Random walks and the effective resistance sum rules

• Published in: Discrete Applied Mathematics Vol. 158; no. 15; pp. 1691 - 1700
• Main Author: Chen, Haiyan
• Format: Journal Article
• Language: English
• Published: Kidlington Elsevier B.V 06-08-2010; Elsevier
• Subjects: Algorithmics. Computability. Computer arithmetics; Applied sciences; Combinatorics; Combinatorics. Ordered structures; Computer science; control theory; systems; Conductance; Effective resistance; Electric networks; Exact sciences and technology; Hitting time; Mathematical analysis; Mathematics; Random walk; Sciences and techniques of general use; Subdivision network; Sum rules; Theoretical computing; Effective resistance; Random walk; Hitting time; Subdivision network; Computer theory; Combinatorics; Optimization
• ISSN: 0166-218X; 1872-6771
• DOI: 10.1016/j.dam.2010.05.020

In this paper, using the intimate relations between random walks and electrical networks, we first prove the following effective resistance local sum rules:
c
i
Ω
i
j
+
∑
k
∈
Γ
(
i
)
c
i
k
(
Ω
i
k
−
Ω
j
k
)
=
2
,
where
Ω
i
j
is the effective resistance between vertices
i
and
j
,
c
i
k
is the conductance of the edge,
Γ
(
i
)
is the neighbor set of
i
, and
c
i
=
∑
k
∈
Γ
(
i
)
c
i
k
. Then we show that from the above rules we can deduce many other local sum rules, including the well-known Foster’s
k
-th formula. Finally, using the above local sum rules, for several kinds of electrical networks, we give the explicit expressions for the effective resistance between two arbitrary vertices.