Journal of Statistical Mechanics: Theory and Experiment
Volume 2006
December 2006
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Michele Leone et al J. Stat. Mech. (2006) P12012 doi:10.1088/1742-5468/2006/12/P12012
Finite size corrections to random Boolean networks
Michele Leone1, Andrea Pagnani1, Giorgio Parisi2 and Osvaldo Zagordi3
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| Figure 1. Factor graph representation of a small Boolean network: circles symbolize the variables, squares the Boolean functions. Arrows stress the directed nature of the graph. Variables x1, x2, x7, x8 are external inputs (non-regulated variables). |
Figure 2. Histograms of the exhaustive algorithm estimates measured on 10 000 samples of
N = 50 (red boxes) and N = 100 (blue boxes) for four different choices of  and  expressed as a percentage. The black arrow is the numerical estimate using BP, that agrees perfectly with the theoretical value
σ = (1–α) ln 2. |
Figure 3. Histogram of cos(θBP–EX) compared with the reference histogram of cos(θEX–π(EX)) (left panel). In the right panel we display the histogram of both
dBP–EX and
dEX–π(EX). Measurements are done at N = 100, M = 90 and a percentage of  functions of X = 20%, 60% over an ensemble of 104 samples. |
| Figure 4. Distribution of the overlaps for 10 000 samples at α = 0.93. X = 0 (left panel), X = 0.5 (central panel), X = 1 (right panel), for N = 50 (thin line) and N = 100 (thick line). |
Figure 5. Plot of C(α, X) for the second moment of the distribution of the number of solutions in the large-N limit. The red uppermost line is the function
e(x–1)/x, diverging for  . |
| Figure 6. Second moment of the number of solutions distribution. Bars are computed by exhaustively counting the solutions for 100 000 samples at α = 1, crosses by evaluating the analytical formula with Mathematica. |
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