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Subcritical random hypergraphs, high-order components, and - - PowerPoint PPT Presentation

Mar 10, 2023 332 likes •601 views

Graph Hypergraph Strategy Remarks Subcritical random hypergraphs, high-order components, and hypertrees Wenjie Fang, Institute of Discrete Mathematics, TU Graz With Oliver Cooley, Nicola del Giudice and Mihyun Kang ANR-FWF-MOST meeting, TU

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Graph Hypergraph Strategy Remarks

Subcritical random hypergraphs, high-order components, and hypertrees

Wenjie Fang, Institute of Discrete Mathematics, TU Graz With Oliver Cooley, Nicola del Giudice and Mihyun Kang ANR-FWF-MOST meeting, TU Wien, 29 October 2018

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Graph Hypergraph Strategy Remarks

Random graphs

Erd˝

  • s–R´

enyi model G(n, p) : n vertices, edges picked with prob. p iid. Their phase transitions have been thoroughly studied.

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Graph Hypergraph Strategy Remarks

Emergence of the giant component

Giant component : a component with a constant fraction of total vertices Phase transition : pc = n−1, with a window O(n−4/3) (Erd˝

  • s–R´

enyi 1960)

p − n−1 ≪ −n−4/3 p − n−1 = cn−4/3 p − n−1 ≫ n−4/3

Everything in “with high probability”.

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Graph Hypergraph Strategy Remarks

Sizes of components

Erd˝

  • s, R´

enyi, Bollob´ as, Luczak, . . . For small ε = ε(n) > 0, let δ=−ε−log(1−ε)∼ ε2/2 and λ = ε3n → ∞. p = (1 − ε)pc: Largest ones are trees with order δ−1 log λ p = (1 + ε)pc: Largest one is far from a tree with order (1 + o(1))2εn Smaller ones are trees with order δ−1 log λ A symmetry!

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Graph Hypergraph Strategy Remarks

Sizes of components

Erd˝

  • s, R´

enyi, Bollob´ as, Luczak, . . . For small ε = ε(n) > 0, let δ=−ε−log(1−ε)∼ ε2/2 and λ = ε3n → ∞. p = (1 − ε)pc: Largest ones are trees with order δ−1

  • log λ − 5

2 log log λ + Op(1)

  • p = (1 + ε)pc: Largest one is far from a tree with order

(1 + o(1))2εn Smaller ones are trees with order δ−1

  • log λ − 5

2 log log λ + Op(1)

  • A symmetry!
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Graph Hypergraph Strategy Remarks

Hypergraphs

Vertex set V = {1, 2, . . . n}. Graph = edges = subsets of V of size 2 k-uniform hypergraph = subsets of V of size k Random k-uniform hypergraph H = Hk(n, p): take each set of k vertices with prob. p.

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Graph Hypergraph Strategy Remarks

Connectedness ...... ?

connected

connected

connected

Need a distinction!

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Graph Hypergraph Strategy Remarks

High-order connectedness

Defined on j-sets: sets of j vertices. j-connectedness: hop by j-sets An example with k = 7, j = 2. Graph: k = 2, j = 1. Inspired by simplicial complexes.

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Graph Hypergraph Strategy Remarks

Previous results

Size: # of hyperedges, Order: # of j-sets For graph (k = 2, j = 1): size = # of edges, order = # of vertices We fix 1 ≤ j < k, c0 = k

j

  • − 1 and ε = ε(n) → 0.

Theorem (Cooley, Kang, Person (2018) & Cooley, Kang, Koch (2018)) Assume that ε3nj → ∞ and ε2n1−2δ → ∞ for some fixed δ > 0. For p∗ = c−1

  • n

k − j −1 Then in H we have (Subcritical) p = (1 − ε)p∗ ⇒ all j-comp. of order O

  • ε−2 log n
  • .

(Supercritical) p = (1 + ε)p∗ ⇒

  • rder of the largest j-comp. = (1 + o(1))2c−1

0 ε

n

j

  • , others o(εnj).

Quite crude... Can we do better?

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Graph Hypergraph Strategy Remarks

Our result

We refined the subcritical case. Theorem (Cooley, F., Del Giudice, Kang) Assume that ε4nj → ∞ and ε2nk−j(log n)−1 → ∞. We take p0 = c−1 n − j k − j −1 . Take p(1 − ε)p0 and any integer m ≥ 1. Let Li be the i-th largest j-comp. by size in H and Li its size. For any 1 ≤ i ≤ m, Li = δ−1

  • log λ − 5

2 log log λ + Op(1)

  • .

Also, Li is a j-hypertree.

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Graph Hypergraph Strategy Remarks

Our result (cont.)

Trees: order = size + 1 j-hypertrees: order = c0 size + 1 Corollary (Cooley, F., Del Giudice, Kang) The previous theorem also holds for order.

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Graph Hypergraph Strategy Remarks

j-hypertrees

Like trees, j-hypertrees are “barely connected”. No wheels, i.e., cyclic paths for j-sets. No “over-sharing”.

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Graph Hypergraph Strategy Remarks

Example of a j-hypertree

All largest components we study look like that!

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Graph Hypergraph Strategy Remarks

j-components and (labeled) two-type graphs

{a, b, c} {b, c, d} {a, c, d} {a, d, e} c d b a e d, e a, e a, d c, d b, d b, c a, b a, c a, c, d a, d, e a, b, c b, c, d

k = 3, j = 2 Bipartite with two types of vertices: type k and type j A vertex of type k has exactly k

j

  • neighbors, all of type j

(Labeled) Type k (resp. j) ⇒ set of k (resp. j) vertices (Labeled) Type k with label K ⇒ neighbors with labels of all j-subsets of K

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Graph Hypergraph Strategy Remarks

Hypertrees and two-type trees

Hypertree ⇒ two-type trees, but !

{a, b, c} {b, c, d} {a, c, d} {a, d, e} d, e a, e a, d c, d b, d b, c a, b a, c a, d, e a, c, d a, b, c b, c, f c d b a e f c, f

A j-comp. is a hypertree iff it corresponds to a two-type tree.

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Graph Hypergraph Strategy Remarks

Strategy

Replace hypergraphs with two-type trees to make it simpler Upper bound: upper coupling with branching process Lower bound: count the hypertrees

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Graph Hypergraph Strategy Remarks

Search process

{a, b, c} {b, c, d} {a, c, d} {a, d, e} c d b a e a, b

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Graph Hypergraph Strategy Remarks

Search process

{a, b, c} {b, c, d} {a, c, d} {a, d, e} c d b a e a, b a, c b, c a, b, c

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Graph Hypergraph Strategy Remarks

Search process

{a, b, c} {b, c, d} {a, c, d} {a, d, e} c d b a e a, b a, c b, c a, d c, d a, c, d a, b, c

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Graph Hypergraph Strategy Remarks

Search process

{a, b, c} {b, c, d} {a, c, d} {a, d, e} c d b a e a, b a, c b, c a, d c, d b, d c, d a, c, d a, b, c b, c, d

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Graph Hypergraph Strategy Remarks

Search process

{a, b, c} {b, c, d} {a, c, d} {a, d, e} c d b a e a, b a, c b, c a, d c, d a, e d, e b, d c, d a, c, d a, d, e a, b, c b, c, d

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Graph Hypergraph Strategy Remarks

Search process and branching process

Branching process for labeled two-type trees on [n]: Start with a node of type j labeled by a j-set J0; Type j node with label J: pick each k-set K ⊃ J with prob. p; Type k node with label K: pick all j-sets J ⊂ K, except the parent’s label.

a, b a, c b, c a, d c, d b, d c, d c, f d, f a, e d, e a, e d, e a, b, c a, c, d b, c, d a, d, e c, d, f a, d, e a, b a, c b, c a, d c, d a, e d, e b, d c, d a, c, d a, d, e a, b, c b, c, d Search process Branching process

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Graph Hypergraph Strategy Remarks

Upper bound

  • Comp. in H “≤”

n

j

  • iid. branching process in terms of size

E of the branching process ⇒ upper bound on comp. sizes of H Trees in branching process ⊆ labeled two-type trees Count the labeled two type trees to get an upper bound TJ(z) = exp (z(1 + TJ(z))c0) − 1. Bs = n j n − j k − j s [zs]TJ(z) = Θ n j n − j k − j s (c0e)s s3/2

  • .
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Graph Hypergraph Strategy Remarks

Lower bound

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Graph Hypergraph Strategy Remarks

Remarks