Abstract. We consider the notion of modulus of families of walks on graphs. We show how Beurling's famous criterion for extremality, that was formulated in the continuous case, can be interpreted on graphs as an instance of the Karush-Kuhn-Tucker conditions. We then develop an algorithm to numerically compute modulus using Beurling's criterion as our guide.

Abstract-This study develops the epidemic hitting time (EHT) metric on graphs measuring the expected time an epidemic starting at node a in a fully susceptible network takes to propagate and reach node b. An associated EHT centrality measure is then compared to degree, betweenness, spectral, and effective resistance centrality measures through exhaustive numerical simulations on several real-world network data-sets. We find two surprising observations: first, EHT centrality is highly correlated with effective resistance centrality; second, the EHT centrality measure is much more delocalized compared to degree and spectral centrality, highlighting the role of peripheral nodes in epidemic spreading on graphs.

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We provide a characterization of countably n-rectifiable measures in terms of σ-finiteness of the integral Menger curvature. We also prove that a finiteness condition on pointwise Menger curvature can characterize rectifiability of Radon measures. Motivated by the partial converse of Meurer's work by Kolasiński we prove that under suitable density assumptions there is a comparability between pointwise-Menger curvature and the sum over scales of the centered β-numbers at a point.

We further develop the relationship between β-numbers and discrete curvatures to provide a new proof that under weak density assumptions, finiteness of the pointwise discrete curvature curv α µ;2 (x, r) at µ-a.e. x ∈ R m implies that µ is C 1,α n-rectifiable.