In the XXI century, network communication generates its own philosophical discourse of thinking. However, network discourse and network interaction appeared at the dawn of axial time, first, in the Eastern philosophy and, eventually, in the Western one. The authors used the information and communication approach, as well as the historical method, to trace the network discourse from traditional Chinese philosophy all the way through Leibnizianism and classical German philosophy. The research also covered the manifestations of triadic network thinking in the Western philosophy of the XX century, including the Russian philosophy of the new and modern history. It featured the Taoist and Confucian categories of trace, void, and dynamic stability, which were rediscovered by Postmodernism. When the network discourse took over the linear one, it opened up prospects for new, digital thinking. The obtained data develop the ontological aspects of network existence and facilitate the applied studies of network communications.