The Competition for Shortest Paths on Sparse Graphs
  Optimal paths connecting randomly selected network nodes and fixed routers are studied analytically in the presence of non-linear overlap cost that penalizes congestion. Routing becomes increasingly more difficult as the number of selected nodes increases and exhibits ergodicity breaking in the case of multiple routers. A distributed linearly-scalable routing algorithm is devised. The ground state of such systems reveals non-monotonic complex behaviors in both average path-length and algorithmic convergence, depending on the network topology, and densities of communicating nodes and routers. 
Chi Ho Yeung and David Saad
cond-mat.dis-nn cond-mat.stat-mech cs.NI physics.soc-ph