1. The document describes a mathematical model of interacting particle systems or cellular automata on graphs and lattices. 2. In the model, sites can change color based on the colors of neighboring sites within a given range. 3. The document analyzes the long-term behavior of the systems, such as whether they reach consensus on one color or fluctuate between colors.

1. Moses Boudourides
University of Patras, Greece
4th Joint Japan-North America Mathematical Sociology Conference
Redondo Beach, CA, May 29 – June 1, 2008

2. • Sites are ether vertices of a finite or an
infinite (but locally finite) graph G or cells
in a lattice Zd (or a hypercube in Zd).
• Each site is in one of κ (≥ 2) colors. Let
ξt(x) denote the color of site x at time t.
• Initially, colors are distributed randomly
(uniformly and independently).
• Rule of interaction (color changes):
If ξt(x) = k, then ξt+1(x) = k + 1 mod κ,
provided that there exist at least θ sites in
the color k + 1 mod κ in the neighborhood
of x within range ρ.

3. Iterations occur according to one of
the following two scenarios:
• Random timing: At each time of a
rate 1 Poisson process, a randomly
chosen site is updated (Interacting
Particle System or IPS).
• Deterministic timing: At each time
(positive integer), all sites are
successively (and independently)
updated (Cellular Automaton or
CA).

4. When κ = 2 and θ = ρ = 1, we have the
(classic) voter model (introduced
independently by Clifford & Sudbury 1973,
Holley & Liggett 1975).
Asymptotic behavior of the IPS on Zd
(Holley & Liggett 1975):
• If d ! 2, then P(ξt(x) = ξt(y)) ! 1
(clustering or complete concensus).
• If d ! 3 and p is the density of the initial
product measure, then ξt
p converges in
distribution to ξ!
p, a one parameter family
of stationary distributions.

5. Let G be a graph of n vertices of one of
the following types:
• Erdös-Rényi random graph
• Barabási-Albert preferential attachment
graph
• Bollobás-Chung or Newman-Watts small-
world graph
Asymptotic behavior of the IPS on G
(Durrett 2007):
The consensus time for ξt
p is O(cpn) (i.e.,
the d ! 3 behavior).

6. IPS: Bramson & Griffeath 1989
CA: Fisch 1990
On Z1, when ρ = 1 and θ = 1,
if κ ! 4, ξt fluctuates,
if κ ≥ 5, ξt fixates,
where ξt fluctuates if, ! x ! Z1,
P(ξt(x) changes color at arbitrarily large t) = 1,
while ξt fixates if, ! x ! Z1,
P(ξt(x) changes color finitely often) = 1.

8. Let δ be a fixed integer in:
1 ! δ ! κ – 1.
If ξt(x) = k, then ξt+1(x) is the
closest color in the set of all colors
at range δ from k:
{k + 1, k + 2, …, k + δ} (mod κ)
that are present in the
neighborhood of x within range ρ.

9. On Z1, when 1 ! δ ! κ – 1 and ρ
≥ 1,
•if θ > ρ, ξt fixates,
•if θ ! ρ, depending on κ and δ,
ξt either fluctuates in one color
(1-fluctuation) or fluctuates
alternating in two colors within
range δ (2-fluctuation).

12. • Square domains LxL,
• with wrap-around boundary
conditions.
• Two types of neighborhoods within
range ρ = 1:
•ρ = 1D, diamonds (von
Newmann nbhd),
•ρ = 1B, boxes (Moore nbhd).

15. • External Forcing: Color λ is pushed
‘externally’ at a site, in the sense
that an m-tuple of extra neighbors
(‘influentials’) in color λ are
appended to the site.
• Internal Propensity: Color µ is
promoted ‘internally’ at a site, i.e., it
is inserted at rank q within the δ
range at that site so that the more
lower q is, the more easily the site is
‘influenced’ by color µ.

16. • The effect of external forcing of any
m-tuples of a color is much lower
than that of internal propensity that
places the same color at the first
upper position.
• This verifies Watts’ & Dodds’
disaffirmation of the ‘influentials
hypothesis.’

18. • The effect of internal propensity
fades away as the placement position
of the preferred color increases in
the δ range.
• Moreover, the preferred color of
internal propensity may become the
initiator of other colors that turn out
to be inadvertently influential.