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T. Antal, (Harvard), P. L. Krapivsky, and S.Redner (Boston University) PRE 72, 036121 (2005), Physica D 224, 130 (2006)
Social Balance on Networks: The Dynamics of Friendship and Hatred
Basic question:How do social networks evolve when both friendly and unfriendly relationships exist?
Partial answers: 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?
(Heider 1944, Cartwright & Harary 1956, Wasserman & Faust 1994)
Socially Balanced States
unfrustrated / balanced frustrated / imbalanced
friendly link
!0
Jack Jill
you
!3
Jack Jill
youunfriendly link
!1
Jack Jill
you
!2
Jack Jill
you
Fritz Heider, 1946
a friend of my friend is my friend;a friend of my enemy is my enemy; an enemy of my friend is my enemy;an enemy of my enemy is my friend.
{Arthashastra, 250 BCE
!1no ⇒
!3no ⇒
Def: A network is balanced if all triads are balanced
Theorem: on a complete graph a balanced state is
- either utopia: everyone likes each other
- or two groups of friends with hate between groups
For not complete graphs there can be more than two cliques
(you are either with us or against us)
Static description
Page 40 of 44
Figure 2. Gang Network
Figure 3. Gang Network with Violent Incidents
_: Gang
_: Ally Relation
_: Enemy Relation
_: Gang
_: Number of Violent Attacks(Thickness is proportional to
the number)
Long Beach Gangs Nakamura, Tita, & Krackhardt (2007)
gang relationslike
hate
violence frequencylow incidence
high incidence
how does violence correlate with relations?
11−p
p
j k
i
Local Triad Dynamics on Arbitrary Networks
1. Pick a random imbalanced (frustrated) triad
p=1/3: flip a random link in the triad equiprobablyp>1/3: predisposition toward tranquilityp
Finite Society Reaches Balance (Complete Graph)
106
104
102
1 4 6 8 10 12 14
T N
N
(a)
104
103
102
101
103102101
T N
N
(b)
2
6
10
14
103102101
T N
N
(c)
p <1
2, TN ∼ e
N2
p =1
2, TN ∼ N
4/3
p >1
2, TN ∼
lnN
2p − 1
Utopia
Two Cliques
Triad Evolution (Infinite Complete Graph)
n+0
n+1
n+2 n
−
1
n−
2
n−
3
N nodes
N(N−1)2 links
N(N−1)(N−2)6 triads
Basic graph characteristics:
n±
k= density of triads of type k attached to a ± link
nk = density of triads of type k
ρ = friendly link density
positive link negative link
N-2 triads
Triad Evolution on the Complete Graph
n±
k= density of triads of type k attached to a ± link
nk = density of triads of type k
dn0
dt= π−n−1 − π
+n+0 ,
dn1
dt= π+n+0 + π
−n−2 − π−n−1 − π
+n+1 ,
dn2
dt= π+n+1 + π
−n−3 − π−n−2 − π
+n+2 ,
dn3
dt= π+n+2 − π
−n−3 .
Master equations:!0 → !1!1 → !0
π+ = (1 − p) n1 flip rate + → −
π− = p n1 + n3 flip rate − → +
1−p
1p
Steady State Triad Densities
0 0.1 0.2 0.3 0.4 0.5p
0
0.2
0.4
0.6
0.8
1
ρ∞
n0
n1
n2
n3
utopia
nj =
(
3
j
)
ρ3−j∞
(1 − ρ∞)j ,
ρ∞ =
1/[√
3(1 − 2p) + 1] p ≤ 1/2;
1 p ≥ 1/2
− → + in #1 + → − in #1 − → + in #3
rate equation for the friendly link density:
dρ
dt= 3ρ2(1 − ρ)[p − (1 − p)] + (1 − ρ)3
= 3(2p − 1)ρ2(1 − ρ) + (1 − ρ)3
ρ(t) ∼
ρ∞ + Ae−Ct p < 1/2;
1 −1 − ρ0
√
1 + 2(1 − ρ0)2tp = 1/2;
1 − e−3(2p−1)t p > 1/2.
rapid onset of frustration
rapid attainment of utopia
slow relaxationto utopia
Triad Evolution (Infinite Complete Graph)
Constrained (Socially Aware) Triad Dynamics
1. Pick a random link
Final state is either balanced or jammed
Jammed state: Imbalanced triads exist, but any update only increases the number of imbalanced triads.
TN ∼ lnN
Outcome: Quick approach to a final static stateTypically:
2. Reverse it only if the total number of balanced triads increases or stays the same
Final state is almost always balanced even though jammed states are much more numerous.
(a)
(b) (c)
Jammed States
N = 9, 11, 12, 13, . . . Jammed states are only for:
N = 9Examples for
(unresolved situations)
101 102 103
N10−5
10−4
10−3
10−2
10−1P j
amρ0=0 1/4 1/2
Likelihood of Jammed States
jammed states unlikely for large N
Final Clique Sizes
balance: two equal size cliques
utopia
≈2/3
〈δ2〉
δ =C1 − C2
N
0 0.2 0.4 0.6 0.8 1ρ0
0
0.2
0.4
0.6
0.8
1
N=256 512 1024 2048
(rough argument gives )ρ0 = 1/2
: 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
3 Emperor’s League 1872-81
GB AH
F
R I
G
Triple Alliance 1882
GB AH
F
R I
G
Entente Cordiale 1904
GB AH
F
R I
G
British-Russian Alliance 1907
GB AH
F
R I
G
German-Russian Lapse 1890
GB AH
F
R I
G
French-Russian Alliance 1891-94
GB AH
F
R I
G
Page 40 of 44
Figure 2. Gang Network
Figure 3. Gang Network with Violent Incidents
_: Gang
_: Ally Relation
_: Enemy Relation
_: Gang
_: Number of Violent Attacks(Thickness is proportional to
the number)
gang relationslike
hate
violence frequencylow incidence
high incidence
Long Beach Gang Lesson Nakamura, Tita, & Krackhardt (2007)
Imbalance implies impulsivenessBalance implies prudence
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 dynamicsjammed states possible but rarely occur
Open questions:
asymmetric relationsallow → several cliques in balanced state
dynamics in real systems, gang control?
infinite network: two cliques always emerge, with utopia for ρ0 ! 2/3