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 Fritz Heider, 1946 friendly link you you you unfriendly link you Jack Jill Jack Jill Jack Jill Jack Jill 0 2 1 3 unfrustrated / balanced frustrated / imbalanced Def: A network is balanced if all triads are balanced a friend of my friend is my friend; no 1 { a friend of my enemy is my enemy; an enemy of my friend is my enemy; no 3 an enemy of my enemy is my friend. Arthashastra, 250 BCE
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) gang relations like 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 i p 1 j k 1 p 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 (Complete Graph) T N 10 6 10 4 10 2 1 (a) Two Cliques p < 1 2, T N e N 2 4 6 8 10 12 14 N T N T N 10 4 10 3 10 2 10 1 14 10 6 1 (b) (c) 10 N 10 2 10 3 p = 1 2, T N N 4/3 p > 1 2, T N ln N 2p 1 2 1 10 N 10 2 10 3 Utopia
Triad Evolution (Infinite Complete Graph) Basic graph characteristics: N nodes N(N 1) 2 links N(N 1)(N 2) 6 triads n ± k ρ = friendly link density n k = density of triads of type k = density of triads of type k attached to a ± link n + 1 n + 2 n 1 n + 0 n 2 n 3 N-2 triads positive link negative link
n ± k Triad Evolution on the Complete Graph n k = density of triads of type k = density of triads of type k attached to a ± link π + = (1 p) n 1 flip rate + π = p n 1 + n 3 flip rate + Master equations: 1 0 1 p p 0 1 1 dn 0 dt dn 1 dt dn 2 dt dn 3 dt = π n 1 π+ n + 0, = π + n + 0 + π n 2 π n 1 π+ n + 1, = π + n + 1 + π n 3 π n 2 π+ n + 2, = π + n + 2 π n 3.
Steady State Triad Densities 1 0.8 0.6 ( ) 3 n j = j ρ = ρ 3 j (1 ρ ) j, 1/[ 3(1 2p) + 1] p 1/2; 1 p 1/2 ρ n 0 0.4 n 1 n 2 utopia 0.2 0 n 3 0 0.1 0.2 0.3 0.4 0.5 p
Triad Evolution (Infinite Complete Graph) rate equation for the friendly link density: + in 1 + in 1 + in 3 dρ dt = 3ρ 2 (1 ρ)[p (1 p)] + (1 ρ) 3 = 3(2p 1)ρ 2 (1 ρ) + (1 ρ) 3 ρ + Ae Ct p < 1/2; rapid onset of frustration ρ(t) 1 1 ρ 0 1 + 2(1 ρ0 ) 2 t p = 1/2; slow relaxation to utopia 1 e 3(2p 1)t 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: T N 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 10 2 ρ 0 =0 1/4 1/2 P jam 10 3 10 4 jammed states unlikely for large N 10 5 10 1 10 2 10 3 N
Final Clique Sizes δ 2 δ = C 1 C 2 N 1 0.8 0.6 N=256 512 1024 2048 utopia 2/3 ρ 0 = 1/2 (rough argument gives ) <δ 2 > 0.4 0.2 0 balance: two equal size cliques 0 0.2 0.4 0.6 0.8 1 ρ 0 : 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 GB AH GB AH F G F G F G R 3 Emperor s League 1872-81 I R Triple Alliance 1882 I R German-Russian Lapse 1890 I GB AH GB AH GB AH F G F G F G R I R I R I French-Russian Alliance 1891-94 Entente Cordiale 1904 British-Russian Alliance 1907
Long Beach Gang Lesson Figure 2. Gang Network Nakamura, Tita, & Krackhardt (2007) gang relations like hate Figure 3. Gang Network with Violent Incidents _: Gang _: Ally Relation _: Enemy Relation Imbalance implies impulsiveness Balance implies prudence 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 ρ 0 2/3 Open questions: allow several cliques in balanced state asymmetric relations dynamics in real systems, gang control?