  The aim of this work is to develop a framework for realising quantum network
algorithms with the use of prior knowledge about the structure of the network.
We seek to obtain computational methods that allows us to locally determine
network properties in a quantum superposition and drive the walk behaviour
accordingly. In particular, we consider a network that consists of different
types of edges, such that the transitions between nodes result in extra
edge-dependent phase shift. We combine amplitude amplification and phase
estimation to develop an algorithm for exploring such networks. In the layered
neural network inspired case we obtain linear increase of the search complexity
with exponential growth of the nodes number. We show that in consequence one is
able to perform quantum search algorithms with exponential speed-up compared to
quantum search that neglects the extra phase shifts.
