On the Upper Tail of Counts of Strictly Balanced Subgraphs ˇ Matas Sileikis Adam Mickiewicz University, Pozna´ n
Berlin-Pozna´ n Seminar on Discrete Mathematics June 24-25, 2011
Notation
Erd˝os-R´enyi binomial random graph G(n, p)
Notation
Erd˝os-R´enyi binomial random graph G(n, p) fixed “small” graph G: vG vertices, eG edges
Notation
Erd˝os-R´enyi binomial random graph G(n, p) fixed “small” graph G: vG vertices, eG edges XG - number of copies of G in G(n, p)
G = C4
XC4 = 2
Notation
density eG /vG maximum density mG = maxH⊆G eH /vH
Notation
density eG /vG maximum density mG = maxH⊆G eH /vH G balanced, if mG = eG /vG G strictly balanced, if mG = eG /vG > eH /vH for all H ( G
Notation
density eG /vG maximum density mG = maxH⊆G eH /vH G balanced, if mG = eG /vG G strictly balanced, if mG = eG /vG > eH /vH for all H ( G p = n−1/mG threshold for existence of G; further for simplicity p ≥ n−1/mG .
Large deviations of XG
Problem: fixed t > 1, good upper bound for P {XG ≥ tE XG }
Large deviations of XG
Problem: fixed t > 1, good upper bound for P {XG ≥ tE XG } Asymptotics of − ln P {XG ≥ tE XG }, n → ∞.
Large deviations of XG
Problem: fixed t > 1, good upper bound for P {XG ≥ tE XG } Asymptotics of − ln P {XG ≥ tE XG }, n → ∞. Works on the upper tail: Spencer, Kim, Vu, Janson, Ruci´ nski, Panchenko. . .
Upper tail sharp up to logarithmic factor
Theorem (Janson, Oleszkiewicz and Ruci´ nski ’04) With a certain function MG = MG (n, p) 1 exp −Ct MG ln p
≤ P {XG ≥ tE XG } ≤ exp {−ct MG }
E.g., for k-regular G MG n2 p k . for star G = K1,r (
MG
n1+1/r p, n2 p r ,
if p ≤ n−1/r , if p > n−1/r .
Filling the gap of ln 1/p
1 exp −Ct MG ln p
≤ P {XG ≥ tE XG } ≤ exp {−ct MG }
Rather trivially for G - matching P {XG ≥ tE XG } ≥ exp {−Ct MG }
Filling the gap of ln 1/p
1 exp −Ct MG ln p
≤ P {XG ≥ tE XG } ≤ exp {−ct MG }
Rather trivially for G - matching P {XG ≥ tE XG } ≥ exp {−Ct MG } No other examples where upper bound is sharp Is otherwise lower bound sharp?
G = K3 = 4, threshold p = 1/n
G = K3 = 4, threshold p = 1/n Theorem (Chatterjee ’11) For p = Ω(ln n/n)
P {X4 ≥ tE X4 } ≤ exp −ct M4 ln
1 p
G = K3 = 4, threshold p = 1/n Theorem (Chatterjee ’11) For p = Ω(ln n/n)
P {X4 ≥ tE X4 } ≤ exp −ct M4 ln
1 p
Theorem (DeMarco and Kahn ’11) For any p
P {X4 ≥ tE X4 } = exp −Θt
1 min{E X4 , M4 ln } p
Note: for p ln n/n E X4 M4 ln
1 p
Lower bound for the upper tail Janson, Oleszkiewicz, Ruci´ nski ’04: 2 2 C M t 4 P {X4 ≥ tE X4 } ≥ p = p Θ(n p )
Lower bound for the upper tail Janson, Oleszkiewicz, Ruci´ nski ’04: 2 2 C M t 4 P {X4 ≥ tE X4 } ≥ p = p Θ(n p ) M4 = min # of edges enough to create E X4 copies of 4 Ct M4 edges enough to create tE X4 copies of 4
Lower bound for the upper tail Janson, Oleszkiewicz, Ruci´ nski ’04: 2 2 C M t 4 P {X4 ≥ tE X4 } ≥ p = p Θ(n p ) M4 = min # of edges enough to create E X4 copies of 4 Ct M4 edges enough to create tE X4 copies of 4 np
G = K3
np
np
Lower bound for the upper tail Janson, Oleszkiewicz, Ruci´ nski ’04: 2 2 C M t 4 P {X4 ≥ tE X4 } ≥ p = p Θ(n p ) M4 = min # of edges enough to create E X4 copies of 4 Ct M4 edges enough to create tE X4 copies of 4 np
G = K3
np
np
DeMarco, Kahn ’11: P {X4 ≥ tE X4 } ≥ P {dtE X4 e disjoint triangles } ≥ exp {−Ct E X4 }
Lower bound for the upper tail Janson, Oleszkiewicz, Ruci´ nski ’04: 2 2 C M t 4 P {X4 ≥ tE X4 } ≥ p = p Θ(n p ) M4 = min # of edges enough to create E X4 copies of 4 Ct M4 edges enough to create tE X4 copies of 4 np
G = K3
np
np
DeMarco, Kahn ’11: P {X4 ≥ tE X4 } ≥ P {dtE X4 e disjoint triangles } ≥ exp {−Ct E X4 } n
Combined: P {X4 ≥ tE X4 } ≥ exp −Ct min{E X4 , M4 ln p1 }
o
Lower bound for the upper tail ˇ ’11+) Theorem (S. For strictly balanced G 1 P {XG ≥ tE XG } ≥ exp −Ct min{E XG , MG ln } p
Lower bound for the upper tail ˇ ’11+) Theorem (S. For strictly balanced G 1 P {XG ≥ tE XG } ≥ exp −Ct min{E XG , MG ln } p
Conjecture (DeMarco and Kahn, RSA’11) For every G 1 P {XG ≥ tE XG } = exp −Θt min{E XG˜ , MG ln } p
,
˜ is a strictly balanced subgraph of G with minimal # of where G vertices.
Lower bound for the upper tail ˇ ’11+) Theorem (S. For strictly balanced G 1 P {XG ≥ tE XG } ≥ exp −Ct min{E XG , MG ln } p
Conjecture (DeMarco and Kahn, RSA’11) For every G 1 P {XG ≥ tE XG } = exp −Θt min{E XG˜ , MG ln } p
,
˜ is a strictly balanced subgraph of G with minimal # of where G vertices. Janson (private communication): conjectured lower bound follows from strictly balanced case
Sharp upper bound
DeMarco and Kahn ’11+: sharp bound for cliques G = Kr for all p
Sharp upper bound
DeMarco and Kahn ’11+: sharp bound for cliques G = Kr for all p ˇ 11+: if G is strictly balanced, E XG ≤ ln n, then S. P {XG ≥ tE XG } ≤ exp {−ct E XG } .
Sharp upper bound
DeMarco and Kahn ’11+: sharp bound for cliques G = Kr for all p ˇ 11+: if G is strictly balanced, E XG ≤ ln n, then S. P {XG ≥ tE XG } ≤ exp {−ct E XG } .
DGe = max # of edge-disjoint copies
Sharp upper bound
DeMarco and Kahn ’11+: sharp bound for cliques G = Kr for all p ˇ 11+: if G is strictly balanced, E XG ≤ ln n, then S. P {XG ≥ tE XG } ≤ exp {−ct E XG } .
DGe = max # of edge-disjoint copies Janson ’90: P {DGe ≥ tE XG } ≤ exp {−ct E XG }
Upper bound for K4 , C4 Janson and Ruci´ nski ’04: deletion method G = C4
Upper bound for K4 , C4 Janson and Ruci´ nski ’04: deletion method G = C4 order of − ln P {XC4 ≥ tE XC4 }: MC4 n−1
MC4 ln1/2 n n−2/3−ε
Cn−1 ln1/4 n MC4 ln n
p
Upper bound for K4 , C4 Janson and Ruci´ nski ’04: deletion method G = C4 order of − ln P {XC4 ≥ tE XC4 }: MC4 ln1/2 n
MC4 n−1
n−2/3−ε
Cn−1 ln1/4 n
p
MC4 ln n ˇ ’11+: S. E XC4 n−1
MC4 ln n n−4/5−ε
n−1 ln1/2 n E XC4
MC4 ln n
p
Upper bound for stars
star G = K1,r ˇ ’11+: S. E XK1,r
MK1,r ln n
n−1−1/r n−1−1/r ln1/(r −1) n E XK1,r MK1,r ln n
n−c(r )−ε
p
ˇ ’11+: G - strictly balanced, E XG ≤ ln n. S. P {XG ≥ tE XG } ≤ exp {−ct E XG } Sketch of proof for G = 4.
ˇ ’11+: G - strictly balanced, E XG ≤ ln n. S. P {XG ≥ tE XG } ≤ exp {−ct E XG } Sketch of proof for G = 4.
d4 = 1, d♦ = 5/4 G =4
F =♦
ˇ ’11+: G - strictly balanced, E XG ≤ ln n. S. P {XG ≥ tE XG } ≤ exp {−ct E XG } Sketch of proof for G = 4.
d4 = 1, d♦ = 5/4 G =4
F =♦
e = max # of edge-disjoint triangles D4
ˇ ’11+: G - strictly balanced, E XG ≤ ln n. S. P {XG ≥ tE XG } ≤ exp {−ct E XG } Sketch of proof for G = 4.
d4 = 1, d♦ = 5/4 G =4
F =♦
e = max # of edge-disjoint triangles D4 e ≥ tE X } + P {X ≥ 1} P {X4 ≥ tE X4 } ≤ P {D4 4 F
ˇ ’11+: G - strictly balanced, E XG ≤ ln n. S. P {XG ≥ tE XG } ≤ exp {−ct E XG } Sketch of proof for G = 4.
d4 = 1, d♦ = 5/4 G =4
F =♦
e = max # of edge-disjoint triangles D4 e ≥ tE X } + P {X ≥ 1} P {X4 ≥ tE X4 } ≤ P {D4 4 F e ≥ tE X } ≤ exp {−c E X } Janson ’90: P {D4 t 4 4
ˇ ’11+: G - strictly balanced, E XG ≤ ln n. S. P {XG ≥ tE XG } ≤ exp {−ct E XG } Sketch of proof for G = 4.
d4 = 1, d♦ = 5/4 G =4
F =♦
e = max # of edge-disjoint triangles D4 e ≥ tE X } + P {X ≥ 1} P {X4 ≥ tE X4 } ≤ P {D4 4 F e ≥ tE X } ≤ exp {−c E X } Janson ’90: P {D4 t 4 4
E XF = O(n−c ).
ˇ ’11+: G - strictly balanced, E XG ≤ ln n. S. P {XG ≥ tE XG } ≤ exp {−ct E XG } Sketch of proof for G = 4.
d4 = 1, d♦ = 5/4 G =4
F =♦
e = max # of edge-disjoint triangles D4 e ≥ tE X } + P {X ≥ 1} P {X4 ≥ tE X4 } ≤ P {D4 4 F e ≥ tE X } ≤ exp {−c E X } Janson ’90: P {D4 t 4 4
E XF = O(n−c ). P {XF ≥ 1} ≤ E XF ≤ exp {−c ln n}
Upper bound for C4 , K4 , stars: proof
Upper bound for C4 , K4 , stars: proof
intersection graph L
Upper bound for C4 , K4 , stars: proof
intersection graph L
Upper bound for C4 , K4 , stars: proof
intersection graph L maximum induced matching
Upper bound for C4 , K4 , stars: proof
intersection graph L maximum induced matching
Upper bound for C4 , K4 , stars: proof
intersection graph L maximum induced matching isolated vertices ⇒ independent set
Upper bound for C4 , K4 , stars: proof
intersection graph L maximum induced matching isolated vertices ⇒ independent set Spencer ’90: vL ≤ αL + 2βL (4L + 1)
Upper bound for C4 , K4 , stars: proof
intersection graph L maximum induced matching isolated vertices ⇒ independent set Spencer ’90: vL ≤ αL + 2βL (4L + 1) e + 2D e (4 + 1) X4 ≤ D4 L
Upper bound for C4 , K4 , stars: proof
intersection graph L maximum induced matching isolated vertices ⇒ independent set Spencer ’90: vL ≤ αL + 2βL (4L + 1) e + 2D e (4 + 1) X4 ≤ D4 L
Janson ’90: P {DGe ≥ x } ≤ exp {−x ln(x /eE XG )}
Upper bound for C4 , K4 , stars: proof
intersection graph L maximum induced matching isolated vertices ⇒ independent set Spencer ’90: vL ≤ αL + 2βL (4L + 1) e + 2D e (4 + 1) X4 ≤ D4 L
Janson ’90: P {DGe ≥ x } ≤ exp {−x ln(x /eE XG )} P {∆t ≥ y } for C4 , K4 , stars: Chernoff’s bound
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