A quantum network has the potential to integrate quantum technologies on a large scale, creating the so-called quantum internet. In this paper, we examine the connectivity of a large-scale quantum network in relation to a target distribution rate and end-to-end fidelity. We consider a network where each link consists of standard quantum photonic channels, characterized by a trade-off between rate and fidelity allowing the nodes in the network to be connected. The network employs a simple entanglement distribution protocol without error correction or purification. We analyze different network topologies, specifically Scale-Free and Erdös-Rényi networks, and provide analytical solutions for Erdös-Rényi in the thermodynamic regime and semi-analytical solutions for Scale-Free and Erdös-Rényi networks in the finite-size regime. Our results show a phase transition in network connectivity as a function of the target distribution rate, which can be continuous or discontinuous depending on the network topology and size. Furthermore, our findings indicate that Erdös-Rényi networks are more prone to discontinuous phase transitions than Scale-Free.