Social Balance on Networks:
The Dynamics of Friendship and Hatred
T. Antal, P. L. Krapivsky, and S. Redner (Boston University)
PRE 72, 036121 (2005)
Dyonet06, Dresden, February 17, 2006
Basic question:
How do social networks evolve when both friendly and
unfriendly relationships exist?
Partial answer: (Heider 1944, Wasserman & Faust 1994)
Social balanced defined; balanced states on a complete
graph must be either paradise or bipolar.
This work:
Endow a network with the simplest dynamics and related work:
Kulakowsi et al.
investigate evolution of relationships.
Main result:
Dynamical phase transition between bipolarity and paradise.
Socially Balanced States
you
you
you
friendly
link
Jack
Jill
!0
Jack
Jill
Jack
!2
unfrustrated/balanced
unfriendly
link
Jill
!1
you
Jack
Jill
!3
frustrated/imbalanced
Social Balance
a friend of my friend is my friend;
an enemy of my enemy is my friend;
a friend of my enemy is my enemy;
an enemy of 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
probability p: unfriendly ! friendly; probability1-p: friendly ! unfriendly
p
i
1
1−p
j
k
Fundamental parameter p:
p=1/3: flip a random link in the triad equiprobably
p>1/3: predisposition toward tranquility
p<1/3: predisposition toward hostility
Triad Evolution on the Complete Graph
Basic graph characteristics:
N nodes
N (N −1)
links
2
N (N −1)(N −2)
6
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
+
n0
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−
1
−
π + n+
0
impose ṅi and π + = π −
,
+ +
− −
− −
= π + n+
0 + π n2 − π n1 − π n1 ,
gives
+
nk
=
−
nk+1
− −
− −
+ +
= π + n+
1 + π n3 − π n2 − π n2 ,
finally, use n±
k =
− −
−
π
n3 .
= π + n+
2
 (3−k)n
k
 3n0 +2n1 +n
2

! "
3
j
(1
−
ρ
)
ρ3−j
nj =
∞ ,
∞
j
ρ∞ =
 $
 1/[ 3(1 − 2p) + 1] p ≤ 1/2;

1
p ≥ 1/2
knk
n1 +2n2 +3n3
Steady State Triad Densities
steady state only for p!"
1
0.8
ρ∞
0.6
n0
paradise
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)
=
3(2p − 1)ρ2 (1 − ρ) + (1 − ρ)3

−Ct
ρ
+
Ae

∞






1 − ρ0
%
1
−
ρ(t) ∼
2t
1
+
2(1
−
ρ
)
0







1 − e−3(2p−1)t
p < 1/2;
rapid onset of
frustration
p = 1/2;
slow relaxation
to paradise
p > 1/2.
rapid attainment
of paradise
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-" is the density of unfriendly links
ln N
∼
2p − 1
N2
∼e
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
Simulations for a Finite Society
TN
106
1
p< ,
2
(a)
4
10
102
1
4
6
8
10
12
N2
TN ∼ e
14
TN
N
104
103
102
10
1
1
p= ,
2
(b)
1
102
10
TN ∼ N 4/3
103
N
TN
14
1
p> ,
2
(c)
10
6
2
1
102
10
N
103
TN
ln N
∼
2p − 1
Constrained (Socially Aware) Triad Dynamics
1. Pick a random imbalanced (frustrated) triad
2. Reverse a random link (p=#) to eliminate a frustrated triad
only if the total number of frustrated triads does not increase
Outcome: Quick approach to a final static state
Typically: TN ∼ ln N
Final state is almost always balanced even though
jammed states are much more numerous.
Jammed state: Imbalanced triads exist, but any update
only increases the number of imbalanced triads.
1
paradise
N=256
512
1024
2048
0.8
0.65
0.6
2
<" > δ = (C1 − C2 )/N
Final Clique Sizes
0.4
0.2
0
balance: two equal
size cliques
0
0.2
0.4
0.6
!0
0.8
1
Origin of the Balance/ParadiseTransition
First consider evolution of an uncorrelated network:
for + !
2!(1"!)
2
(1"!)
!2
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
!+
2!(1"!)
2
(1"!)
!2
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
Conclusion: only negative links flip, except when "!!
flow diagram for ":
0
!
1
Instability near "=!
intraclique relationship evolution
1+
ρ1 =
2
1+
ρ2 =
2
C1
for
1−
β=
2
! + , we need:
−
−
−
+
n
+
n
>
n
n− =
n−
2 , with !
3
0
1
! "# $
! "# $
frustrated
%
C2
[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
, 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
G
I
R
3 Emperor’s League 1872-81
GB
AH
I
Entente Cordiale 1904
F
G
I
German-Russian Lapse 1890
French-Russian Alliance 1891-94
GB
G
AH
R
I
Triple Alliance 1882
F
R
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, with the time until balance
strongly dependent on p
infinite network: phase transition between paradise and
social balance at p=!
Global triad dynamics (p=#):
jammed states possible but never occur
infinite network: two cliques always emerge, with paradise
when "0 0.65 (rough argument gives "0 =!)
Open questions:
incomplete graphs, indifference, continuous interactions
asymmetric relations
allow