  We study the inverse problem of determining the conductivity matrix of an
electrical network from the prescribed knowledge of the magnitude of the
induced current along the edges coupled with the imposed voltage or injected
current on the boundary nodes. This problem leads to a weighted $l^1$
minimization problem for the corresponding voltage potential. We also
investigate the problem of determining the transition probabilities of random
walks on graphs from the prescribed net number of times the walker passes along
the edges of the graph. We also show that a mass preserving flow $J=(J_{i.j})$
on a network can be uniquely recovered from the knowledge of $|J|=(|J_{i,j}|)$
and the flux of the flow on the boundary nodes, where $J_{i,j}$ is the flow
from node $i$ to node $j$ and $J_{i,j}=-J_{j,i}$. Convergent numerical
algorithms for solving such problems are also presented.
