SLIDE 1 Greedy maximal independent sets via local limits
Peleg Michaeli
Tel Aviv University
The 31st International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA2020) September 2020 Joint work with Michael Krivelevich, Tamás Mészáros and Clara Shikhelman
SLIDE 2
Parking cars on a cycle
SLIDE 3
Parking cars on a cycle
SLIDE 4
Parking cars on a cycle
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Parking cars on a cycle
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Parking cars on a cycle
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Parking cars on a cycle
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Parking cars on a cycle
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Parking cars on a cycle
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Parking cars on a cycle
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Parking cars on a cycle
SLIDE 12
Independent sets
An independent set is a set of vertices in a graph, no two of which are adjacent. Finding maximum independent sets is very hard Finding maximal independent sets is very easy
SLIDE 13
Independent sets
An independent set is a set of vertices in a graph, no two of which are adjacent. Finding maximum independent sets is very hard Finding maximal independent sets is very easy
SLIDE 14
Independent sets
An independent set is a set of vertices in a graph, no two of which are adjacent. Finding maximum independent sets is very hard Finding maximal independent sets is very easy
SLIDE 15
Greedy MIS
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Greedy MIS
1 2 3 4 5 6 7 8 9
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Greedy MIS
1 2 3 4 5 6 7 8 9
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Greedy MIS
1 2 3 4 5 6 7 8 9
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Greedy MIS
1 2 3 4 5 6 7 8 9
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Greedy MIS
1 2 3 4 5 6 7 8 9
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Greedy MIS
1 2 3 4 5 6 7 8 9
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Greedy MIS
1 2 3 4 5 6 7 8 9
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Greedy MIS
1 2 3 4 5 6 7 8 9
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Greedy MIS
1 2 3 4 5 6 7 8 9
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Greedy MIS
1 2 3 4 5 6 7 8 9
SLIDE 26
Greedy MIS — performance
SLIDE 27
Greedy MIS — performance
1 2 3 4 5 6 7 8
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Greedy MIS — performance
1 2 3 4 5 6 7 8
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Greedy MIS — performance
1 2 3 4 5 6 7 8
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Greedy MIS — performance
1 2 3 4 5 6 7 8
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Greedy MIS — performance
1 2 3 4 5 6 7 8 2 1 3 4 5 6 7 8
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Greedy MIS — performance
1 2 3 4 5 6 7 8 2 1 3 4 5 6 7 8
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Greedy MIS — performance
1 2 3 4 5 6 7 8 2 1 3 4 5 6 7 8
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Greedy MIS — performance
1 2 3 4 5 6 7 8 2 1 3 4 5 6 7 8
SLIDE 35
Random greedy MIS — sequential
SLIDE 36
Random greedy MIS — sequential
1 2 3 4 5 6 7 8 9
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Random greedy MIS — sequential
1 2 3 4 5 6 7 8 9
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Random greedy MIS — sequential
1 2 3 4 5 6 7 8 9
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Random greedy MIS — sequential
1 2 3 4 5 6 7 8 9
SLIDE 40
Random greedy MIS — sequential
1 2 3 4 5 6 7 8 9
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Random greedy MIS — sequential
1 2 3 4 5 6 7 8 9
SLIDE 42
Random greedy MIS — sequential
1 2 3 4 5 6 7 8 9
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Random greedy MIS — sequential
1 2 3 4 5 6 7 8 9
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Random greedy MIS — sequential
1 2 3 4 5 6 7 8 9
SLIDE 45
Random greedy MIS — sequential
1 2 3 4 5 6 7 8 9
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Random greedy MIS — sequential
1 2 3 4 5 6 7 8 9
SLIDE 47 Greedy independence ratio — previous work
Let I(G) be the yielded independent set, and let ι(G) = |I(G)|/|V (G)|. random variable Flory ’39, Page ’59 McDiarmid ’84 log Wormald ’95 Lauer–Wormald ’07 (same for
- regular graphs with girth
) BJL ’17, BJM ’17 random graphs with given degree sequence
SLIDE 48 Greedy independence ratio — previous work
Let I(G) be the yielded independent set, and let ι(G) = |I(G)|/|V (G)|. random variable Flory ’39, Page ’59 McDiarmid ’84 log Wormald ’95 Lauer–Wormald ’07 (same for
- regular graphs with girth
) BJL ’17, BJM ’17 random graphs with given degree sequence
SLIDE 49 Greedy independence ratio — previous work
Let I(G) be the yielded independent set, and let ι(G) = |I(G)|/|V (G)|. random variable Flory ’39, Page ’59 ι(Pn) → 1
2(1 − e−2)
McDiarmid ’84 log Wormald ’95 Lauer–Wormald ’07 (same for
- regular graphs with girth
) BJL ’17, BJM ’17 random graphs with given degree sequence
SLIDE 50 Greedy independence ratio — previous work
Let I(G) be the yielded independent set, and let ι(G) = |I(G)|/|V (G)|. random variable Flory ’39, Page ’59 ι(Pn) → 1
2(1 − e−2)
McDiarmid ’84 ι(G(n, d/n)) → log(1 + d)/d Wormald ’95 Lauer–Wormald ’07 (same for
- regular graphs with girth
) BJL ’17, BJM ’17 random graphs with given degree sequence
SLIDE 51 Greedy independence ratio — previous work
Let I(G) be the yielded independent set, and let ι(G) = |I(G)|/|V (G)|. random variable Flory ’39, Page ’59 ι(Pn) → 1
2(1 − e−2)
McDiarmid ’84 ι(G(n, d/n)) → log(1 + d)/d Wormald ’95 ι(Gn,d) → 1
2
( 1 − (d − 1)−2/(d−2)) Lauer–Wormald ’07 (same for
- regular graphs with girth
) BJL ’17, BJM ’17 random graphs with given degree sequence
SLIDE 52 Greedy independence ratio — previous work
Let I(G) be the yielded independent set, and let ι(G) = |I(G)|/|V (G)|. random variable Flory ’39, Page ’59 ι(Pn) → 1
2(1 − e−2)
McDiarmid ’84 ι(G(n, d/n)) → log(1 + d)/d Wormald ’95 ι(Gn,d) → 1
2
( 1 − (d − 1)−2/(d−2)) Lauer–Wormald ’07 (same for d-regular graphs with girth → ∞) BJL ’17, BJM ’17 random graphs with given degree sequence
SLIDE 53 Greedy independence ratio — previous work
Let I(G) be the yielded independent set, and let ι(G) = |I(G)|/|V (G)|. random variable Flory ’39, Page ’59 ι(Pn) → 1
2(1 − e−2)
McDiarmid ’84 ι(G(n, d/n)) → log(1 + d)/d Wormald ’95 ι(Gn,d) → 1
2
( 1 − (d − 1)−2/(d−2)) Lauer–Wormald ’07 (same for d-regular graphs with girth → ∞) BJL ’17, BJM ’17 random graphs with given degree sequence
SLIDE 54
(Random) greedy MIS — parallel
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(Random) greedy MIS — parallel
1 2 3 4 5 6 7 8 9
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(Random) greedy MIS — parallel
1 2 3 4 5 6 7 8 9
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(Random) greedy MIS — parallel
1 2 3 4 5 6 7 8 9
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(Random) greedy MIS — parallel
1 2 3 4 5 6 7 8 9
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Random labelling
SLIDE 60 Random labelling
0.05 0.06 0.09 0.21 0.24 0.35 0.37 0.41 0.99
SLIDE 61 Random labelling
0.05 0.06 0.09 0.21 0.24 0.35 0.37 0.41 0.99
SLIDE 62 Random labelling
0.05 0.06 0.09 0.21 0.24 0.35 0.37 0.41 0.99
SLIDE 63 Random labelling
0.05 0.06 0.09 0.21 0.24 0.35 0.37 0.41 0.99
SLIDE 64 Random labelling
0.05 0.06 0.09 0.21 0.24 0.35 0.37 0.41 0.99 0.05 0.06 0.08 0.10 0.15 0.25 0.50 0.75 0.85 0.90 0.92 0.94 0.95
SLIDE 65 Random labelling
0.05 0.06 0.09 0.21 0.24 0.35 0.37 0.41 0.99 0.76 0.77 0.68 0.35 0.42 0.54 0.83 0.57 0.56 0.90 0.45 0.63 0.87
➩
SLIDE 66 Random labelling
0.05 0.06 0.09 0.21 0.24 0.35 0.37 0.41 0.99 0.76 0.77 0.68 0.35 0.42 0.54 0.83 0.57 0.56 0.90 0.45 0.63 0.87
➩
SLIDE 67 Random labelling
0.05 0.06 0.09 0.21 0.24 0.35 0.37 0.41 0.99 0.76 0.77 0.68 0.35 0.42 0.54 0.83 0.57 0.56 0.90 0.45 0.63 0.87
➩
SLIDE 68 Random labelling
0.05 0.06 0.09 0.21 0.24 0.35 0.37 0.41 0.99 0.76 0.77 0.68 0.35 0.42 0.54 0.83 0.57 0.56 0.90 0.45 0.63 0.87
➩
SLIDE 69 Random labelling
0.05 0.06 0.09 0.21 0.24 0.35 0.37 0.41 0.99 0.76 0.77 0.68 0.35 0.42 0.54 0.83 0.57 0.56 0.90 0.45 0.63 0.87
➩
SLIDE 70 Random labelling
0.05 0.06 0.09 0.21 0.24 0.35 0.37 0.41 0.99 0.76 0.77 0.68 0.35 0.42 0.54 0.83 0.57 0.56 0.90 0.45 0.63 0.87
➩
SLIDE 71
General framework
Let Gn be a graph sequence satisfying |Gn| → ∞. We wish to calculate the asymptotics of ι(Gn). We approximate I for chosen uniformly. We hope that this is determined by a small neighbourhood of . Decay of correlation = a.a.s. This local view of is captured by the local limit of . Develop a machinery to calculate the probability that the root of the local limit is red.
SLIDE 72
General framework
Let Gn be a graph sequence satisfying |Gn| → ∞. We wish to calculate the asymptotics of ι(Gn). We approximate E[ι(Gn)] = P(ρn ∈ I(Gn)) for ρn chosen uniformly. We hope that this is determined by a small neighbourhood of . Decay of correlation = a.a.s. This local view of is captured by the local limit of . Develop a machinery to calculate the probability that the root of the local limit is red.
SLIDE 73
General framework
Let Gn be a graph sequence satisfying |Gn| → ∞. We wish to calculate the asymptotics of ι(Gn). We approximate E[ι(Gn)] = P(ρn ∈ I(Gn)) for ρn chosen uniformly. We hope that this is determined by a small neighbourhood of ρn. Decay of correlation = a.a.s. This local view of is captured by the local limit of . Develop a machinery to calculate the probability that the root of the local limit is red.
SLIDE 74
General framework
Let Gn be a graph sequence satisfying |Gn| → ∞. We wish to calculate the asymptotics of ι(Gn). We approximate E[ι(Gn)] = P(ρn ∈ I(Gn)) for ρn chosen uniformly. We hope that this is determined by a small neighbourhood of ρn. Decay of correlation = ⇒ ι(Gn) ∼ E[ι(Gn)] a.a.s. This local view of is captured by the local limit of . Develop a machinery to calculate the probability that the root of the local limit is red.
SLIDE 75
General framework
Let Gn be a graph sequence satisfying |Gn| → ∞. We wish to calculate the asymptotics of ι(Gn). We approximate E[ι(Gn)] = P(ρn ∈ I(Gn)) for ρn chosen uniformly. We hope that this is determined by a small neighbourhood of ρn. Decay of correlation = ⇒ ι(Gn) ∼ E[ι(Gn)] a.a.s. This local view of ρn is captured by the local limit of Gn. Develop a machinery to calculate the probability that the root of the local limit is red.
SLIDE 76
General framework
Let Gn be a graph sequence satisfying |Gn| → ∞. We wish to calculate the asymptotics of ι(Gn). We approximate E[ι(Gn)] = P(ρn ∈ I(Gn)) for ρn chosen uniformly. We hope that this is determined by a small neighbourhood of ρn. Decay of correlation = ⇒ ι(Gn) ∼ E[ι(Gn)] a.a.s. This local view of ρn is captured by the local limit of Gn. Develop a machinery to calculate the probability that the root of the local limit is red.
SLIDE 77
Local limits (a.k.a. Benjamini–Schramm Limits)
We say that a (random) graph sequence Gn converges locally to a (random) rooted graph (U, ρ) if for every r ≥ 0 the ball BGn(ρn, r) converges in distribution to BU(ρ, r), where ρn is a uniform vertex of Gn.
SLIDE 78
Local limits (a.k.a. Benjamini–Schramm Limits)
We say that a (random) graph sequence Gn converges locally to a (random) rooted graph (U, ρ) if for every r ≥ 0 the ball BGn(ρn, r) converges in distribution to BU(ρ, r), where ρn is a uniform vertex of Gn.
SLIDE 79
Local limits (a.k.a. Benjamini–Schramm Limits)
We say that a (random) graph sequence Gn converges locally to a (random) rooted graph (U, ρ) if for every r ≥ 0 the ball BGn(ρn, r) converges in distribution to BU(ρ, r), where ρn is a uniform vertex of Gn.
SLIDE 80
Local limits (a.k.a. Benjamini–Schramm Limits)
We say that a (random) graph sequence Gn converges locally to a (random) rooted graph (U, ρ) if for every r ≥ 0 the ball BGn(ρn, r) converges in distribution to BU(ρ, r), where ρn is a uniform vertex of Gn.
SLIDE 81
Local limits (a.k.a. Benjamini–Schramm Limits)
We say that a (random) graph sequence Gn converges locally to a (random) rooted graph (U, ρ) if for every r ≥ 0 the ball BGn(ρn, r) converges in distribution to BU(ρ, r), where ρn is a uniform vertex of Gn.
SLIDE 82
Local limits (a.k.a. Benjamini–Schramm Limits)
We say that a (random) graph sequence Gn converges locally to a (random) rooted graph (U, ρ) if for every r ≥ 0 the ball BGn(ρn, r) converges in distribution to BU(ρ, r), where ρn is a uniform vertex of Gn. · · ·
SLIDE 83 Local limits (a.k.a. Benjamini–Schramm Limits)
We say that a (random) graph sequence Gn converges locally to a (random) rooted graph (U, ρ) if for every r ≥ 0 the ball BGn(ρn, r) converges in distribution to BU(ρ, r), where ρn is a uniform vertex of Gn. · · ·
➪
SLIDE 84 Local limits (a.k.a. Benjamini–Schramm Limits)
We say that a (random) graph sequence Gn converges locally to a (random) rooted graph (U, ρ) if for every r ≥ 0 the ball BGn(ρn, r) converges in distribution to BU(ρ, r), where ρn is a uniform vertex of Gn. · · ·
➪
➪
➪
SLIDE 85 Local limits (a.k.a. Benjamini–Schramm Limits)
We say that a (random) graph sequence Gn converges locally to a (random) rooted graph (U, ρ) if for every r ≥ 0 the ball BGn(ρn, r) converges in distribution to BU(ρ, r), where ρn is a uniform vertex of Gn.
Examples
Pn, Cn
loc
− → Z [n]d loc − → Zd G(n, d/n) loc − → Td, a Galton–Watson Pois(d) tree Gn,d
loc
− → the d-regular tree Uniform random tree Tn
loc
− → ˆ T1, a size-biased GW Pois(1) tree Finite d-ary balanced tree loc − → the canopy tree
SLIDE 86 Convergence of the greedy independence ratio
Say that Gn has subfactorial path growth if the expected number of paths from a typical vertex is subfactorial in their length. (bounded degree subfactorial path growth)
Theorem (Krivelevich, Mészáros, M., Shikhelman ’20)
Suppose has subfactorial path growth. If
loc
then a.a.s. is red
SLIDE 87 Convergence of the greedy independence ratio
Say that Gn has subfactorial path growth if the expected number of paths from a typical vertex is subfactorial in their length. (bounded degree ⊊ subfactorial path growth)
Theorem (Krivelevich, Mészáros, M., Shikhelman ’20)
Suppose has subfactorial path growth. If
loc
then a.a.s. is red
SLIDE 88 Convergence of the greedy independence ratio
Say that Gn has subfactorial path growth if the expected number of paths from a typical vertex is subfactorial in their length. (bounded degree ⊊ subfactorial path growth)
Theorem (Krivelevich, Mészáros, M., Shikhelman ’20)
Suppose Gn has subfactorial path growth. If Gn
loc
− → (U, ρ) then ι(Gn) → ι(U, ρ) a.a.s. is red
SLIDE 89 Convergence of the greedy independence ratio
Say that Gn has subfactorial path growth if the expected number of paths from a typical vertex is subfactorial in their length. (bounded degree ⊊ subfactorial path growth)
Theorem (Krivelevich, Mészáros, M., Shikhelman ’20)
Suppose Gn has subfactorial path growth. If Gn
loc
− → (U, ρ) then ι(Gn) → ι(U, ρ) a.a.s. P(ρ is red)
SLIDE 90
Exploration algorithms / decay of correlation
SLIDE 91
Exploration algorithms / decay of correlation
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Exploration algorithms / decay of correlation
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Exploration algorithms / decay of correlation
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Exploration algorithms / decay of correlation
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Exploration algorithms / decay of correlation
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Exploration algorithms / decay of correlation
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Exploration algorithms / decay of correlation
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Exploration algorithms / decay of correlation
SLIDE 99
Exploration algorithms / decay of correlation
SLIDE 100
Exploration algorithms / decay of correlation
SLIDE 101
Exploration algorithms / decay of correlation
SLIDE 102
Exploration algorithms / decay of correlation
SLIDE 103
Exploration algorithms / decay of correlation
SLIDE 104
Locally tree-like
We need to calculate ι(U, ρ), but even is still unknown... Let us therefore restrict ourselves to locally tree-like graph sequences, i.e., graph sequences for which is almost surely a tree. Assuming the children of are roots to independent subtrees, and conditioning on the label of , children of the past are roots to independent processes.
SLIDE 105
Locally tree-like
We need to calculate ι(U, ρ), but even ι(Z2) is still unknown... Let us therefore restrict ourselves to locally tree-like graph sequences, i.e., graph sequences for which is almost surely a tree. Assuming the children of are roots to independent subtrees, and conditioning on the label of , children of the past are roots to independent processes.
SLIDE 106
Locally tree-like
We need to calculate ι(U, ρ), but even ι(Z2) is still unknown... Let us therefore restrict ourselves to locally tree-like graph sequences, i.e., graph sequences for which (U, ρ) is almost surely a tree. Assuming the children of are roots to independent subtrees, and conditioning on the label of , children of the past are roots to independent processes.
SLIDE 107 Locally tree-like
We need to calculate ι(U, ρ), but even ι(Z2) is still unknown... Let us therefore restrict ourselves to locally tree-like graph sequences, i.e., graph sequences for which (U, ρ) is almost surely a tree. ρ
u1 u2
· · ·
ud
Assuming the children of are roots to independent subtrees, and conditioning on the label of , children of the past are roots to independent processes.
SLIDE 108 Locally tree-like
We need to calculate ι(U, ρ), but even ι(Z2) is still unknown... Let us therefore restrict ourselves to locally tree-like graph sequences, i.e., graph sequences for which (U, ρ) is almost surely a tree. ρ
u1 u2
· · ·
ud
Assuming the children of ρ are roots to independent subtrees, and conditioning on the label of ρ, children of the past are roots to independent processes.
SLIDE 109
Systems of ordinary difgerential equations
Let (U, ρ) be a single-type branching process. I I I I
SLIDE 110
Systems of ordinary difgerential equations
Let (U, ρ) be a single-type branching process. y(x) = P(ρ ∈ I(U[Pρ]) ∧ σρ < x) I I I
SLIDE 111
Systems of ordinary difgerential equations
Let (U, ρ) be a single-type branching process. y(x) = P(ρ ∈ I(U[Pρ]) ∧ σρ < x) = x · P(ρ ∈ I(U[Pρ]) | σρ < x) = ∫ x P(ρ ∈ I(U[Pρ]) | σρ = z)dz I
SLIDE 112
Systems of ordinary difgerential equations
Let (U, ρ) be a single-type branching process. y(x) = P(ρ ∈ I(U[Pρ]) ∧ σρ < x) = x · P(ρ ∈ I(U[Pρ]) | σρ < x) = ∫ x P(ρ ∈ I(U[Pρ]) | σρ = z)dz y′(x) = P(ρ ∈ I(U[Pρ]) | σρ = x)
SLIDE 113 Systems of ordinary difgerential equations
Let (U, ρ) be a single-type branching process. y(x) = P(ρ ∈ I(U[Pρ]) ∧ σρ < x) = x · P(ρ ∈ I(U[Pρ]) | σρ < x) = ∫ x P(ρ ∈ I(U[Pρ]) | σρ = z)dz y′(x) = P(ρ ∈ I(U[Pρ]) | σρ = x) Thus, if y is a unique solution of y′(x) = ∑
ℓ∈N
P(ξ[< x] = ℓ) ( 1 − y(x) x )ℓ , y(0) = 0, then, ι(U, ρ) = y(1).
SLIDE 114 Systems of ordinary difgerential equations
Let (U, ρ) be a single-type branching process. y(x) = P(ρ ∈ I(U[Pρ]) ∧ σρ < x) = x · P(ρ ∈ I(U[Pρ]) | σρ < x) = ∫ x P(ρ ∈ I(U[Pρ]) | σρ = z)dz y′(x) = P(ρ ∈ I(U[Pρ]) | σρ = x) Thus, if y is a unique solution of y′(x) = ∑
ℓ∈N
P(ξ[< x] = ℓ) ( 1 − y(x) x )ℓ , y(0) = 0, then, ι(U, ρ) = y(1).
SLIDE 115 Systems of ordinary difgerential equations
Let (U, ρ) be a (simple) multitype branching process. yk(x) = P(ρ ∈ I(U[Pρ]) ∧ σρ < x | τ = k) = x · P(ρ ∈ I(U[Pρ]) | σρ < x, τ = k) = ∫ x P(ρ ∈ I(U[Pρ]) | σρ = z, τ = k)dz y′
k(x) = P(ρ ∈ I(U[Pρ]) | σρ = x, τ = k)
Thus, if y is a unique solution of y′
k(x) =
∑
ℓ∈NT
∏
j∈T
P ( ξk→j[< x] = ℓj )( 1 − yj(x) x )ℓj , yk(0) = 0, then, ι(U, ρ) = E[yk(1)].
SLIDE 116 Application: paths and cycles
Pn and Cn converge locally to Z, which can be thought of as a 2-type branching process.
1
2
3
= = Thus
SLIDE 117 Application: paths and cycles
Pn and Cn converge locally to Z, which can be thought of as a 2-type branching process.
1
2
3
= = Thus
SLIDE 118 Application: paths and cycles
Pn and Cn converge locally to Z, which can be thought of as a 2-type branching process.
1
2
3
y′
b(x) = 1 − yb(x)
= ⇒ yb(x) = 1 − e−x, y′
r(x) = (1 − yb(x))2 = e−2x
= ⇒ yr(x) = 1 2 ( 1 − e−2x) . Thus
SLIDE 119 Application: paths and cycles
Pn and Cn converge locally to Z, which can be thought of as a 2-type branching process.
1
2
3
y′
b(x) = 1 − yb(x)
= ⇒ yb(x) = 1 − e−x, y′
r(x) = (1 − yb(x))2 = e−2x
= ⇒ yr(x) = 1 2 ( 1 − e−2x) . Thus ι(Pn), ι(Cn) → ι(Z) = y2(1) = 1 2 ( 1 − e−2) .
SLIDE 120 Application: paths and cycles
Pn and Cn converge locally to Z, which can be thought of as a 2-type branching process.
1
2
3
y′
b(x) = 1 − yb(x)
= ⇒ yb(x) = 1 − e−x, y′
r(x) = (1 − yb(x))2 = e−2x
= ⇒ yr(x) = 1 2 ( 1 − e−2x) . Thus ι(Pn), ι(Cn) → ι(Z) = y2(1) = 1 2 ( 1 − e−2) . α(Pn)/n, α(Cn)/n → 1/2.
SLIDE 121 Application: binomial random graphs
Easy fact: G(n, d/n) converges locally to the Pois(d) branching process. y′(x) =
∞
∑
ℓ=0
(dx)ℓ edxℓ! ( 1 − y(x) x )ℓ = e−dy(x). hence y(x) = log(1 + dx)/d. Thus log log
SLIDE 122 Application: binomial random graphs
Easy fact: G(n, d/n) converges locally to the Pois(d) branching process. y′(x) =
∞
∑
ℓ=0
(dx)ℓ edxℓ! ( 1 − y(x) x )ℓ = e−dy(x). hence y(x) = log(1 + dx)/d. Thus ι(G(n, d/n)) → ι(Td) = y(1) = log(1 + d) d . log
SLIDE 123 Application: binomial random graphs
Easy fact: G(n, d/n) converges locally to the Pois(d) branching process. y′(x) =
∞
∑
ℓ=0
(dx)ℓ edxℓ! ( 1 − y(x) x )ℓ = e−dy(x). hence y(x) = log(1 + dx)/d. Thus ι(G(n, d/n)) → ι(Td) = y(1) = log(1 + d) d . α(G(n, d/n))/n → 2 log d/d · (1 + od(1)).
SLIDE 124
Size-biased Galton–Watson branching processes
Grimmett ’80: the sequence of uniform random trees converges locally to the size-biased Galton–Watson Pois(1) tree.
SLIDE 125
Size-biased Galton–Watson branching processes
Grimmett ’80: the sequence of uniform random trees converges locally to the size-biased Galton–Watson Pois(1) tree.
SLIDE 126
Size-biased Galton–Watson branching processes
Grimmett ’80: the sequence of uniform random trees converges locally to the size-biased Galton–Watson Pois(1) tree.
SLIDE 127
Size-biased Galton–Watson branching processes
Grimmett ’80: the sequence of uniform random trees converges locally to the size-biased Galton–Watson Pois(1) tree.
SLIDE 128
Size-biased Galton–Watson branching processes
Grimmett ’80: the sequence of uniform random trees converges locally to the size-biased Galton–Watson Pois(1) tree.
SLIDE 129
Size-biased Galton–Watson branching processes
Grimmett ’80: the sequence of uniform random trees converges locally to the size-biased Galton–Watson Pois(1) tree.
SLIDE 130
Size-biased Galton–Watson branching processes
Grimmett ’80: the sequence of uniform random trees converges locally to the size-biased Galton–Watson Pois(1) tree.
SLIDE 131
Size-biased Galton–Watson branching processes
Grimmett ’80: the sequence of uniform random trees converges locally to the size-biased Galton–Watson Pois(1) tree.
SLIDE 132
Size-biased Galton–Watson branching processes
Grimmett ’80: the sequence of uniform random trees converges locally to the size-biased Galton–Watson Pois(1) tree.
SLIDE 133
Size-biased Galton–Watson branching processes
Grimmett ’80: the sequence of uniform random trees converges locally to the size-biased Galton–Watson Pois(1) tree.
SLIDE 134 Application: uniform random trees
Let s be the type of a vertex on the spine, and t be the type of a vertex on
- ne of the hanging trees. We have already seen
yt(x) = log(1 + x), and
s s t s
hence
s
, and we get
s
SLIDE 135 Application: uniform random trees
Let s be the type of a vertex on the spine, and t be the type of a vertex on
- ne of the hanging trees. We have already seen
yt(x) = log(1 + x), and y′
s(x) = (1 − ys(x))y′ t(x) = 1 − ys(x)
1 + x , hence ys(x) = 1 − (1 + x)−1, and we get ι(Tn) → ι( ˆ T1) = ys(1) = 1 2.
SLIDE 136 Application: uniform random trees
Let s be the type of a vertex on the spine, and t be the type of a vertex on
- ne of the hanging trees. We have already seen
yt(x) = log(1 + x), and y′
s(x) = (1 − ys(x))y′ t(x) = 1 − ys(x)
1 + x , hence ys(x) = 1 − (1 + x)−1, and we get ι(Tn) → ι( ˆ T1) = ys(1) = 1 2. α(Tn)/n → W0(1) ≈ 0.56714...
SLIDE 137
Simulations don’t lie
red: 125 (50%), green: 92 ( 37%), blue: 32 ( 13%), black: 1
SLIDE 138
Simulations don’t lie
red: 125 (50%), green: 92 (≈ 37%), blue: 32 (≈ 13%), black: 1
SLIDE 139
Simulations don’t lie (but I do)
red: 125 (50%), green: 92 (≈ 37%), blue: 32 (≈ 13%), black: 1
SLIDE 140 Greedy independence ratio — results
Flory ’39, Page ’59 ι(Pn) → 1
2(1 − e−2)
McDiarmid ’84 ι(G(n, d/n)) → log(1 + d)/d Wormald ’95 ι(Gn,d) → 1
2
( 1 − (d − 1)−2/(d−2)) Lauer–Wormald ’07 (d-regular graphs with girth → ∞) KMMS ’20 (same for functional digraphs)
SLIDE 141 Greedy independence ratio — results
Flory ’39, Page ’59 ι(Pn) → 1
2(1 − e−2)
✓
McDiarmid ’84 ι(G(n, d/n)) → log(1 + d)/d Wormald ’95 ι(Gn,d) → 1
2
( 1 − (d − 1)−2/(d−2)) Lauer–Wormald ’07 (d-regular graphs with girth → ∞) KMMS ’20 (same for functional digraphs)
SLIDE 142 Greedy independence ratio — results
Flory ’39, Page ’59 ι(Pn) → 1
2(1 − e−2)
✓
McDiarmid ’84 ι(G(n, d/n)) → log(1 + d)/d
✓
Wormald ’95 ι(Gn,d) → 1
2
( 1 − (d − 1)−2/(d−2)) Lauer–Wormald ’07 (d-regular graphs with girth → ∞) KMMS ’20 (same for functional digraphs)
SLIDE 143 Greedy independence ratio — results
Flory ’39, Page ’59 ι(Pn) → 1
2(1 − e−2)
✓
McDiarmid ’84 ι(G(n, d/n)) → log(1 + d)/d
✓
Wormald ’95 ι(Gn,d) → 1
2
( 1 − (d − 1)−2/(d−2)) ✓ Lauer–Wormald ’07 (d-regular graphs with girth → ∞) KMMS ’20 (same for functional digraphs)
SLIDE 144 Greedy independence ratio — results
Flory ’39, Page ’59 ι(Pn) → 1
2(1 − e−2)
✓
McDiarmid ’84 ι(G(n, d/n)) → log(1 + d)/d
✓
Wormald ’95 ι(Gn,d) → 1
2
( 1 − (d − 1)−2/(d−2)) ✓ Lauer–Wormald ’07 (d-regular graphs with girth → ∞) ✓ KMMS ’20 (same for functional digraphs)
SLIDE 145 Greedy independence ratio — results
Flory ’39, Page ’59 ι(Pn) → 1
2(1 − e−2)
✓
McDiarmid ’84 ι(G(n, d/n)) → log(1 + d)/d
✓
Wormald ’95 ι(Gn,d) → 1
2
( 1 − (d − 1)−2/(d−2)) ✓ Lauer–Wormald ’07 (d-regular graphs with girth → ∞) ✓ KMMS ’20 ι(Tn) → 1
2
✸
(same for functional digraphs)
SLIDE 146 Greedy independence ratio — results
Flory ’39, Page ’59 ι(Pn) → 1
2(1 − e−2)
✓
McDiarmid ’84 ι(G(n, d/n)) → log(1 + d)/d
✓
Wormald ’95 ι(Gn,d) → 1
2
( 1 − (d − 1)−2/(d−2)) ✓ Lauer–Wormald ’07 (d-regular graphs with girth → ∞) ✓ KMMS ’20 ι(Tn) → 1
2
✸
(same for functional digraphs)
✸
SLIDE 147
Paths are the worst trees
Theorem (Krivelevich, Mészáros, M., Shikhelman ’20)
If is a tree on vertices, then .
SLIDE 148 Paths are the worst trees
ι(Pn) → 1
2
( 1 − e−2) ≈ 0.43233...
Theorem (Krivelevich, Mészáros, M., Shikhelman ’20)
If is a tree on vertices, then .
SLIDE 149 Paths are the worst trees
ι(Pn) → 1
2
( 1 − e−2) ≈ 0.43233... ι(Sn) → 1
Theorem (Krivelevich, Mészáros, M., Shikhelman ’20)
If is a tree on vertices, then .
SLIDE 150 Paths are the worst trees
ι(Pn) → 1
2
( 1 − e−2) ≈ 0.43233... ι(Sn) → 1
≤
Theorem (Krivelevich, Mészáros, M., Shikhelman ’20)
If is a tree on vertices, then .
SLIDE 151 Paths are the worst trees
ι(Pn) → 1
2
( 1 − e−2) ≈ 0.43233... ι(Sn) → 1
≤
Theorem (Krivelevich, Mészáros, M., Shikhelman ’20)
If is a tree on vertices, then .
SLIDE 152 Paths are the worst trees
ι(Pn) → 1
2
( 1 − e−2) ≈ 0.43233... ι(Sn) → 1
≤
Theorem (Krivelevich, Mészáros, M., Shikhelman ’20)
If T is a tree on n vertices, then E[ι(Pn)] ≤ E[ι(T)].
SLIDE 153
What’s next?
Graph sequences that are not locally tree-like Better/other local rules Other colours ???
SLIDE 154
Thank You!
