Social Balance on Networks:
The Dynamics of Friendship and Hatred
T. Antal, (Harvard), P. L. Krapivsky, and S.Redner (Boston University)
PRE 72, 036121 (2005), Physica D 224, 130 (2006)
Basic question:
How do social networks evolve when both friendly and
unfriendly relationships exist?
Partial answers:
(Heider 1944, Cartwright & Harary 1956, Wasserman & Faust 1994)
Social balance as a static concept: a state without frustration.
This work:
Endow a network with the simplest dynamics and
investigate evolution of relationships.
Main questions:
Is balance ever reached? What is the final state like?
Or does the network evolves forever?
Socially Balanced States
you
Fritz Heider, 1946
you
you
friendly
link
Jack
Jill
Jack
Jill
!0
!2
unfrustrated / balanced
Jack
unfriendly
link
Jill
you
Jack
!1
!3
frustrated / imbalanced
Def: A network is balanced if all triads are balanced
a friend of my friend is my friend;
a friend of my enemy is my enemy;
no !1 ⇒
an enemy of my friend is my enemy;
an enemy of my enemy is my friend.
no !3 ⇒
{
Arthashastra, 250 BCE
Jill
Theorem: on a complete graph a balanced state is
- either utopia:
everyone likes each other
- or two groups of friends
with hate between groups
(you are either with us or against us)
For not complete graphs there can be more than two cliques
Static description
Long Beach Gangs
Figure 2. Gang Network
Nakamura,Tita, &
Krackhardt (2007)
like
gang relations
hate
_: Gang
_: Ally Relation
_: Enemy Relation
how does violence correlate with relations?
Figure 3. Gang Network with Violent Incidents
violence frequency
low incidence
high incidence
_: Gang
_: Number of Violent Attacks
(Thickness is proportional to
the number)
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
Finite Society Reaches Balance
Two Cliques
6
10
TN
(Complete Graph)
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
Utopia
TN
ln N
∼
2p − 1
Triad Evolution (Infinite 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 Triad Densities
1
! "
3
j
(1
−
ρ
)
ρ3−j
nj =
∞ ,
∞
j
0.8
ρ∞ =
 $
 1/[ 3(1 − 2p) + 1] p ≤ 1/2;

0.6
1
ρ∞
p ≥ 1/2
n0
n1
utopia
0.4
n2
0.2
n3
0
0
0.1
0.2
0.3
p
0.4
0.5
Triad Evolution (Infinite Complete Graph)
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 utopia
p > 1/2.
rapid attainment
of utopia
Constrained (Socially Aware) Triad Dynamics
1. Pick a random link
2. Reverse it only if
the total number of balanced triads increases or stays the same
Outcome: Quick approach to a final static state
Typically: TN ∼ ln N
Final state is either balanced or jammed
Jammed state: Imbalanced triads exist, but any update
only increases the number of imbalanced triads.
Final state is almost always balanced even though
jammed states are much more numerous.
Jammed States
Examples for
N =9
(a)
(b)
(c)
Jammed states are only for:
N = 9, 11, 12, 13, . . .
(unresolved situations)
Likelihood of Jammed States
10
−1
ρ0=0
1/4
1/2
Pjam
10
10
−2
−3
10
−4
10
−5
jammed states unlikely for large N
10
1
10
N
2
3
10
Final Clique Sizes
1
utopia
N=256
512
1024
2048
2
!δ " 0.8
C1 − C2
δ=
N
≈2/3
(rough argument gives
ρ0 = 1/2 )
2
<δ >
0.6
0.4
0.2
balance: two equal
size cliques
0
0
0.2
0.4
0.6
0.8
1
ρ0 : initial density of friendly links
Related theoretical works
Kulakowsi et al. ‘05:
- Continuous measure of friendship
Meyer-Ortmanns et al. ‘07:
- Diluted graphs
- Spatial effects
- k-cycles
- ~Spin glasses ~Sat problems
A Historical Lesson
GB
AH
F
GB
G
R
F
I
GB
G
R
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
GB
G
R
3 Emperor’s League 1872-81
GB
AH
AH
F
G
R
I
British-Russian Alliance 1907
Long Beach Gang Lesson
Nakamura, Tita, &
Krackhardt (2007)
Figure 2. Gang Network
like
gang relations
hate
_: Gang
_: Ally Relation
_: Enemy Relation
Imbalance implies impulsiveness
Balance implies prudence
Figure 3. Gang Network with Violent Incidents
violence frequency
low incidence
high incidence
_: Gang
_: Number of Violent Attacks
(Thickness is proportional to
the number)
Summary & Outlook
Local triad dynamics:
finite network: social balance, with the time until balance
strongly dependent on p
infinite network: phase transition at p=½ between utopia
and a dynamical state
Constrained triad dynamics
jammed states possible but rarely occur
infinite network: two cliques always emerge,
with utopia for ρ0 ! 2/3
Open questions:
allow
→ several cliques in balanced state
asymmetric relations
dynamics in real systems, gang control?