Proof of uniform sampling of binary matrices with fixed row sums and column sums for the fast Curveball algorithm

Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

Volumn 91, Issue 4, 2015, Pages

  Carstens, C J ^a  

^a RMIT UNIVERSITY   (Australia)

Author keywords

[No Author keywords available]

Indexed keywords

BIOINFORMATICS;

BINARY MATRIX; COMPUTATIONAL BIOLOGY; DEGREE SEQUENCE; DIRECTED NETWORK; QUANTITATIVE TOOL; RANDOM MATRICES; UNIFORM DISTRIBUTION; UNIFORM SAMPLING;

ALGORITHMS;

ALGORITHM; PROBABILITY;

ALGORITHMS; MARKOV CHAINS;

EID: 84929190456     PISSN: 15393755     EISSN: 15502376     Source Type: Journal    
DOI: 10.1103/PhysRevE.91.042812     Document Type: Article

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15. From now on we refer to the "switch and hold" method from Ref. [9] as "the switching method".
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18. Depending on the dimensions of the matrix it may be faster to make a list of the columns, and depending on the number of zeros and ones in the matrix it may be faster to make lists of the zeros.
19. This explicit description of step (d) is based on the implementation of the Curveball algorithm as can be downloaded from http://www.nature.com/ncomms/2014/140611/ncomms5114/extref/ncomms5114-s6.txt.
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21. These formulas for the transition probabilities are not in correspondence with those presented in Ref. [17] for the small example of matrices with row sums and column sums equal to (1,2,1). However, these formulas correspond to the algorithm as found in the Supplementary Material code of Ref. [17]. The probabilities presented in Ref. [17] correspond to the good-shuffle Curveball algorithm discussed in Sec. IIIA.
22. An implementation of the good-shuffle Curveball algorithm can be found at http://github.com/queenBNE/Curveball.