Random Electrical Networks Research Articles

Percolation concepts in solid state ionics

Structural disorder is an inherent property of solid materials, which can support a macroscopic ionic current. Many transport phenomena in these solid ionic conductors appear to be related to concepts from percolation theory. We demonstrate this for three classes of materials, namely (i) dispersed ionic conductors, which show conductance properties that can be related to random electrical networks, (ii) ion-doped network glasses, whose concentration-dependent diffusion properties are accessible by critical path analysis, and (iii) polymer ionic conductors. For the latter we discuss Monte Carlo simulations which indicate the applicability of dynamic bond percolation theory.

Random Walk on Periodic Trees

Following Lyons (1990, Random Walks and Percolation on Trees) we define a periodic tree, restate its branching number and consider a biased random walk on it. In the case of a transient walk, we describe the walk-invariant random periodic tree and calculate the asymptotic rate of escape (speed) of the walk. This is achieved by exploiting the connections between random walks and electric networks.

The electrical resistance of a graph captures its commute and cover times

View ann-vertex,m-edge undirected graph as an electrical network with unit resistors as edges. We extend known relations between random walks and electrical networks by showing that resistance in this network is intimately connected with thelengths of random walks on the graph. For example, thecommute time between two verticess andt (the expected length of a random walk froms tot and back) is precisely characterized by the effective resistanceR st betweens andt: commute time=2mR st . As a corollary, thecover time (the expected length of a random walk visiting all vertices) is characterized by the maximum resistanceR in the graph to within a factor of logn:mR<-cover time<-O(mRlogn). For many graphs, the bounds on cover time obtained in this manner are better than those obtained from previous techniques such as the eigenvalues of the adjacency matrix. In particular, we improve known bounds on cover times for high-degree graphs and expanders, and give new proofs of known results for multi-dimensional meshes. Moreover, resistance seems to provide an intuitively appealing and tractable approach to these problems.

A Note on Electrical Networks and the Inverse Gaussian Distribution

When a random electrical network has the structure of a rooted tree and the edge resistances are either inverse Gaussian or reciprocal inverse Gaussian random variables then, subject to some restrictions, the overall resistance of the network is shown to follow a reciprocal inverse Gaussian distribution.

A Note on Electrical Networks and the Inverse Gaussian Distribution

When a random electrical network has the structure of a rooted tree and the edge resistances are either inverse Gaussian or reciprocal inverse Gaussian random variables then, subject to some restrictions, the overall resistance of the network is shown to follow a reciprocal inverse Gaussian distribution.

Random walk on the infinite cluster of the percolation model

We consider random walk on the infinite cluster of bond percolation on ℤ d . We show that, in the supercritical regime whend≧3, this random walk is a.s. transient. This conclusion is achieved by considering the infinite percolation cluster as a random electrical network in which each open edge has unit resistance. It is proved that the effective resistance of this network between a nominated point and the points at infinity is almost surely finite.

Random walks and the effective resistance of networks

In this article we present an interpretation ofeffective resistance in electrical networks in terms of random walks on underlying graphs. Using this characterization we provide simple and elegant proofs for some known results in random walks and electrical networks. We also interpret the Reciprocity theorem of electrical networks in terms of traversals in random walks. The byproducts are (a) precise version of thetriangle inequality for effective resistances, and (b) an exact formula for the expectedone-way transit time between vertices.

Transport in random composite materials

Percolative transport processes in special types of random composites are reviewed. By means of a random electrical network model, effects of interface percolation in composite ionic conductors are...

Random Walks and Electric Networks. By Peter G. Doyle and J. Laurie Snell

(1987). Random Walks and Electric Networks. By Peter G. Doyle and J. Laurie Snell. The American Mathematical Monthly: Vol. 94, No. 2, pp. 202-205.

Random Walks and Electric Networks.

Probability theory, like much of mathematics, is indebted to physics as a source of problems and intuition for solving these problems. Unfortunately, the level of abstraction of current mathematics often makes it difficult for anyone but an expert to appreciate this fact. Random Walks and Electric Networks looks at the interplay of physics and mathematics in terms of an example — the relation between elementary electric network theory and random walks —where the mathematics involved is at the college level.

Two-Phase Flow in Random Network Models of Porous Media

Abstract To explore how the microscopic geometry of a pore space affects the macroscopic characteristics of fluid flow in porous media, we have used approximate solutions of the Navier-Stokes equations to calculate the flow of two fluids in random networks. The model pore space consists of an array of pores of variable radius connected to a random number of nearest neighbors by throats of variable length and radius. The various size and connectedness distributions may be arbitrarily assigned, as are the wetting characteristics of the two fluids in the pore space. The fluids are assumed to be incompressible, immiscible, Newtonian, and of equal viscosity. In the calculation, we use Stokes flow results for the motion of the individual fluids and incorporate microscopic capillary force by using the Washburn approximation. At any time, the problem is mathematically identical to a random electrical network of resistors, batteries, and diodes. From the numerical solution of the latter, we compute the fluid velocities and saturation rates of change and use a discrete timestepping procedure to follow the subsequent motion. The scale of the computation has restricted us so far to networks of hundreds of pores in two dimensions (2D). Within these limitations, we discuss the dependence of residual oil saturations and interface shapes on network geometry and flow conditions.

ONE- AND TWO-PHASE FLOW IN NETWORK MODELS OF POROUS MEDIA

Abstract We discuss the low Reynolds number flow of one or two immiscible Newtonian fluids in network models of microscopically random porous media. For the case of a single fluid, we reduce the flow problem to an analog random electrical resistor problem and use an 'effective medium theory' to express the permeability of such networks in terms of the pore space geometry. For the flow of two fluids we use the Washburn approximation to incorporate capillary pressure differences, and show that this problem may also be formulated as a random electrical network. In this case, the capillary menisci correspond to moving batteries, and we follow the motion of the fluid-fluid interface (the ensemble of analog batteries) by a time-step procedure. We study the time evolution of the interface and the dynamics of blobs of one fluid contained in the other, as a function of the network geometry.’

Percolation theory and resistance of random electrical networks

Percolation theory and resistance of random electrical networks

Stochastic models with electrical analogues

1. Introduction. It has long been known that a connexion exists between reversible random walks and electrical networks ((2), (4) pp. 303–310, (6)). In section 2 we outline this connexion. More recently Kingman (5) has exhibited an electrical analogy for a flow model. In section 3 we show that this analogy follows from the results on random walks and we extend the analogy to cover the transient behaviour of Kingman's model. In so far as this model may be regarded as a naive description of the mechanism by which electrons move in a conductor it provides a physical explanation of the connexion between random walks and electrical networks. In section 4 we describe an invasion model which was inspired by the work of Clifford and Sudbury (1) on models to represent competition between species. Adapting their method we display the relationship of this model to random walks and hence obtain an electrical analogy for another class of stochastic processes.