# Random walks and electric networks

@article{Doyle1987RandomWA, title={Random walks and electric networks}, author={Peter G. Doyle and James Laurie Snell}, journal={American Mathematical Monthly}, year={1987}, volume={94}, pages={202} }

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. In this work we will look at the interplay of physics and mathematics in terms of an example where the mathematics involved is at the college level. The example is the relation between elementary electric network theory and…

## 1,686 Citations

### Electric Networks and Commute Time

- Mathematics
- 2011

The equivalence of random walks on weighted graphs with reversible Markov chains has long been known. Another such correspondence exists between electric networks and these random walks. Results in…

### Random Walks on Graphs: a Survey

- Mathematics

Dedicated to the marvelous random walk of Paul Erd} os through universities, c ontinents, and mathematics Various aspects of the theory of random walks on graphs are surveyed. In particular,…

### Random Walks on Graphs: a Survey

- Mathematics
- 1993

Dedicated to the marvelous random walk of Paul Erd} os through universities, continents, and mathematics Various aspects of the theory of random walks on graphs are surveyed. In particular, estimates…

### Random Walks on Graphs : A SurveyL

- Mathematics
- 2007

Dedicated to the marvelous random walk of Paul Erd} os through universities, continents, and mathematics Various aspects of the theory of random walks on graphs are surveyed. In particular, estimates…

### OPERATORS, AND THE TRANSIENCE OF RANDOM WALKS

- Mathematics
- 2010

3 Abstract. A resistance network is a connected graph (G; c). The conductance function cxy weights the edges, which are then interpreted as resistors of possibly varying strengths. The relationship…

### A Physics Perspective on the Resistance Distance for Graphs

- PhysicsMath. Comput. Sci.
- 2019

A brief review of the concept and a physics perspective on resistance distance is provided, highlighting some useful analytical methods for computing it and discussing the concept in the context of the Weisfeiler–Leman stabilization.

### A Physics Perspective on the Resistance Distance for Graphs

- PhysicsMathematics in Computer Science
- 2018

The notion of resistance distance as a convenient metric for graphs was introduced in Klein (J Math Chem 12:81–95, 1993). It is inspired by the concept of equivalent resistance for electrical…

### An Electric Network for Nonreversible Markov Chains

- MathematicsAm. Math. Mon.
- 2016

It is proved that absorption probabilities, escape probabilities, expected number of jumps over edges, and commute times can be computed from electrical properties of the network as in the classical case.

### Probability on Trees and Networks

- Mathematics
- 2017

Starting around the late 1950s, several research communities began relating the geometry of graphs to stochastic processes on these graphs. This book, twenty years in the making, ties together…

### Random Walks and Current Flow

- Mathematics
- 2013

The branching number of an arbitrary infinite locally finite tree is, conceptually, the "average" number of branches extending from a given vertex. A random walk performed on such a tree with known…

## References

SHOWING 1-10 OF 38 REFERENCES

### Random walk and electric currents in networks

- MathematicsMathematical Proceedings of the Cambridge Philosophical Society
- 1959

ABSTRACT Let G be a locally finite connected graph and c be a positive real-valued function defined on its edges. Let D(ξ) denote the sum of the values of c on the edges incident with a vertex ξ. A…

### Reversibility and Stochastic Networks

- Economics
- 1979

This classic in stochastic network modelling broke new ground when it was published in 1979, and it remains a superb introduction to reversibility and its applications. The book concerns behaviour in…

### The Mathematical Theory of Electricity and Magnetism

- PhysicsNature
- 1921

SINCE the third edition of this volume was published in 1915, the theory of relativity has been developed. It is now recognised that Maxwell's theory that the ultimate seat of electromagnetic and…

### V. On the theory of resonance

- EngineeringPhilosophical Transactions of the Royal Society of London
- 1871

Although the theory of aërial vibrations has been treated by more than one generation of mathematicians and experimenters, comparatively little has been done towards obtaining a clear view of what…

### The Mathematical Theory of Electricity and Magnetism

- PhysicsNature
- 1926

THE fifth edition of an established text-book calls for little comment. The previous edition was marked by the introduction of a new chapter on the theory of relativity. The present volume has yet…

### A Treatise on Electricity and Magnetism

- HistoryNature
- 1873

IN his deservedly celebrated treatise on “Sound,” the late Sir John Herschel felt himself justified in saying, “It is vain to conceal the melancholy truth. We are fast dropping behind. In Mathematics…

### Denumerable Markov chains

- Mathematics
- 1969

This textbook provides a systematic treatment of denumerable Markov chains, covering both the foundations of the subject and topics in potential theory and boundary theory. It is a discussion of…

### Finite Markov chains

- Mathematics
- 1960

This lecture reviews the theory of Markov chains and introduces some of the high quality routines for working with Markov Chains available in QuantEcon.jl.

### An Introduction to Probability Theory and Its Applications

- Mathematics
- 1950

Thank you for reading an introduction to probability theory and its applications vol 2. As you may know, people have look numerous times for their favorite novels like this an introduction to…

### Reciprocal Relations in Irreversible Processes. II.

- Materials Science
- 1931

A general reciprocal relation, applicable to transport processes such as the conduction of heat and electricity, and diffusion, is derived from the assumption of microscopic reversibility. In the…