Two fundamental problems for extraterrestrial intelligences (ETIs) attempting to establish interstellar communication are timing and energy consumption. Humanity's study of exoplanets via their transit across the host star highlights a means of solving both problems. An ETI ‘A’ can communicate with ETI ‘B’ if B is observing transiting planets in A's star system, either by building structures to produce artificial transits observable by B, or by emitting signals at B during transit, at significantly lower energy consumption than typical electromagnetic transmission schemes. This can produce a network of interconnected civilizations, establishing contact via observing each other's transits. Assuming that civilizations reside in a Galactic Habitable Zone (GHZ), I conduct Monte Carlo Realization simulations of the establishment and growth of this network, and analyse its properties in the context of graph theory. I find that at any instant, only a few civilizations are correctly aligned to communicate via transits. However, we should expect the true network to be cumulative, where a ‘handshake’ connection at any time guarantees connection in the future via e.g. electromagnetic signals. In all our simulations, the cumulative network connects all civilizations together in a complete network. If civilizations share knowledge of their network connections, the network can be fully complete on timescales of order a hundred thousand years. Once established, this network can connect any two civilizations either directly, or via intermediate civilizations, with a path much less than the dimensions of the GHZ.