Ripple Algorithm to Evaluate the Importance of Network Nodes

International Journal on Recent and Innovation Trends in Computing and Communication ISSN: 2321-8169
Volume: 5 Issue: 12 72 – 75
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72
IJRITCC | December 2017, Available @ http://www.ijritcc.org
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Ripple Algorithm to Evaluate the Importance of Network Nodes
WANG Yan-Peng
School of Information and Communication Engineering
Dalian University of Technology
Dalian, LiaoningProvince, China
378875773@qq.com
YU Ming
School of Information and Communication Engineering
Dalian University of Technology
Dalian, LiaoningProvince, China
yu_ming1111@dlut.edu.cn.
Abstract—Inthis paper raise the ripples algorithm to evaluate the importance of network node was proposed, its principle is based onthe direct
influence of adjacent nodes, and affect farther nodes indirectlyby closer ones just like the ripples on the water. Then we defined two
judgments,the discriminationof node importance and the accuracy of key node selecting, to verify its efficiency. The greater degree of
discriminationand higher accuracy means better efficiency of algorithm. At last we performed experiment on ARPA network, to compare the
efficiency of different algorithms, closeness centricity, node deletion, node contraction method, algorithm raised by Zhou Xuan etc. and ripple
method. Results show that ripple algorithm is better than the other measures in the discrimination of node importance and the accuracy of key
node selecting.
Keywords-network Nodes, Ripple Algorithm, Discrimination of Node Importance, Accuracy of Key Node Selecting
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I. INTRODUCTION
With the further research on network reliability and anti-
destructive, it is found that under random attack, scale-free
networks are more reliable than random networks.And under
the selective attack, scale-free networks are more vulnerable
than random networks[1].Therefore, to evaluate the importance
of nodes, and protect the principle nodes in network protection
is particularly important.
There are many methods to evaluate the importance of
nodes in networks, most of which are based on graph
theory.However, each algorithm has its limitations and may not
be applied to all kinds of networks.
The traditional methodsfor judging the importance nodes in
networks are a lot. Such as, Degree Centrality, Closeness
Centrality [2], BetweennessCentrality, etc.
Kitsak et al. [3-4] raised K - Shell Decomposition.
Stripping nodes of peripherallayers until the center nodes of
inner layers, and judge nodes of inner layers have got more
influence.The effect of this method is remarkable for the
analysis of spreading. But shortcomings are obvious when this
method are used on the analysis of nodes evaluations in
communication networks.Firstly, the analysis result is too
coarse graining. Secondly, it is the default that nodes in the
same layerlink the same number of neighbors in the outer layer,
which leading to the error.
Node Deletion method analyze network after target node is
deleted, observing the change of network topology and some
properties, and judge node important degree on the basis of the
parameters changing.For instance, the Spanning Tree is widely
used in this method. Node Deletion method is limited by the
topology of network.If there are multiple nodes, any one of
whichare deleted maylead to the network separating, the
importance of these nodes will not be distinguished.Aiming at
the shortcomings of the Node Deletion method, TanYuejinet al.
raised theNode Contraction method [1], After a node and its
neighbor nodes shrink into a new node, if the condensation of
the network become better, the node is considered to be more
important.
In addition, there are many methods to determine the
importance of nodes, but most of them are improvements or
integration of existing methods.Such as Chen Jing, SunLinfu[5]
consider Closeness Centrality as the global importance of
nodes, and Betweennessofa node within the its neighborhood
as the local importance, then multiply the two results to get the
node important degree.Wu Guo, Fang Liguoand Li Zhongput
forward a method based on D - S evidence theory,which
evaluate the importance of nodes of complex network
bycomprehend multi-index[6].Comprehend degree, Closeness,
Betweenness, tenacity, and Condensation degree by using the
D-S evidence theory, so as to get the node important
degree.Shasha Wang et al. raised Efficiency Centrality method,
evaluate the importance of a node by comparing the efficiency
changing after the node is deleted[7].Junyi Wang et al. put
forward a new algorithm that take the weight of neighbor nodes
into account to evaluate the important degree of the node. This
method is more accurate in the importance of nodes in local
nodescalculation[8], Zhong-KuiBao et al. comprehend
thelength and number of the shortest path between nodes of a
network, and transmission rates, then put forward a new
algorithm for network node evaluation[9].Zhou Xuan
etc.definednode efficiency and node importance evaluation
matrix[10], proposed an evaluation method forfinding out the
key node in the complex network by importance evaluation
matrix.This method comprehend node efficiency, degree, and
the importance of neighbornodes. The contribution of a node
for the importance of its neighbor is calculated with its
efficiency and degree.However, the calculation of its node
efficiency is similar to that of the Closeness Centrality, the
distance of every pair of nodes in the network are need to be
calculated, which lead to higher computational complexity.
II. RIPPLEALGORITHM
The changing of the importance of a node in the network
will cause a series of changing of other nodes in the network
which leading to a chain reaction.According to this
characteristic, this paper proposes a method for judging node
importance names ripple algorithm. The effect of a nodeto
others is transported with iterative calculation like ripple

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layers spreading to the edge of the network.Finally select out
the important nodes in the network. We can represent the
importance of a node with the weight of its neighborsby a
function in a communication network.
There are several characteristics of the backbone routing
network of communication network has the following
comparing with other networks：
1) There is information exchange between adjacent nodes,
and the amount of information exchanged is not necessarily
equal.
2) The information transmitted in the network must be sent
from a source node to a destination node;
3) The information in the network is from the outside;
4) The information will be sent back to the outside after
reaching the destination node.
Based on the above characteristics, we should do the
following provisions while evaluating the importance of node
in network:
1) The probability that a node vi sends information to any
other node in the network is equal to
1
n−1
；
2) During a unit time, the amount of information received
from the outside through any node vi in the network is equal to
the amount of information sent to the outside world through
node vi, all of which are u.
On the basis of the provisions mentioned above, we can
use the weight of the node to represent the importance of the
node, and make the following definition for the node weight:
Definition: the weight of node vi is the sum of the amount
of information received from other nodes and the input
information from outsideduring the unit time, which is
represented bywi.
wi = pj,i wj − uj
′
n
j=1
+ ui(1)
Among Eq.(1),pi,j represents the probability that node vi
sends information to node vj;ui is the amount of information
input to the network through node vi in the unit time.uj
′
isthe
amount of information that vjsend to the outside world during
the unit time.
P in Eq.(1)could be a fixed value or variable.When p is
constant value, the value of w can be calculated immediately.
When p is variable, the equations could be calculated by
iterative algorithm, and iterative equations are as follows:
wi
k+1
= pj,i
k
wj
k
− uj
′
n
j=1
+ ui,
k = 1,2, ⋯ m (2)
Eq.(2) can be represented as matrix form:
w1
(k+1)
w2
(k+1)
⋮
wn
(k+1)
=
p1,1
(k)
，p2,1
(k)
， ⋯ ，pn,1
(k)
p1,2
(k)
，p2,2
(k)
， ⋯ ，pn,2
(k)
⋮ ⋮ ⋱ ⋮
p1,n
(k)
，p2,n
(k)
， ⋯ ，pn,n
(k)
w1
k
− u1
′
w2
k
− u2
′
⋮
wn
k
− un
′
+
u1
u2
⋮
un
，
k = 1,2, ⋯ m (3)
In the upper formula, the initial value of the weight vector
can be set in advance.The step transfer probability of node vi
to node vj can be estimated by the importance degree of node
vj relative to node vi, and two errors should be taken into
account, as shown in Figure1:
1) In computing the relative importance of node v2to node
v1,we should get rid of the influence of the node v1 to v2.Or the
information may spread back and forth between the two nodes,
leading to the values of the two nodes beover evaluated;
2) Node v8is a common neighbor of v1 and v2. The three
nodes make up a triangle. We should get rid of the influence of
the node v8 when evaluating the relative importance of v2 to v1,
because the connection of v1 and v8makes the connection of v2
and v8 no sense to v1.
Figure 1. Simple Network
Then we give the following calculation formula:
pi,j
k+1
=
𝑎𝑖,𝑗 wj
k+1
− ej,i
k
𝑎𝑖 W(𝑘+1) − ei
k
𝑖 = 1,2, ⋯ , 𝑛
𝑗 = 1,2, ⋯ , 𝑛
𝑘 = 1,2, ⋯ , 𝑚
(4)
Where, ai,j is an element in the adjacency matrix A;ai is a
row vector of the i’throw of the adjacency matrix A;W is the
column vector of weights of all nodes in the network;ei,j is the
error generated when calculating the importance of node vj
relative to node vi;ei is the i’th column vector of the matrix
ei,j.Let's discuss how to getei,j, for every node of every
network, we can list the following formula:
ej,i
（k）
= pi,j
（k）
wi
（k）
− ui
′
+ ai,tpt,j
k
wt
k
− ut
′
(5)n
t=1
Plug ei,jfrom Eq.(5) into Eq. (4), calculate the value of
pi,j
k+1
, and plug it into Eq.(3) to get a new weight of the node.
III. EXPERIMENTAL RESULTS
In order to facilitate the analysis of the performance of
different network node importance evaluation methods, this
paper analyzes two aspects. The first one is the ability for
distinguishing the importance degree of each node in different
types of network. The second is the accuracy in searching the
specific node with the maximum degree of
importance.Therefore, the definition of the importance of
network node is given as the following words:
Definition: the specific value of the number of nodes with
different importance degree and the number of nodes in the
whole network, which is called the discrimination degree of
the algorithm in calculating the network.
I =
𝑁𝑑
N
(6)
N is the total number of nodes in the network;Nd is the
number of nodes with different importance evaluation.
For complex networks, if not for a particular purpose, it is
strictly to prove which one or several nodes arethe most
important ones in the network.So this papersuggests the

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method of "vote" to determine the most important node in the
network:
Definition: evaluate the node in the network with several
methods. Each method will select one or more nodes to be the
most important nodes in the network. And the node which is
selected by the most methods should be voted to be the most
valuable one.Then according to the outcome of the vote, in
turn, calculate the voting accuracy of each algorithm.Voting
accuracy calculation formula:
C =
𝑛 𝑐
𝑁 𝑐
𝑛 𝑣
(7)
In Eq.(7), C is the voting accuracy of
algorithm;ncrepresents the number of nodes that the algorithm
"vote" for and is elected as most important ones in the
network;Nc is the number of the most important nodes in the
network;nv is the total number of this algorithm votes for.
Figure 2shows the network of ARPA(Advanced Research
Project Agency), which is a commonly used network topology
in testing the evaluation algorithms.The network is
experimentally studied by Closeness Centrality, Contraction
method, Deletion method, algorithm of literature [10] and
Ripple algorithm.
Figure 2. Network Topologyof ARPA
TABLE I. THE NODE IMPORTANCE EVALUATION RESULTS OF
ARPANETWORK TOPOLOGY
node Closeness Contraction Deletion literature[10] Ripple
v1 0.0127 0.1270 0.6262 0.1528 2.2295
v2 0.0149 0.2514 0.9721 0.2987 10.1247
v3 0.0179 0.3080 0.9930 0.2984 10.5049
v4 0.0159 0.1911 0.8387 0.1562 2.4741
v5 0.0147 0.1911 0.8387 0.1090 1.2307
v6 0.0147 0.2550 0.9836 0.1261 1.2167
v7 0.0127 0.1835 0.8797 0.0935 1.1356
v8 0.0115 0.1835 0.8797 0.0634 1.1304
v9 0.0116 0.1835 0.8797 0.0624 1.1303
v10 0.0127 0.1835 0.8797 0.0680 1.1308
v11 0.0143 0.1835 0.8797 0.1062 1.1370
v12 0.0169 0.2615 0.9780 0.1815 1.2300
v13 0.0159 0.1911 0.8051 0.1839 1.1929
v14 0.0159 0.2754 0.9864 0.2369 1.9184
v15 0.0139 0.1855 0.8787 0.2522 2.7991
v16 0.0137 0.1255 0.6639 0.1978 2.5803
v17 0.0154 0.1484 0.6977 0.2214 2.7185
v18 0.0167 0.1667 0.7701 0.1970 2.5061
v19 0.0172 0.2308 0.9671 0.1845 1.3303
v20 0.0149 0.1499 0.8279 0.1115 1.1435
v21 0.0135 0.1499 0.8279 0.1023 1.1364
the number shown in bold is the maximum value;
the underlined value is nodes has one or more similar node with the same value.
the number in brackets is the ranking of the node's importance in the network.
Have a look at the experimental results, literature [10] and
the ripple algorithm have the maximum discrimination
degree.Closeness, Contraction and Deletion method
discriminate the nodes into fewer parts, with 16, 15 and 15
different important values respectively.In terms of voting
accuracy, four of the five methods consider node v3 as the
most important node. Only the literature[10] select node v2 as
the most important node, turning out that the voting accuracy
of it is low.
TABLEⅡshows the discrimination of node importance
evaluation of the five node importance evaluation method.
TABLE Ⅲ shows the node with the max valueof node
importance evaluation and accuracy of voting respectively of
the five node importance evaluation method.
TABLE II. THE DISCRIMINATION OF NODE IMPORTANCE EVALUATION
Network
Topology
Clo. Con. Del.
Lit.
[10]
Rip.
ARPA 0.7619 0.7143 0.7143 1 1
TABLE III. THE NODE WITH THE MAX VALUEOF NODE IMPORTANCE
EVALUATION AND ACCURACY OF VOTING
Network
Topology
Clo. Con. Del.
Lit.
[10]
Rip.
ARPA
V3 V3 V3 V2 V3
1 1 1 0 1
TABLEⅡ and TABLE Ⅲshow the compares of the
discrimination degreeand voting accuracy of the algorithms
experimented on ARPA network.The literature [10] and
ripples have a good perform at discrimination, but the
literature [10] performs worse at voting accuracy. Therefore,
considering algorithm at discrimination degree and voting
accuracy, the ripple algorithm has the best performance in this
experiment.
IV. ALGORITHM PERFORMANCE ANALYSIS
We conduct experiments on simple network, mesh
network, fat tree network, ARPA network and random
network of different sizes, and obtained the following data:
TABLE IV. THE AVERAGE DISCRIMINATION OF NODE IMPORTANCE
EVALUATION
Algorithm Clo. Con. Del. Lit. [10] Rip.
Average
Discrimination
0.629 0.634 0.467 0.667 0.771
TABLE V. THE NODE WITH THE MAX VALUEOF NODE IMPORTANCE
EVALUATION AND AVERAGE ACCURACY OF VOTING
Algorithm Clo. Con. Del. Lit. [10] Rip.
Average
Voting
Accuracy
0.667 0.667 0.417 0.833 0.936
TABLE Ⅳshows that, the average discrimination degree of
the Ripple algorithm is the highest in these 5 methods, and the
Betweenness method is the lower, and the Deletion method is
the lowest.As can be seen from TABLE Ⅴ, the Ripple
algorithm has the highest accuracy in searching the most
important nodes in the network, the Betweenness method is
the lower, and the Deletion is the least accurate. So the
conclusion of the two forms is, Ripple algorithm perform the
best, followed by the method of Betweenness, Closeness and
Contraction method, Deletion method and literature [10]
algorithm integrated performance is poorer.The calculation
speed of the Ripple algorithm and theBetweenness method is
compared in the following figure.

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Figure 3. the Elapsed Time of the Algorithm Varies With the Network
Diameter
Figure 3 shows the elapsed time of the Ripple and the
Betweenness algorithm varies with the network diameter.The
elapsed time of the Betweenness method increases
exponentially with the increase of network diameter.The
elapsed time of the ripple algorithm increases linearly with the
increase of network diameter.
Based on the experimental data above, the Ripples
algorithm can balance the location of the node in the network,
local importance, bridge characteristics and the contribution of
the network robustness index to evaluate network node
importance degree. So the Ripple algorithms has several
advantages such as high degree of discrimination, excellent
selection accuracy of importance node, short elapsed time and
wide applicable scope.
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