Voltage Distribution in Growing Conducting Networks

We investigate by random-walk simulations and a mean-field theory how growth by biased addition of nodes affects flow of the current through the emergent conducting graph, representing a digital circuit. In the interior of a large network the voltage varies with the addition time $s<t$ of the node as $V(s)\sim \ln (s)/s^\theta$ when constant current enters the network at last added node $t$ and leaves at the root of the graph which is grounded. The topological closeness of the conduction path and shortest path through a node suggests that the charged random walk determines these global graph properties by using only {\it local} search algorithms. The results agree with mean-field theory on tree structures, while the numerical method is applicable to graphs of any complexity.