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In 1985, Frieze showed that the expected sum of the edge weights of theminimum spanning tree (MST) in the uniformly weighted graph converges to$\zeta(3)$. Recently, Hino and Kanazawa extended this result to a uniformlyweighted simplicial complex, where the role of the MST is played by itshigher-dimensional analog -- the Minimum Spanning Acycle (MSA). Our work goesbeyond and describes the histogram of all the weights in this random MST andrandom MSA. Specifically, we show that their empirical distributions convergeto a measure based on a concept called the shadow. The shadow of a graph is theset of all the missing transitive edges, and, for a simplicial complex, it is arelated topological generalization. As a corollary, we obtain a similar claimfor the death times in the persistence diagram corresponding to the aboveweighted complex, a result of interest in applied topology.

The Shadow knows: Empirical Distributions of Minimum Spanning Acycles and Persistence Diagrams of Random Complexes