Theoretical Bounds for the Number of Inferable Edges in Sparse Random Networks

The inference of a network structure from experimental data providing dynamical information about the underlying system of investigation is an important and still outstanding problem if the number of nodes within a network is not small. For example, high-throughput data from gene networks of, e.g., metabolic, signaling or transcriptional regulatory networks, provide information of thousands of genes or products thereof. Theoretically, the graph-theoretical measure dseparation provides a criteria to recover the network structure edge-by-edge by calculating the partial correlation. However, practically, for large networks it is not possible to estimate the partial correlation up to an arbitrary order because the number of possible d-separating sets grows exponential with the number of nodes in the network. In this paper, we determine numerically theoretical bounds for the number of inferable edges in directed (possible cyclic) sparse random networks if the maximal size of d-separating sets is restricted to nmax. Under ideal experimental conditions these bounds correspond to the maximal precision an unknown network structure can be recovered utilizing partial correlation of order up to nmax.

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