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

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.

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.

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.

<-- T

IME

ɹ = 5, ɩ = 1, ɥ = 1, Ĺ = 1

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ɹ = 3, ɩ = 1, ɥ = 1, Ĺ = 1

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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).

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.

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.

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.’

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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olor

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