The recent development of small-scale quantum networks poses the question of which network-level mechanisms could enable quantum communication to operate at scale in a future Quantum Internet. The question can be answered within a well-established framework where a quantum network is mapped into a classical weighted graph, and communication reliability is assessed using methods proper of network theory. Previous work relied on this framework to determine sufficient conditions for viable network-wide communication only on special network topologies, such as regular lattices, and to devise communication protocols applicable only to small- and moderate-sized real networks. Here, we overcome both limitations by introducing a family of quantum communication protocols that scale to networks with hundreds of millions of nodes. We show that the competition between increasing communication distance and path redundancy determines their large-scale behavior, and that degree heterogeneity can provide sufficient redundancy to sustain communication in small-world networks. These conclusions are supported by a systematic analysis of both real network topologies and synthetic networks, and an analytical treatment in the limit of infinitely large random scale-free graphs.