/far-order via the (quasi-Euclidean) lattice metric given on Z 2 is a simple example.…”

The Statistical Mechanics of Microscopic Long-Range Bulk-Boundary Dependence in Black-Hole Physics and Holography

“…One should also note that closely related phenomena do occur in the small-world scenario (see e.g. [62] for a rigorous discussion and more references).…”

Foreland determination for containership and general cargo ports in Europe (2007–2011)

“…There has been much more work done on l describing its theoretical relationship with the small world network it characterizes [16,17,18,19,20]. Small-world networks have been analyzed using percolation theory and mean field theory among others to attempt to understand the exact nature of the transition from a "large" to a small world network.…”