Abstract
AbstractWe introduce and study analogues of expander and hyperfinite graph sequences in the context of directed acyclic graphs, which we call ‘extender’ and ‘hypershallow’ graph sequences, respectively. Our main result is a probabilistic construction of non-hypershallow graph sequences.
Highlights
Explicitly introduced in [2]
In this article we study the analogues of hyperfinite and expander graph sequences in the context of oriented graphs, directed acyclic graphs
Our main result is a stochastic construction of graph sequences which are not hypershallow
Summary
3. If f is an N-valued random variable and α is its law, we define H(f ) := H(α). (a) If f and g are random variables with values in the same space Y , with laws α and β respectively, the law of f. It allows us to say the following: if f and g are N-valued random variables with laws α and β respectively, and (x, y) ∈ N2 is chosen according to the law α × β, either it is roughly as probable that x > y as it is that y > x, or the entropy of f ⊔ g is substantially larger than the average of the entropies of f and g. Let f and g be N-valued random variables with laws α and β, respectively.
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