Making Networks Less Amplifiers Under Resource Constraints

In a network invaded by a mutant according to the birth–death updating rule and uniform initialization, in order to minimize the amplifying effect of any directed network, the adjacency matrix is modified at each time step up to a given time horizon, subject to resource constraints. The fixation probability of an invasive mutant is deduced from the first eigenvector of the resulting modified Markov transition matrix. Large-scale minimization is solved numerically for a representative sample of directed graphs of dimensions 6 to 8. The effects of the determinants of the optimal reduction of the fixation probability are estimated using a Heckman selection model. The number of neighbors, the heterogeneity of the incoming edge weights, and the homogeneity of the outgoing edge weights of the initial network increase the likelihood that the graphs are amendable. Among the amended networks, the reduction in the fixation probability is greater when the outgoing edge weights of the initial network are heterogeneous, those of its incoming edges are homogeneous, and the sequence of modifications increases the variance of the outgoing edge weights and decreases that of the incoming edge weights, thereby creating a trade-off, which is estimated numerically.