Identification of influential nodes in network of networks

0.17 MB·English

by  Meizhu Li

Identification of influential nodes in network of networks
Meizhu Li,1 Qi Zhang,1 Qi Liu,2,3 and Yong Deng1,4,5,∗
1School of Computer and Information Science, Southwest University, Chongqing 400715, China
2Center for Quantitative Sciences, Vanderbilt University School of Medicine, Nashville, TN 37232, USA
3Department of Biomedical Informatics, Vanderbilt University School of Medicine, Nashville, TN 37232, USA
4School of Automation, Northwestern Polytechnical University, Xian, Shaanxi 710072, China
5School of Engineering, Vanderbilt University, Nashville, TN, 37235, USA
(Dated: January 26, 2015)
Thenetworkofnetworks(NON)researchisfocusedonstudyingthepropertiesofninterdependent
networks which is ubiquitous in the real world. Identifying the influential nodes in the network of
networks is theoretical and practical significance. However, it is hard to describe the structure
property of the NON based on traditional methods. In this paper, a new method is proposed to
5 identifytheinfluentialnodesinthenetworkofnetworksbaseontheevidencetheory. Theproposed
1 method can fuse different kinds of relationship between the network components to constructed a
0 comprehensivesimilaritynetwork. Thenodeswhichhaveabigvalueofsimilarityaretheinfluential
2 nodes in the NON. The experiment results illustrate that the proposed method is reasonable and
significant.
n
a
J PACSnumbers: 89.20.-a,05.10.-a,02.50.-r,02.10.-v
Keywords: Networkofnetworks, Evidencetheory, Influential nodes
3
2

I. INTRODUCTION The rest of this paper is organised as follows. Section
]
I II introduces some preliminaries of this work. In section
S
III, a new method is proposed to identify the influential
Complex networksdescribe a wide rangeofsystemsin
.
s natureandsociety,ithasbeenwidelyusedinmanyfields nodes in the network of network. The application based
c
on the cancer gene expression networks is illustrated in
[ [1–3]. In the real world, a large amount of systems can
be described as the complex networks,such as the inter- section IV. Conclusion is given in Section V.
1
net, airline routes, electric power grids and the protein
v
interaction networks. The function of all these networks
4 II. PRELIMINARIES
relies on the connectivity between the network compo-
1
7 nents. However in the real world, numbers of networks
A. The network of networks
5 havethepropertythatthenodesofthenetworkhavedif-
0 ferent kinds of relationship based on different principles,
1. such as the protein interaction networks and the cancer Thenetworkofnetworks(NON),sometimescalledmul-
0 gene expression network [4–6]. This kind of networks is tilayer networks or multiplex, has attracted more and
5 called the network of networks (NON) [7]. moreattention. Duetothefastgrowthofthisfield,there
1 Compared to the traditional complex networks with are many definitions of different types of NON, such as
v: singlerelationship,the networkofnetworksis morediffi- interdependent networks, interconnected networks, mul-
i cult to illuminate the structure property of it [8–10]. In tilayerednetworks,multiplex networks andso on. There
X
the NON, identification of the influential nodes is theo- exist many datasets that can be represented as NON,
r retical and practical significance. In this paper, a new such as flight networks, reliway networks and road net-
a
method is proposed to identify the influential nodes in works,networkofbiologicalnetworksincludinggenereg-
the network of networks based on the evidence theory. ulationnetworks,metabolic networkandprotein-protein
Dempster-Shafer theory of evidence [11, 12], is used interacting network.
to deal with uncertain information and has been widely In this paper, we focus onthe networkwhich have the
used in many fields [13–17]. Here the combination rules same nodes and different kinds of relationship between
of evidence theory are used to fuse the influence of each the components in the networks.
node in different single networks. The nodes which have
a big value of similarity are the influential nodes in the
B. The evidence theory
NON. One of the advantages of the proposed method
is that the more the type of the interrelation between
networks components is, the more accurate the results Dempster-Shafer theory [11, 12] is often regarded as
are. anextension ofthe bayesiantheory. For completeness of
the explanation, a few basic concepts are introduced as
follows.
Definition 1. Let Ω be a set of mutually exclusive and
∗ [email protected];[email protected] collectively exhaustive, indicted by
2
the structure of it becomes more complex. Based on the
Ω={E1,E2,···,Ei,···,EN} (1) evidence theory, a new method is proposed to identify
the influential nodes in NON by making a combination
The set Ω is called frame of discernment. The power among the networks divided from the NON.
set of Ω is indicated by 2Ω, where
Ω
2 ={∅,{E1},···,{EN},{E1,E2},···,Ω} (2)
If A∈2Ω, A is called a proposition.
Definition 2. For a frame of discernment Ω, a mass
function is a mapping m from 2Ω to [0,1], formally de- FIG. 1. Division of NON.
fined by:
As shown in the subfigure (a) of Fig.1, our research
Ω
m: 2 →[0,1] (3) is focus on the NON which has the same nodes but dif-
ferent edges. In the proposed method, one NON can be
which satisfies the following condition: divided into numbers of single networks based on differ-
entprinciples,whichisshowninthesubfigre(b)ofFig.1.
BasedonthesinglenetworksdividedfromNON,aseries
m(∅)=0 and m(A)=1 (4) of similarity networks can be established. According to
AX∈2Ω the similarity networks, a comprehensive networks can
be constructed by fusing the similarity networks based
on the combination rules of evidence theory. Compared
In Dempster-Shafer theory, a mass function is also
toothernodesinthe comprehensivenetworks,thenodes
called a basic probability assignment (BPA).
have a large value of similarity are the influential nodes
ConsidertwopiecesofevidenceindicatedbytwoBPAs
in the NON.
m and m on the frame of discernment Ω, Dempster’s
1 2 Tointroducetheproposedmethodindetails,foursteps
rule of combination is used to combine them. This rule
areessentialintheidentificationresearch,whichisshown
assumes that these BPAs are independent.
as follows.
Definition 3. Dempster’s rule of combination, also
Step 1: Based on the different significance of the edges
called orthogonal sum, denoted by m = m ⊕m , is de-
1 2 , divide the NON into numbers of single networks.
fined as follows
Step 2: According to the distance between each node,
1
1−K m1(B)m2(C), A6=∅; establish the distance matrix D of the single net-
m(A)=(cid:26)0, B∩PC=A A=∅. (5) works. Each single network has a distance matrix
D to describe the similarity between each node.
with
The details of the distance matrix D are shown as
follows:
K= m1(B)m2(C) (6)
BX∩C=∅ d ... d
11 1n
where B and C are also elements of 2Ω, and K is a D = ... ... ... 
constant to show the conflict between the two BPAs. dn1 ··· dnn 
 
Note that the Dempster’s rule of combination is only
applicable to such two BPAs which satisfy the condition
d =max(d ),(1≤i≤n,1≤j ≤n,)
K <1. max ij
Where d is the shortest distance between node i
ij
and node j.
III. NEW METHODS TO IDENTIFY THE
INFLUENTIAL NODES IN THE NETWORK OF Based on the distance matrix, the similarity net-
NETWORKS work SN can be defined as follows:
S ... S
To identify the influential nodes in the complex net- 11 1n
works is one of the important directions in network sci- SN = ... ... ... 
ence. In the single network, many methods have been  Sn1 ··· Snn 
proposed to identify the influential nodes, such as the  
degree centrality, the betweenness centrality, the local
d
structure entropy and so on. However,in the network of S =1− ij
ij
networks,morethantwonetworksdependoneachother, dmax
3
Step 3: Accordingtosimilaritynetwork,thebasicprob- Thesimilaritymatrixofthesinglenetwork(c):
abilityassignment(BPA)canbeconstructed,which
1.00 0.51 0.51 0.75 0.26 0.51 0.51 0.26 0.01 0.26
isanessentialconceptintheevidencetheory. Each 0.51 1.00 0.51 0.75 0.26 0.51 0.51 0.26 0.01 0.26
 
elementinthesimilaritynetworkhasacorrespond- 0.51 0.51 1.00 0.75 0.26 0.51 0.51 0.26 0.01 0.26
ing BPA, which is defined as follows.  0.75 0.75 0.75 1.00 0.51 0.75 0.75 0.51 0.26 0.51
 0.26 0.26 0.26 0.51 1.00 0.26 0.75 0.51 0.26 0.51
Definition 4. Given an N ×N similarity network SN,  0.51 0.51 0.51 0.76 0.26 1.00 0.51 0.26 0.01 0.26
theframe of discernment of thenetwork is {Y,N},where  0.51 0.51 0.51 0.76 0.76 0.51 1.00 0.76 0.51 0.75
 
Y represents similarity and N represents dissimilarity.  0.26 0.26 0.26 0.51 0.51 0.26 0.76 1.00 0.26 0.51
The BPA of element SNij is:  00..2061 00..2061 00..2061 00..5216 00..5216 00..2061 00..7561 00..5216 01..7060 10..0706
mij(Y)= |SNij−min(SN)| (7) Thesimilaritymatrixofthesinglenetwork(d):
SUMM
1.00 0.34 0.84 0.51 0.01 0.67 0.17 0.01 0.51 0.34
0.34 1.00 0.51 0.84 0.01 0.67 0.17 0.01 0.51 0.34
 
|SNij−max(SN)| 0.84 0.51 1.00 0.67 0.17 0.84 0.34 0.17 0.67 0.51
mij(N)= (8) 0.51 0.84 0.67 1.00 0.17 0.84 0.34 0.17 0.67 0.51
SUMM  
 0.01 0.01 0.17 0.17 1.00 0.34 0.84 0.67 0.51 0.67
 0.67 0.67 0.84 0.84 0.34 1.00 0.51 0.34 0.84 0.67
 
|SNij−(max(SN)+min(SN))/2|  0.18 0.18 0.34 0.34 0.84 0.51 1.00 0.84 0.67 0.84
mij(Y,N)= (9)  
SUMM  0.01 0.01 0.17 0.17 0.67 0.34 0.84 1.00 0.51 0.67
 0.51 0.51 0.67 0.67 0.51 0.84 0.67 0.51 1.00 0.84
Where max(SN) represents the maximum element in  0.34 0.34 0.51 0.51 0.67 0.67 0.84 0.67 0.84 1.00
the similarity network, except the diagonal elements.
According to Definition 4, the BPA of eachelement in
min(SN) represents the minimum element in the sim-
the similarity networks can be constructed. Then using
ilarity network.
the combination rules of evidence theory, fuse the corre-
spondingBPAofeachelementinthesimilaritynetworks.
Basedonthecombinationrulesoftheevidencethe- Here an example is shown to fuse the element SN of
34
ory,fusetheBPAofcorrespondingelementsinsim- network (b), network (c) and network (c).
ilaritynetworkintoacomprehensivesimilaritynet- Based on the three single networks above, the values
work. of the element SN in the networks can be shown as
34
follows.
Step 4: The nodes which have a big value of similarity SNb =0.29, SNc =0.75, SNd =0.67.
34 34 34
withothernodesinthefusedsimilaritynetworkare According to Definition 4, the BPA of elements SN
34
the influential nodes in the NON. in the networks can be constructed.
mb34(Y)=0.2814, mb34(N)=0.5729, mb34(Y,N)=0.1457.
Hereanetworkofnetworks,whichisshowninFig.2,is mc34(Y)=0.6637, mc34(N)=0.0090, mc34(Y,N)=0.3273.
constructed to show the details of the combination pro- md34(Y)=0.6140, md34(N)=0.1581, md34(Y,N)=0.2279.
Aftercombinationbasedonevidencetheory,thefusion
cess.
result of element SN is:
34
The NONin the Fig.2 canbe dividedinto three single
m34(Y)=0.7874, m34(N)=0.1878, m34(Y,N)=0.0248.
networks. The details of the network (b), network (c)
So the value of similarity between node 3 and node
and network (d) are shown as follows. 4 in the network and networks is 0.7874. The order of
the influential nodes in the example networks is shown
Vb=Vc=Vd={1,2,3,4,5,6,7,8,9,10} in Table I:
Eb={{1,2},{1,3},{3,6},{4,5},{5,8},{6,7},{7,8},{7,9},{8,10}}
Ec={{1,4},{2,4},{3,4},{4,6},{4,7},{5,7},{7,8},{7,10},{9,10}}
Ed={{1,3},{2,4},{3,6},{4,6},{5,7},{6,9},{7,8},{7,10},{9,10}} TABLE I. The influential nodes in the example network of
networks
The similarity matrix of the three single networks is Order number 1 2 3 4 5 6 7 8 9 10
shown as follows. Node number 7 6 4 10 3 9 8 5 1 2
The similarity matrix of the single network (b):
1.00 0.86 0.86 0.15 0.29 0.72 0.58 0.43 0.43 0.29
0.86 1.00 0.72 0.01 0.15 0.58 0.43 0.29 0.29 0.15
  IV. APPLICATION OF THE NEW METHOD IN
0.86 0.72 1.00 0.29 0.43 0.86 0.72 0.58 0.58 0.43
THE CANCER GENE EXPRESSION NETWORKS
0.15 0.01 0.29 1.00 0.86 0.43 0.28 0.72 0.43 0.58
 
 0.29 0.15 0.43 0.86 1.00 0.58 0.72 0.86 0.58 0.72 
 0.72 0.58 0.86 0.43 0.58 1.00 0.86 0.72 0.72 0.58  In order to illuminate the usefulness of the pro-
 0.58 0.43 0.72 0.58 0.72 0.86 1.00 0.86 0.86 0.72 
  posedmethod, fourcancergeneexpressionnetworks,the
 0.43 0.29 0.58 0.72 0.86 0.72 0.86 1.00 0.72 0.86 
 0.43 0.29 0.58 0.43 0.58 0.72 0.86 0.72 1.00 0.58  glioblastomamultiforme(GBM),thebreastinvasivecar-
 
 0.29 0.15 0.43 0.58 0.72 0.58 0.72 0.86 0.58 1.00  cinoma(BIC),thekidneyrenalclearcellcarcinoma(KR-
CCC) and the lung squamous cell carcinoma (LSCC),
4
SUMM =|SNij−max(SN)|+|SNij−min(SN)|+|SNij−(max(SN)+min(SN))/2| (10)
(a)GBM (b)BIC
FIG.2. ExampleofNONandthecorrespondingdiviedsingle
networks.
have been applied as cases. Each cancer gene expression
networkhasthreekindsofexpressionnetworks,theDNA
methylationnetwork,themRNAexpressionnetworkand
(c)KRCCC (d)LSCC
the miRNA expression network.
Thenodesinthenetworksrepresentthepatients. The
relationship in these expression networks are the simi- FIG. 3. Influentialnodes of four NON
larity between each patient. Based on the definition of
thenetworkofnetworks,thefourcancergeneexpression
to deal with uncertain information and has been widely
networks can treated as four NON. In order to find the
used in many fields. In this paper the combination rules
influential patients in the NON, a comprehensive simi-
of evidence theory are used to fuse the influence of each
larity expression network is constructed by the combi-
node in different single networks. The proposed method
nation rules of evidence theory. Each NON in this case
can fuse different kinds of relationship between the net-
hasthreesinglenetworks,theDNAmethylationnetwork,
work components to constructed a comprehensive simi-
the mRNA expression network and the miRNA expres-
laritynetwork. Thenodeswhichhaveabigvalueofsim-
sion network.
ilarity are the influential nodes in the NON. One of the
The experiment results are shown in the Table II and
advantages of the proposed method is that the more the
Fig. 3. In the Table II, the number of most influential
type of the interrelation between networks components
nodes in the four NON are shown. For example, in the
is, the more accurate the results are. The experiment
network GBM, the first influential node is the node 116.
results illustrate that the proposedmethod is reasonable
IntheFig. 3,thenodeswithbiggersizeanddeepercolor
and significant.
arethemoreinfluentialnodes. Theresultsillustratethat
the proposed method is reasonable and significant.
VI. ACKNOWLEDGMENT
TABLE II. The influential nodes in the four cancer gene ex-
pression networks
Ordernumber 1 2 3 4 5 6 7 8 9 10 The work is partially supported by National High
GBM 116 60 190 179 50 42 139 72 194 209 Technology Research and Development Program of
BIC 106 34 71 49 68 7 55 76 100 51 China (863 Program) (Grant No. 2013AA013801),
KRCCC 104 15 113 117 110 50 41 96 77 87 National Natural Science Foundation of China (Grant
LSCC 37 90 91 51 93 33 88 28 92 38 No. 61174022), Specialized Research Fund for
the Doctoral Program of Higher Education (Grant
No. 20131102130002), R&D Program of China
(2012BAH07B01), the open funding project of State
V. CONCLUSION Key Laboratory of Virtual Reality Technology and Sys-
tems, Beihang University (Grant No.BUAA-VR-14KF-
Many real systems in the real world can be treated as 02). Fundamental Research Funds for the Central
thenetworkofnetworks. Identifyingtheinfluentialnodes Universities No. XDJK2015D009. Chongqing Grad-
in the network of networks is theoretical and practical uate Student Research Innovation Project (Grant No.
significance. Dempster-Shafer theory of evidence is used CYS14062).
5
[1] M. E. Newman, SIAMreview 45, 167 (2003). Z.Wang, andM.Zanin,PhysicsReports544,1(2014).
[2] M. Newman, A.-L. Barab´asi, and D. J. Watts, The [9] F.Battiston,V.Nicosia, andV.Latora,PhysicalReview
structure and dynamics of networks (Princeton Univer- E 89, 032804 (2014).
sity Press, 2006). [10] Q. Zhang, C. Luo, M. Li, Y. Deng, and S. Mahade-
[3] D.J. Wattsand S. H. Strogatz, nature 393, 440 (1998). van, Physica A: Statistical Mechanics and its Applica-
[4] V.Nicosia,G.Bianconi, V.Latora, andM.Barthelemy, tions 419, 707 (2015).
Physical review letters 111, 058701 (2013). [11] A. P. Dempster, The annals of mathematical statistics
[5] B. Wang, A. M. Mezlini, F. Demir, M. Fiume, Z. Tu, 38, 325 (1967).
M.Brudno,B.Haibe-Kains, andA.Goldenberg,Nature [12] G. Shafer, A mathematical theory of evidence, Vol. 1
methods11, 333 (2014). (Princeton university press Princeton, 1976).
[6] D. Y. Kenett, J. Gao, X. Huang, S. Shao, I. Vodenska, [13] I. Bloch, Pattern Recognition Letters 17, 905 (1996).
S.V.Buldyrev,G.Paul,H.E.Stanley, andS.Havlin,in [14] F. Cuzzolin, Systems, Man, and Cybernetics, Part C:
Networks of Networks: The Last Frontier of Complexity ApplicationsandReviews,IEEETransactionson38,522
(Springer,2014) pp. 3–36. (2008).
[7] J. Gao, S. V. Buldyrev, S. Havlin, and H. E. Stanley, [15] T. Denœux,Fuzzy sets and systems 183, 72 (2011).
Physical Review Letters 107, 195701 (2011). [16] Y. Deng, R. Sadiq, W. Jiang, and S. Tesfamariam, Ex-
[8] S. Boccaletti, G. Bianconi, R. Criado, C. Del Genio, pert Systemswith Applications 38, 15438 (2011).
J. Go´mez-Garden˜es, M. Romance, I. Sendin˜a-Nadal, [17] M. Li, X. Lu, Q. Zhang, and Y. Deng,
Mathematical Problems in Engineering 2014, Article ID 319264, 6 pages