  Biological networks often encapsulate promotion/inhibition as signed
edge-weights of a graph. Nodes may correspond to genes assigned expression
levels (mass) of respective proteins. The promotion/inhibition nature of
co-expression between nodes is encoded in the sign of the corresponding entry
of a sign-indefinite adjacency matrix, though the strength of such
co-expression (i.e., the precise value of edge weights) cannot typically be
directly measured. Herein we address the inverse problem to determine network
edge-weights based on a sign-indefinite adjacency and expression levels at the
nodes. While our motivation originates in gene networks, the framework applies
to networks where promotion/inhibition dictates a stationary mass distribution
at the nodes. In order to identify suitable edge-weights we adopt a framework
of ``negative probabilities,'' advocated by P.\ Dirac and R.\ Feynman, and we
set up a likelihood formalism to obtain values for the sought edge-weights. The
proposed optimization problem can be solved via a generalization of the
well-known Sinkhorn algorithm; in our setting the Sinkhorn-type ``diagonal
scalings'' are multiplicative or inverse-multiplicative, depending on the sign
of the respective entries in the adjacency matrix, with value computed as the
positive root of a quadratic polynomial.
