Spectra of Random Regular Hypergraphs Yizhe Zhu University of - PowerPoint PPT Presentation

Spectra of Random Regular Hypergraphs Yizhe Zhu University of California, San Diego G2D2 Conference Yichang, China August 21, 2019 Joint work with Ioana Dumitriu Yizhe Zhu G2D2 Conference 1 / 12

Hypergraph H = ( V , E ), V : vertex set, E : hyperedge set. d -regular: the degree of each vertex is d . k -uniform: each hyperedge is of size k . 1 2 3 e 1 ( d , k ) -regular : both k -uniform and d -regular. 4 5 6 e 2 k = 2: d -regular graphs. e 4 e 5 e 6 7 8 9 e 3 Yizhe Zhu G2D2 Conference 2 / 12

Eigenvalues of the Adjacency Matrix A ∈ Z n × n Introduced in Feng-Li (1996). A ij = number of hyperedges containing i , j . Yizhe Zhu G2D2 Conference 3 / 12

Eigenvalues of the Adjacency Matrix A ∈ Z n × n Introduced in Feng-Li (1996). A ij = number of hyperedges containing i , j . λ 1 = d ( k − 1), since A 󰂔 e = d ( k − 1) 󰂔 e with 󰂔 e = (1 , . . . , 1) . Yizhe Zhu G2D2 Conference 3 / 12

Eigenvalues of the Adjacency Matrix A ∈ Z n × n Introduced in Feng-Li (1996). A ij = number of hyperedges containing i , j . λ 1 = d ( k − 1), since A 󰂔 e = d ( k − 1) 󰂔 e with 󰂔 e = (1 , . . . , 1) . What about λ 2 ? Yizhe Zhu G2D2 Conference 3 / 12

Eigenvalues of the Adjacency Matrix A ∈ Z n × n Introduced in Feng-Li (1996). A ij = number of hyperedges containing i , j . λ 1 = d ( k − 1), since A 󰂔 e = d ( k − 1) 󰂔 e with 󰂔 e = (1 , . . . , 1) . What about λ 2 ? For k = 2, many results for (random) d -regular graphs in (random) graph theory/ random matrix theory/ theoretical computer science. Yizhe Zhu G2D2 Conference 3 / 12

Theorem (Feng-Li 1996) Let G n be any sequence of connected ( d , k ) -regular hypergraphs with n vertices. Then 󰁴 λ 2 ( A n ) ≥ k − 2 + 2 ( d − 1)( k − 1) − 󰂄 n . with 󰂄 n → 0 as n → ∞ . Yizhe Zhu G2D2 Conference 4 / 12

Theorem (Feng-Li 1996) Let G n be any sequence of connected ( d , k ) -regular hypergraphs with n vertices. Then 󰁴 λ 2 ( A n ) ≥ k − 2 + 2 ( d − 1)( k − 1) − 󰂄 n . with 󰂄 n → 0 as n → ∞ . k = 2: Alon-Boppana bound for d -regular graphs. Ramanujan √ graphs: | λ | ≤ 2 d − 1 for all λ ∕ = d . Yizhe Zhu G2D2 Conference 4 / 12

Theorem (Feng-Li 1996) Let G n be any sequence of connected ( d , k ) -regular hypergraphs with n vertices. Then 󰁴 λ 2 ( A n ) ≥ k − 2 + 2 ( d − 1)( k − 1) − 󰂄 n . with 󰂄 n → 0 as n → ∞ . k = 2: Alon-Boppana bound for d -regular graphs. Ramanujan √ graphs: | λ | ≤ 2 d − 1 for all λ ∕ = d . Li-Sol´ e (1996): Ramanujan hypergraphs. For all eigenvalues λ ∕ = d ( k − 1), 󰁴 | λ − ( k − 2) | ≤ 2 ( d − 1)( k − 1) . Yizhe Zhu G2D2 Conference 4 / 12

Theorem (Feng-Li 1996) Let G n be any sequence of connected ( d , k ) -regular hypergraphs with n vertices. Then 󰁴 λ 2 ( A n ) ≥ k − 2 + 2 ( d − 1)( k − 1) − 󰂄 n . with 󰂄 n → 0 as n → ∞ . k = 2: Alon-Boppana bound for d -regular graphs. Ramanujan √ graphs: | λ | ≤ 2 d − 1 for all λ ∕ = d . Li-Sol´ e (1996): Ramanujan hypergraphs. For all eigenvalues λ ∕ = d ( k − 1), 󰁴 | λ − ( k − 2) | ≤ 2 ( d − 1)( k − 1) . Algebraic construction: Mart´ ınez-Stark-Terras (2001), Li (2004), Sarveniazi (2007). Yizhe Zhu G2D2 Conference 4 / 12

Spectral Gap Yizhe Zhu G2D2 Conference 5 / 12

Spectral Gap Random regular hypergraphs : uniformly chosen from all ( d , k )-regular hypergraphs on n vertices. Yizhe Zhu G2D2 Conference 5 / 12

Spectral Gap Random regular hypergraphs : uniformly chosen from all ( d , k )-regular hypergraphs on n vertices. Theorem (Dumitriu-Z. 2019) Let G n be a random ( d , k ) -regular hypergraphs with n vertices. Then with high probability for any eigenvalue λ ∕ = d ( k − 1) , 󰁴 | λ ( A n ) − ( k − 2) | ≤ 2 ( d − 1)( k − 1) + 󰂄 n with 󰂄 n → 0 . Yizhe Zhu G2D2 Conference 5 / 12

Spectral Gap Random regular hypergraphs : uniformly chosen from all ( d , k )-regular hypergraphs on n vertices. Theorem (Dumitriu-Z. 2019) Let G n be a random ( d , k ) -regular hypergraphs with n vertices. Then with high probability for any eigenvalue λ ∕ = d ( k − 1) , 󰁴 | λ ( A n ) − ( k − 2) | ≤ 2 ( d − 1)( k − 1) + 󰂄 n with 󰂄 n → 0 . A matching upper bound to Feng-Li (1996). Yizhe Zhu G2D2 Conference 5 / 12

Spectral Gap Random regular hypergraphs : uniformly chosen from all ( d , k )-regular hypergraphs on n vertices. Theorem (Dumitriu-Z. 2019) Let G n be a random ( d , k ) -regular hypergraphs with n vertices. Then with high probability for any eigenvalue λ ∕ = d ( k − 1) , 󰁴 | λ ( A n ) − ( k − 2) | ≤ 2 ( d − 1)( k − 1) + 󰂄 n with 󰂄 n → 0 . A matching upper bound to Feng-Li (1996). A generalization of Alon’s conjecture (1986) proved by Friedman (2008) and Bordenave (2015) for random d -regular graphs. Yizhe Zhu G2D2 Conference 5 / 12

Spectral Gap Random regular hypergraphs : uniformly chosen from all ( d , k )-regular hypergraphs on n vertices. Theorem (Dumitriu-Z. 2019) Let G n be a random ( d , k ) -regular hypergraphs with n vertices. Then with high probability for any eigenvalue λ ∕ = d ( k − 1) , 󰁴 | λ ( A n ) − ( k − 2) | ≤ 2 ( d − 1)( k − 1) + 󰂄 n with 󰂄 n → 0 . A matching upper bound to Feng-Li (1996). A generalization of Alon’s conjecture (1986) proved by Friedman (2008) and Bordenave (2015) for random d -regular graphs. Almost all regular hypergraphs are almost Ramanujan. Yizhe Zhu G2D2 Conference 5 / 12

Spectral Gap Random regular hypergraphs : uniformly chosen from all ( d , k )-regular hypergraphs on n vertices. Theorem (Dumitriu-Z. 2019) Let G n be a random ( d , k ) -regular hypergraphs with n vertices. Then with high probability for any eigenvalue λ ∕ = d ( k − 1) , 󰁴 | λ ( A n ) − ( k − 2) | ≤ 2 ( d − 1)( k − 1) + 󰂄 n with 󰂄 n → 0 . A matching upper bound to Feng-Li (1996). A generalization of Alon’s conjecture (1986) proved by Friedman (2008) and Bordenave (2015) for random d -regular graphs. Almost all regular hypergraphs are almost Ramanujan. What does λ 2 tell us about H ? Yizhe Zhu G2D2 Conference 5 / 12

Expander Mixing Lemma Yizhe Zhu G2D2 Conference 6 / 12

Expander Mixing Lemma Theorem (Dumitriu-Z. 2019) Let H be a ( d , k ) -regular hypergraph and λ = max { λ 2 , | λ n |} . Then the following holds: for any subsets V 1 , V 2 ⊂ V , 󰁷 󰀐 󰀐 󰀖 󰀗 󰀖 󰀗 󰀐 󰀐 󰀐 e ( V 1 , V 2 ) − d ( k − 1) 1 − | V 1 | 1 − | V 2 | 󰀐 󰀐 | V 1 | · | V 2 | 󰀐 ≤ λ | V 1 | · | V 2 | . n n n Yizhe Zhu G2D2 Conference 6 / 12

Expander Mixing Lemma Theorem (Dumitriu-Z. 2019) Let H be a ( d , k ) -regular hypergraph and λ = max { λ 2 , | λ n |} . Then the following holds: for any subsets V 1 , V 2 ⊂ V , 󰁷 󰀐 󰀐 󰀖 󰀗 󰀖 󰀗 󰀐 󰀐 󰀐 e ( V 1 , V 2 ) − d ( k − 1) 1 − | V 1 | 1 − | V 2 | 󰀐 󰀐 | V 1 | · | V 2 | 󰀐 ≤ λ | V 1 | · | V 2 | . n n n e ( V 1 , V 2 ) : number of hyperedges between V 1 , V 2 with multiplicity | e ∩ V 1 | · | e ∩ V 2 | for any hyperedge e . Yizhe Zhu G2D2 Conference 6 / 12

Non-backtracking Random Walks (NBRWs) Yizhe Zhu G2D2 Conference 7 / 12

Non-backtracking Random Walks (NBRWs) a non-backtracking walk of length ℓ in a hypergraph is a sequence w = ( v 0 , e 1 , v 1 , e 2 , . . . , v ℓ − 1 , e ℓ , v ℓ ) such that v i ∕ = v i +1 , { v i , v i +1 } ⊂ e i +1 and e i ∕ = e i +1 for 1 ≤ i ≤ ℓ − 1. Yizhe Zhu G2D2 Conference 7 / 12

Non-backtracking Random Walks (NBRWs) a non-backtracking walk of length ℓ in a hypergraph is a sequence w = ( v 0 , e 1 , v 1 , e 2 , . . . , v ℓ − 1 , e ℓ , v ℓ ) such that v i ∕ = v i +1 , { v i , v i +1 } ⊂ e i +1 and e i ∕ = e i +1 for 1 ≤ i ≤ ℓ − 1. a NBRW of length ℓ from v 0 : a uniformly chosen member of all non-backtracking walks of length ℓ starting at v 0 . Yizhe Zhu G2D2 Conference 7 / 12

Non-backtracking Random Walks (NBRWs) a non-backtracking walk of length ℓ in a hypergraph is a sequence w = ( v 0 , e 1 , v 1 , e 2 , . . . , v ℓ − 1 , e ℓ , v ℓ ) such that v i ∕ = v i +1 , { v i , v i +1 } ⊂ e i +1 and e i ∕ = e i +1 for 1 ≤ i ≤ ℓ − 1. a NBRW of length ℓ from v 0 : a uniformly chosen member of all non-backtracking walks of length ℓ starting at v 0 . How fast does the NBRW converge to a stationary distribution? Mixing rate: 󰀐 󰀐 1 / ℓ 󰀐 󰀐 󰀐 ( P ( ℓ ) ) ij − 1 󰀐 󰀐 ρ ( H ) := lim sup max . 󰀐 n i , j ∈ V ℓ →∞ Yizhe Zhu G2D2 Conference 7 / 12

Mixing Rate Theorem (Dumitriu-Z. 2019) 󰀖 󰀗 √ 1 2 √ λ ρ ( H ) = ( d − 1)( k − 1) ψ , where λ := max { λ 2 , | λ n |} and ( k − 1)( d − 1) 󰀬 √ x 2 − 1 if x ≥ 1 , x + ψ ( x ) := 1 if 0 ≤ x ≤ 1 . Yizhe Zhu G2D2 Conference 8 / 12

Mixing Rate Theorem (Dumitriu-Z. 2019) 󰀖 󰀗 √ 1 2 √ λ ρ ( H ) = ( d − 1)( k − 1) ψ , where λ := max { λ 2 , | λ n |} and ( k − 1)( d − 1) 󰀬 √ x 2 − 1 if x ≥ 1 , x + ψ ( x ) := 1 if 0 ≤ x ≤ 1 . k = 2: Alon-Benjamini-Lubetzky-Sodin (2007) for d -regular graphs. Proof by Chebyshev polynomials of the second kind. Yizhe Zhu G2D2 Conference 8 / 12