The global minima of the communicative energy of natural communication systems

doi:10.1088/1742-5468/2007/06/P06009

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Journal of Statistical Mechanics: Theory and Experiment

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Journal of Statistical Mechanics: Theory and Experiment Volume 2007 June 2007 Create an alert RSS this journal

Ramon Ferrer i Cancho and Albert Díaz-Guilera J. Stat. Mech. (2007) P06009 doi:10.1088/1742-5468/2007/06/P06009

The global minima of the communicative energy of natural communication systems

Ramon Ferrer i Cancho and Albert Díaz-Guilera

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Table 1. Summary of results about the definitions of various entropies for models A (p(r_j) = ω_j/M) and B (p(r_j) = 1/m with ω_j ≥ 1). S and R are, respectively, the set of signals and the set of stimuli. H(S, R) is the joint entropy of S and R. H(R|S) is the conditional entropy of R when S is known and H(S|R) is the conditional entropy of S when R is known. H(S) and H(R) are, respectively, the entropies of S and R. $b_i=\sum_{k=1}^m a_{ik}/\omega_k$ .

	Model A: p(r_j) = ω_j/M	Model B: p(r_j) = 1/m (with ω_j ≥ 1)
H(S, R)	log M	$({1}/{m})\sum_{j=1}^m ({\log (m \omega_j)}/{\omega_j})$
H(R\|S)	$({1}/{M})\sum_{i=1}^{n} \mu_i \log \mu_i$	$({1}/{m}) \sum_{i=1}^n b_i H(R\|s_i)$
H(R\|s_i)	log μ_i	$\log b_i+({1}/{b_i}) \sum_{j=1}^m ({a_{ij}}/{\omega_j})\log \omega _j$
H(S\|R)	$({1}/{M})\sum_{j=1}^{m} \omega_j \log \omega_j$	$({1}/{m})\sum_{j=1}^{m} \log \omega_j$
H(S\|r_j)	log ω_j	log ω_j
H(S)	H(S, R)–H(R\|S)	$\log M-({1}/{M})\sum_{i=1}^{n} \mu_i \log \mu_i$
H(R)	$\log M-({1}/{M})\sum_{j=1}^{m} \omega_j \log \omega_j$	log m

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The global minima of the communicative energy of natural communication systems

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