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Ground state entropy of the network source location problem is evaluated atboth the replica symmetric level and one-step replica symmetry breaking levelusing the entropic cavity method. The regime that is a focus of this study, isclosely related to the vertex cover problem with randomly quenched coverednodes. The resulting entropic message passing inspired decimation andreinforcement algorithms are used to identify the optimal location of sourcesin single instances of transportation networks. The conventional beliefpropagation without taking the entropic effect into account is also compared.We find that in the glassy phase the entropic message passing inspireddecimation yields a lower ground state energy compared to the beliefpropagation without taking the entropic effect. Using the extremal optimizationalgorithm, we study the ground state energy and the fraction of frozen hubs,and extend the algorithm to collect statistics of the entropy. The theoreticalresults are compared with the extremal optimization results.

The network source location problem: ground state energy, entropy and effects of freezing