Optimal Network Topologies for Local Search with Congestion

The problem of searchability in decentralized complex networks is of great importance in computer science, economy, and sociology. We present a formalism that is able to cope simultaneously with the problem of search and the congestion effects that arise when parallel searches are performed, and we obtain expressions for the average search cost both in the presence and the absence of congestion. This formalism is used to obtain optimal network structures for a system using a local search algorithm. It is found that only two classes of networks can be optimal: starlike configurations, when the number of parallel searches is small, and homogeneous-isotropic configurations, when it is large.

Figure 1

Optimal structures for local search with congestion. Top: starlike configuration optimal for $\rho <{\rho }^{*}$ (left) and homogeneous-isotropic configuration optimal for $\rho >{\rho }^{*}$ (right). Bottom: polarization of the optimal structure as a function of $\rho$, for networks of size $S=32$ and different number of links $L$.