Definition of graph-dependent measures for arbitrary graph topologies

Develop a formal definition of the graph-dependent measure associated with arbitrary graphs, not only series-parallel graphs, in a way that extends the series-parallel construction and accommodates the absence of a simple recursive decomposition.

Background

The paper defines graph-dependent probability measures recursively for series-parallel graphs. The recursive construction uses series and parallel decompositions and is connected to free additive and multiplicative convolutions; for such graphs, the measure describes the properly rescaled limiting eigenvalue distribution associated with a random tensor network.

The paper notes that the same recursive relation formally remains relevant for general graphs, but arbitrary graphs need not decompose into trivial graphs through series and parallel operations. Consequently, the existing construction does not provide a formal measure for arbitrary graph topologies, and the authors state this as a significant open question requiring new techniques or frameworks.