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The Giant Component in a Random Subgraph of a Given Graph 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<pc, almost surely there is no giant component ■ 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 .
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