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Gels self-assembled from colloidal nanoparticles (NPs) translate the size-dependent properties of nanostructures to materials with macroscale volumes. Large spanning networks of NP chains provide high interconnectivity within the material necessary for a wide range of properties from conductivity to viscoelasticity. However, a great challenge for nanoscale engineering of such gels lies in being able to accurately and quantitatively describe their complex non-crystalline structure that combines order and disorder. The quantitative relationships between the mesoscale structural and material properties of nanostructured gels are currently unknown. Here, it is shown that lead telluride NPs spontaneously self-assemble into a spanning network hydrogel. By applying graph theory (GT), a method for quantifying the complex structure of the NP gels is established using a topological descriptor of average nodal connectivity that is found to correlate with the gel's mechanical and charge transport properties. GT descriptions make possible the design of non-crystalline porous materials from a variety of nanoscale components for photonics, catalysis, adsorption, and thermoelectrics.