Social Balance on Networks:
The Dynamics of Friendship and Hatred
T. Antal, P. L. Krapivsky, and S. Redner
PRE 72, 036121 (2005); physics/0605183
Basic question:
How do social networks evolve when both friendly and
unfriendly relationships exist?
Partial answer: (Heider 1944, Wasserman & Faust 1994)
Social balance; on a complete graph socially-balanced
states must be either utopia or bipolar.
This talk:
Endow a network with the simple social dynamics related work:
Kulakowsi et al.
and investigate the evolution of relationships.
Main result:
Dynamical phase transition between bipolarity and utopia.
Socially Balanced States
you
you
you
you
Hollywood
divorce
friendly
link
Brad
Jen
!0
Brad
Jen
Brad
!2
unfrustrated/balanced
Jen
Brad
!1
!3
frustrated/imbalanced
Social Balance Defined
a friend of my friend
an enemy of my enemy
a friend of my enemy
an enemy of my friend
Jen
} is my friend;
} is my enemy.
Local Triad Dynamics on Arbitrary Networks
(social graces of the clueless)
1. Pick a random imbalanced (frustrated) triad
2. Reverse a single link so that the triad becomes balanced
free parameter:
p
i
1
1−p
j
probability p: unfriendly → friendly;
probability1-p: friendly → unfriendly
k
Triad Evolution on the Complete Graph
Basic graph characteristics:
N nodes
1
2 N (N
− 1) links
1
6 N (N
− 1)(N − 2) triads
ρ = friendly link density
nk = density of triads of type k
±
nk = density of triads of type k attached to a ± link
+
n1
+
n2
−
n1
n+
0
N-2 triads
positive link
negative link
−
n2
n−
3
Triad Evolution on the Complete Graph
nk = density of triads of type k
n±
k = density of triads of type k attached to a ± link
+
π = (1 − p) n1
π − = p n 1 + n3
flip rate + → −
flip rate − → +
Master equations:
dn0
dt
dn1
dt
dn2
dt
dn3
dt
1−p
!1 → !0
=
− −
π n1
=
+ +
π n0
=
+ +
π n1
=
+ +
π n2
p
1
!0 → !1
−
+ +
π n0
+
− −
π n2
,
−
− −
π n1
+
− −
π n3
−
− −
π n2
−
− −
π n3
.
−
+ +
π n1
,
−
+ +
π n2
,
Steady State Solution
dn0
dt
dn1
dt
dn2
dt
dn3
dt
+ +
= π − n−
−
π
n0 ,
1
=
π + n+
0
=
π + n+
1
+
π − n−
2
+
π − n−
3
−
π − n−
1
−
π − n−
2
−
π + n+
1
−
π + n+
2
+
,
,
impose ṅi and π = π
gives
+
nk
=
−
−
nk+1
− −
−
π
n3 .
= π + n+
2
then use n±
k =
 (3−k)n
k
 3n0 +2n1 +n
2

to give
knk
n1 +2n2 +3n3
 $
 1/[ 3(1 − 2p) + 1] p ≤ 1/2;
! "
3
j
(1
−
ρ
)
ρ3−j
nj =
∞ , ρ∞ =
∞

j
1
p ≥ 1/2
Steady State Triad Densities
p≤½, steady state; p>½, utopia
1
ρ∞ =
0.8
!
"
1/[ 3(1 − 2p) + 1] p ≤ 1/2;
1
p ≥ 1/2
! "
3
j
(1
−
ρ
)
ρ3−j
nj =
∞ ,
∞
j
0.6
n0
utopia
n1
0.4
n2
0.2
n3
0
0
0.1
0.2
0.3
p
0.4
0.5
The Evolving State
rate equation for the friendly link density:
− → + in #1
dρ
dt
=
+ → − in #1
− → + in #3
2
3
3ρ (1 − ρ)[p − (1 − p)] + (1 − ρ)
Solution:

−Ct
ρ
+
Ae

∞






1 − ρ0
%
1
−
ρ(t) ∼
2t
1
+
2(1
−
ρ
)
0







1 − e−3(2p−1)t
p < 1/2;
quick
frustration
p = 1/2;
slow relaxation
to utopia
p > 1/2.
rapid attainment
of utopia
Fate of a Finite Society
p<1/2: effective random walk picture
balance
(N 3 /6 balanced triads)
±N
D∝N
balance
N
2
v∝N
vLN /D
→ TN ∼ e
N2
∼e
Fate of a Finite Society
TN
106
1
p< ,
2
(a)
4
10
102
(from TN = e
4
6
8
10
12
14
TN
N
(b)
1
102
10
103
N
14
TN
TN ∼ e
N ·N 3 /N 2
1
104
103
102
10
1
N2
(c)
10
6
2
1
102
10
N
103
)
Fate of a Finite Society
p<1/2: effective random walk picture
balance
(N 3 /6 balanced triads)
±N
D∝N
balance
N
2
v∝N
vLN /D
→ TN ∼ e
p>1/2: inversion of the rate equation
u∼
−3(2p−1)t ≈ N −2 → TN
e
u ≡ 1−ρ = unfriendly link density
ln N
∼
2p − 1
N2
∼e
Fate of a Finite Society
TN
106
1
p< ,
2
(a)
4
10
102
N2
TN ∼ e
N ·N 3 /N 2
(from TN = e
1
4
6
8
10
12
)
14
TN
N
104
103
102
10
1
(b)
1
102
10
103
N
TN
14
1
p> ,
2
(c)
10
6
TN
ln N
∼
2p − 1
(from 1−ρ ∼ e−3(2p−1)t =
2
1
102
10
N
103
1
N2 )
p=1/2
naive rate equation estimate:
u≡1−ρ∝t
−1/2
≈N
→ TN ∼ N
−2
4
incorporating fluctuations as balance is approached:
ln u
U
−1/2
N
√
= Lu + L η
√ 1/4
L
∼ √ + Lt
t
equating the 2 terms in U:
−2
2/3
TN ∼ L
ln t
N
4/3
N
4
∼N
4/3
Fate of a Finite Society
TN
106
1
p< ,
2
(a)
4
10
102
N2
TN ∼ e
N ·N 3 /N 2
(from TN = e
1
4
6
8
10
12
)
14
TN
N
104
103
102
10
1
1
p= ,
2
(b)
TN ∼ N
(from 1−ρ ∼
1
2
10
3
10
10
1
√
t
=
4/3
1
N2
+ fluctuations)
N
TN
14
1
p> ,
2
(c)
10
6
TN
ln N
∼
2p − 1
(from 1−ρ ∼ e−3(2p−1)t =
2
1
102
10
N
103
1
N2 )
Constrained (Socially Aware) Triad Dynamics
1. Pick a random imbalanced (frustrated) triad.
2. Reverse a random link (p=⅓) to eliminate a frustrated triad
only if the total number of frustrated triads does not increase.
only free parameter:
initial density of friendly links ρ0
Outcome: Quick approach to a final static state.
Typically: TN ∼ ln N
Final Clique Sizes
1
utopia
N=256
512
1024
2048
2
!δ " 0.8
≈2/3
2
<δ >
C1 − C2
δ=
N
0.6
0.4
0.2
bipolarity: two equal
size cliques
0
0
0.2
0.4
0.6
ρ0
0.8
1
Bipolarity/Utopia Transition
First consider evolution of an uncorrelated network:
for + →
2ρ(1−ρ)
2
(1−ρ)
ρ2
N-2
we need:
+
+
+
2
2
>
n
,
with
!
n
=
[ρ
,
2ρ(1−ρ),
(1−ρ)
, 0]
n+
+
n
+
n
+
0
1
2
3
! "# $
! "# $
frustrated
unfrustrated
→ 1 − 4ρ(1 − ρ) < 0, impossible, so + links never flip
for
2ρ(1−ρ)
→+
2
(1−ρ)
ρ2
N-2
we need:
−
−
−
2
2
>
n
,
with
!
n
=
[0,
ρ
,
2ρ(1−ρ),
(1−ρ)
]
n−
+
n
+
n
−
0
1
2
3
! "# $
! "# $
frustrated
unfrustrated
→ 1 − 4ρ(1 − ρ) > 0, valid when ρ #= 1/2
Only negative links flip, except when ρ→½
flow diagram for ρ:
0
½
1
Instability near ρ=½
intraclique relationship evolution
1+
ρ1 =
2
1+
ρ2 =
2
C1
for
1−
β=
2
C2
→ + within cliques, we need:
%
−
−
−
+
n
+
n
>
n
n− =
n−
2 , with !
3
0
1
! "# $
! "# $
frustrated
[0, ρ2i , 2ρi (1−ρi ), (1−ρi )2 ] intraclique
[0, β 2 , 2β(1−β), (1−β)2 ]
interclique
unfrustrated
→ C1 [1 − 4ρi (1 − ρi )]+C2 [1 − 4β(1 − β)] > 0, always true
negative intraclique links disappear
increased cohesiveness within cliques
interclique relationship evolution
1+
ρ1 =
2
1+
ρ2 =
2
C1
for + →
1−
β=
2
C2
between cliques , we need:
+
+
+
>
n
n+
n+ = [βρi , β(1 − ρi ) + ρi (1 − β), (1 − β)(1−ρi ), 0]
+
n
+
n
0
1
2 , with !
3
! "# $
! "# $
frustrated
unfrustrated
→ [C1 (2ρ1 − 1) + C2 (2ρ2 − 1)](1 − 2β) > 0, true if ρ1 , ρ2 > 1/2, β < 1/2
positive interclique links disappear
increased emnity between cliques
A Historical Lesson
GB
AH
F
GB
G
R
F
I
GB
GB
G
I
French-Russian Alliance 1891-94
I
Entente Cordiale 1904
I
German-Russian Lapse 1890
GB
G
R
G
R
AH
F
AH
F
I
Triple Alliance 1882
AH
F
R
G
R
3 Emperor’s League 1872-81
GB
AH
AH
F
G
R
I
British-Russian Alliance 1907
Summary & Outlook
If we can’t all love each other → social balance
Local triad dynamics:
finite network: social balance is achieved, with the
time until balance strongly dependent on p
infinite network: phase transition between utopia
and bipolarity at p=½
Global triad dynamics (p=⅓):
jammed states possible but never occur
infinite network: two cliques emerge, with utopia
when ρ0 2/3 (rough argument gives ρ0 =½)
Open questions:
incomplete graphs/indifference, continuous interactions
allow
→ Machiavellian society
asymmetric relations