The Giant Component in a Random Subgraph of a Given Graph

Linyuan Lu University of South Carolina

Coauthors: Fan Chung Graham, Paul Horn, Xing Peng

Atlanta Lecture Series in Combinatorics & Graph Theory IV Georgia State University, November 5-6, 2011. Outline

■ Percolation on graphs ◆ Motivations ◆ Previous results ◆ Examples ◆ Our results ◆ Methods

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 2 / 49 Outline

■ Percolation on graphs ◆ Motivations ◆ Previous results ◆ Examples ◆ Our results ◆ Methods

■ Ongoing projects on hypergraphs ◆ Laplacians of hypergraphs ◆ Random hypergraphs

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 2 / 49 Part I: Graph percolation

■ G: a connected graph on n vertices ■ p: a probability (0 p 1) ≤ ≤ Gp: a random spanning subgraph of G, obtained as follows: for each edge f of G, independently,

Pr(f is an edge of Gp) = p.

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 3 / 49 Part I: Graph percolation

■ G: a connected graph on n vertices ■ p: a probability (0 p 1) ≤ ≤ Gp: a random spanning subgraph of G, obtained as follows: for each edge f of G, independently,

Pr(f is an edge of Gp) = p.

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 3 / 49 Percolation threshold pc

■ For p

■ For p>pc, almost surely there is a giant component.

pc

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 4 / 49 Motivations

■ Graph theory: random graphs ■ Theoretical physics: crystals melting ■ Sociology: the spread of disease on contact networks

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 5 / 49 The case G = Kn

For G = Kn, Gp = G(n,p): Erd˝os-R´enyi random graphs

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 6 / 49 The case G = Kn

For G = Kn, Gp = G(n,p): Erd˝os-R´enyi random graphs - n nodes

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 6 / 49 The case G = Kn

For G = Kn, Gp = G(n,p): Erd˝os-R´enyi random graphs - n nodes - For each pair of vertices, create an edge independently with probability p.

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 6 / 49 The case G = Kn

For G = Kn, Gp = G(n,p): Erd˝os-R´enyi random graphs - n nodes - For each pair of vertices, create an edge independently with probability p. 1 An example G(3, 2):

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 6 / 49 The evolution of G(n, p)

Let p 1/n + µ/n. ∼ ■ If µ < 0, the largest component has size 1 (µ log(1 + µ))− log n + O(log log n). −

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 7 / 49 The evolution of G(n, p)

Let p 1/n + µ/n. ∼ ■ If µ < 0, the largest component has size 1 (µ log(1 + µ))− log n + O(log log n). − ■ If µ = 0, the largest component has size of order n2/3.

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 7 / 49 The evolution of G(n, p)

Let p 1/n + µ/n. ∼ ■ If µ < 0, the largest component has size 1 (µ log(1 + µ))− log n + O(log log n). − ■ If µ = 0, the largest component has size of order n2/3. ■ If µ > 0, there is a unique giant component of size αn 1 where µ = α− log(1 α) 1. − − −

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 7 / 49 The evolution of G(n, p)

Let p 1/n + µ/n. ∼ ■ If µ < 0, the largest component has size 1 (µ log(1 + µ))− log n + O(log log n). − ■ If µ = 0, the largest component has size of order n2/3. ■ If µ > 0, there is a unique giant component of size αn 1 where µ = α− log(1 α) 1. − − − ■ Bollob´as showed that a component of size at least n2/3 in Gn,p is almost always unique if p exceeds 1/2 4/3 1/n + 4(log n) n− . (Later he removed the log n-factor.)

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 7 / 49 Percolation of Zd

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 8 / 49 Percolation of Zd

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 9 / 49 Percolation of Zd

Z2 1 Kesten (1980): pc( ) = 2.

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 9 / 49 Percolation of Zd

Z2 1 Kesten (1980): pc( ) = 2. Lorenz and Ziff (1997, simulation): p (Z3) 0.2488126 0.0000005 if it exists. c ≈ ±

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 9 / 49 Percolation of Zd

Z2 1 Kesten (1980): pc( ) = 2. Lorenz and Ziff (1997, simulation): p (Z3) 0.2488126 0.0000005 if it exists. c ≈ ± Kesten (1990): p (Zd) 1 as d . c ∼ 2d → ∞ TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 9 / 49 d-regular graphs

Alon, Benjamini, Stacey (2004): Suppose d 2 and let ≥ (Gn) be a sequence of d-regular expanders with girth(G ) , then n → ∞ 1 p = + o(1). c d 1 −

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 10 / 49 Dense graphs

Bollob´as, Borgs, Chayes, and Riordan (2008): Suppose that G is a dense graph (i.e., average degree d = Θ(n)). Let µ be the largest eigenvalue of the adjacency matrix of G. Then 1 p . c ≈ µ

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 11 / 49 Dense graphs

Bollob´as, Borgs, Chayes, and Riordan (2008): Suppose that G is a dense graph (i.e., average degree d = Θ(n)). Let µ be the largest eigenvalue of the adjacency matrix of G. Then 1 p . c ≈ µ Remark: The requirement of “dense graph” is essential. Their methods can not be extended to sparse graphs.

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 11 / 49 Questions

Is p 1? c ≈ µ

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 12 / 49 Questions

Is p 1? c ≈ µ “Yes” for some regular graphs and for dense graphs.

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 12 / 49 Questions

Is p 1? c ≈ µ “Yes” for some regular graphs and for dense graphs. “No” for general graphs.

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 12 / 49 Questions

Is p 1? c ≈ µ “Yes” for some regular graphs and for dense graphs. “No” for general graphs. We ask ■ Is p 1 ? c ≥ µ ■ Under what conditions, p 1 ? c ≈ µ

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 12 / 49 Notations

■ Degrees: d1,d2,...,dn.

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 13 / 49 Notations

■ Degrees: d1,d2,...,dn.

■ d = (d1,d2,...,dn)∗.

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 13 / 49 Notations

■ Degrees: d1,d2,...,dn.

■ d = (d1,d2,...,dn)∗. ■ The volume: vol(S) = i S di. ∈ P

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 13 / 49 Notations

■ Degrees: d1,d2,...,dn.

■ d = (d1,d2,...,dn)∗. ■ The volume: vol(S) = i S di. ∈ ■ k The k-th volume: volk(S) = i S di . P ∈ P

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 13 / 49 Notations

■ Degrees: d1,d2,...,dn.

■ d = (d1,d2,...,dn)∗. ■ The volume: vol(S) = i S di. ∈ ■ k The k-th volume: volk(S) = i S di . P ∈ ■ vol(G) Average degree: d = n . P

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 13 / 49 Notations

■ Degrees: d1,d2,...,dn.

■ d = (d1,d2,...,dn)∗. ■ The volume: vol(S) = i S di. ∈ ■ k The k-th volume: volk(S) = i S di . P ∈ ■ vol(G) Average degree: d = n . P ■ ˜ vol2(G) Second order average degree: d = vol(G)

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 13 / 49 Notations

■ Degrees: d1,d2,...,dn.

■ d = (d1,d2,...,dn)∗. ■ The volume: vol(S) = i S di. ∈ ■ k The k-th volume: volk(S) = i S di . P ∈ ■ vol(G) Average degree: d = n . P ■ ˜ vol2(G) Second order average degree: d = vol(G)

■ ˜˜ vol3(G) Third order average degree: d = vol2(G).

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 13 / 49 Notations

■ Degrees: d1,d2,...,dn.

■ d = (d1,d2,...,dn)∗. ■ The volume: vol(S) = i S di. ∈ ■ k The k-th volume: volk(S) = i S di . P ∈ ■ vol(G) Average degree: d = n . P ■ ˜ vol2(G) Second order average degree: d = vol(G)

■ ˜˜ vol3(G) Third order average degree: d = vol2(G).

A connected component is giant if its volume is Θ(vol(G)).

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 13 / 49 Yes, p 1 . c ≥ µ

1 Chung, Lu, Horn 2008: For p < µ, almost surely every connected component in Gp has volume at most O( vol2(G)g(n)), where g(n) is any slowly growing function as n . p → ∞

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 14 / 49 Yes, p 1 . c ≥ µ

1 Chung, Lu, Horn 2008: For p < µ, almost surely every connected component in Gp has volume at most O( vol2(G)g(n)), where g(n) is any slowly growing function as n . p → ∞ Proof: Let A be the event that there exists a component S in Gp with vol(S) >C vol2(G). 1 Claim A: Pr(A) C2(1ppµ). ≤ −

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 14 / 49 Yes, p 1 . c ≥ µ

1 Chung, Lu, Horn 2008: For p < µ, almost surely every connected component in Gp has volume at most O( vol2(G)g(n)), where g(n) is any slowly growing function as n . p → ∞ Proof: Let A be the event that there exists a component S in Gp with vol(S) >C vol2(G). 1 Claim A: Pr(A) C2(1ppµ). ≤ − - u, v: two random vertices selected with probability proportional to their degrees.

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 14 / 49 Yes, p 1 . c ≥ µ

1 Chung, Lu, Horn 2008: For p < µ, almost surely every connected component in Gp has volume at most O( vol2(G)g(n)), where g(n) is any slowly growing function as n . p → ∞ Proof: Let A be the event that there exists a component S in Gp with vol(S) >C vol2(G). 1 Claim A: Pr(A) C2(1ppµ). ≤ − - u, v: two random vertices selected with probability proportional to their degrees. k - k-path from u to v in G 1∗A 1 . |{ }| ≤ u v

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 14 / 49 Yes, p 1 . c ≥ µ

1 Chung, Lu, Horn 2008: For p < µ, almost surely every connected component in Gp has volume at most O( vol2(G)g(n)), where g(n) is any slowly growing function as n . p → ∞ Proof: Let A be the event that there exists a component S in Gp with vol(S) >C vol2(G). 1 Claim A: Pr(A) C2(1ppµ). ≤ − - u, v: two random vertices selected with probability proportional to their degrees. k - k-path from u to v in G 1∗A 1 . |{ }| ≤ u v k - The probability of a k-path survived in Gp is p .

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 14 / 49 Proof of Claim A:

Pr(u, v are in the same component of Gp)

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 15 / 49 Proof of Claim A:

Pr(u, v are in the same component of Gp) n du dv k k p 1∗A 1 ≤ vol(G) vol(G) u v u,v X Xk=0

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 15 / 49 Proof of Claim A:

Pr(u, v are in the same component of Gp) n du dv k k p 1∗A 1 ≤ vol(G) vol(G) u v u,v k=0 Xn X 1 k k = p d∗A d vol(G)2 Xk=0

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 15 / 49 Proof of Claim A:

Pr(u, v are in the same component of Gp) n du dv k k p 1∗A 1 ≤ vol(G) vol(G) u v u,v k=0 Xn X 1 k k = p d∗A d vol(G)2 Xk=0 k k ∞ p µ vol2(G) ≤ vol(G)2 Xk=0

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 15 / 49 Proof of Claim A:

Pr(u, v are in the same component of Gp) n du dv k k p 1∗A 1 ≤ vol(G) vol(G) u v u,v k=0 Xn X 1 k k = p d∗A d vol(G)2 Xk=0 k k ∞ p µ vol2(G) ≤ vol(G)2 Xk=0 d˜ . ≤ (1 pµ)vol(G) −

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 15 / 49 Continue

Pr(u, v are in the same component of Gp)

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 16 / 49 Continue

Pr(u, v are in the same component of Gp) Pr(A and u, v S) ≥ ∈

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 16 / 49 Continue

Pr(u, v are in the same component of Gp) Pr(A and u, v S) ≥ ∈ C vol (G) C vol (G) > Pr(A) 2 2 pvol(G) pvol(G)

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 16 / 49 Continue

Pr(u, v are in the same component of Gp) Pr(A and u, v S) ≥ ∈ C vol (G) C vol (G) > Pr(A) 2 2 pvol(G) pvol(G) C2d˜ = Pr(A) . vol(G)

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 16 / 49 Continue

Pr(u, v are in the same component of Gp) Pr(A and u, v S) ≥ ∈ C vol (G) C vol (G) > Pr(A) 2 2 pvol(G) pvol(G) C2d˜ = Pr(A) . vol(G)

d˜ C2d˜ Thus, > Pr(A) (1 pµ)vol(G) vol(G) −

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 16 / 49 Continue

Pr(u, v are in the same component of Gp) Pr(A and u, v S) ≥ ∈ C vol (G) C vol (G) > Pr(A) 2 2 pvol(G) pvol(G) C2d˜ = Pr(A) . vol(G)

d˜ C2d˜ Thus, > Pr(A) (1 pµ)vol(G) vol(G) − 1 Pr(A) < .  C2(1 pµ) −

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 16 / 49 The target graphs

■ Not dense graphs. ■ Unevenly distributed degree sequence, like power law graphs. ■ Some bounds on spectra, like expanders.

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 17 / 49 Model G(w1, w2,...,wn)

Random graph model with given expected degree sequence

- n nodes with weights w1, w2,...,wn.

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 18 / 49 Model G(w1, w2,...,wn)

Random graph model with given expected degree sequence

- n nodes with weights w1, w2,...,wn. - For each pair (i,j), create an edge independently with 1 probability pij = wiwjρ, where ρ = n . i=1 wi P

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 18 / 49 Model G(w1, w2,...,wn)

Random graph model with given expected degree sequence

- n nodes with weights w1, w2,...,wn. - For each pair (i,j), create an edge independently with 1 probability pij = wiwjρ, where ρ = n . i=1 wi - The graph H has probability P

p (1 p ). ij − ij ij E(H) ij E(H) ∈Y 6∈Y

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 18 / 49 Model G(w1, w2,...,wn)

Random graph model with given expected degree sequence

- n nodes with weights w1, w2,...,wn. - For each pair (i,j), create an edge independently with 1 probability pij = wiwjρ, where ρ = n . i=1 wi - The graph H has probability P

p (1 p ). ij − ij ij E(H) ij E(H) ∈Y 6∈Y

- The expected degree of vertex i is wi.

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 18 / 49 Model G(w1, w2,...,wn)

Random graph model with given expected degree sequence

- n nodes with weights w1, w2,...,wn. - For each pair (i,j), create an edge independently with 1 probability pij = wiwjρ, where ρ = n . i=1 wi - The graph H has probability P

p (1 p ). ij − ij ij E(H) ij E(H) ∈Y 6∈Y

- The expected degree of vertex i is wi. - Erd˝os-R´enyi model G(n,p) = G(np,...,np).

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 18 / 49 An example: G(w1, w2, w3, w4)

✓✏ ✓✏ ✒✑❅❅ ✒✑ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ✓✏ ✓✏ ✒✑ ✒✑

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 19 / 49 An example: G(w1, w2, w3, w4)

✓✏w3 ✓✏w4 ✒✑❅❅ ✒✑ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ✓✏w1 ✓✏w2 ✒✑ ✒✑

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 19 / 49 An example: G(w1, w2, w3, w4)

✓✏w3 ✓✏w4 ✒✑❅❅ ✒✑ ❅ ❅ w1w3ρ ❅ ❅ w2w3ρ ❅ w1w4ρ ❅ ❅ ✓✏w1 ✓✏w2 w1w2ρ ✒✑ ✒✑

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 19 / 49 An example: G(w1, w2, w3, w4)

✓✏1 w3w4ρ ✓✏ w3 − w4 qqqqqqq ✒✑❅❅ ✒✑ ❅ q ❅ q ❅ q w1w3ρ 1 w2w4ρ ❅ w2w3ρ q − ❅ q w1w4ρ ❅ q ❅ q ✓✏w1 ✓✏w2 w1w2ρ ✒✑ ✒✑

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 19 / 49 An example: G(w1, w2, w3, w4)

2 1 w3ρ q q − q q ✓✏q 1 w3w4ρ ✓✏q 2 q w3 − w4 1q w4ρ q qqqqqqq q − ✒✑❅❅ ✒✑ ❅ q ❅ q ❅ q w1w3ρ 1 w2w4ρ ❅ w2w3ρ q − ❅ q w1w4ρ ❅ q q ❅ 2 ✓✏w1 ✓✏w2 1 w2ρ q w1w2ρ q − q ✒✑ ✒✑q q q 2 q q 1q w1ρ q −

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 19 / 49 An example: G(w1, w2, w3, w4)

2 1 w3ρ q q − q q ✓✏q 1 w3w4ρ ✓✏q 2 q w3 − w4 1q w4ρ q qqqqqqq q − ✒✑❅❅ ✒✑ ❅ q ❅ q ❅ q w1w3ρ 1 w2w4ρ ❅ w2w3ρ q − ❅ q w1w4ρ ❅ q q ❅ 2 ✓✏w1 ✓✏w2 1 w2ρ q w1w2ρ q − q ✒✑ ✒✑q q q 2 q q 1q w1ρ q − The probability of the graph is

4 w3w2w2w ρ4(1 w w ρ) (1 w w ρ) (1 w2ρ). 1 2 3 4 − 2 4 × − 3 4 − i i=1 TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan LuY (University of South Carolina) – 19 / 49 A example: G(1, 2, 1)

Loops are omitted here.

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 20 / 49 Notations

For G = G(w1,...,wn), let 1 n - d = i=1 wi n n 2 i=1 wi - d˜= n . wi PPi=1 - The volumeP of S: vol(S) = i S wi. ∈ P

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 21 / 49 Notations

For G = G(w1,...,wn), let 1 n - d = i=1 wi n n 2 i=1 wi - d˜= n . wi PPi=1 - The volumeP of S: vol(S) = i S wi. ∈ We have P d˜ d ≥ “=” holds if and only if w = = w . 1 · · · n

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 21 / 49 Notations

For G = G(w1,...,wn), let 1 n - d = i=1 wi n n 2 i=1 wi - d˜= n . wi PPi=1 - The volumeP of S: vol(S) = i S wi. ∈ We have P d˜ d ≥ “=” holds if and only if w = = w . 1 · · · n A connected component S is called a giant component if

vol(S) = Θ(vol(G)).

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 21 / 49 Connected components

Chung and Lu (2001) For G = G(w1,...,wn), ■ If d˜ < 1 ǫ, then almost surely, all components have volume at− most O(√n log n).

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 22 / 49 Connected components

Chung and Lu (2001) For G = G(w1,...,wn), ■ If d˜ < 1 ǫ, then almost surely, all components have volume at− most O(√n log n). ■ If d > 1 + ǫ, then almost surely there is a unique giant component of volume Θ(vol(G)). All other components have size at most

log n 1 2 d 1 log d ǫd if 1 ǫ < d < 1 ǫ − − −log n − 4 − ( 1+log d log 4+2 log(1 ǫ) if d > e(1 ǫ)2 . − − −

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 22 / 49 Volume of Giant Component

Chung and Lu (2004) If the average degree is strictly greater than 1, then almost surely the giant component in a graph G in G(w) has 3.5 log n volume (λ0 + O(√n vol(G) ))vol(G), where λ0 is the unique positive root of the following equation:

n n wiλ w e− = (1 λ) w . i − i i=1 i=1 X X

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 23 / 49 Percolation on G(w1,w2,...,wn)

Gp = G(w1p,...,wnp).

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 24 / 49 Percolation on G(w1,w2,...,wn)

Gp = G(w1p,...,wnp). We have ■ If p < 1, there is no giant component. d˜

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 24 / 49 Percolation on G(w1,w2,...,wn)

Gp = G(w1p,...,wnp). We have ■ If p < 1, there is no giant component. d˜ ■ 1 If p > d, there is a giant component.

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 24 / 49 Percolation on G(w1,w2,...,wn)

Gp = G(w1p,...,wnp). We have ■ If p < 1, there is no giant component. d˜ ■ 1 If p > d, there is a giant component. ■ For 1 < p < 1 , no conclusion. d˜ d

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 24 / 49 Sub-dense graphs

For any big constant C and any ǫ 0, there exists a graph n → G = Gn satisfying

■ d = Ω(ǫnn)

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 25 / 49 Sub-dense graphs

For any big constant C and any ǫ 0, there exists a graph n → G = Gn satisfying

■ d = Ω(ǫnn) ■ µ Θ(√ǫ n) ≈ n

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 25 / 49 Sub-dense graphs

For any big constant C and any ǫ 0, there exists a graph n → G = Gn satisfying

■ d = Ω(ǫnn) ■ µ Θ(√ǫ n) ≈ n ■ C For p = µ , the volume of all components is at most O(ǫnn). 1 Conclusion: In general, µ is not the threshold function for the giant component appearing.

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 25 / 49 d˜ and µ

1/2 1/2 ■ = I D− AD− : normalized Laplacian L −

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 26 / 49 d˜ and µ

1/2 1/2 ■ = I D− AD− : normalized Laplacian L − ■ 0 = λ0 λ1 ..., λn 1: Laplacian spectra ≤ ≤ ≤ −

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 26 / 49 d˜ and µ

1/2 1/2 ■ = I D− AD− : normalized Laplacian L − ■ 0 = λ0 λ1 ..., λn 1: Laplacian spectra ≤ ≤ ≤ − n 1 ■ σ = max − 1 λ : spectral bound i=1 | − i|

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 26 / 49 d˜ and µ

1/2 1/2 ■ = I D− AD− : normalized Laplacian L − ■ 0 = λ0 λ1 ..., λn 1: Laplacian spectra ≤ ≤ ≤ − n 1 ■ σ = max − 1 λ : spectral bound i=1 | − i| ■ ∆: maximum degree

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 26 / 49 d˜ and µ

1/2 1/2 ■ = I D− AD− : normalized Laplacian L − ■ 0 = λ0 λ1 ..., λn 1: Laplacian spectra ≤ ≤ ≤ − n 1 ■ σ = max − 1 λ : spectral bound i=1 | − i| ■ ∆: maximum degree Lemma: µ d˜ σ∆. | − |≤

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 26 / 49 Admissible conditions

A set U is a (ǫ, M)-admissible set if

(i) vol2(U) (1 ǫ)vol2(G). (ii) vol (U)≥ Md−vol (G) 3 ≤ 2

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 27 / 49 Admissible conditions

A set U is a (ǫ, M)-admissible set if

(i) vol2(U) (1 ǫ)vol2(G). (ii) vol (U)≥ Md−vol (G) 3 ≤ 2 We say G is (ǫ, M)-admissible.

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 27 / 49 Admissible conditions

A set U is a (ǫ, M)-admissible set if

(i) vol2(U) (1 ǫ)vol2(G). (ii) vol (U)≥ Md−vol (G) 3 ≤ 2 We say G is (ǫ, M)-admissible.

“∆ Md” implies “d˜˜ Md”. ≤ ≤

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 27 / 49 Admissible conditions

A set U is a (ǫ, M)-admissible set if

(i) vol2(U) (1 ǫ)vol2(G). (ii) vol (U)≥ Md−vol (G) 3 ≤ 2 We say G is (ǫ, M)-admissible.

“∆ Md” implies “d˜˜ Md”. ≤ ≤ “d˜˜ Md” implies “G is (ǫ, M)-admissible”. ≤

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 27 / 49 Our result (II)

Chung, Lu, Horn (2008): Suppose p 1+c for some c 1 . Suppose G satisfies ≥ d˜ ≤ 20 d˜ d√n κ ∆ = o(σ ), ∆ = o(log n) and σ = o(n− ) for some κ > 0, and cκ G is (10, M)-admissible. Then almost surely there is a unique giant connected component in Gp with volume Θ(vol(G)), and no other component has volume more than max(2d log n,ω(σ vol(G))). p

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 28 / 49 Our result (II)

Chung, Lu, Horn (2008): Suppose p 1+c for some c 1 . Suppose G satisfies ≥ d˜ ≤ 20 d˜ d√n κ ∆ = o(σ ), ∆ = o(log n) and σ = o(n− ) for some κ > 0, and cκ G is (10, M)-admissible. Then almost surely there is a unique giant connected component in Gp with volume Θ(vol(G)), and no other component has volume more than max(2d log n,ω(σ vol(G))). p 1 1 If ∆ = O(d) and σ = o(log n), then pc = (1 + o(1))µ.

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 28 / 49 Neighborhood expansion in Gp

2 If vol(S) Θ(σ vol(G)), and vol2(T ) > (1 δ)vol2(G), then ≥ − vol(Γ(S) T ) (1 δ)pd˜vol(S). ∩ ≥ −

T S

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 29 / 49 Neighborhood expansion in Gp

2 If vol(S) Θ(σ vol(G)), and vol2(T ) > (1 δ)vol2(G), then ≥ − vol(Γ(S) T ) (1 δ)pd˜vol(S). ∩ ≥ −

T S

Since p > 1+c, (1 δ)pd˜ > 1 for small δ. d˜ −

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 29 / 49 The difficulty

The spectral bound gives a good control of neighborhood expansion of S if vol(S) = Ω(σ2vol(G)).

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 30 / 49 The difficulty

The spectral bound gives a good control of neighborhood expansion of S if vol(S) = Ω(σ2vol(G)). Question: How to find a large connect set S with vol(S) = Ω(σ2vol(G))?

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 30 / 49 Main ideas

2 ■ Choose S0 with vol(S0) = Θ(σ vol(G)).

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 31 / 49 Main ideas

2 ■ Choose S0 with vol(S0) = Θ(σ vol(G)).

■ Grow the neighborhood of S0 so that the second volume of exposed vertices is about xσvol2(G). By the 1 2 2 admissible condition, its volume reaches C x σ vol(S).

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 31 / 49 Main ideas

2 ■ Choose S0 with vol(S0) = Θ(σ vol(G)).

■ Grow the neighborhood of S0 so that the second volume of exposed vertices is about xσvol2(G). By the 1 2 2 admissible condition, its volume reaches C x σ vol(S). ■ Pick up vertices in big components with volume about 2 Θ(σ vol(G)) and call this set S1.

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 31 / 49 Main ideas

2 ■ Choose S0 with vol(S0) = Θ(σ vol(G)).

■ Grow the neighborhood of S0 so that the second volume of exposed vertices is about xσvol2(G). By the 1 2 2 admissible condition, its volume reaches C x σ vol(S). ■ Pick up vertices in big components with volume about 2 Θ(σ vol(G)) and call this set S1. ■ Repeat this process until we we find the giant connected component.

S S S S 2 3 0 1

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 31 / 49 Analysis

Let f(S) be the number of connected pieces in S. We have

σ2 f(S ) n 0 ≤ √ǫ 2 f(S1) f(S0)Cx− ≤ 2 f(S2) f(S1)Cx− ≤. .

log n After at most t = log(x2/C) step, St is connected

2 vol(St) = Θ(σ vol(G)).

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 32 / 49 Continue

The unexposed vertices have second volume at least

(1 txσ)vol (G) > (1 δ)vol (G). − 2 − 2 1 Here we assume σ = o(log n). Apply the neighborhood expansion lemma once again. We find the giant connected component. 

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 33 / 49 Summary

■ 1 If p < µ, almost surely Gp has no giant component. ■ Suppose p 1+c for some c 1 . Suppose G satisfies ≥ d˜ ≤ 20 d˜ d√n κ ∆ = o(σ ), ∆ = o(log n) and σ = o(n− ) for some κ > 0, cκ and G is (10, M)-admissible. Then almost surely Gp has giant components. ■ If ∆ = O(d) and σ = o( 1 ), then p 1 . log n c ≈ µ

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 34 / 49 Summary

■ 1 If p < µ, almost surely Gp has no giant component. ■ Suppose p 1+c for some c 1 . Suppose G satisfies ≥ d˜ ≤ 20 d˜ d√n κ ∆ = o(σ ), ∆ = o(log n) and σ = o(n− ) for some κ > 0, cκ and G is (10, M)-admissible. Then almost surely Gp has giant components. ■ If ∆ = O(d) and σ = o( 1 ), then p 1 . log n c ≈ µ Question: For any dense graph G, is it true that there exists a dense subgraph H with V (H) >ǫ V (G) and σ <ǫ? | | | | H

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 34 / 49 Part II: Hypergraphs

H = (V, E) is an r-uniform hypergraph (r-graph, for short). ■ V : the set of vertices ■ E: the set of edges, each edge has cardinality r.

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 35 / 49 Part II: Hypergraphs

H = (V, E) is an r-uniform hypergraph (r-graph, for short). ■ V : the set of vertices ■ E: the set of edges, each edge has cardinality r.

A 3-uniform loose cycle A 3-uniform tight cycle

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 35 / 49 Ongoing projects

■ How to define “connected components” for hypergraphs?

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 36 / 49 Ongoing projects

■ How to define “connected components” for hypergraphs? ■ How to define “Laplacians” for hypergraphs?

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 36 / 49 Ongoing projects

■ How to define “connected components” for hypergraphs? ■ How to define “Laplacians” for hypergraphs? ■ Phase transition for random hypergraphs?

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 36 / 49 Ongoing projects

■ How to define “connected components” for hypergraphs? ■ How to define “Laplacians” for hypergraphs? ■ Phase transition for random hypergraphs? ■ Percolation on hypergraphs? The remaining talk will focus on Question 1 and 2.

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 36 / 49 Connectivities in 3-graphs

■ Vertex to Vertex

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 37 / 49 Connectivities in 3-graphs

■ Vertex to Vertex

■ Pair to Pair

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 37 / 49 s-walks on hypergraphs

For 1 s r 1, an s-walk on H consists of ≤ ≤ − ■ a vertex sequence: v1, v2,...,v(k 1)(r s)+r − − ■ an edge sequence: F1,F2,...,Fk satisfying Fi = v(r s)(i 1)+1, v(r s)(i 1)+2,...,v(r s)(i 1)+r for 1 i{ k−. − − − − − } ≤ ≤

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 38 / 49 s-walks on hypergraphs

For 1 s r 1, an s-walk on H consists of ≤ ≤ − ■ a vertex sequence: v1, v2,...,v(k 1)(r s)+r − − ■ an edge sequence: F1,F2,...,Fk satisfying Fi = v(r s)(i 1)+1, v(r s)(i 1)+2,...,v(r s)(i 1)+r for 1 i{ k−. − − − − − } ≤ ≤ F F = s | i ∩ i+1|

v v v v v v1 v2 3 4 5 6 7

A 1-walk in a 3-graph

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 38 / 49 s-walks on hypergraphs

For 1 s r 1, an s-walk on H consists of ≤ ≤ − ■ a vertex sequence: v1, v2,...,v(k 1)(r s)+r − − ■ an edge sequence: F1,F2,...,Fk satisfying Fi = v(r s)(i 1)+1, v(r s)(i 1)+2,...,v(r s)(i 1)+r for 1 i{ k−. − − − − − } ≤ ≤ F F = s | i ∩ i+1|

v v v v1 v2 3 4 5 v6

A 2-walk in a 3-graph

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 38 / 49 s-walks on hypergraphs

For 1 s r 1, an s-walk on H consists of ≤ ≤ − ■ a vertex sequence: v1, v2,...,v(k 1)(r s)+r − − ■ an edge sequence: F1,F2,...,Fk satisfying Fi = v(r s)(i 1)+1, v(r s)(i 1)+2,...,v(r s)(i 1)+r for 1 i{ k−. − − − − − } ≤ ≤ F F = s | i ∩ i+1|

v v v v v v v1 v2 3 4 5 6 7 8

A 2-walk in a 4-graph

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 38 / 49 Loose random walks

Loose walk: 1 s r . ≤ ≤ 2

v v v v v v v1 v2 3 4 5 6 7 8

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 39 / 49 Loose random walks

Loose walk: 1 s r . ≤ ≤ 2

v v v v v v v1 v2 3 4 5 6 7 8

Observation: an s-th random walk on H is essentially a random walk on an auxiliary weighted graph G(s).

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 39 / 49 Loose random walks

Loose walk: 1 s r . ≤ ≤ 2

v v v v v v v1 v2 3 4 5 6 7 8

Observation: an s-th random walk on H is essentially a random walk on an auxiliary weighted graph G(s). s V - Vertex set V (G ) = s - Weight function w: V V Z: s
× s →
0
if S T = w(S, T ) = ∩ 6 ∅ d[S] [T ] if S T = .  ∪ ∩ ∅

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 39 / 49 Laplacians of hypergraph (I)

For 1 s r/2, the s-th Laplacian of H, denoted by (s), is defined≤ as≤ the Laplacian of G(s). L

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 40 / 49 Laplacians of hypergraph (I)

For 1 s r/2, the s-th Laplacian of H, denoted by (s), is defined≤ as≤ the Laplacian of G(s). L

- λ(s): the smallest non-trivial eigenvalue of (s). 1 L

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 40 / 49 Laplacians of hypergraph (I)

For 1 s r/2, the s-th Laplacian of H, denoted by (s), is defined≤ as≤ the Laplacian of G(s). L

- λ(s): the smallest non-trivial eigenvalue of (s). 1 L (s) (s) - λmax: the largest eigenvalue of . L

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 40 / 49 Laplacians of hypergraph (I)

For 1 s r/2, the s-th Laplacian of H, denoted by (s), is defined≤ as≤ the Laplacian of G(s). L

- λ(s): the smallest non-trivial eigenvalue of (s). 1 L (s) (s) - λmax: the largest eigenvalue of . L (s) (s) (s) - λ¯ : the spectral bound max 1 λ , λmax 1 . {| − 1 | | − |}

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 40 / 49 Laplacians of hypergraph (I)

For 1 s r/2, the s-th Laplacian of H, denoted by (s), is defined≤ as≤ the Laplacian of G(s). L

- λ(s): the smallest non-trivial eigenvalue of (s). 1 L (s) (s) - λmax: the largest eigenvalue of . L (s) (s) (s) - λ¯ : the spectral bound max 1 λ , λmax 1 . {| − 1 | | − |} (1) is the same as the Laplacian of hypergraph introduced byL Rodr´ıguez [2009].

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 40 / 49 Tight random walks

Tight walk: r < s r 1. 2 ≤ −

v v v v1 v2 3 4 5 v6

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 41 / 49 Tight random walks

Tight walk: r < s r 1. 2 ≤ −

v v v v1 v2 3 4 5 v6

Observation: an s-th random walk on H is “essentially” a random walk on an auxiliary directed graph D(s).

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 41 / 49 Tight random walks

Tight walk: r < s r 1. 2 ≤ −

v v v v1 v2 3 4 5 v6

Observation: an s-th random walk on H is “essentially” a random walk on an auxiliary directed graph D(s). ■ Vertex set V (G(s))=Vs ■ For x = (x1,...,xs) and y = (y1,...,ys), xy is a directed edge if

- xr s+j = yj for 1 j 2s r. − ≤ ≤ − - x1,...,xs, y2s r+1, ys is an edge of H. { − }

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 41 / 49 Laplacians of hypergraph (II)

For r/2 < s r 1, the s-th Laplacian of H, denoted by (s), is defined≤ as− the Laplacian of D(s). L

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 42 / 49 Laplacians of hypergraph (II)

For r/2 < s r 1, the s-th Laplacian of H, denoted by (s), is defined≤ as− the Laplacian of D(s). L ■ D(s) is Eulerian, i.e., indegree=outdegree at any vertex.

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 42 / 49 Laplacians of hypergraph (II)

For r/2 < s r 1, the s-th Laplacian of H, denoted by (s), is defined≤ as− the Laplacian of D(s). L ■ D(s) is Eulerian, i.e., indegree=outdegree at any vertex. ■ Chung [2005] defined the Laplacian of directed graphs. 1/2 1/2 Let ~ = T − AT − . Define the Laplacian L + ′ (s) = L L . L 2

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 42 / 49 Laplacians of hypergraph (II)

For r/2 < s r 1, the s-th Laplacian of H, denoted by (s), is defined≤ as− the Laplacian of D(s). L ■ D(s) is Eulerian, i.e., indegree=outdegree at any vertex. ■ Chung [2005] defined the Laplacian of directed graphs. 1/2 1/2 Let ~ = T − AT − . Define the Laplacian L + ′ (s) = L L . L 2

(r 1) ■ − is close related to the Laplacian of a regular Lhypergraph introduced by Chung [1993].

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 42 / 49 Laplacians of hypergraph (II)

For r/2 < s r 1, the s-th Laplacian of H, denoted by (s), is defined≤ as− the Laplacian of D(s). L ■ D(s) is Eulerian, i.e., indegree=outdegree at any vertex. ■ Chung [2005] defined the Laplacian of directed graphs. 1/2 1/2 Let ~ = T − AT − . Define the Laplacian L + ′ (s) = L L . L 2

(r 1) ■ − is close related to the Laplacian of a regular Lhypergraph introduced by Chung [1993]. (s) (s) ¯(s) λ1 , λmax, and λ are defined in the same way.

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 42 / 49 Application: Edge expansion

V V - S s , T t . - E(⊆S, T ): the⊆ set of edges containing x y for some x S,
y T with
x y = . ∪ ∈ vol(∈ S) ∩ vol(∅T ) - e(S) := V , e(T ) := V . vol(( s )) vol(( t )) E(S,T ) - e(S, T ) := | V V| . E( , ) | ( s ) ( t ) |

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 43 / 49 Application: Edge expansion

V V - S s , T t . - E(⊆S, T ): the⊆ set of edges containing x y for some x S,
y T with
x y = . ∪ ∈ vol(∈ S) ∩ vol(∅T ) - e(S) := V , e(T ) := V . vol(( s )) vol(( t )) E(S,T ) - e(S, T ) := | V V| . E( , ) | ( s ) ( t ) | Theorem [Lu-Peng 2011]: If 1 t s r/2, then ≤ ≤ ≤ e(S, T ) e(S)e(T ) λ¯(s) e(S)e(T )e(S¯)e(T¯). | − |≤ q

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 43 / 49 Connections of different (s) L

Theorem [Lu, Peng 2011] We have the following inequalities for the “loose” Laplacian eigenvalues.

(1) (2) ( r/2 ) λ1 λ1 . . . λ1⌊ ⌋ ; (1) ≥ (2) ≥ ≥ ( r/2 ) λ λ . . . λ ⌊ ⌋ . max ≤ max ≤ ≤ max

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 44 / 49 r Complete hypergraph Kn

Theorem: For 1 s r/2, the s-th Laplacian eigenvalues r ≤ ≤ r of Kn is the eigenvalues of s-th Laplacian of Kn are given by

i n s i) ( 1) −s −i n n 1 − n s − with multiplicity − − i − i 1 s
   −  for 0 i s.
≤ ≤

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 45 / 49 r Complete hypergraph Kn

Theorem: For 1 s r/2, the s-th Laplacian eigenvalues r ≤ ≤ r of Kn is the eigenvalues of s-th Laplacian of Kn are given by

i n s i) ( 1) −s −i n n 1 − n s − with multiplicity − − i − i 1 s
   −  for 0 i s.
≤ ≤ ′ Observation: G(s) is essentially the Kneser graph K(n, s). The vertices are s-sets in [n]. A pair of s-sets S and T forms an edge of S T = . ∩ ∅

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 45 / 49 Laplacians of Hr(n, p)

Random r-uniform random hypergraph Hr(n,p): - n: the number of vertices. - p: the probability; each r-set is an edge with probability p independently. λ(s)(H) : the s-the Laplacian spectrum of H. { k }

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 46 / 49 Laplacians of Hr(n, p)

Random r-uniform random hypergraph Hr(n,p): - n: the number of vertices. - p: the probability; each r-set is an edge with probability p independently. λ(s)(H) : the s-the Laplacian spectrum of H. { k } Theorem [Lu and Peng 2011+]: For 1 s r/2, if 4 2 log n log n ≤ ≤ p(1 p) n and 1 p n2 , the almost surely for − ≫ (s) − ≫ n 0 k s 1, ≤(s) ≤r − (s) r 1 p λk (H (n,p)) λk (Kn) (3 + o(1)) n−−s . |
− |≤ (r−s)p q

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 46 / 49 Semicircle Law

1 r - Fn(x): n of the number of s-th Laplacian of H (n,p) (s) less than x. 1 p - R := (2 + o(1)) r−s −n−s . ( s )(r−s)p q

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 47 / 49 Semicircle Law

1 r - Fn(x): n of the number of s-th Laplacian of H (n,p) (s) less than x. 1 p - R := (2 + o(1)) r−s −n−s . ( s )(r−s)p q Theorem [Lu and Peng 2011+]: log n F(x) For 1 s r/2, if p(1 p) nr−s , then almost≤ ≤ surely − ≫ 1−R1 1+R p F (x) F (x). n →

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 47 / 49 Semicircle Law

1 r - Fn(x): n of the number of s-th Laplacian of H (n,p) (s) less than x. 1 p - R := (2 + o(1)) r−s −n−s . ( s )(r−s)p q Theorem [Lu and Peng 2011+]: log n F(x) For 1 s r/2, if p(1 p) nr−s , then almost≤ ≤ surely − ≫ 1−R1 1+R p F (x) F (x). n → Previously known similar results: - on G(n,p) [F¨uredi-Koml´os 1981]. - on G(w1,...,wn) Chung-Lu-Vu 2002.

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 47 / 49 Phase transition of Hr(n, p)

■ Vertex to vertex : Karo´nski-Luczak (2002) determines the threshold 1 pc n 1 . ≈ (r 1) r−1 − −

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 48 / 49 Phase transition of Hr(n, p)

■ Vertex to vertex : Karo´nski-Luczak (2002) determines the threshold 1 pc n 1 . ≈ (r 1) r−1 − −
■ Pair to pair? In general, s-tuple to s-tuple? (Ongoing project with Peng.)

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 48 / 49 Thank you.

TheGiantComponentinaRandomSubgraphofaGivenGraph Linyuan Lu (University of South Carolina) – 49 / 49