A Lévy family of graphs consists of finite connected graphs with their uniform probability measures and graph distances divided by their diameters, whose concentration of measure functions vanish at every fixed positive normalized radius. Equivalently, every family occupying at least half the vertices has a radius- neighbourhood occupying a proportion tending to one, uniformly over such families. Harper theorem and binomial distribution tail bounds prove this for the hypercube graphs.
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The Lévy family of graphs is a concept in the field of probability theory and statistics, particularly in the context of Lévy processes. A Lévy process is a type of stochastic process that generalizes random walks and is characterized by stationary increments and continuity in probability. In particular, the Lévy family of graphs refers to the collection of parametric forms that describe the characteristic functions (or Laplace transforms) of Lévy processes.