The main contribution of this work is a fast algorithm to compute the barycentre of a set of networks based on a Laplacian spectral pseudo-distance. The core engine for the estimation of the barycentre is an algorithm that explores the large library of Soules bases, and returns the best Soules basis that leads to the reconstruction of a weighted network whose spectrum is the sample mean spectrum, and whose geometry matches that of the sample mean adjacency matrix. We prove that when the networks are random realizations of stochastic block models our algorithm reconstructs the population mean adjacency matrix. In addition to the theoretical analysis, we perform Monte Carlo simulations to validate the theory. This work is significant because it opens the door to the design of new spectral-based network synthesis that have theoretical guarantees.