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Social influencing and associated random walk models: Asymptotic consensus times on the complete graph
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10.1063/1.3598450
/content/aip/journal/chaos/21/2/10.1063/1.3598450
http://aip.metastore.ingenta.com/content/aip/journal/chaos/21/2/10.1063/1.3598450

Figures

Image of FIG. 1.
FIG. 1.

(Color online) Consensus time for voter model on complete graph. The vertical axis represents the consensus time (normalized by N). The horizontal axis is the number of nodes in complete graph. Each star point is an average of 10 runs of numerical simulations of voter model and the solid straight line consists of the solutions of the linear equation for each N value.

Image of FIG. 2.
FIG. 2.

(Color online) Vector field of the random walk coarse-grained from 2-word naming game. Each vector is the expected drift of the random walk at macrostates . The network is 10 nodes complete graph and the domain of random walk is the lower triangle of the square lattice.

Image of FIG. 3.
FIG. 3.

(Color online) Consensus time (normalized by N) as a function of the logarithm of the network size N for 2-word naming game on complete graph. Each star point is an average of 10 runs of numerical simulations of 2-word naming game and the solid straight line consists of the solutions of the linear equation for each N.

Image of FIG. 4.
FIG. 4.

(Color online) The expected time spent on each macrostate before consensus in the 2-word NG on a complete graph with N = 100 nodes. The vertical axis T(n A ,n B ) is the expected time that the random walk spends in macrostate (n A ,n B ) before consensus, starting from the (n A (0),n B (0)) = (50,50) initial macrostate.

Image of FIG. 5.
FIG. 5.

(Color online) Vector field representing the drift of the coarse-grained random walk for four different central influence levels on a complete graph with N=20. Increasing f corresponds to a progressively stronger biased flow toward state A. The length of each vector has been rescaled to its square root to avoid cluttering the whole graph.

Image of FIG. 6.
FIG. 6.

(Color online) Expected time spent on each macrostate before consensus T(n A ,n B ) in the 2-word NG on complete graph with N = 100 with central influence f = 0.05, starting from a unbiased initial macrostate (n A (0),n B (0)) = (50,50).

Image of FIG. 7.
FIG. 7.

(Color online) Probability of all A consensus P A with different external influence level f starting from macrostate (n A ,n B ) = (n 0,N − n 0) on 100-node complete graph.

Image of FIG. 8.
FIG. 8.

(Color online) Probability of all B consensus 1 − P A starting from macrostate (n A ,n B ) = (N/2,N/2) as a function of network size N with different external influence level f ’s.

Image of FIG. 9.
FIG. 9.

(Color online) Expected normalized consensus time () as a function of the initial macrostate (n A ,n B ) on the complete graph with N = 100 nodes. (a) When the fraction of committed agents is q = 0.06 < q c and (b) when q = 0.12 > q c .

Image of FIG. 10.
FIG. 10.

(Color online) Expected time spent in each macrostate before consensus T(n A ,n B ) on the complete graph with N = 100 nodes, starting from the (n A (0),n B (0)) = (n q ,N − n q ) initial macrostate, (a) for q = 0.06 < q c and (b) for q = 0.12 > q c .

Image of FIG. 11.
FIG. 11.

(Color online) Normalized time spent near the consensus state before consensus as a function of network size N for different fraction of committed agents q, including cases for both q < q c and q > q c . Note the logarithmic scales on the horizontal axis.

Image of FIG. 12.
FIG. 12.

(Color online) Normalized time spent near the meta-stable state as a function of network size N for different fraction of committed agents q, (a) for q < q c and (b) for q > q c . The behavior for q = 0.08 is shown in both (a) and (b), corresponding to our rough estimate of the critical fraction of committed agents, q c ≈ 0.08 ± 0.01.

Tables

Generic image for table
Table I.

Update events for the voter model and the associated random walk transition probabilities.

Generic image for table
Table II.

Update events for the 2-word naming game and the associated random walk transition probabilities.

Generic image for table
Table III.

Update events for the 2-word naming game with central influence and the associated random walk transition probabilities.

Generic image for table
Table IV.

Update events for the 2-word naming game with committed agents and the associated random walk transition probabilities.

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/content/aip/journal/chaos/21/2/10.1063/1.3598450
2011-06-28
2014-04-18
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Social influencing and associated random walk models: Asymptotic consensus times on the complete graph
http://aip.metastore.ingenta.com/content/aip/journal/chaos/21/2/10.1063/1.3598450
10.1063/1.3598450
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