In this work, we consider a class of recursively grown fractal networks G(t) whose topology is controlled by two integer parameters, t and n. We first analyse the structural properties of G(t) (including fractal dimension, modularity, and clustering coefficient), and then we move to its transport properties. The latter are studied in terms of first-passage quantities (including the mean trapping time, the global mean first-passage time, and Kemeny's constant), and we highlight that their asymptotic behavior is controlled by the network's size and diameter. Remarkably, if we tune n (or, analogously, t) while keeping the network size fixed, as n increases (t decreases) the network gets more and more clustered and modular while its diameter is reduced, implying, ultimately, a better transport performance. The connection between this class of networks and models for polymer architectures is also discussed.
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