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In the emerging quantum internet, complex network topology could lead to efficient quantum communication and enhanced robustness against failures. However, there are some concerns about complexity in quantum communication networks, such as potentially limited end-to-end transmission capacity. These challenges call for model systems in which the feasibility and impact of complex network topology on quantum communication protocols can be explored. Here, we present a theoretical model for complex quantum communication networks on a lattice of spins, wherein entangled spin clusters in interacting quantum spin systems serve as communication links between appropriately selected regions of spins. Specifically, we show that ground state Greenberger-Horne-Zeilinger clusters of the two-dimensional random transverse Ising model can be used as communication links between regions of spins, and we show that the resulting quantum networks can have complexity comparable to that of the classical internet. Our work provides an accessible generative model for further studies towards determining the network characteristics of the emerging quantum internet.