In recent years, there has been a surge in demand for statistical tools for analyzing dynamic networks involving point processes. This has been driven largely by the availability of high-dimensional data in a number of application areas such as systems neuroscience and stochastic finance. Given some network data, an extremely challenging problem is to infer the causal dependencies between nodes to reconstruct the network. The high volume and decentralized storage of such data poses new challenges for analysis. Here we develop for the first time a distributed optimization algorithm based on the point process likelihood for dynamic networks of interacting Hawkes processes. We demonstrate the algorithm using genomic data to construct a transcriptional regulatory network in embryonic stem cells and tick-level financial data to construct an influence map of major currencies.