Extended Cyclic Cellular Automata: Emulating Social Influence. By Moses A. Boudourides SunBelt XXVIII (January 22-27, 2008, St. Pete Beach, FL)

1. EXTENDED CYCLIC
CELLULAR AUTOMATA:
Emulating Social
Influence
Moses Boudourides
University of Patras, Greece
Sunbelt 28 Social Networks Conference
Jan 22-27, 2008, St. Pete Beach, FL

2. Cyclic Cellular Automata
• Each site in Zd (or a hypercube in Zd)
is in one of κ (≥ 2) colors.
• Initially, colors are distributed
randomly (uniformly and
independently).
• A site of color k will change its color
to k + 1 mod κ, if there are already
at least θ sites of that color in its
neighborhood within range ρ.

3. Extended Cyclic Cellular
Automata
A site of color k will change its color
to 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 its neighborhood
(within range ρ), where
1 ! δ ! κ – 1.

4. CCA on Z1
Bramson & Griffeath (Fisch):
On Z1, for any κ ≥ 2 (δ = 1), when ρ
= 1 and θ = 1,
§ if κ ! 4, CCA fluctuates,
§ if κ ≥ 5, CCA fixates,
where fluctuation means that, for
any time, color changes are
occurring at some sites, while
fixation means that no color
changes may occur after some time.

5. <--TIME
ɹ = 5, ɩ = 1, ɥ = 1, Ĺ = 1
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<--TIME ɹ = 3, ɩ = 1, ɥ = 1, Ĺ = 1
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6. ECCA on Z1
On Z1, for any κ ≥ 2, when 1 ! δ !
κ – 1 and ρ ≥ 1,
• if θ > ρ, ECCA fixates,
• if θ ! ρ, depending on κ and δ,
ECCA either fluctuates in one
color (1-fluctuation) or
fluctuates by turns in two colors
within range δ (2-fluctuation).

9. ECCA on squares in Z2
• 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).
• δ = 1 means CCA.
• δ ≥ 2 means ECCA.

12. Two Scenarios of Influence
• 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.

13. Findings I
• 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.’

15. Findings II
• 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 become
inadvertently influential.

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%Color
ɩ = 1D, L = 100, ɹ = 10, ɥ = 1-2, Ĺ = 1-9, intpos = 3-7, intcol = 10, extpos = 0, extcol = 0