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.