Characterize aggregation-invariant random graph ensembles beyond independent edges

Determine the most general class of random graph probability distributions that remain invariant in functional form under arbitrary node aggregations across hierarchical levels—given the coarse-graining rule that a superedge exists if any constituent pair is connected—thereby extending the multiscale renormalization framework beyond the specific solution for edge-independent models.

Background

The multiscale network renormalization program seeks random graph ensembles whose probability laws are preserved (up to parameter renormalization) under arbitrary node aggregations induced by coarse-graining. Concretely, one looks for probabilities P(A|Θ) that are functional fixed points of the renormalization flow for all partitions, with parameters mapped between levels.

In the edge-independent case, a unique nontrivial fixed point (the MultiScale Model) has been identified, with connection probabilities of the form p_{ij}=1−exp(−δ x_i x_j f(d_{ij})), which remain invariant under aggregation with explicit parameter flows. The authors note that the general solution, potentially allowing dependencies among edges, is not yet known, motivating the explicit open problem to characterize all aggregation-invariant ensembles.