Local negative circuits and fixed points in non-expansive Boolean networks
Given a Boolean function F:{0,1}n?{0,1}n F : { 0 , 1 } n ? { 0 , 1 } n , and a point x x in {0,1}n { 0 , 1 } n , we represent the discrete Jacobian matrix of F F at point x x by a signed directed graph GF(x) G F ( x ) . We then focus on the following open problem: Is the absence of a negative circuit in GF(x) G F ( x ) for every x x in {0,1}n { 0 , 1 } n a sufficient condition for F F to have at least one fixed point? As result, we give a positive answer to this question under the additional condition that F F is non-expansive with respect to the Hamming distance.
negative circuit fixed point boolean network discrete jacobian matrix interaction graph