Foundations of Data Science

i
↑
↓
↑
↓
↑
j
↑
=⇒
i
⇐=
↑
↓
↑
↓
↑
j
↑
Insert current at each vertex
equal to degree of the vertex.
Extract 2m at vertex j. v ij = h ij
(a)
↑ ↑
i
j
=⇒
↓
↓
↓ ↓
Reverse currents in (b).
For new voltages −v ji = h ji.
Since −v ji = v ij , h ji = v ij .
(c)
Extract current from i instead of j.
For new voltages v ji = h ji .
(b)
2m i
j
2m
=⇒ =⇒
Superpose currents in (a) and (c).
2mr ij = v ij = h ij + h ji = commute(i, j)
(d)
Figure 5.8: Illustration of proof that commute(x, y) = 2mr xy where m is the number of
edges in the undirected graph and r xy is the effective resistance between x and y.
Corollary 5.11 For vertices x and y in an n vertex graph, the commute time, commute(x, y),
is less than or equal to n 3 .
Proof: By Theorem 5.9 the commute time is given by the formula commute(x, y) =
2mr xy where m is the number of edges. In an n vertex graph there exists a path from x
to y of length at most n. Since the resistance can not be greater than that of any path
from x to y, r xy ≤ n. Since the number of edges is at most ( n
2
( n
commute(x, y) = 2mr xy ≤ 2 n
2)
∼ = n 3 .
)
Again adding edges to a graph may increase or decrease the commute time. To see
this consider three graphs: the graph consisting of a chain of n vertices, the graph of
Figure 5.7, and the clique on n vertices.
Cover time
The cover time, cover(x, G) , is the expected time of a random walk starting at vertex x
in the graph G to reach each vertex at least once. We write cover(x) when G is understood.
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