new networks evolve when both friendly and · 2008. 8. 12. · unfriendly you link! 1 jack jill...
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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)
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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
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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
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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
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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
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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
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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
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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
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− → + 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)
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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.
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(a)
(b) (c)
Jammed States
N = 9, 11, 12, 13, . . . Jammed states are only for:
N = 9Examples for
(unresolved situations)
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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
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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
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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
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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
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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
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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
top related