Ternary Social Networks: Dynamic Balance and Self-Organized Criticality

0.64 MB·English

by  Qing-Kuan Meng

Ternary Social Networks: Dynamic Balance and Self-Organized Criticality
Qing-Kuan Meng,1,∗ Wei Liu,2,† and Jian-Yang Zhu1,‡
1Department of Physics, Beijing Normal University, Beijing 100875, China
2School of Science, Xi’an University of Science and Technology, Xi’an 710054, China
(Dated: January 4, 2010)
Antal et al. [Phys. Rev. E 72, 036121 (2005)] have studied the balance dynamics on the
social networks. In this paper, based on the model proposed by Antal et al., we improve it and
generalize the binary social networks to the ternary social networks. When the social networks get
dynamicallybalanced,weobtainthedistributionsofeachrelationandthetimeneededfordynamic
balance. Besides,westudytheself-organizedcriticalityontheternarysocialnetworksbasedonour
0
model. For the ternary social networks evolving to the sensitive state, any small disturbance may
1
result in an avalanche. The occurrence of the avalanche satisfies the power-law form both spatially
0
and temporally. Numerical results verify our theoretical expectations.
2

n PACSnumbers: 05.65.+b,87.23.Ge,89.75.-k
a
J

2 I. INTRODUCTION networks, there being two opposing opinions is extreme,
we introduce the neutral opinion. So the values of each

]
nodearegeneralizedto+1, 0and-1, andthemetastable
h Antal et al. [1] have studied the balance dynamics on
p the social networks based on the notion of social balance triadrelationsappear. Wefindthedensitiesofeachtriad
- relationassociatedwiththeHamiltonianandthegeomet-
c [2, 3]. In the networks, each node is connected to all the
rical temperature which is determined by the structure
o others, representing each person knows all the others in
of the network without manual setting. Based on our
s the society. Each edge in the networks has two values,
. model, we obtain the distributions of each triad relation
s +1 and -1. If the edge is +1, it means the two persons
c and the time needed for dynamic balance both for the
are friendly with each other. If the edge is -1, it means
i binary and ternary cases.
s the two persons are hostile towards each other. At every
y
step, they choose a triangle randomly from the network.
h
p If the product of the three edges is +1, the triangle is
[ stable. Otherwise, iftheproductis-1, thetriangleisun- In the latter half of this paper, we study the self-

stable. For the stable triangle, it satisfies (i) the friend organized criticality (SOC) [6–8] on the ternary social
1
ofmyfriendbeingmyfriend; (ii)theenemyofmyfriend networks. The previous studies of the SOC on complex
v
9 being my enemy; (iii) the friend of my enemy being my networks are mainly based on the BTW model [6, 9–14].
0 enemy;and(iv)theenemyofmyenemybeingmyfriend. In this paper, we establish our model with some differ-
3 The unstable triangles always try to be stable, but the ences. Asweknow,inthesociety,ifonerelationchanges
0 final state of the network depends on the edge flipping from one state into anther, it may affect other relations
1. probability p, which is set manually. If p ≥ 21, the net- associating with it. The effect may be big or small, de-
0 workwillreachthestateof”paradise”, withallrelations pending on many internal factors. So in this paper, we
0 being friendly. Several studies around the balance dy- simplify this phenomenon, with an eye to the triad bal-
1 namics have been carried out, including the studies of ancedynamics,andmaketwomodificationscomparedto
:
v the university class of triad dynamics [4] and the satisfi- Sec. II. We find under specific conditions the SOC to be
i ability problem of computer science [5], etc. observable. Besides,forthesmall-worldnetwork[15,16],
X
In the first half of this paper, at first based on Ref. we find the way of construction have an influence on the
ar [1], we map the edge relations to the node relations with occurrence of the avalanche. We analyze it theoretically.
somerelaxation,sotherearetwoopposingopinionsinthe Numerical results verify our theoretical expectations.
network. Wesupposeforeachtriadrelation inthesocial
networks,itwillchangefromstabletounstableorthere-
verse, caused by the change of some persons’ opinions in
This paper is organized as follows. In Sec. II, we map
it. While for the change, the person’s opinion depending
theedgerelationstothenoderelationsandgeneralizethe
both on the other opinions in the triad relation and the
binarynetworkstotheternaryones. Forboththebinary
opinions all around him. Next, considering in the social
and the ternary networks, we obtain the distributions
of each triad relation and the time needed for dynamic
balance. In Sec. III, we study the SOC on the regular
network and the small-world network respectively, and
∗Electronicaddress: [email protected]
find out the small-world effect on the occurrence of the
†Electronicaddress: [email protected]
‡Authortowhomcorrespondenceshouldbeaddressed;Electronic avalanche. We end this paper with conclusions in Sec.
address: [email protected] IV.
2
nodes from the selected triangle at random, then let the
selected node do spin flipping. The spin flipping proba-
bility is defined as
p(σ →σ′)= e−βH∆′ , (2)
i i Z
where H′ = −α(σ′σ +σ σ +σ′σ ) stands for the fi-
∆ i j j k i k
nal Hamiltonian after the spin flipping, with σ′ the final
i
value of the selected spin, and σ , σ the values of the
j k
other spins unchanged. It is noted that the spin flipping
probability depends on the final value, independent of
the initial one [18]. In addition, Z = σ′=+1,−1e−βH∆′
i
is the normalizing factor and β is a geometrical temper-
P
ature determined by the structure of the network. Since
β is not a real temperature, and in order to embody the
FIG.1: a1−a4 standforeachtypeofthetriadrelations,and majority principle, we propose
the corresponding densities in addition. a1 and a4 are stable
relations, while a2 and a3 are unstable relations. β =1/| σ /(N −1)−σ′|≃1/|M −σ′|, (3)
j i i
j6=i
X
II. DYNAMIC BALANCE
where N is the size of the network, and M =
hσ (t)i/N is the average value of all spins. So the
A. The Node Model i i
spin flipping probability depends both on the triad rela-
P
tion and the surroundings [18, 19].
Firstwedefinethenoderelations inanalogy with Ref.
Thenumbersofpositiveandnegativenodesinthenet-
[1] with some relaxation. We propose each node in the
work are as follows,
network has two values, +1 and -1, standing for each
person’s two opposite standpoints in the society, or two C3 (3a +2a +a ) N
opposing opinions. For a triad relation, if all nodes have N+ = N 1C2 2 3 = 3 (3a1+2a2+a3),
the same value, i.e., all persons have the same opinion, N
the relation is stable. If there are different opinions in N− = CN3 (a2+2a3+3a4) = N (a +2a +3a )(.4)
a triad relation, conflicts are easily to occur, so the re- C2 3 2 3 4
N
lation is unstable. The stable and unstable triad rela-
tions are given in Fig. 1. In our model, the stable and where N is the number of total nodes and ai stands for
unstable triad relations have the probabilities to change the triangle density, satisfying 4i=1ai = 1. The densi-
into the other, which is more realistic for the social net- ties of each triangle attached to a positive node are
P
works. As we know in the society, stable relations may
become unstable because of conflicts and the surround- n+ = N3Ciai = Ciai , (5)
ings, and unstable relations may become stable because ai N+ 3a +2a +a
1 2 3
of the same interest and the surroundings. Considering
each node has two values which have the probabilities to where i=1,2,3, and Ci is the number of positive nodes
change into the other, we study the social networks in in ai. In a similar way, the densities of each triangle
association with the spin systems. In the following con- attached to a negative node are
text,weestablishourmodelinassociationwiththeIsing
model and the generalized Glauber dynamics [17, 18]. n− = N3Djaj = Djaj , (6)
When associated with the spin systems, each node in aj N− a +2a +3a
2 3 4
the network standing for a spin. We propose the Hamil-
tonian of each triangle to be where j = 2,3,4, with Dj the number of negtive nodes
in a .
j
H∆ =−α(σiσj +σjσk+σiσk), (1) For each node, the spin flipping probabilities from one
value to the other are given by Eq. (2), i.e.,
where σ ,σ and σ stand for the three spins of the tri-
i j k
aonfgtlhee. tIwnothvealbuiensa,r+y1caasned, σ-1i,.σjBaesniddeσs,kαca>n t0akise aeitrheearl p(σ →σ′)= e|M−ασi′|(σi′σj+σjσk+σi′σk) .
parameter, standing for the strength of coupling. So the i i e−(1+αM)(−σj+σjσk−σk)+e(1−αM)(σj+σjσk+σk)
HamiltonianistheformoftheIsingmodel. Theevolving (7)
rules are as follows. At every step we choose a triangle Sothetotal flippingprobabilities foreachnodefromone
from the network randomly, and choose one of the three value to the other are as follows,
3
4
C a
i i
p(1→−1) = p(1→−1)
3
i=1
X
e−α/(1+M) 2a e−α/(1+M) a e3α/(1+M)
2 3
= a + + , (8)
1e−α/(1+M)+e3α/(1−M) 3 e−α/(1+M)+e−α/(1−M) 3 e3α/(1+M)+e−α/(1−M)
4
D a
i i
p(−1 → 1)= p(−1→1)
3
i=1
X
a e3α/(1−M) 2a e−α/(1−M) e−α/(1−M)
2 3
= + +a , (9)
3 e3α/(1−M)+e−α/(1+M) 3 e−α/(1−M)+e−α/(1+M) 4e−α/(1−M)+e3α/(1+M)

0 0 0
=0.1
=2.0
TN
+ a5 + + a6 - - a7 -
0 0 0
101 4 0 a8 + 0 a9 - 0 a10 0
N 10
FIG. 2: A plot of TN with N in the log-log coordinate. FIG. 3: a5 and a7 represent the stable relations, a6 the un-
M(0) = −1, N is the size of the network, and TN is the stable relation, while a8−a10 the metastable relations.
time needed for dynamic balance. The numerical results are
obtained by taking the average of 5×102 simulations on the
networks with size from 2×103 to 2×104.
B. The Generalized Node Model
where a to a stand for each triangle density, and C
1 4 i
and D the number of positive and negative nodes in a
i i
respectively. When the network gets dynamically bal-
anced, the densities of each triad relation are as follows, There being only two opposing opinions in the social
networks is extreme. We propose the neutral opinion
1 3 3 1 shouldexist,whichmeansthepersonholdingthisopinion
a1 = , a2 = , a3 = , a4 = . (10) is indifferent to other opinions. So we propose each node
8 8 8 8
should have three values, +1, 0 and -1, standing for the
threeopinions. Iftherearethesame opinion inthe triad
We note the network reaches an explicit state, indepen-
relationexcepttheneutralopinion,therelationisstable.
dent of the parameter α. And in the dynamically bal-
Iftherearedifferentopinionsinthetriadrelationexcept
anced state n+ = n−, where n+ and n− stand for the
the neutral opinion, the relation is unstable. Otherwise
densities of positive and negative nodes respectively.
therelationismetastable. Thesimplestwayisbyjudging
The time needed for dynamic balance is
the Hamiltonian (see Eq. (1)). If H < 0, it is stable.
∆
If H > 0, it is unstable. If H = 0, it is metastable.
∆ ∆
T ∼NC(α), (11) There are four stable relations, three unstable relations,
N
and three metastable relations in the network, as shown
by Fig. 1 and Fig. 3.
where N is the size of the network, and C(α)>0, being
a function of α. The numerical results are given in Fig.
2. More details are given in Appendix A1. Compared with Eq. (4), the numbers of positive, neu-
4
tral and negative nodes in the network are as follows, wherej =5,6,7,8,9,10, andF isthenumberofneutral
j
nodes in a . The densities of each triangle attached to a
j
N
N+ = (3a +2a +a +2a +a +a ), negative node are as follows,
1 2 3 5 6 8
3
N
N0 = (a5+a6+a7+2a8+2a9+3a10), n− = Gkak , (15)
3 ak a +2a +3a +a +2a +a
2 3 4 6 7 9
N
N− = (a +2a +3a +a +2a +a ), (12)
2 3 4 6 7 9
3 where k = 2,3,4,6,7,9, with G being the the number
k
of negtive nodes in a .
wherea -a standforthedensitiesofeachtriangle. The k
1 10
densities of each triangle attached to a positive node are We study the ternary network in association with the
as follows, Potts spin systems. Since the values of each spin are
morethantwo,thespinflippingmechanismchangesinto
n+ = Eiai , (13) the spin transition mechanism. For the spin transition
ai 3a +2a +a +2a +a +a mechanism, the spin takes one of the three values, +1, 0
1 2 3 5 6 8
and-1,dependingbothonthetriadrelationandthesur-
where i=1,2,3,5,6,8, and E is the number of positive
i roundings [18, 19]. The Hamiltonian, the spin transition
nodesina . Inasimilarway,thedensitiesofeachtriangle
i probabilities and the geometrical temperature are illus-
attached to a neutral node are as follows,
trated by Eqs. (1), (2) and (3) respectively. The total
probabilities from one value to the others for each node
F a
n0 = j j , (14) are as follows,
aj a +a +a +2a +2a +3a
5 6 7 8 9 10
a2 e|Mα| 2a3 e−|Mα| e|Mα|
p(−1→0) = + +a
3 e|Mα| +e−1+αM +e1−3αM 3 e−|Mα| +e−1+αM +e−1−αM 4e|Mα| +e1+3αM +e−1−αM
a 1 2a 1 a
6 7 9
+ + + ,
3 1+e−1+αM +e1−αM 3 1+e1+αM +e−1−αM 9
a2 e1−3αM 2a3 e−1−αM e−1−αM
p(−1→1) = + +a
3 e|Mα| +e−1+αM +e1−3αM 3 e−|Mα| +e−1+αM +e−1−αM 4e|Mα| +e1+3αM +e−1−αM
a6 e1−αM 2a7 e−1−αM a9
+ + + ,
3 1+e−1+αM +e1−αM 3 1+e1+αM +e−1−αM 9
a5 e−1+αM a6 e−1+αM a7 e1+3αM
p(0→−1) = + +
3 e−1+αM +e|Mα| +e1−3αM 3 e−1+αM +e−|Mα| +e−1−αM 3 e1+3αM +e|Mα| +e−1−αM
2a8 e−1+αM 2a9 e1+αM a10
+ + + ,
3 e−1+αM +1+e1−αM 3 e1+αM +1+e−1−αM 3
a5 e1−3αM a6 e−1−αM a7 e−1−αM
p(0→1) = + +
3 e−1+αM +e|Mα| +e1−3αM 3 e−1+αM +e−|Mα| +e−1−αM 3 e1+3αM +e|Mα| +e−1−αM
2a8 e1−αM 2a9 e−1−αM a10
+ + + ,
3 e−1+αM +1+e1−αM 3 e1+αM +1+e−1−αM 3
e−1+αM 2a2 e−1+αM a3 e1+3αM
p(1→−1) = a + +
1e−1+αM +e|Mα| +e1−3αM 3 e−1+αM +e−|Mα| +e−1−αM 3 e1+3αM +e|Mα| +e−1−αM
2a5 e−1+αM a6 e1+αM a8
+ + + ,
3 e−1+αM +1+e1−αM 3 e1+αM +1+e−1−αM 9
e|Mα| 2a2 e−|Mα| a3 e|Mα|
p(1→0) = a + +
1e−1+αM +e|Mα| +e1−3αM 3 e−1+αM +e−|Mα| +e−1−αM 3 e1+3αM +e|Mα| +e−1−αM
2a 1 a 1 a
5 6 8
+ + + . (16)
3 e−1+αM +1+e1−αM 3 e1+αM +1+e−1−αM 9
But Eq. (16) is correct only for M 6= 0. When M → 0, the transition probabilities are defined as (the explana-
5

0.5
=0.1
=2.0
0.4
0.3 TN

0.2
n+ 101
0.1 n-
0
n
0.0
4
0 2 4 6 8 10 N 10
FIG. 5: A plot of TN with N in the log-log coordinate.
FIG. 4: The numerical results are obtained by taking the
averageof20simulationsonthenetworkwith104 nodes. Ini- M(0) = −1/2, y(0) = 1/2. N is the size of the network.
tially M(0) = −1/2, y(0) = 1/2, and the network takes a TreNsuilststahreetoimbteainneeeddebdyftoarkidnygntahmeicavbearalagnecoef. 5T×he10n2umsimeruiclaa-l
long enough time to get dynamically balanced. tions on the networks with size from 2×103 to 2×104.
tions are given in Appendix A1)
The numerical results of the time needed for dynamic
balance are given in Fig. 5. More details are given in
p(−1→0) = 0, p(1→0)=0,
Appendix A2.
1 a a a 2a 2a
5 6 7 8 9
p(0→1) = + + + + +a ,
10
2 3 3 3 3 3
(cid:18) (cid:19)
1 a a a 2a 2a III. SELF-ORGANIZED CRITICALITY
5 6 7 8 9
p(0→−1) = + + + + +a ,
10
2 3 3 3 3 3
(cid:18) (cid:19)
1 2a a 2a a a As is well known, SOC [6–8] is studied on many com-
2 3 5 6 8
p(1→−1) = a1+ + + + + , plex systems, including on the random graph [9], the
2 3 3 3 3 3
(cid:18) (cid:19) scale-free network [10–12] and the small-world network
1 a 2a a 2a a
p(−1→1) = 2 + 3 +a + 6 + 7 + 9 . [13], etc, which are mainly based on the BTW model [6].
4
2 3 3 3 3 3 Besides,therearethestudiesoftheSOConcomplexnet-
(cid:18) (cid:19)
(17) works based on other models, which we refer the readers
to Ref. [14] for details. In this paper, we study the SOC
Since the transition probabilities are not continued at on the ternary social networks based on our model and
M =0,wecannotgivetheanalyticalresultsofn+,n−,n0 the evolving rules, along with two modifications com-
withα, soweturntothenumericalmethod. Theresults pared to Sec. II. The first modification is the struc-
are shown by Fig. 4. We note when α → 0, n+ = n− = ture of the network. Because for a completely connected
n0 = 1/3. And when α → +∞,n0 = 0, which means network, any disturbance is globe, with no propagation,
the neutral opinion disappears, and the ternary network so the occurrence of the avalanche will not follow the
comes back to the binary network. power-law form either spatially or temporally. Besides,
Whenthenetworkgetsdynamicallybalanced, thedis- compared to the completely connected network, the reg-
tributions of each triangle are as follows, ularnetworkorthesmall-worldnetwork[15,16]iscloser
to the real structure of the society. So we take the latter
a ≃C xl+nym, (18)
i lmn forms for study. The second modification is the judge-
mentofatriadrelation. Inordertoembodythemajority
where n+ ≃ n− = x and n0 = y. Besides, i =
principle, we take the judgement as the gauge invariance
1,2,3,...,10, and l,m,n are the respective numbers of
or the gauge variance [19–22], which is explained as fol-
positive, neutral, negative nodes in a , along with C
i lmn lows. Under this judgement, the triad relation evolves
the corresponding combinatorial number. For example
from one state to the other by changing node’s value.
C =3.
201
If the value of a node being +1, it means the person
For the time needed for dynamic balance, we expect it
shows the friendly face in the triad relation, and always
to satisfy
tires to keep the relation stable. If the value being -1, it
T ∼ND(α), (19) meansthepersonshowsthehostileface,andalwaystries
N
to destroy the stability of the triad relation. While the
where N is the size of the network and D(α)>0, being valuebeing0,itmeansthepersondoesnotcarewhether
a function of α. the triad relation is stable or not. So we propose if the
6
friendly faces are in the majority, the triad relation is in
one state, or the gauge invariance state. While the hos-
tile faces are in the majority, the triad relation is in the
other state, or the gauge variance state. The explicit ex-
pressions of the gauge invariance and the gauge variance
states [19–22] are as follows.
For each triad relation, we propose if
σ +σ +σ ≥0 or σ =σ =σ =0, (20)
1 2 3 1 2 3
it satisfies the gauge invariance. If
σ +σ +σ <0 or σ 6=σ 6=σ =0, (21)
1 2 3 1 2 3
the triad relation does not satisfy the gauge invariance.
Hereσ1,σ2andσ3arethevaluesofthethreenodesofthe FIG.6: Thereare9×104nodesinthetwo-dimensionallattice
triangle. Soa1,a2,a5,a8,a10 aregaugeinvariancestates, with equal length and width. It is one of the simplest two-
while a3,a4,a6,a7,a9 are gauge variance ones, as shown dimensional regular networks satisfying the triad relations.
by Fig. 1 and Fig. 3.
We study the ternary network in association with the
Potts spin systems. Since each node has three values, (2) Choose all the stored triangles successively at ran-
the way of spin changing is the spin transition [18]. Be- dom. Because of the interaction, first we judge whether
sides, we propose the spin transition depends both on the triangle is gauge invariance. If it is gauge variance,
itself and the surroundings [18, 19]. Considering the nothing happens. Otherwise, let one of the three nodes
SOC is determined by the structure of the system, we do spin transition. If the final triangle is gauge invari-
set the geometric temperature as β = 1 > 0, where ance, nothing happens, otherwise we find out all the
T
T =| σ /n −σ′|,withσ′ thefinalvalueofi,and gauge invariance triangles attached to the selected tri-
<i,j> j i i i
σ /n the average value of all neighboring spins angle.
<i,jP> j i
of i. Based on these considerations, we give the spin (3) Find out all the gauge invariance triangles by the
tPransition probability of each node as follows, combinedactionofthestoredtrianglesinstep(2). Ifthe
number of gauge invariance triangles to store is nonzero,
1
W(σi →σi′)= Ze−βH(σi′,Σ<i,j>σj). (22) gotFoorsteeapch(2a)v,aolathnechrwe,isweegdoetofinsetetphe(1s)iz.easthenumberof
trianglesstoredinstep(2)changedfromthegaugeinvari-
In Eq. (22), H(σ′, σ ) = −σ′ σ , where
i <i,j> j i <i,j> j ance state to the other state, and plus the one changed
σ is the sum of values of all neighboring spins
<i,j> j P P in step (1). Besides, we define the time of avalanche as
oPf i, and Z = σ′=−1,0,+1eβσi′Σ<i,j>σj is the normaliz- follows. If the number of stored triangles in step (2)
i
ing factor. In order to avoid the denominator of β being changed from the gauge invariance state to the other
P
zero, we propose if the final value of the spin results in state is nonzero, the time is increased by one, and plus
T =0, it happens with probability zero, while the other the one changed in step (1).
values happen with probability one. In this way, what- Atfirst,westudytheSOContheregularnetwork. The
ever the initial condition is, the network will evolve to structure of the regular network is given in Fig. 6. The
the sensitive state. size distribution and the time distribution are given in
Theevolvingrulesareasfollows. Wesetalltriadrela- Fig. 7(a)and7(b)respectively. Inthefollowingcontent,
tions in the network to satisfy gauge invariance initially. wekeepthenumberofnodesinboththeregularnetwork
The simplest way is to set all spins to be zero, which andthesmall-worldnetworkconstantas9×104. Besides,
stands for all persons in the society being neutral with the numerical results are obtained by taking the average
each other initially. After a long enough time of evolu- of 100 simulations. For each simulation, we execute 9×
tion,thenetworkwillreachthesensitivestate. Thenany 106 time of disturbance.
small disturbance may result in an avalanche. In Fig. 7 (a) and Fig. 7 (b), the diagrams are curved
For any small disturbance: when s and t are small. Because we set all spins to be
(1)Chooseatriangleatrandom. Ifitisgaugevariance, zero initially. It takes a long time for the network to
nothing happens. If it is gauge invariance, we choose reach the sensitive state. While the diagrams curved at
one of the three nodes at random, and let the node do the ends is caused by the finite size effect.
spin transition according to Eq. (22). If after the spin Next, we study the SOC on the small-world network
transition, the triangle is still gauge invariance, nothing [15, 16]. The structure of the small-world network is
happens. Otherwise, we store all the gauge invariance given in Fig. 8.
triangles attached to the selected triangle, and goto step When a triangle changes from gauge invariance to
(2). gauge variance, the value of the selected node is de-
7
104 104
3
10
103 (a) 103 (b) 103 (a) (b)
2
102 102 102 10
P(s)101 P(t)1 01 1P(s)01 1P(s) 01
0
100 100 100 10
10-1 NP(=s9)~*1s0-14.76 10-1 PN(=t)9~*t1 -02.438 10-1 NP(=s9)~*1s0-14.9,1 q=0.05 10-1 PN(=s9)~*1s0-24.7,9 q=0.3
10-2100 101 102 10-2100 101 102 10-2100 101 102 103 10-2100 101 102
s t s s
FIG.7: PlotsofP(s)versussandP(t)versustontheregular FIG.9: AplotofP(s)versussonthesmall-worldnetwork. q
network. (a)sisthesizeofavalanche. P(s)isthedistribution is the edge adding probability. s is the size of the avalanche.
function,satisfyingthepower-lawform,withexponent-1.76. P(s)isthedistributionfunction. (a)q=0.05. Theexponent
(b)tisthetimeofavalanche. P(t)isthedistributionfunction, of the power-law distribution is -1.91. (b) q = 0.3. The
satisfying the power-law form, with exponent -2.38. exponent of the power-law distribution is -2.79

4 4
10 10
3 3
10 10
(a) (b)
2 2
P(t)10 1P(t)0
1 1
10 10
b c
0 0
10 10
-1 -1
a 10 N=9*104, q=0.05 10 N=9*104, q=0.3
-2.65
-2 P(t)~t -2
10 10
0 1 2 0 1 2
10 10 10 10 10 10
t t
FIG. 10: A plot of P(t) versus t on the small-world network.
q is the edge adding probability. t is the time of avalanche.
FIG. 8: From a two-dimensional lattice with equal length
P(t)isthedistributionfunction. (a)q=0.05. Theexponent
and width, for each node in the network, it sends out an
of the power-law distribution is -2.65. (b) q=0.3.
edgewithprobabilityq,andattachestoanothernodeatran-
dom, then attaches to one neighbor of the selected node at
random. Multi-connection and self-connection are avoided.
Thus,byaddingedges,moretrianglesarecreated. Theman- when the edge adding probability q is big enough, the
ner of adding random edges is shown by nodes a,b,c. structure of the network becomes random. We note the
diagrams are curved in Fig. 9 (a) and Fig. 9 (b) when s
issmall, buttheytakedifferentforms. Itmeansthatthe
creased. Because of the small-world effect, it may result random networks are easier for the spread in the initial
in the triangles far away compared to the regular net- conditions.
work becoming of gauge variance. If the triangles far Then there is the time distribution of the avalanche
away compared to the regular network become of gauge on the small-world network, which is similar to the size
variance,itwillblockthespread. Soweexpectthesizeof distribution of the avalanche, as given in Fig. 10 (a)
avalanchetodecrease,andtheexponentofthepower-law and Fig. 10 (b) for comparison. But we note, when q
distributiontodecreaseaccordingly. Basedontheseanal- is big enough, with the structure of the network becom-
ysis, we set the edge adding probability q of the small- ing random, the time distribution no longer satisfies the
worldfromq =0.05toq =0.3,andobservetheexponent power-law form. Besides, there is condensation at the
of the size distribution decrease from -1.91 to -2.79, as end of the diagram, which we want to study in the later
given in Fig. 9 (a) and Fig. 9 (b) respectively. work.
As is well known, for the small-world network[15, 16], At last, for the construction of the small-world net-
8

104 104 104 104
103 103 103 103
(a) (b) (a) (b)
P(s)110012 P(s)11 0012 11P(t)0012 11 P(t)0012
0
100 100 10 100
-1
10-1 δN==19.0*1, 0P4(,s q)~=s0-.10.855 10-1 Nδ==49.0*1, 0P4(,s q)~=s0-.10.659 10-2 Nδ==19.0*1, 0P4(,t )q~=t0-2..5065 10-1 δN==49.0*1, 0P4(,t )q~=t0-2..2025
10-2100 101 s 102 103 10-2100 101 s 102 103 10 100 101 t 102 10-2100 101t 102
FIG. 11: A plot of P(s) versus s on the small-world network FIG. 12: A plot of P(t) versus t on the small-world network
considering the distance effect. q is the edge adding prob- considering the distance effect. q is the edge adding prob-
ability. s is the size of avalanche. P(s) is the distribution ability. t is the time of avalanche. P(t) is the distribution
function. (a) q = 0.05, δ = 1.0, and the exponent of the function. (a) q = 0.05, δ = 1.0, and the exponent of the
power-lawdistributionis-1.85. (b)q=0.05,δ=4.0,andthe power-lawdistributionis-2.56. (b)q=0.05,δ=4.0,andthe
exponent of the power-law distribution is -1.69. exponent of the power-law distribution is -2.22.
work, when considering the factor of distance, the small-
world effect on the triangles far away compared to the
IV. CONCLUSIONS
regularnetworkisweakened,whichmaybeeasierforthe
spread. Besides, the number of local triangles are in-
creased. So we expect the size of avalanche to increase,
We have studied the social networks based on our
and the exponent of the size distribution to increase ac-
model and obtained meaningful results. First, in Sec.
cordingly. Forthesmall-worldnetwork,whenconsidering
II of this paper, based on Ref. [1], we map the edge re-
the distance effect, it is constructed as follows. From a
lations to the node relations with some relaxation, along
regular network, as shown by Fig. 6, for each node in
with introducing the neutral relation. So the social net-
the network, it sends out an edge with probability q and
works change from the binary networks into the ternary
attachestoanothernodeinthenetwork. Theprobability
ones. We suppose the change of each triad relation de-
of attachment between nodes i and j is
pendingbothonitselfandthesurroundings. Sobasedon
ourmodel,whenthesocialnetworksgetdynamicallybal-
ℓ−δ
p = i,j . (23) anced, we obtain the distributions of each triad relation
ij m<nℓ−mδ,n and the time needed for dynamic balance. Besides, we
find under the extreme condition α → +∞, the ternary
P
Here δ > 0, and ℓi,j stands for the smallest distance networks come back to the binary ones. Second, in Sec.
between i and j on the regular network. Then it at- IIIofthispaper, westudytheSOContheternarysocial
taches to one neighbor of the selected node at random. networksbasedonourmodel. Becausewesuppose,ifone
Multi-connection and self-connection are forbidden. In triad relation changes, it will affect other triad relations
this manner, more triangles are created. For the small- associating with it. The effect may be big or small, de-
worldnetwork,wekeepq =0.05constant,andsetδ =1.0 pending on many internal factors. While compared with
and δ = 4.0 for comparison. For the size distribution of Sec. II of this paper, we make two modifications for our
theavalanche,theexponentincreasesfrom-1.85to-1.69, node model as follows. One is the structure of the social
as shown by Fig. 11 (a) and Fig. 11 (b). Meanwhile, for networks, which takes the form of regular or small-world
the time distribution of the avalanche, the exponent in- network. The other is the judgement of a triad relation,
creases from -2.56 to -2.22, as shown by Fig. 12 (a) and whichtakestheformofthegaugeinvarianceorthegauge
Fig. 12 (b). variance. When the ternary social networks evolving to
Inaword,thesizedistributionoftheavalanchealways thesensitivestate,anysmalldisturbancemayresultinan
satisfies the power-law form, no matter on the regular avalanche. Sincethenumberoftriadrelationschangedin
network, the small-world network or the random net- the avalanche being no scale preferred, it should satisfy
work. But for the time distribution of the avalanche, the power-law distributions both spatially and tempo-
it deviates from the power-law form on the random net- rally. Third, for the small-world network, we find out
work, which we want to do further study in the later thesmall-world effectontheoccurrenceoftheavalanche
work. both theoretically and numerically.
9
Acknowledgments From Eqs. (A1), (A2) and (A3), along with the normal-
ization 4 a =1, we obtain
i=1 i
WethankH.Zhuforhelpfuldiscussions. Theworkwas
P
supportedbytheNationalNaturalScienceFoundationof
China (No. 10875012). 1 3 3 1
a = ,a = ,a = ,a = . (A4)
1 2 3 4
8 8 8 8
Appendix A: Distributions of each triad relation and
the time needed for dynamic balance We note the distributions of each triad relation indepen-
dentoftheparameterα,andn+ =n−inthedynamically
balanced state.
1. The node model
Defining at any time the density of positive nodes to
Forthenodemodel,eachnodehastwospinvalues,+1 be ρ, we obtain
and -1. We define one step of time as executing N spin
flipping successively at random [1]. Because each spin
flipping is linked up with (N−1)(N−2)/2 variations of a = Cjρj(1−ρ)3−j,
i 3
the triangles, the variations of the densities of each triad M = n+−n− =2ρ−1,
relation with time are as follows,
1
ρ = (1+M)/2, ρ(∞)= , (A5)
da 2
1 = p(−1→1)n− −p(1→−1)n+,
dt a2 a1
da
2 = p(1→−1)n+ +p(−1→1)n− where j is the number of positive nodes in a and Cj is
dt a1 a3 i 3
the combinatorial number. We have the new equations
−p(1→−1)n+ −p(−1→1)n−,
a2 a2 as follows,
da
3 = p(1→−1)n+ +p(−1→1)n−
dt a2 a4
−p(1→−1)n+ −p(−1→1)n−, dn+
a3 a3 = p(−1→1)−p(1→−1),
dt
da
4 = p(1→−1)n+ −p(−1→1)n−. (A1) dn−
dt a3 a4 = p(1→−1)−p(−1→1),
dt
When the network gets dynamically balanced, we have dM dn+ dn−
= −
dt dt dt
da da da da
1 2 3 4
= = = =0, (A2) = 2[p(−1→1)−p(1→−1)]. (A6)
dt dt dt dt
and
Substitute Eqs. (8), (9) and (A5) into Eq. (A6), we
p(−1→1)=p(1→−1). (A3) obtain
dM 1+M 2 1−M e3α/(1−M) 1+M 1−M 2 e−α/(1−M)
= 2 +2
dt "(cid:18) 2 (cid:19) (cid:18) 2 (cid:19)e3α/(1−M)+e−α/(1+M) (cid:18) 2 (cid:19)(cid:18) 2 (cid:19) e−α/(1−M)+e−α/(1+M)
1−M 3 e−α/(1−M) 1+M 3 e−α/(1+M)
+ −
2 e−α/(1−M)+e3α/(1+M) 2 e−α/(1+M)+e3α/(1−M)
(cid:18) (cid:19) (cid:18) (cid:19)
1+M 2 1−M e−α/(1+M) 1+M 1−M 2 e3α/(1+M)
−2 − (A7)
(cid:18) 2 (cid:19) (cid:18) 2 (cid:19)e−α/(1+M)+e−α/(1−M) (cid:18) 2 (cid:19)(cid:18) 2 (cid:19) e3α/(1+M)+e−α/(1−M)#
For Eq. (A7), when M(t) ∼ 0, after some calculations, For the dynamic balance of the network, we propose
we obtain the approximate result,
M(t)∼t−1/C(α), (A8)
1
|M(T )|∼O . (A9)
where C(α)>0, being a function of α. N N
(cid:18) (cid:19)
10
So the time needed for dynamic balance is network will reach the dynamically balanced state.
T ∼NC(α). (A10)
N
Inthesimulations,bothforthebinaryandtheternary
2. The generalized model
cases, if the final value σ′ results in the denominator
i
of β (see Eq. (3)) being zero, we propose it happens
withprobabilityzero,whiletheothervalueshappenwith When each node has three spin values, the variations
probability one (see Eq. (17) for example). In this man- of the densities of each triad relation with time are as
ner, no matter what M(0) (or M(0) and ρ(0)) is, the follows,
da
1 = p(−1→1)n− +p(0→1)n0 −[p(1→0)+p(1→−1)]n+,
dt a2 a5 a1
da
2 = p(1→−1)n+ +p(−1→1)n− +p(0→−1)n0 +p(0→1)n0 −[p(1→0)+p(1→−1)]n+
dt a1 a3 a5 a6 a2
−[p(−1→0)+p(−1→1)]n−,
a2
.
.
.
da
9 = p(1→0)n+ +p(−1→0)n− +p(1→−1)n+ +p(0→−1)n0 −[p(0→−1)+p(0→1)]n0
dt a6 a7 a8 a10 a9
−[p(−1→0)+p(−1→1)]n−,
a9
da
10 = p(1→0)n+ +p(−1→0)n− −[p(0→−1)+p(0→1)]n0 . (A11)
dt a8 a9 a10
Whenthenetworkgetsdynamicallybalanced,itsatisfies we expect n+ ≃ n− and M(∞) ≃ 0 in the dynamically
balancedstate. Definingn+ ≃n− =xandn0 =y inthe
da
i dynamically balanced state, the densities of each type of
=0, (A12)
dt the triangles are
where i=1,2,3,...,10, along with the normalization
a ≃C xl+nym, (A14)
i lmn
10
where l,m and n are the numbers of positive, neutral
a =1. (A13)
i
andnegative nodesina respectively. C mnisthecorre-
i=1 i l
X sponding combinatorial number. For example C = 3.
201
Although Eqs. (A11) are very hard to solve, from ap- Because near M = 0, there are fluctuations. The spin
pendixA1andthesymmetricspintransitionmechanism, transition probabilities are (M 6=0)
2x2y 2x2y xy2
p(−1→0) ≃ x3+x3+ + + ,
1+eα+e−α 1+eα+e−α 3
2x2yeα 2x2ye−α xy2
p(−1→1) ≃ x3+ + + ,
1+eα+e−α 1+eα+e−α 3
2xy2e−α 2xy2eα y3
p(0→−1) ≃ x2y+ + + ,
1+eα+e−α 1+eα+e−α 3
2xy2e−α 2xy2eα y3
p(0→1) ≃ x2y+ + + ,
1+eα+e−α 1+eα+e−α 3
2x2ye−α 2x2yeα xy2
p(1→−1) ≃ x3+ + + ,
1+eα+e−α 1+eα+e−α 3
2x2y 2x2y xy2
p(1→0) ≃ x3+x3+ + + ,
1+eα+e−α 1+eα+e−α 3
(A15)