Chaos, Solitons and Fractals 110 (2018) 20–27

Contents lists available at ScienceDirect

Chaos, Solitons and Fractals

Nonlinear Science, and Nonequilibrium and Complex Phenomena

journal homepage: www.elsevier.com/locate/chaos

Frontiers

Evolving the attribute flow for dynamical clustering in signed

networks

Hui-Jia Li a , e , ∗, Zhan Bu

b , Yulong Li a , Zhongyuan Zhang

c , Yanchang Chu

d , e , Guijun Li a , Jie Cao

b

a School of Management Science and Engineering, Central University of Finance and Economics, Beijing 10 0 080, China b Jiangsu Provincial Key Laboratory of E-Business, Nanjing University of Finance and Economics, Nanjing 210 0 03, China c School of Statistics and Mathematics, Central University of Finance and Economics, Beijing 10 0 080, China d Economics and Management College, Civil Aviation University Of China, Tianjin 30 030 0, China e The Research Center of Beijing-Tianjin-Hebei Civil Aviation Coordinate Development, Civil Aviation University Of China, Tianjin 30 030 0, China

a r t i c l e i n f o

Article history:

Received 28 October 2017

Revised 7 February 2018

Accepted 8 February 2018

Keywords:

Signed networks

Attribute flow

Clustering algorithm

Dynamical systems

Convergence and divergence

a b s t r a c t

In real networks, clustering is of great value to the analysis, design, and optimization of numerous com-

plex systems in natural science and engineering, e.g. power supply systems ,modern transportation net-

works, and real-world networks. However, the majority of them simply pay attention to the density of

edges rather than the signs of edges as the attributes to cluster, which usually suffer a high-level com-

putational complexity. In this paper, a new rule is proposed to update the attributes flow, which can

guarantee network clustering reach a state of optimal convergence. The positive and negative update rule

we introduced, represent the cooperative and hostile relationship, and the attribute configuration will con-

vergence and one can identify the reasonable cluster configuration automatically. An algorithm with high

efficiency is proposed: a nearly linear relationship is found between the time complexity and the size in

sparse networks. Finally, we conduct the verification of the algorithmic performance by a representative

simulations on Correlates of War data.

© 2018 Elsevier Ltd. All rights reserved.

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1. Introduction

In the real world, social networks are used in modeling of

numerous complex systems [1–3] . We can use G = { V, E} as a

graph to define a social network, where the set of vertices is de-

fined as V = { 1 , . . . , n } , and the set of edges which connecting

pairs of vertices is defined as E . An example is that vertices indi-

cate agents/individuals ,and edges indicate relations/links between

nodes in an interpersonal network. The signed social networks,

which is used to describe the social networks with positive and

negative links, where “friendship”as an example of “positive rela-

tionship” is denoted by positive links ,and equally a “negative re-

lationship” such as “hostility” may be indicated by negative links

[4–12] . An example can be cited in the Gahuku-Gama subtribes

network is that the “political alliance relation” are represented by

the positive links, while “political opposition relation” are repre-

sented by the negative links [13] . We can create the signed net-

∗ Corresponding author at: School of Management Science and Engineering, Cen-

tral University of Finance and Economics, Beijing 10 0 080, China.

E-mail addresses: hjli@amss.ac.cn , lihuu20 0 0@126.com (H.-J. Li), buzhan@

nuaa.edu.cn (Z. Bu), caojie690929@163.com (J. Cao).

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https://doi.org/10.1016/j.chaos.2018.02.009

0960-0779/© 2018 Elsevier Ltd. All rights reserved.

ork by the relationships,and obtain more learning about the at-

ributes of the social networks such as the cluster configuration

14,15] from the analysis and mining on it. Clusters in networks

efer to the phenomenon when nodes of the network can be natu-

ally grouped into sets such that each set is densely connected in-

ernally, thus dividing the network into smaller groups with dense

nternal connections and sparser external connections. The links

hose density and signs combine to define the signed network

lusters. Qualitatively, clusters in signed network are defined as

ubgraphs that the positive links are within the nodes in each

roup and the negative link are connected between the different

roups of nodes [16,17] . For the discovery of the hidden cluster

onfiguration, it is anything but simple to find the optimal and

teady partition of the network.

Despite many clustering techniques have been presented for

nalysis in the field of complex network, the majority of them sim-

ly pay attention to the density of edges rather than the signs of

dges as the attributes to cluster [18–20] . If an accuracy within ac-

eptable limits is obtained based on comparison between the inter-

al and external cohesion of a subgraph by the traditional heuris-

ic methods, the complexity of computation is usually high-level

21,22] . In this paper, a new rule is proposed to update attributes,

hich can guarantee network clustering reach a state of optimal

H.-J. Li et al. / Chaos, Solitons and Fractals 110 (2018) 20–27 21

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onvergence. By introducing the positive and negative update rule

o represent the cooperative and hostile relationship [23] , one can

rove the convergence and divergence of the attribute evolution

n mean and find its conditions. Unlike the heuristic method, the

umber clusters is not specified by user, meanwhile, the reason-

ble partition can be automatically distinguished by it. An algo-

ithm with high efficiency is proposed: a nearly linear relation-

hip is found between the time complexity and the size in sparse

etworks. Finally, we conduct the verification of the algorithmic

erformance by a representative simulations on Correlates of War

ata.

. Materials and methods

.1. Signed network

An undirected connected social network without self loops can

e considered as G = (V, E) , where the set of vertices is defined

s V = { 1 , . . . , n } , and the set of edges which connecting pairs

f vertices is defined as E . For a signed network , we mark these

ositive and negative edges with a plus sign “+ ” and a minus

ign “−” respectively in E , where “+ ” indicates a collaboration or

riend relationship and “−” indicates an opponent or enemy re-

ationship. In order to analyse, E is divided into the positive and

egative edge collections,which are separately denoted as E pst and

neg . The positive and the negative networks are separately de-

oted as G pst = (V, E pst ) and G neg = (V, E neg ), where E pst ∩ E neg = Ø

nd E pst ∪ E neg = E. In General, it can be considered that the neg-

tive edge collection E neg is nonempty. For signed network G , the

ollowing definition is used to define a k -clusters configuration.

efinition 1. Let’s call V = V 1 ∪ V 2 . . . ∪ V k is k -clusters configura-

ion in a signed network G . The label of each edge within V i or

etween different V i is positive or negative, respectively. Here, ev-

ry V i is nonempty and any two different V i satisfies disjoint.

Intuitively, cutting all the negative links, which makes dis-

inguishing clusters in partitionable or balanced signed networks

ompleted easily. The positive links will be merely included in the

ubgraphs we achieved and clusters will take shape. However, the

iscrimination assignment becomes extraordinary owing to some

cenarios : 1) the signed social networks cannot be partitioned and

) despite it is feasible to cut the signed social networks by parti-

ioning, merely cutting negative links cannot achieve the optimal

artition or the most natural partition. Actually before cutting out

ll negative links, some large subgraphs with some isolated nodes

rising from large subgraphs in a great number. In order to iden-

ify more natural clusters, the steady partitions of subgraphs in an

ncreasing number are supposed to be reasonably ignored as well

s keeping some positive and negative links (or reasonably reduce

heir effect).

What we find especially interesting is that the definition of

luster configuration can be related to the balance theory in signed

etwork [23,24] .

efinition 2. A signed graph is defined as G = (V, E) . Then

(i) G is a weak equilibrium if it satisfies that there is an integer

≥ 2 and a k -way partition V = V 1 ∪ V 2 . . . ∪ V k , where V 1 , . . . , V k are

onempty and disjoint with each other, in this way any edge be-

ween different V i or within each V i is negative or positive, respec-

ively.

(ii) G is a strong equilibrium when it satisfies that itself is a weak

quilibrium and k = 2 .

.2. Attribute dynamics

The attribute , or opinion of the nodes can be denoted as a real

nd scalar value when the nodes initiate interactions. For each

ode at time k , its attribute vector is defined as x (k ) ∈ R n . The at-

ribute of nodes will be updated according to the situation that

odes interact with their collaborators or opponents. Particularly,

nly two nodes { i, j } are chosen and the following rules are used

o update their attribute at each time k .

• ( Positive Update Rule ) If { i, j } ∈ E pst , we update the attribute of

ode m ∈ { i, j } as

m

(k + 1) = x m

(k ) + α(x −m

(k ) − x m

(k ))

= (1 − α) x m

(k ) + αx −m

(k ) , (1)

here −m ∈ { i, j}\{ m } and 0 ≤α ≤ 1.

• ( Negative Update Rule ) If { i, j } ∈ E neg , we update the attribute of

ode m ∈ { i, j } as

m

(k + 1) = x m

(k ) − β(x −m

(k ) − x m

(k ))

= (1 + β) x m

(k ) − βx −m

(k ) , (2)

here β ≥ 0

For the positive update rule, nodes update their attributes

ased on the previous attributes of nodes and of their neighbors

hich are regarded as a convex combination. The cooperative or

nsuspecting relationships are shown naturally in this update. The

ositive update rule, considered as the attraction of the attributes,

hich tends to drive node attributes closer to each other.

On the other hand, there are controversy over the dynamics on

he negative edges in the literature. Substantial efforts have been

aken to characterize these suspecting or hostile relationships. The

roposed negative update rule, is the contrary of the positive up-

ate rule, which enforces attribute differences between interacting

odes. Note that the negative update rule satisfies the following

laborations:

• Node i tries to trick her negative neighbors j , by turning to

he opposite sign of her true attribute (i.e., x i ( k ) to −x i (k ) ) before

howing it to j ;

• Node i distinguishes j as her negative neighbor and upon ob-

erving x j ( k ) which is j ’s true attribute, she attempts to get closer

o the opposite view of j since x i (k + 1) is a convex combination of

i ( k ) and −x j (k ) .

.3. The mean convergence and divergence

Let define the (random)vector of attributes at time k resulting

rom the node interactions as x (k ) = (x 1 (k ) , . . . , x n (k )) , k = 0 , 1 , . . . .

he initial attributes x (0) is also denoted as x 0 and deemed to be

eatured in determinacy. In this section, we make a thorough in-

estigation into the mean evolution of the attributes. We present

he following definition.

efinition 3. (i) The expected attribute convergence is obtained if

im k →∞

E { x i (k ) − x j (k ) } = 0 for all i and j .

(ii) The expected attribute divergence is obtained if

im sup k →∞

max i, j | E { x i (k ) − x j (k ) }| = ∞ .

.3.1. Node pair selection

The actual interactions are selected using the following model:

odes interact with each other in the moment of a rate-one Pois-

on process and a node was selected randomly to interact with the

thers in each of these moments. Under this model, only one node

r none originates an interaction at a given time. Ordering interac-

ion events promptly and concentrating on modeling the node pair

hich is selected at interaction times. The node selection process

s characterized by an n × n stochastic matrix P = [ p i j ] , complying

ith the graph G in the sense that p ij > 0 always implies { i, j } ∈ for i = j ∈ V . We use p ij to denote the probability that node i in-

eracts with node j immediately after the node pair selection is

xecuted as follows.

22 H.-J. Li et al. / Chaos, Solitons and Fractals 110 (2018) 20–27

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Definition 4. At each interaction event k ≥ 0,

(i) a node i ∈ V is drawn uniformly at random, i.e., with proba-

bility 1/ n ;

(ii) node i picks node j with probability p ij . In this case, we say

that the unordered node pair { i, j } is selected.

We suppose that the process of selecting node pairs is identical

distribution and independent distribution (i.i.d.); i.e., it is identi-

cally distributed and independent for the nodes to initiate an in-

teraction and select node pairs over k ≥ 0. Ultimately, the follow-

ing probability spaces can analyze the process of selecting nodes.

Define the probability space as (E, R , μ) , where R is the discrete

σ -algebra on E and μ is the probability measure. For all { i, j } ∈ E,

μ is defined by μ({ i, j} ) = (p i j + p i j ) /n . In the product probability

space ( �, R , P ), � = E N = { ω = (ω 0 , ω 1 , . . . , ) : ∀ k, ω k ∈ E} , F = R

N ,

and the unique product probability measure P is defined by the

following: for any finite subset K ⊂ N , P ((ω k ) k ∈ K ) = �k ∈ K μ(ω k ) for

any ∈ E | K | . For any k ∈ N , the coordinate mapping G k : �← E is

defined by G k (ω) = ω k , for all ω ∈ � (note that P (G k = ω k )), and

( G k , k = 0 , 1 , . . . ) is referred to as the node pair selection process.

Further, F k = σ (G 0 , . . . , G k ) is referred to as the σ -algebra captur-

ing the k + 1 first interactions of the selection process. The pro-

cess of selecting nodes can be considered as a random event in

the product probability space.

Based on the process of selecting node pairs, we write the at-

tribute dynamics as

x (k + 1) = W (k ) x (k ) , (3)

where W (k ) , k = 0 , 1 , . . . are i.i.d. random matrices satisfying

P (W (k ) = W

+ i j

: = I − α(e i − e j )(e i − e j ) ′ )

= (p i j + p ji ) /n, { i, j} ∈ E pst ,

P (W (k ) = W

−i j

: = I − β(e i − e j )(e i − e j ) ′ )

= (p i j + p ji ) /n, { i, j} ∈ E neg , (4)

and the m th element of the n -dimensional unit vector e m

=(0 . . . 0 , 1 , 0 . . . 0) ′ is 1. In this section, spectral properties of the lin-

ear system Eq. (3) is used to study convergence and divergence in

the mean manner.

2.3.2. Convergence/divergence conditions

Firstly, convergence conditions are offered, which imply diver-

gence. Then, the existence of a phase transition for convergence

is established by utilizing these conditions when the parameter βof the negative update advances. At the end of this subsection, we

will illustrate these outcomes. The following assumption is general,

which is adopt on account of technical reasons.

Assumption 1. There holds either

(i) p ii ≥ 1/2 for all i ∈ V or

(ii) P = [ p i j ] is doubly stochastic with n ≥ 4.

Denote P † = (P + P ′ ) /n . Then P † = P † pst + P

† neg can be written,

where P † pst and P

† neg represent the positive and negative graphs,

respectively. Particularly, [ P † pst ] = [ P † ] i j if { i, j } ∈ E pst and [ P

† pst ] i j =

0 otherwise, whereas [ P † neg ] i j = [ P † ] i j if { i, j } ∈ E neg and [ P

† neg ] i j =

0 otherwise. Further, the degree matrix D

† pst = diag( d +

1 . . . d + n ) of

the positive graph is proposed, where d + i

=

∑ n j =1 , j = i [ P

† neg ] i j . After

that L † pst = D

† pst − P

† pst and L

† neg = D

† neg − P

† neg indicate the (weighted)

Laplacian matrices of the positive graph G pst , and negative graph

G neg , respectively. From Eq. (4) , it is effortless to infer that

E { W (k ) } = I − αL † pst + βL † neg . (5)

Clearly, 1 ′ E { W (k ) } = E { W (k ) } 1 = 1 where 1 = (1 . . . 1) ′ denotes

the n × 1 vector of all ones, but E { W (k ) } is not necessarily a

stochastic matrix since it may contain negative entries.

We write y i (k ) = x i (k ) − ∑ n s =1 x s (k ) /n and y (k ) =

(y 1 (k ) . . . y n (k ) ′ ) . Define U : 11 ′ / n and note that y (k ) = (1 − U) x (k ) ;

oreover, for all possible realizations of W ( k ), (1 − U) W (k ) = (k )(1 − U) = W (k ) − U . Thus, E { y (k ) } evolves linearly:

{ y (k + 1) } = E { (I − U) W (k ) x (k ) } = E { (I − U) W (k )(I − U) x (k ) } = (E { W (k ) } − U) E { y (k ) } . (6)

The following elementary inequalities

E { x i (k ) − x j (k ) }| ≤ | E y i (k ) | + | E y j (k ) | , | E y i (k ) | ≤ 1

n

E

n ∑

s =1

| x i (k ) − x s (k ) | (7)

ignify that expected attribute convergence is equivalent to

im k →∞

| E { y (k ) }| = 0 , and attribute divergence is equivalent

o lim sup k →∞

| E { y (k ) }| = ∞ . Therefore the spectral radius of

{ W (k ) − U} determines attribute convergence or divergence. With

ssumption 1 , there always holds that d + i

=

∑ n j =1 , j = i [ P

† pst ] i j ≤

n j =1 , j = i (p i j + p ji ) /n ≤ 1 / 2 .

As a result, Gerihgorin’s Circle Theorem [25] ensures that each

igenvalue of I − αL † pst is nonnegative. It then follows that each

igenvalue of I − αL † pst − U is nonnegative since L

† pst U = UL

† pst = 0

nd the two matrices I − αL † pst and U share the same eigenvector 1

or eigenvalue one. Moreover, it is well known in algebraic graph

heory that L † pst and L

† neg are positive semidefinite matrices. As a

esult, Weyl’s inequality [25] further ensures that each eigenvalue

f E { W (k ) } − U is also nonnegative. To summarize, we have shown

he following:

roposition 1. Let Assumption 1 hold. For all initial values, the ex-

ected convergence of attribute can be obtained if λmax (I − αL † pst +

L † neg − U) < 1 ; for almost all initial values, the expected divergence

f attribute can be obtained if λmax (I − αL † pst + βL

† neg − U) > 1 .

Proposition as described above and below, the largest eigen-

alue of the real symmetric matrix M is denoted as λmax (M) ;

y “nearly all initial conditions,” our meaning is that the prop-

rty holds for any initial condition y (0) only when y (0) is com-

letely orthogonal to the eigenspace of E { W (k ) } − U correspond-

ng to its maximal eigenvalue λmax (I − αL † pst + βL

† neg − U) . Hence

he set of initial conditions where the property does not hold has

ero Lebesgue measure.

The Courant–Fischer Theorem [25] implies

(I − αL † pst + βL † neg − U)

= sup

| z=1 | z ′ (I − αL † pst + βL † neg − U) z

= 1 + sup

| z=1 |

[

−α∑

{ i, j}∈ E pst

[ P † ] i j (z i − z j ) 2

+ β∑

{ i, j}∈ E pst

[ P † ] i j (z i − z j ) 2 − 1

n

(

n ∑

i =1

z i

) 2 ⎤

⎦ . (8)

rom Eq. (8) , observing that the impact of G pst and G neg on the

ean attribute convergence/divergence are separated : links in E pst

onduce to attribute convergence, whereas links in E neg conduce to

ttribute divergence.

.3.3. Phase transition

Our next research is that the influence of update parameters

and β on the expected convergence. f (α, β) := λmax (I − αL † pst +

H.-J. Li et al. / Chaos, Solitons and Fractals 110 (2018) 20–27 23

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L † neg − U) is defined. The function f has the following properties

nder Assumption 1:

(i) ( Convexity ) f ( α, β) is the spectral norm of I − αL † pst + βL

† neg −

since both L † pst and L

† neg are symmetric. Because every matrix

orm is convex, we have

f (γ (α1 , β1 ) + (1 − γ )(α2 , β2 ))

≤ γ f (α1 , β1 ) + (1 − γ ) f (α2 , β2 ) (9)

or all γ ∈ [0, 1] and α1 , α2 , β1 , β2 ∈ R . This indicates that f ( α, β)

s convex in ( α, β).

(ii) ( Monotonicity ) From Eq. (8) , in α for fixed β or in β for

xed α, f ( α, β) is nonincreasing or nondecreasing, respectively.

herefore, the fastest convergence can be provided by setting α = 1

henever the expected convergence of attribute is obtained (for a

iven fixed β). It is worth noting that two nodes merely switch

heir attributes when α = 1 and they interact.

When G pst is linked, the second smallest eigenvalue of L † pst is

ositive and defined as λ2 (L † pst ) . f (α, 0) = 1 − αλ2 (L

† pst ) < 1 is eas-

ly seen. From Eq. (8) , we also have f ( α, β) → ∞ as β → ∞ suppos-

ng that G neg is nonempty. We can draw the following conclusion

rom our observations and the monotonicity of f :

roposition 2. We suppose G pst is linked and let Assumption 1 hold.

hen for any fixed α ∈ (0, 1], there exists a threshold value β (that

epends on α) such that

(i) For all initial values, the expected convergence of attribute can

e obtained if 0 ≤β < β ;

(ii) For almost all initial values, the expected divergence of at-

ribute can be obtained if β > β .

We need to point out that attribute divergence can only occur

or nearly all initial values since if the initial attributes of all the

odes are identical, they do not evolve over time.

.4. Dynamical clustering algorithm

.4.1. Asymmetric constrained model

Although this symmetric and unconstrained attribute update

ule in Eqs. (1) and (2) is plausible for ideal social network models,

n reality these assumptions might not hold: when { i, j } is selected,

t might happen that only one of the two nodes in i and j up-

ates its attribute; there might be a hard constraint on attributes:

i (k ) ∈ [ −A, A ] for all i and k and for some A > 0. In this section,

or nodes i and j , a new model is introduced for the updates of the

ttribute:

x i (k + 1) = f A ((1 − θ ) x i (k ) + θx j (k )) and

x j (k + 1) = x j (k ) , with probability a ;x j (k + 1) = f A ((1 − θ ) x j (k ) + θx i (k )) and

x i (k + 1) = x i (k ) , with probability b; m

(k + 1) = f A ((1 − θ ) x m

(k ) + θx m

(k )) ,

m ∈ { i, j} , with probability c. (10)

here

f A (z) =

{ −A, i f z < −A ;z, i f z ∈ [ −A, A ] ;A, i f z > A ;

(11)

ere, let a, b, c > 0 be three positive real numbers such that a + + c = 1 , and define the function θ : E → R so that θ ({ i, j} ) = α if

i, j } ∈ E pst and θ ({ i, j} ) = −β if { i, j } ∈ E neg . Assume that node i in-

eracts with node j at time k , nodes i and j update their attributes

sing Eq. (10) .

Based on the fundamental model of the world, performing the

ttribute within the interval [ −A, A ] can be seen as a decision of

social member. From Eq. (11) , one can notice that the boundary

alue A (or −A ) can be used to determine the size of clusters, i.e.

arger A means larger attribute boundary. We can enlarge A to get a

ore coarse-grain level partition with fewer clusters. The dynam-

cs become intrinsically nonlinear because of asymmetric and con-

trained attribute evolution, which brings new challenges in the

nalysis. P is still used to denote the overall probability measure

apturing the randomness of the updates in the asymmetric con-

trained model.

.4.2. Balanced graphs and clustering

The results showed that clustering arises for the social network

ttributes when the underlying graph reaches a certain balance.

he following definition is introduced.

efinition 5. (i) Let G be a strong equilibrium and obey partition

= V 1 ∪ V 2 . Then almost sure clustering using Eq. (10) for the ini-

ial value x 0 can be obtained if two random variables B † 1 (x 0 ) and

† 2 (x 0 ) exist and take values in {−A, A } , such that

(lim

k →∞

x i (k ) = B

† 1 (x 0 ) , i ∈ V 1 ;

lim

k →∞

x i (k ) = B

† 2 (x 0 ) , i ∈ V 2

)= 1

(12)

ii) Let G be a weak equilibrium and obey partition V = V 1 ∪ 2 . . . V m

for some m ≥ 2. Then almost sure clustering using

q. (10) for the initial value x 0 can be obtained if m random vari-

bles B � 1 (x 0 ) , . . . , B � m

(x 0 ) exist and each of them takes values in

−A, A } , such that

(lim

k →∞

x i (k ) = B

�

j (x 0 ) , i ∈ V j , j = 1 , . . . , m

)= 1 . (13)

Under circumstance of strongly balanced graphs, when β is

arge enough, we can show that attributes are clustered in an

symptotic state, as stated in the following theorem.

heorem 1. We suppose that G is a strong equilibrium under parti-

ion V = V 1 ∪ V 2 , meanwhile, G V 1 and G V 2

are linked. For any α ∈ (0,

) �{1/2}, when β is large enough, for almost all initial values X

0 , al-

ost sure the ideal clustering result is obtained under the update rule

f Eq. (10) .

The fact is that there holds B � 1 (x 0 ) + B �

2 (x 0 ) = 0 almost surely

sing the update rule of Eq. (10) for strongly balanced social net-

orks. For strongly balanced social networks, attributes are finally

ivided into the two attribute boundaries in the form of polariza-

ion, which is stated in theorem 1. Next, as stated in the following

heorem, attributes of weakly balanced graphs are clustered again.

heorem 2. We suppose that G is a weakly balanced graph un-

er the partition V = V 1 ∪ V 2 . . . V m

with m ≥ 2 . Further assume that

V j , j = 1 , . . . , m are connected. For any α ∈ [0, 1) �{1/2}, when β is

ufficiently large, almost sure the ideal clustering result is obtained

nder the update rule of Eq. (10) .

The proof of Theorems 1 and 2 is obtained by establishing suit-

ble separation events which happen inevitably, i.e., the node at-

ributes for a subnetwork become group polarized (either larger

r smaller than the remaining node’ attributes). Obviously from

he analysis, the occurrence trend of such events is more easy

or small subnetworks in the partition of (strongly or weakly) bal-

nced social networks. For another, the proposed method of clus-

ering follows quickly after the separation event, even in the pres-

nce of large subgroups. For a large subgroup, the proposed clus-

ering method to a consensus for its members is more a conse-

uence of the “push” by the already separated small subgroups

ather than the trustful interactions therein. This means relatively

mall subgroups contribute to faster occurrence of the clustering

24 H.-J. Li et al. / Chaos, Solitons and Fractals 110 (2018) 20–27

Fig. 1. Clustering in signed network. (a) A signed network that composed by positive and negative links. (b) Two clusters in the signed network. Qualitatively, clusters in

signed network are defined as subgraphs that the positive links are within the nodes in each group and the negative link are connected between the different groups of

nodes. (c) The dynamical attribute update rule, i.e. X(t + 1) = f (X(t)) , is proposed. For sectors of different colors there are different cluster configurations X in squares. We

use a new attribute dynamic rule that guaranteeing the cluster configuration converges to an optimal state. It can be noticed that the cluster configuration in the rightmost

circle is strongest one.

(

t

f

r

m

t

c

i

A

s

t

t

l

k

s

W

p

A

a

a

w

a

p

u

c

c

a

c

t

t

of the entire social network attributes. In this way, the initial at-

tributes make an impact on the final attribute limit, which is either

−A or A .

2.4.3. Computational complexity

In General, the dynamical algorithm has a fast speed in com-

putation. First, the initial value of x i for each node is set in O ( V )

time. Then, the attribute of a node will be consecutively updated

according to Eqs. (10) and (11) . The updating process ends when

n nodes all update its attribute. We call this a round and the to-

tal cost of the time in one round is O ( E ). The solution converges

to a stable state before the process of update is repeat in finite

rounds, which is independent on the graph scale n and merely re-

lies on the iterative rounds r . That is to say, the total time complex-

ity is O ( r · E ), where r is the number of iteration rounds. Specially,

for a sparse network, the total complexity in computation O ( r · E )

will dip to O ( r · N ) since a linear relationship is found between the

number of nodes n and of edges m . Therefore, for the size of net-

works, our approach is extensible in the computational time, and

it is perfectly applicable for very large-scale networks.

3. Results and discussion

The performance of our approach applied in an empirical net-

work will be shown by way of analyzing international relations,

the data set arises from the Correlates of War [26,27] from 1993

to 2001, where positive links “+ ” and negative links “−” represent

military alliances and disputes respectively. Diverse disputes are in

the data set, several example can be cited to prove it:the border

condition between Venezuela and Colombia remains tense, China

deploys submarines to the Japanese archipelagoes, and Turkish sol-

diers entered Iraq. For the disputes, the hostility levels from “no

militarized action” to “interstate war” are designated. We encode

the alliances with one of (1) entente, (2) non-aggression pact, or

3) defense pact. The disputes and alliances are assigned as nega-

ive links and positive links,respectively. 161 nodes and 2517 link

orm a largest connected component of the world,where nodes

epresent countries and links represent conflicts and alliances.

The result of the analysis is shown in Fig. 2 . The same cluster

ore properly known as a power bloc is the set of countries in

he same color (or pattern) in this context. The following identifi-

ations of the power blocs are that: (1) the West; (2) Latin Amer-

ca; (3) Muslim World; (4) Asia; (5) West Africa; and, (6) Central

frica.

The result we obtained is similar with the deployment de-

cribed in The Clash of Civilizations [28] written by Hunting-

on,such as South Korea and South Africa belong to the group of

he West, while Pakistan and Iran belong to the group of the Mus-

im World. However, there are also a few notable exceptions.The

ey distinction for Huntington is that China, Japan or India them-

elves are not a separate bloc. Furthermore, the power bloc of the

est African is lacking in the deployment of Huntington, our result

rovides an additional insight.

If setting A = 6 in the algorithm, Latin America and North

merica merge into a power bloc while Europe is independently

ggregated into a cluster, and the Middle East, North Africa, China

nd Russia become allies.Conversely, when running the algorithm

ith A = 2 , former Soviet countries expect Russia independently

ggregate into a cluster after seceding from the Soviet Union.In the

artition, a variety of layers can be detected by using diverse val-

es of A in range. Not only the collisions of power blocs are in our

onfiguration, such as the collisions of Russia and Georgia, and the

ollisions of Rwanda and Democratic Republic of the Congo, which

re still respectively classified into a power bloc. Under these cir-

umstances, the collisions of different groups were overcome by

he alliances, the pattern of international ties is proved to be larger

han the total bilateral ties.

H.-J. Li et al. / Chaos, Solitons and Fractals 110 (2018) 20–27 25

Fig. 2. Map of the cluster configuration in the conflict and alliance network found using the algorithm described in the text.

f

s

p

4

s

u

o

t

c

e

t

w

A

v

p

t

(

C

f

n

H

P

G

A

f

L

p

i

Z

{

t

J

f

P

c

p

L

E

ϑ

P

P

j

g

T

n

g

η

s

a

J

v

w

Z

P

(

f

L

i

P

Generally speaking, the configuration of civilizations comes

rom Huntington’s work is seemingly quite robust although it’s

till not exactly right, and which with a few notable exceptions is

roved by analyzing.

. Conclusions

In this paper, we proposed an efficient clustering algorithm in

igned networks based on a new positive and negative attribute

pdate rule. In the future work, it is worthwhile taking account

f the improvement of the algorithm efficiency and wide applica-

ion on the extremely large networks. Besides, clearly interpreting

lusters appeared in the signed real-life networks, e.g., the knowl-

dge of physics and sociology, and the dynamical evolution of clus-

er configuration could be introduced into the model we proposed,

hich maybe more significative in practice.

cknowledgments

We expressed our appreciations for the anonymous re-

iewers offering their valuable suggestions. This work was

artially supported by the Beijing Natural Science Founda-

ion ( 9182015 ), National Natural Science Foundation of China

71473285 , 61502222 , 71401194 ), Young Elite Teacher Project of

entral University of Finance and Economics ( QYP 1603 ), Program

or Innovation Research in Central University of Finance and Eco-

omics, Open program of the Research Center of Beijing-Tianjin-

ebei Civil Aviation Coordinate Development (RCCA 02), and Key

rogram of National Natural Science Foundation of China under

rant 91646204 .

ppendix A. The proof of Theorem 1

We first state and prove intermediate lemmas that will be use-

ul for the proofs of theorems 1 and theorem 2.

emma 1. Assume that α ∈ (0, 1) . Let i 1 . . . i k be a path in the

ositive graph; i.e., { i s , i s +1 } ∈ G pst , s = 1 , . . . , k − 1 . Take a node

∗ ∈ { i 1 , . . . , i k } . Then for any ε > 0, there always exists an integer

( ε) ≥ 1, such that we can select a sequence of node pairs from

i s , i s +1 } , s = 1 , . . . , k − 1 under asymmetric updates, which guaran-

ees

i ∗i s (Z ) ≤ 2 Aε, s ∈ { 1 , . . . , k } (A.1)

or all initial condition x i s (0) , s = 1 , . . . , k .

roof. The proof is easy and an appropriate sequence of node pairs

an be built just observing that J i ∗i s ≤ 2 A for all s ∈ { 1 , . . . , k } . The

roof is end. �

emma 2. Fix α ∈ (0, 1) with α = 1/2 . Under attribute dynamics in

q. (10) in the main text, there exist an integer Z 0 ≥ 1 and a constant

0 > 0 such that

(∃{ i ∗, j ∗} ∈ G neg s.t. J i ∗ j ∗ (Z 0 ) ≥ 1

2 n

χ(0)) ≥ ϑ 0 (A.2)

roof. We can always uniquely divide V into m 0 ≥ 1 mutually dis-

oint sets V 1 , . . . , V m 0 such that G pst (V k ) , k = 1 , . . . , m 0 are connected

raphs, where G pst ( V k ) is the induced graph of G pst by node set V k .

he idea is to treat each G pst ( V k ) as a super node. Since G is con-

ected and G neg is nonempty, these super nodes form a connected

raph whose edges are negative.

One can readily show that there exist two distinct nodes η1 ,

2 ∈ V with ηi ∈ V v i , i = 1 , 2 ( V v 1 and V v 2 can be the same, of course)

uch that there is at least one negative edge between V v 1 and V v 2 nd such that

η1 η2 (0) ≥ 1

m 0 χ(0) . (A.3)

Now select v 1 ∈ V v 1 and v 2 ∈ V v 2 such that { v 1 , v 2 } ∈ E neg . In

iew of Lemma 1 and observing that asymmetric updates happen

ith a strictly positive probability, we can always find ϑ0 > 0 and

0 ≥ 1 (both functions of ( α, n, a, b, c )) such that

(x v 1 (Z 0 ) = x v i (0) , J u i v i (Z 0 ) ≤ 1

4 n

χ(0) , i = 1 , 2) ≥ ϑ 0 (A.4)

because G pst (V v i ) , i = 1 , 2 are connected graphs). Eq. (A.2) follows

rom Eqs. (A.3) and (A.4) since m 0 ≤ n . The proof is end. �

emma 3. Fix α ∈ (0,1) with α = 1/2 . Under attribute dynamics

n Eq. (10) in the main text, there exists β♦( α) > 0 such that

( lim sup χ(k ) = 2 A ) = 1 for almost all initial opinions if β > β♦.

k →∞

26 H.-J. Li et al. / Chaos, Solitons and Fractals 110 (2018) 20–27

T

p

A

p

P

c

χ

w

i

E

w

f

a

e

m

h

|

a

x

x

q

s

u

(

e

e

o

q

a

a

E

L

1

R

Proof. In view of Lemma 2, we have

P

(χ(Z 0 +t) ≥min

{(β+1) t

2 n

χ(0) , 2 A

})≥

(cp ∗n

)t

ϑ 0 , t = 0 , 1 , . . .

(A.5)

We can conclude that

P

(lim sup

k →∞

χ(k ) = 2 A

)+ P

(lim sup

k →∞

χ(k ) = 0

)= 1 (A.6)

as long as β > 0.

With Eq. (A.6) , we have

P (χ(z 0 + 1) ≥ β + 1

2 n

χ(0)) ≥ cp ∗n

ϑ 0 (A.7)

conditioned on χ(0) ≤ 4 An/ (1 + β) . Therefore, we can conclude

that there exists β♦( α) > 0 such that

P

(lim sup

k →∞

χ(k ) ≥ 4 An/ (1 + β)

)= 1 (A.8)

for all β > β♦( α). Combining Eqs. (A.6) and (A.8) , we get the de-

sired result. The proof is end. �

Lemma 4. Assume that the graph is strongly balanced under par-

tition V = V 1 ∪ V 2 and that G V 1 and G V 2

are connected. Let α ∈ (0,

1) �{1/2} . Fix the initial opinions x 0 . Then under attribute dynam-

ics in Eq. (10) in the main text, there are two random variables,

B † 1 (x 0 ) , B †

2 (x 0 ) both taking value in {−A, A } , such that

P

(lim

k →∞

x i (k ) = B

† 1 , i ∈ V 1 ; lim

k →∞

x i (k ) = B

† 2 , i ∈ V 2 | E sep (ε)

)= 1 (A.9)

for all ε > 0, where by definition, E sep (ε) is the ε − separation event

E sep (ε) : = { lim sup

k →∞

max i ∈ V 1 , j∈ V 2

| x i (k ) − x j (k ) | ≥ ε} . (A.10)

Proof. Suppose x i 1 (0) − x i 2 (0) ≥ ε > 0 for i 1 ∈ V 1 and i 2 ∈ V 2 . By as-

sumption, G V 1 and G V 2

are connected. Thus, from Lemma 1 , there

exist an integer Z 1 ≥ 1 and a constant p̄ (both depending on ε, n,

α, a, b ) such that

min

i ∈ V 1 x i (Z 1 ) − max

i ∈ V 2 x i (Z 1 ) ≥ ε

2

(A.11)

happens with probability at least p̄ . Intuitively, Eq. (A.11) char-

acterizes the event where the opinions in the two sets V 1 and

V 2 are completely separated. Since all edges between the two

sets are negative, conditioned on Eq. (A.11) , it is then straightfor-

ward to see that almost surely we have lim k →∞

x i (k ) = A, i ∈ V 1 , and

lim k →∞

x i (k ) = − A, i ∈ V 2 .

Given E sep (ε) , {∃ i 1 ∈ V 1 , i 2 ∈ V 2 s.t. x i 1 (k ) − x i 2 (k ) ≥ ε f or

in f initely many k} is an almost sure event. Based on our pre-

vious discussion and by a simple stopping time argument, the

Borel–Cantelli Lemma implies that the complete separation event

happens almost surely given E sep (ε) . This completes the proof. �

Lemma 5. Assume that the graph is strongly balanced under parti-

tion V = V 1 ∪ V 2 and that G V 1 and G V 2

are connected. Suppose α ∈ (0,

1) �{1/2} . Then under attribute dynamics in Eq. (10) in the main text,

there exists β sufficiently large such that P (E (A/ 2)) = 1 for almost all

initial opinions.

Proof. Let us first focus on a fixed time instant k . Suppose x i (k ) −x j (k ) ≥ A for some i, j ∈ V . If i and j belong to different sets

V 1 and V 2 , we already have max i ∈ V 1 , j∈ V 2 | x i (k ) − x i (k ) | ≥ A . Other-

wise, say i, j ∈ V 1 . There must be another node l ∈ V 2 . We have

max i ∈ V 1 , j∈ V 2 | x i (k ) − x j (k ) | ≥ A/ 2 since either | x j (k ) − x l (k ) | ≥ A/ 2

must hold. Therefore, we conclude that

χ(k ) ≥ A ⇒ max i ∈ V 1 , j∈ V 2

| x i (k ) − x j (k ) | ≥ A/ 2 . (A.12)

hen the desired conclusion follows directly from Lemma 3 . The

roof is end. �

Theorem 1 is a direct consequence of Lemmas 4 and 5.

ppendix B. The proof of Theorem 2

The proof is similar to that of Theorem 1. We just

rovide the main arguments. First by Lemma 3 we have

( lim sup k →∞

χ(k ) = 2 A ) = 1 for almost all initial values with suffi-

iently large β . Then as for Eq. (A.12) , we have

(k ) ≥ A ⇒ max i ∈ V s , j∈ V t ,s = t∈{ 1 , ... ,m }

| x i (k ) − x j (k ) | ≥ A

m

. (B.1)

here m ≥ 2 comes from the definition of weak balance. Therefore,

ntroducing

∗sep (ε) : =

{lim sup

k →∞

max i ∈ V s , j∈ V t ,s = t∈{ 1 , ... ,m }

| x i (k ) − x j (k ) ≥ ε| }

, (B.2)

e can show that P (E

∗sep (A/m )) = 1 for almost all initial opinions,

or sufficiently large β .

Next, suppose there exist a constant η > 0 and two node sets V i 1 nd V i 2 with i 1 , i 2 ∈ { 1 , . . . , m } such that the complete separation

vent

in

i ∈ V i 1 x i (k ) − max

i ∈ V i 2 x i (k ) ≥ η (B.3)

appens. Then if (β + 1) η ≥ 2 A, we can always select Z ∗ : = | V i 1 | + V i 2 | negative edges between nodes in the sets V i 1 and V i 2 so that

fter the corresponding updates

i (k + Z ∗) = A, i ∈ V i 1 ,

i (k + Z ∗) = −A, i ∈ V i 2 . (B.4)

One can easily see that we can continue to build the (finite) se-

uence of edges for updates such that nodes in V k will hold the

ame opinion in {−A, A } , for all k = 1 , . . . , m . After this sequence of

pdates, the opinions held at the various nodes remain unchanged

two nodes with the same opinion cannot influence each other,

ven in presence of a negative link; and two nodes with differ-

nt opinions are necessarily enemies). To summarize, conditioned

n the complete separation event in Eq. (B.3) , we can select a se-

uence of node pairs under which opinion clustering is reached,

nd this clustering state is an absorbing state.

Finally, the Borel–Cantelli Lemma and P (E

∗sep (A/m )) = 1 guar-

ntee that almost surely the complete separation event in

q. (B.3) happens an infinite number of times if η= A/ 2 m in view of

emma 1 . The end of the proof is then done as in that of Theorem

.

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