One goal of persistent homology is to recover meaningful information from point-cloud data by examining long-lived topological features of filtered simplicial complexes built over the point-cloud. Motivated by real-world applications, the classic setting for this approach has been on finite metric spaces where many suitable complexes can be defined, and a natural filtration exists via sublevel sets of the metric. We consider the extension of persistent homology to dissimilarity networks equipped with a relaxed metric that does not assume symmetry nor the triangle inequality, by computing persistent homology on the directed clique complex defined over weighted directed graphs induced from a dissimilarity network and filtered by an adapted Rips filtration. We characterize digraph maps that induce maps on homology, describe a procedure to lift any digraph map to one that does induce maps on homology, and present a homotopy classification that provides a condition for two such digraph maps to induce the same map at the homology level. We also prove functoriality of directed clique homology and describe filtrations of digraphs induced by digraph maps. We then prove stability of persistent directed clique homology by showing that the persistence modules of a digraph and that of an admissible perturbation are interleaved. These admissible perturbations include perturbing dissimilarity measures in the network that either preserve the digraph structure or collapse series of arrows. We also explore similar constructions for maps between digraphs that allow reversal of arrows and show that while such maps, in general, produce unstable persistence barcodes, one can recover stability by inducing a reverse filtration and truncating at an appropriate threshold. Finally, we present an application of persistent directed clique homology to trace patterns and shapes embedded in migration and remittance networks.