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Formation of Political Networks:Bifurcations, Patterns, and
Universality
Eli Ben-Naim
Theoretical Division, Los Alamos National Lab
I Motivation
II Continuum: Numerics & Scaling
III Discrete: Theory & General Features
with: Paul Krapivsky, Sidney Redner (Boston)
How many political parties?
0 5 10 15 20number of political parties
0
0.05
0.1
0.15
0.2
freq
uenc
y
Israel
Ukraine
• Data: CIA world factbook 2002
• 120 countries with multiparty senates
• Average=5.8, Variance=2.9
Simple model?
The Compromise Model
• Opinion measured by continuum variable
−∆ < x < ∆
• Compromise: via pairwise interactions
(x1, x2)→(
x1 + x22
,x1 + x22
)
• Conviction: restricted interaction range
|x1 − x2| < 1
• Initial conditions: uniform distribution
P (x, t = 0) =
{
1 |x| < ∆,
0 |x| > ∆.
• Minimal, one parameter model
• Mimics competition between compromiseand conviction
Consensus
• Nonlinear rate equations
∂P (x, t)
∂t=
∫ ∫
|x1−x2|<1
dx1dx2P (x1, t)P (x2, t)
×[
2 δ
(
x− x1 + x22
)
− δ(x− x1)− δ(x− x2)
]
• Integrable for ∆ < 1/2:
〈x2(t)〉 = 〈x2(0)〉 e−∆t
• Final state: localized
P∞(x) = 2∆δ(x)
• Time dependence: similarity solution
Φ(z) =2∆
π
1
(1 + z2)2z =
x
〈x2(t)〉1/2
Generally, what is nature of final state?
Diversity
−4 −2 0 2 4x
t=9
t=6
t=3
t=0.5
•√
Numerical integration of rate equations
• Monte Carlo simulation of random process
• Final state:
P∞(x) =
N∑
i=1
mi δ(x− xi)
Multiple political networks (parties)
Bifurcations and Patterns
0 2 4 6 8 10∆
−10
−5
0
5
10
x
majorcentralminor
• Periodic sequence of bifurcationsx(∆) = x(∆ + L)
• Alternating major-minor pattern• Clusters are equally spaced• Period → cluster mass, separation
L = 2.155
Self-similar structure, universality
Cluster masses, bifurcation types
0 2 4 6 8 10∆
−10
−5
0
5
10
x
majorcentralminor
1 2 3
0 2 4 6 8 10∆
10−8
10−6
10−4
10−2
m
0
1
2
3
4
m
• Masses are periodic as well
m(∆) = m(∆ + L)
• 3 types of bifurcations:
1. ∅ → {−x, x} Nucleation of 2 minor branches2. {0} → {−x, x} Nucleation of 2 major branch’s3. ∅ → {0} Nucleation of central cluster
• Bifurcations occur near origin• Major: M → 2.15, Minor: m→ 3× 10−4
Central cluster may or may not exist
Near critical behavior
−1 1 1+ε−1−ε
• Perturbation theory: ∆ = 1 + ε
• Central cluster: mass M , x(∞) = 0• Minor cluster: mass m, x(∞) = 1 + ε/2
dm
dt= −mM → m(t) ∼ ε e−t
• Process stops when x ∼ e−tf/2 ∼ ε
• Final minor cluster massm(∞) ∼ m(tf) ∼ ε3
• Argument generalizes to type 3 bifurcations
m ∼ (∆−∆c)α α =
3 type 1
4 type 3
Masses vanish algebraically near type 1, 3 bif
Discrete Opinions
1
11/2
1/2
2P
1P
• Basic process: (i− 1, i+ 1)→ (i, i)
• Rate equation:
d
dtPi = 2Pi−1Pi−1 − Pi(Pi−2 + Pi+2)
• Example: 6 states, Pi = PN−i
• Initial conditions determine final state
• Isolated fixed points, lines of fixed points
General Features
0 10 20 30 40 50 60N
−30
−20
−10
0
10
20
30
x
• Dissipative system: volume contracts
• Lyapunov (energy) function exists 〈x2〉
• No cycles or strange attractors
• Uniform state is unstable: Pi = 1 + φi
φt +(
φ+ aφxx + bφ2)
xx= 0
Discrete case yields useful insights
Exponential initial conditions
0 2 4 6 8 10∆
−10
−5
0
5
10
x
0 2 4 6 8 10∆
10−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
101
m
• Bifurcations induced at the boundary
• Periodic structure
• Two types of bifurcations
1. Nucleation of major branch2. Nucleation of minor branch
Central party is stable
Conclusions
• Networks form via bifurcations
• Periodic structure
• Alternating major-minor pattern
• Central party not always exists
• Power-law behavior near transitions
Outlook
• Role of initial conditions? Classification?
• Role of spatial dimension? Correlations?
• Add disorder, inhomogeneities
• Tiling/Packing in 2D,3D?
E. Ben-Naim, P.L. Krapivsky, S. Redner, it cond-mat/0212313, Physica D, submitted.G. Weisbuch, cond-mat/0111494.
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