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

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

Graph Hypergraph Strategy Remarks Random graphs Erd˝ os–R´ enyi model G ( n, p ) : n vertices, edges picked with prob. p iid. Their phase transitions have been thoroughly studied.

Graph Hypergraph Strategy Remarks Emergence of the giant component Giant component : a component with a constant fraction of total vertices Phase transition : p c = n − 1 , with a window O ( n − 4 / 3 ) (Erd˝ os–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”.

Graph Hypergraph Strategy Remarks Sizes of components Erd˝ os, R´ enyi, Bollob´ as, � Luczak, . . . For small ε = ε ( n ) > 0 , let δ = − ε − log(1 − ε ) ∼ ε 2 / 2 and λ = ε 3 n → ∞ . p = (1 − ε ) p c : Largest ones are trees with order δ − 1 log λ p = (1 + ε ) p c : Largest one is far from a tree with order (1 + o (1))2 εn Smaller ones are trees with order δ − 1 log λ A symmetry!

Graph Hypergraph Strategy Remarks Sizes of components Erd˝ os, R´ enyi, Bollob´ as, � Luczak, . . . For small ε = ε ( n ) > 0 , let δ = − ε − log(1 − ε ) ∼ ε 2 / 2 and λ = ε 3 n → ∞ . p = (1 − ε ) p c : Largest ones are trees with order � log λ − 5 � δ − 1 2 log log λ + O p (1) p = (1 + ε ) p c : Largest one is far from a tree with order (1 + o (1))2 εn Smaller ones are trees with order � log λ − 5 � δ − 1 2 log log λ + O p (1) A symmetry!

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

Graph Hypergraph Strategy Remarks Connectedness ...... ? connected connected connected Need a distinction!

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.

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 � k � We fix 1 ≤ j < k , c 0 = − 1 and ε = ε ( n ) → 0 . j Theorem (Cooley, Kang, Person (2018) & Cooley, Kang, Koch (2018)) Assume that ε 3 n j → ∞ and ε 2 n 1 − 2 δ → ∞ for some fixed δ > 0 . For � − 1 � n p ∗ = c − 1 0 k − j Then in H we have ε − 2 log n � � (Subcritical) p = (1 − ε ) p ∗ ⇒ all j -comp. of order O . (Supercritical) p = (1 + ε ) p ∗ ⇒ order of the largest j -comp. = (1 + o (1))2 c − 1 � n � , others o ( εn j ) . 0 ε j Quite crude... Can we do better?

Graph Hypergraph Strategy Remarks Our result We refined the subcritical case. Theorem (Cooley, F., Del Giudice, Kang) Assume that ε 4 n j → ∞ and ε 2 n k − j (log n ) − 1 → ∞ . We take � − 1 � n − j p 0 = c − 1 . 0 k − j Take p (1 − ε ) p 0 and any integer m ≥ 1 . Let L i be the i -th largest j -comp. by size in H and L i its size. For any 1 ≤ i ≤ m , � � log λ − 5 L i = δ − 1 2 log log λ + O p (1) . Also, L i is a j -hypertree.

Graph Hypergraph Strategy Remarks Our result (cont.) Trees: order = size + 1 j -hypertrees: order = c 0 size + 1 Corollary (Cooley, F., Del Giudice, Kang) The previous theorem also holds for order.

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”.

Graph Hypergraph Strategy Remarks Example of a j -hypertree All largest components we study look like that!

Graph Hypergraph Strategy Remarks j -components and (labeled) two-type graphs a, b a, b, c { a, b, c } b a { b, c, d } a, c c { a, c, d } d, e b, c e { a, d, e } d a, d, e a, d a, c, d b, c, d a, e c, d b, d k = 3 , j = 2 Bipartite with two types of vertices: type k and type j � k � A vertex of type k has exactly neighbors, all of type j 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

Graph Hypergraph Strategy Remarks Hypertrees and two-type trees Hypertree ⇒ two-type trees, but � ! a, b a, b, c { a, b, c } b a { b, c, d } c { a, c, d } a, c d, e b, c f e { a, d, e } d a, d, e a, d a, c, d b, c, f a, e c, d c, f b, d A j -comp. is a hypertree iff it corresponds to a two-type tree.

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

Graph Hypergraph Strategy Remarks Search process a, b { a, b, c } b a { b, c, d } c { a, c, d } e { a, d, e } d

Graph Hypergraph Strategy Remarks Search process a, b a, b, c a, c b, c { a, b, c } b a { b, c, d } c { a, c, d } e { a, d, e } d

Graph Hypergraph Strategy Remarks Search process a, b a, b, c a, c b, c { a, b, c } b a { b, c, d } a, c, d c { a, c, d } e { a, d, e } a, d c, d d

Graph Hypergraph Strategy Remarks Search process a, b a, b, c a, c b, c { a, b, c } b a { b, c, d } a, c, d b, c, d c { a, c, d } e { a, d, e } a, d c, d c, d b, d d

Graph Hypergraph Strategy Remarks Search process a, b a, b, c a, c b, c { a, b, c } b a { b, c, d } a, c, d b, c, d c { a, c, d } e { a, d, e } a, d c, d c, d b, d d a, d, e a, e d, e

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 J 0 ; 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, b a, b, c a, b, c a, c a, c b, c b, c a, c, d b, c, d a, c, d b, c, d a, d c, d c, d b, d a, d c, d c, d b, d a, d, e a, d, e c, d, f a, e a, e d, e d, e c, f d, f a, d, e a, e d, e Search process Branching process

Graph Hypergraph Strategy Remarks Upper bound � n � Comp. in H “ ≤ ” iid. branching process in terms of size j 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 T J ( z ) = exp ( z (1 + T J ( z )) c 0 ) − 1 . � s ( c 0 e ) s � s � n �� n − j �� n �� n − j � [ z s ] T J ( z ) = Θ B s = . s 3 / 2 j k − j j k − j

Graph Hypergraph Strategy Remarks Lower bound

Graph Hypergraph Strategy Remarks Remarks