The maximum likelihood threshold (MLT) of a graph G is the minimum number of samples required to almost surely guarantee existence of the maximum likelihood estimate in the corresponding Gaussian graphical model. Recently the MLT was characterised in terms of rigidity-theoretic properties of G. This allows us to use combinatorial rigidity techniques to compute the MLT precisely in certain special cases. In this talk I will present recent work on the MLT for topological families of graphs. This is joint work with Sean Dewar and Eleftherios Kastis.