Energy and Power E ngineering, 2013, 5, 597-602
doi:10.4236/epe.2013.54B115 Published Online July 2013 (http://www.scirp .o rg/journal/epe)
An Energy-Based Centrality for Electric al Net works
Ruiyuan Kong1, Congying Han2, Tiande Guo1, Wei Pei3
1Scho ol of Mathematical S ciences, U niversity o f Chinese Acade my of Sciences , Beijin g, China
2College of Humanities and Social Sciences , Universit y of Ch i nese Academy of Sciences, Beijing, China
3Insti tute of Electri cal Engin eer ing, Chin es e Academy of Sciences, Beij ing, Chin a
Received December, 2012
ABSTRACT
We present an energy-based method to estimate centrality in electrical networks. Here the energy between a pair of ver-
tices denotes by the effective resistance between them. If there is only one generation and one load, then the centrality
of an edge in our method is the difference between the energy of network after deleting the edge and that of the original
network. Compared with the local current-flow betweenness on the IEEE 14-bus system, we have an interesting dis-
covery that our proposed centrality is closely related to it in the sense of that the significance of edges under the two
measures are very si milar.
Keywords: Centra lity; Energy; Effective Re si stance; Current-flow Betweenness
1. Introduction
The electrical network is one of the most critical and
complex infrastructure networks in modern society.
There are some important issues which are keys to the
performance of the network. Reliable electric power
supply, for example, is crucial for many devices and its
disturbances may disrupt the devices or even paralyze the
network. This brings the concern about reliability and
resilience to disturbances and failures of various types of
infrastructure systems, and a corresponding demand for
methods of analyzing the vulnerabilities of the electrical
network [1]. Moreover, the blackouts of the North
American and Italian electric power grids in 2003 ex-
posed the weaknesses of the electrical network. The
weakness and vulnerable analysis about the electrical
network have been widely stud ied in the past years [1-4].
With recent advances in network and graph theory,
many researchers have applied centrality measures to
complex networks in order to study network properties.
Various centrality measures have been defined. They
draw links between the structure of networks and the
vulnerability to certain types of failures, and are used to
identify the most vulnerable ele ments of a network. Tra-
ditionally, there are four centrality measures within net-
work analysis, i.e., degree centrality, betweenness,
closeness, and eigenvector centrality. The degree metric
utilizes the local i nformation. Close ness and betweenness
utilize the shortest pa th information. And the eigenvecto r
metric rely on the Laplacian matrix of the group. All of
them consider only the topological properties but not the
actual physical flow through the power system. Moreo-
ver, the betweenness and closeness centrality postulate
that the information or flow transfer along the shortest
path, but this is not true for the current in the electrical
network. A series of centrality measures considering the
physical flow are proposed. [5] proposed a so-called
random-walk betweenness, counting how often a node is
traversed by a random walk between two other nodes.
This centrality is known to be useful for finding vertices
of high centralit y that d o not lie on the shortest p ath. Ac-
tuall y, the r andom-walk betweenness is closely related to
the current-flow betweenness proposed in [6]. The paper
derives the metric straightforward from the electrical
current and proves that the current-flow closeness is in
fact identical with the information centrality [7]. Some
papers proposed their measures which are actually of no
difference with the current-flow betweenness though
they didn’t point t hat directly. For example, [8] proposed
an electrical centrality measure based on the impedance
matrix which is similar to the current-flow centrality.
Besides, they pointed out the differences of the topology
of power grids from that of Erdos-Renyi random graphs,
the “small-world” networks or “scale-free” networks but
the power networks appear to have a scale-free network
structure under their proposed measure. However, as the
indication of [9], the proposed electrical centrality meas-
ure in [8] was defined incorrectly. But a simple analysis
shows that the revised measure was the right current-
flow betweenness.
The betweenness above needs to take into account all
pairs of nodes in the networks. [10] considered only the

R. Y. KONG ET AL.
598
pairs of generations and load nodes. Besides, they consi-
dered some other features of power systems such as
power transfer distribution and line flow limits, and got
that according to the un-served energy after the network
being attacked the nodes ranked highly is more vulnera-
ble.
The centralities defined before are not so easy to un-
derstand. This paper will propose an easy-understanding
method which is based on the effective resistance. The
effective resistance between a pair of vertices s and t is
the potential difference between them ensuring a current
of size 1 from s to t, and can be seen as the total energy
in the system. The effective resistance is local in some
sense. Its global form is the Kirchhoff index, which is
based on the resistance-distance matrix introduced in
1993 by Klein and Randic and defined as the effective
resistance between pairs of vertices [11]. The Kirchhoff
index is often used to quantify the str uctural attr ibutes of
the graph. See [12,13] for more information.
This paper is organized as follows. In Section 2, we
introduce some preliminary concepts about centrality
measures, the effective resistance and so on. In section 3,
the definition of our measure based on energy and the
variation of current-flow betweenness, that is, the local
current-flow betweenness is given. Section 4 provides
the comparisons with the current-flow betweenness cen-
trality and other centrality measures. Conclusions are
drawn in Section 5.
2. Preliminaries
2.1. Betweenness
Vertex betweenness, first introduced by Freeman in 1977
[14], is one of the most used centrality measure. It re-
flects the occurrence degree of a node on the shortest
path between any pair of nodes. Given a undirected graph
(,)
GV E, whe r e V is the set of vertices and E is the set of
edges, the betweenness of a node v is defined by:
( )/
() ,
(1)(2)/ 2
st st
svtV
b
v
Cv nn
σσ
≠ ≠∈
=−−
∑
where st
σ
and
()
st v
σ
are the number of shortest paths
from s to t and the number of shortest paths from s to t
through v. Girva n and Newm an [15] generalized the ver-
tex betweenness to edges and proposed edge between-
ness which is defined as the number of shortest paths
between pairs of vertices that run along it and used to
find which edges are most important. If there is more
than one shortest path between a pair of vertices, then
take t hem as one path. The edge betwee nnes s is given b y:
( )/
() .
(1) / 2
st st
stV
b
e
Cv nn
σσ
≠∈
=−
∑
It is found that the removal of the nodes or edges with
large bet weenness will put th e network at high risk to b e
disconnected.
2.2. Current-Flow Betweenness
The current-flow betweenness here is based on the defi-
nition of [6]. An electrical network is a graph
N=
(, ,)(,),
VEc Gc=together with a function
:cR
+
→
,
where (e)
e
cc= is the reciprocal of the resistance of the
edge e. Given a supply of size 1 from a source s to a sink
t, the throughput of an edge e and an inner vertex v is
defined by:
:
()|() |,
1
()|() |.
2
st
st ev e
e we
v we
τ
τ
∈
=
=∑


Define the edge and vertex current-flow betweenness
respectively by:
,
,
1
() (),
( 1)
1
() ().
( 1)(2)
CB st
st V
CB st
st V
ce e
nn
cv v
nn
τ
τ
∈
∈
=−
=−−
∑
∑
The current-flow betweenness is reasonable for that
the current is uni que by Lemma 1 of [6].
Besides, Brandes proposed also the current-flow
closeness centrality which is a variation of closeness
centrality. It is defined by:
() () ()
C
CC st st
ts
n
cs ps pt
≠
=−
∑
for all s in V, where
()ps
refers to the voltage in the
vertex s. Moreover, Brandes proved that the current-
flow closeness centrality equals informatio n centrality.
2.3. Effective Res istan ces , E nergy and Ki rchh o ff
Index
Now we give the definition of effective resistances. The
effective resistance is the potential difference between s
and t ensur ing a c urrent of siz e 1 fr om s to t a nd denoted
by
()
st
RG
.
The total energy in a network is defined as:
22
() (),
xyx yxyx yxy
xy Exy Exy E
w VVcVVw
∈∈ ∈
=−=−
∑∑ ∑
where x V is absolute potential and
()
xyxy xy
wVVc= −
is the energy in the edge between x and y. By Lemma 2
of [6], there are unique potentials. So the definition of the
total energy is reasonable.
Kirchhoff index is the sum of the effective resistances
over all pairs of vertices:
() ,
ij
ij
Kf Gr
<
=
∑
where
ij
r
is the effective resistance between i and j. It is

R. Y. KONG ET AL.
599
proved that Kirchhoff index satisfies
1
1
1
() ,
n
ii
Kf G
λ
−
=
=∑
where
{ ,1,...,1}
iin
λ = −
are the nonzero eigenvalues of
the Laplacian matrix of G.
3. Centrality Metric Based on Energy
It is known that the total energy in an electric current
with size 1 from s to t is the effective resistance [16]. If
there is o nl y one so urce s and one s in k t, t hen t he c ha nge
in the energy is the change in the effective resistance
between s and t. Therefore to measure the influence of
removing one edge we can define a ‘metric’ based on the
variant of energy by:
( )(\)(),
st st
EeRGe RG∆= −
where
\Ge
is the graph deleted by the edge e. Though
()Ee∆
is not a strict definition of a metric, we regard it
as a metric since we only focus on the results under the
metric but not their values. The larger
()Ee∆
, the greater
the risk for the network to be damaged when dele ti ng t he
edge e. If there is no connection between s and t after
deleting the edge e, t hen
()Ee
∆=∞
. In other words, the
edge e with
()Ee
∆=∞
is very important to the network.
By the monotonicity principle, Corollary 7 of [16],
()Ee∆
is nonnega tive for all edges.
For the network with sources S and sinks T where
ST∩=∅
, the metric based on energy is defined by
,
( \)()
() ,
| || |
st st
sSt T
RGe RG
Ee ST
∈∈
−
∆=
∑
where
||S
denotes the cardinality of the set S. The
energy-based centrality can be seen as a local metric to
measure the importance of an edge. Analogous to the
definition of betweenness, we consider the whole impor-
tance of an edge, that is,
( \)()
'( ).
(1) / 2
st st
st
RGe RG
Ee nn
≠
−
∆= −
∑
(1)
If we define
( \)()
() ,
(1) / 2
Kf GeKf G
Kf enn
−
∆= −
then it is easy to check that
() '()Kf eEe∆=∆
by the
definition of the Kirchhoff index. So we call the case
defined by Equation (1) the edge Kirchhoff-based cen-
trality. Analo gous to the defi nit ion o f verte x cur ren t-flow
betweenness, we can also define the vertex energy-based
centrality and vertex Kirchhoff-based centrality by
() ()
() ,
|( )|
eu
v
Ee
Eu u
∈Γ ∆
∆=
Γ
∑
() ()
() ,
|( )|
eu
v
Kf e
Kf uu
∈Γ ∆
∆=
Γ
∑
where
()uΓ
denotes by the adjacent edge s of the vertex
u. Next we take Theorem 1 of [17] as a lemma which
will be used to compute the energy-based centrality.
Lemma 1. Let G be a connected graph on
3n≥
ver-
tices and
1i jn≤≠ ≤
. Let
()
Li be the submatrix ob-
tained from the Laplacian matrix L of the graph G by
deleting its ith row and ith column and
(, )Li j
be the
submatrix obtained from the Laplacian matrix L by de-
leting its ith and jth rows and the ith and jth columns.
Then the effective resistance
ij
r
between i and j satisfies
det(( ,)).
det(( ))
ij
Li j
rLi =
(2)
Note that the gra ph G in the lemma above can be seen
as a graph with one unit resistance on each edge of the
network. Following the steps of its proof, we can easily
check t hat the result holds for the graph G with different
resistances on the edges and the Laplacian matrix L being
replaced by the admittance matrix of the network.
Though in general the complexity of computing the ef-
fective resistance using equation (2) is3
()
On , the re-
markably simple expression is still very valuable.
Recall the argument of the reference [10] in the intro-
duction. The authors utilized a local idea considering the
electrical betweeness only between the pairs of genera-
tions and loads with some other restriction, but they
didn’t point out that clearly and didn’t give the corres-
ponding simulation results. Thus this paper gives the
definition and simulation clearly, and calls it the local
current-f low betwe enness. Analogo us to the current -flow
betweenness, for the network with so urces S and sinks T
we define the edge local curre nt-flow b et weenness by
,()
() .
| || |
st
sSt T
b
e
LC eST
σ
∈∈
=∑
The vertex local current-flow betweenness is defined
similarly and denoted by
()
v
LCu.
4. Numerical Analysis
In this section, the edge (vertex) energy-based metric and
the edge (vertex) Kirchhoff-based metric are compared
with the edge (vertex) current-flow betweenness, the
edge (vertex) local curr e nt-flow betweenness and the
closeness centrality on the IEEE 14-bus. The IEEE 14-
bus consists of 20 lines and 14 buses including 2 genera-
tors and 2 loads, as shown in Figure 1. And Figure 2 is
the graph representation of IEEE 14-bus transmission
network. The circles with labeled G represent the gene-
rator nodes, the circles with labeled L represent the load

R. Y. KONG ET AL.
600
Figure 1 . Transmissio n netw ork IEEE 14-bus.
Figure 2 . The graph representation of IEEE 14-bus transmissio n network.
nodes and the ellipses represent the transmission nodes.
It is known that for the high-voltage transmission net-
work in a power grid the reactance is usually the domi-
nant component of a line impedance. Thus for the pur-
pose of simplicity, we take the reactance as the edge
weights. And the bus 8 will not affect the effective resis-
tance between the generations and loads, thus we don’t
consider it. To keep in accord, we compute other central-
ity metrics based on the case above. Besides, the bus 8 is
of low betweenness centrality by [18].
Table 1 ranks the edges according to the four edge
centralities. The edge current-flow betweenness, edge lo-
cal current-flow betweenness, edge energy-based meas-
ure a nd ed ge Kir chho ff-based measure are abbreviated as
B, Local-B, E-based and Kf-based respectively. All the
four methods rank the edge 5-6 first, which says that the
edge 5-6 is very possible to be the most important b ranch
in this network. It shows that the edge betweenness and
the edge Kirchhoff-based centrality are quite different
with each other and with the other two measures.

R. Y. KONG ET AL.
601
Table 1. The importance order of edges from high to low.
Order B Local-B E-based Kf -based
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
5-6
1-2
7-9
9-10
4-7
4-5
6-11
10-11
9-14
6-13
13-14
1-5
2-4
2-5
2-3
3-4
12-13
6-12
4-9
5-6
6-13
9-14
1-2
13-14
7-9
4-7
2-4
2-5
1-5
4-9
3-4
2-3
12-13
6-12
9-10
10-11
6-11
4-5
5-6
9-14
13-14
6-13
1-2
4-7
7-9
1-5
4-9
2-4
2-5
6-12
12-13
9-10
10-11
6-11
2-3
3-4
4-5
5-6
9-10
9-14
7-9
6-11
10-11
4-7
13-14
6-13
1-2
3-4
12-13
6-12
4-5
4-9
2-3
1-5
2-4
2-5
Table 2. The importanc e order of nodes f rom high to lo w.
Order V-B VLocal-B C VE-based VKf-based
1
2
3
4
5
6
7
8
9
10
11
12
13
4
5
6
9
2
7
1
13
10
11
14
3
12
2
13
14
1
6
5
4
9
7
3
12
10
11
12
14
3
11
13
1
10
7
2
6
9
5
4
14
6
9
13
5
7
1
2
4
12
10
11
3
10
9
7
14
11
6
5
13
12
4
3
1
2
But there is a strong correlation between the edge local
current-flow betweenness and our edge energy-based
centrality. The first 5 important edges are the same and
their orders are of little difference. Moreover, the edges
ranking from 5 to 11 are also consistent with their or ders
being little differe nt. I n fact, f or a ny edge e in the f irst 11
edges, the difference between the ranking of e in t he t wo
cases is at most 2. However, the complexity of compu-
ting the edge energy-based ce ntralit y is lower than tha t of
computing the edge local current-flo w b et weenne ss. And
the complexities for computing the two centrality are
both
3
()On
. But the latter has a much more clear ex-
pression. In practice, we can use both to measure the
importance of two edges from different perspectives. If
we give the edge ranking first the score 19, the edge
ranki n g se co nd the score 18, …, the e d ge r an ki ng la st the
score 1, and denote
ij−
by the edge bet ween the ver-
tex i and j. Then given the ordered sequence {1-2, 1-5,
2-3, 2-4, 2-5, 3-4, 4-5, 4-7, 4-9, 5-6, 6-11, 6-12, 6-13, 7-9,
9-10, 9-14, 10-11, 12-13, 13-14}, denote by
1
X
and
2
X the ranking lists for energy-based centrality and
edge local current-flow betweenness respectively. Using
linear regression to check whether
1
X
and 2
X are
relevant, we get that the adjusted 2
R
equals 0.814 and
the p-value is very close to 0. This shows that they are
strongly correlated.
5. Conclusions
This paper considers measures of centrality that are used
to rank the importance of the nodes or edges in an elec-
trical network. New methods of centrality are defined
from the perspective of the energy of a network. More
specifically, we use the variant of the effective resis-
tances between the generations and loads after deleting
an edge or a node to measure its importance, similar to
whi ch we a lso define a Kirchhoff-based measure with the
effective resistances being replaced the Kirchhoff index.
Besides, we propose the local current-flow betweenness
in the most simple way more clear ly.
Based on defined measures, experiments are performed
on IEEE 14-bus and some interesting results are discov-
ered. It has been found that our proposed edge energy-
based measure is very similar to the local current-flow
bet wee nne s s, i n the sen se t ha t the i mpo r ta nce r a nki n gs of
the edges in the two measures are of little difference.
While the expression of computing the ener gy-based
measure is very simple and clear. Besides, from the ex-
periments we get that the current-flow betweenness is
very different fro m the local current-flo w. Ho wever, it is
difficult to j udge which are mo re accurate. Moreover, we
verify that the current-flow betweenness is closely re-
lated to the closeness centrality in our experimen ts. H ow -
ever, more tests and analysis need to be done in order to
validate the proposed measure, to find the most effective
measure and to dig deep to see which nodes or edges are
the real most important nodes or edges.
6. Acknowled gements
This work is supported by CAS Knowledge Innovation
Program (grant number KGCX2-RW-329) and National
Natural Science Foundation of P.R. China (grant nu mber
10831006, 71271204).
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