IEEE Inl. Conf. Neural Networks 8 Signal Processing
Nanjing, China, December 14-17, 2003
FINDING THE MOST VITAL NODE WITH RESPECT TO THE NUMBER OF
SPANNING TREES
Yong Chen, Ai-qun Hu, Kun- Wah Ep, Jun Hu, and Zi-guo Zhong
Deparhnent of Radio Engineering, Southeast University,
Nanjing, Jiangsu 210096, P.R. China
ABSTRACT
'An evaluation method for finding the most vital node with
respect to the number of spanning trees in communication
networks is proposed. For a given node v in the graph C,
G-v is the graph with the node deleted, where the edge
denotes a link and the vertex denotes a node respectively.
The relative importance of two nodes in the graph can be
compared with each other with respect to the number of
spanning trees. The most vital node in G is a node whose
removal with its incident links most impacts system
reliability. Moreover, the concise generalized expression is
given. Experimental results show that the method can
identify the most vital node in a network efficiently.
1,
a, =<-I,
1. INTRODUCTION
0,
Reliability measure in communication networks has
caused much attention. Network models adopted at present
mainly involve edge reliability rather than node reliability.
The problem of finding a most vital node of a shortest path
has been defined and motivated by Nardelli [I] and Corley
[Z]. A most vital node of a given shortest path from the
source r to the destination s is a node (other than r and s)
whose removal from an undirected, connected graph G
results in the largest increase of the distance from r to s.
But this method is merely suitable for 2-terminal
connected graph rather than all-node graph.
In this paper, a novel evaluation method for node
reliability in communication networks is proposed, which
can rank the node importance in terms of overall network.
Moreover, the concise generalized expression for node
reliability is given.
2. NETWORK MODEL
This work was supported "1 part by the National
High-Technology Research and Development h o g " (863
Program), P. R. China, through grants (No. 2002AA143010 &
2003AA143040), and in part by a funding from Education
Ministry OutstandingTeachen Support Program.
0-7803-7702-8/03/ $17.00 02003 IEEE
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if the jth edge is incident on the
ith vertex and oriented away from it;
if the jth edgeis incident on the
ith vertex and oriented toward it;
if the jth edge is not incident on
the ith vertex.
(1)
z(G)=det(AA')
(2)
where r(G) is the number of spanning trees of G .
The generalized expression of the i-th node is
4 = 1- r ( G - v j ) / r ( G )
(3)
where ri is the reliability measure of the i-th node,
r(G--Vi) is the number of spanning trees of graph
G - vi and t ( G ) is the number of spanning trees of
graph G The relative importance of two nodes in the graph
can be compared with each other with eqn. 3. The smaller
the number of spanning trees corresponding to a specified
node is, the more vital the node is. If the number of
spanning trees corresponding to the i-th node equals to 0,
the remaining graph is then disconnected after removing
the i-th node and its incident edges. In this case, we can
safely conclude that node vi is the most important in
topology and corresponding generalized result is 1.
A brief description of the algorithm is as follows:
Input: the all-node incidence matrix A, = [a,] of G .
Output: the number of spanning trees or the generalized
results by removing from G the node V i , i = 1,2;..,n
and its incident edges, respectively.
Step I : Initialization of the all-node incidence matrix
A, and i = l .
Step 2: Remove corresponding columns from matrix
A, if there are nonzero entries in its i-th row. Then
remove the i-th row. The new matrix is called matrix B .
Step 3: Matrix A is obtained by removing from matrix
B any one of the rows.
Step 4: Calculate the number of spanning trees
corresponding to the i-th node using (2) and (3). Then,
i = i 1, If i > n the algorithm terminates, otherwise,
retum to step 2.
+
4. EXPERIMENTAL RESULTS
Fig. 1. ARPA network from [3]
Fig. 2.
Directed graph obtained by assigning arbitrary orientations to the edges ofARPA network
example, consider the graph G shown in Fig. 1. A
directed graph obtained by assigning arbitrary orientations
As an
to the edges of G is shown in Fig. 2. The all-node
incidence matix of this directed graph is given by (4).
1671
0 - 1 1
0
0
0
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0
1
0
0
0
0
0
0
0
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0
0
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0 0 - I 1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 - I 1 0
0
0
0
0
0 - I 1 0
0 0 0 0 0 - 1 1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
- l l 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0
0 - I 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0
0
0
0
0
0
0
0
0
0
0
-
I
0
0
0
0
0
0
0
0
0
-
l
1
0
0
0
0
0
0
0
0
0
0
0
0
0 0 0
0
I
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 - I 1 0
0
0 0 - 1 1 0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
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0
0
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0
-
1
0 -
0
0
0
0
0
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0
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I 0 0
I 1 0
0
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0
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0 0 0 0 - 1 0
0
0
0
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0
0 0 0 0 0 0 0 0 0 0 0 0 0 - l 1 0 0 - 1 - l 0 0 0 0 0 0 0
0
0
0
0
I
0
0
0
0
0
0
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0
0
0
0
0
0
0
0
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0
0
0
0
0
0
0
0
0
0
0
0
0
0
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sP3-v:)
1
2
5856
437
110
2527
257
1884
344
3053
213
1901
5266
4736
3602
3
4,s
6
7,8,9,10,11
12
13
14
15
16
17
18
0
0
0
0
1.
0
1
0
0.6262
0
0
0
0
-
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l
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0
(4)
45, the runtime is still less than 1 second (average value is
720 ms). When the number of nodes is not larger than 80,
the algorithm performs in approximately linear time with
respect to the number of nodes.
0.9721
0.9930
0.8387
0.9836
0.8797
0.9780
0.8051
t m = 2 U 5
4 - m
= 3U5
0.9864
0.8787
0.6639
0.6977
0.7701
I9
515
0.9671
20,21
2691
0.8279
Table 1 gives the simulation results of node reliability
for the graph denoted by Fig. 1. The generalized result
corresponding to the third node is larger than any one of
the other nodes. which means the thiid node is the most
important. Also, the results show that node vz0 and vZI
have equal importance. The proposed algorithm is
implemented on a
pc Pentium in Maaah and it,s
performed within less than
1
0
0 - 1 1 0
0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 - 1 1 l 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 - 1 1 0
0
0
0 0 0 0 0 0 0 . 0 0 0 0 0 0 0 0 -1-1
0
Table 1. Generalized results for node importance with
given graph
V:
-
IO ms for given graph.
A
graph has
=
(n - ') edges' Fig'
shows simulation runtime in Matlab for several graphs
with given number of nodes and edges. In general, a large
communication network consists of 25 to 30 backbone
nodes. When the number of nodes of a complete graph is
M
1w
im
the number of nodes
Fig. 3. Runtime for different graph with given number of
nodes and edges
5. CONCLUSIONS
An evaluation method for node importance in
communication networks is proposed. The relative
importance of two nodes in the graph can be compared
with each other with respect to the number of spanning
trees. The most vital node is the one whose removal with
its adjacent edges
&astically &creases the number of
the node importance
trees. niS
in terms of
M
~ the concise
~
~
generalized expression is given. Experimental results show
1672
~
that the method can identify the most vital node in a
network efficiently.
6.REFERENCES
[l] E. Nardelli, G Proietti, P. Widmayer, ‘Tinding the most
vital node of a shortest path,” LNCS 2108, Springer-Verlag,
pp. 278-287, Aug. 2001.
[2] H. W.Corky, D. Y. Sha, “Most vital links and nodes in
weighted networks,” Operations Research Letters, Vol.1,
pp. 157-160, S e p . 1982.
[3] L. B. Page, J. E. Perry, “Reliability polynomials and
link importance’innetworks,” IEEE Trans. Reliability, Vol.
43,No. 1,pp. 9 - 5 8 , Mar. 1994.
[4] M. N. S. Swamy, K. Thulasiraman, Graphs, Networks,
and,Algorithms,New York John Wiley & Sons Inc, 1981.
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