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

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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