### Citations

229 |
Exchangeability and related topics. In École d’été de probabilités de Saint-Flour
- Aldous
- 1985
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Citation Context ...xdy. Note that it follows from the uniqueness result (1.1) that entropy is a function of the underling graph limit, and it does not depend on the choice of the graphon representing it. It is shown in =-=[Ald85]-=- and [Jan, Theorem D.5] that lim n→∞ Ent(G(n,W ))( n 2 ) = Ent(W ). (1.2) A graphon is called random-free if it is {0, 1}-valued almost everywhere. Note that a graphonW is randomfree if and only if En... |

18 | Extremal graph theory, volume 11 of London Mathematical Society Monographs - Bollobás - 1978 |

13 | cut norm and distance, couplings and rearrangements
- Janson, Graphons
- 2010
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Citation Context ...m-free and it has nO(1) steps, |supp(G(n,W ′))| = nO(n). Consequently Ent(G(n,W ′)) = O(n log n). 4. Concluding remarks 1. Note that if W is a random-free stepfunction, then Ent(G(n,W )) = O(n). In =-=[Jan]-=- it is conjectured that the converse is also true. That is Ent(G(n,W )) = O(n) if and only if W is equivalent to a random-free stepfunction. The following simple example disproves this conjecture. Let... |

10 |
Lovász: Moments of Two-Variable Functions and the Uniqueness of Graph Limits
- Borgs, Chayes, et al.
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Citation Context ...: If W1 and W2 are two different graphons representing the same graph limit, then there exists a measure-preserving map σ : [0, 1]→ [0, 1] such that W1(x, y) = W2(σ(x), σ(y)), (1.1) almost everywhere =-=[BCL10]-=-. With these considerations, sometimes we shall not distinguish between the graph limits and their corresponding graphons. We define the δ1 distance of two graphons W1 and W2 as δ1(W1,W2) = inf ‖W1 −W... |

4 | Efficient testing of bipartite graphs for forbidden induced subgraphs
- Alon, Fischer, et al.
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Citation Context ...h property Q is random-free if and only if there exists a bigraph H such that pb(H ;W ) = 0 for all W ∈ Q. The following lemma is due to Alon, Fischer, and Newman (See [AFN07, Lemma 1.6]). Lemma 3.2. =-=[AFN07]-=- Let k be a fixed integer and let δ > 0 be a small real. For every graph G, either there exists stepfunction graphon W ′ with r ≤ ( k δ )O(k) steps such that δ1(WG,W ′) ≤ δ, or for every bigraph H on ... |