Critical percolation on networks with given degrees. Souvik Dhara

Transcription

1 Critical percolation on networks with given degrees Souvik Dhara Microsoft Research and MIT Mathematics Montréal Summer Workshop: Challenges in Probability and Mathematical Physics June 9,

2 Remco van der Hofstad Johan van Leeuwaarden Shankar Bhamidi Sanchayan Sen 2

3 Percolation as a dynamic process Percolation: Keep each edge in the graph with probability p, independently 3

4 Percolation as a dynamic process Percolation: Keep each edge in the graph with probability p, independently As a dynamic process (Harris coupling): Associate independent uniform [0,1] weights U e to each edge e p as time: Keep edge e, U e p at time p, and then increase p 3

5 Percolation as a dynamic process Percolation: Keep each edge in the graph with probability p, independently As a dynamic process (Harris coupling): Associate independent uniform [0,1] weights U e to each edge e p as time: Keep edge e, U e p at time p, and then increase p 3

6 Percolation as a dynamic process Percolation: Keep each edge in the graph with probability p, independently As a dynamic process (Harris coupling): Associate independent uniform [0,1] weights U e to each edge e p as time: Keep edge e, U e p at time p, and then increase p 3

7 Percolation as a dynamic process Percolation: Keep each edge in the graph with probability p, independently As a dynamic process (Harris coupling): Associate independent uniform [0,1] weights U e to each edge e p as time: Keep edge e, U e p at time p, and then increase p 3

8 Percolation as a dynamic process Percolation: Keep each edge in the graph with probability p, independently As a dynamic process (Harris coupling): Associate independent uniform [0,1] weights U e to each edge e p as time: Keep edge e, U e p at time p, and then increase p 3

9 Percolation as a dynamic process Percolation: Keep each edge in the graph with probability p, independently As a dynamic process (Harris coupling): Associate independent uniform [0,1] weights U e to each edge e p as time: Keep edge e, U e p at time p, and then increase p 3

10 Percolation as a dynamic process Percolation: Keep each edge in the graph with probability p, independently As a dynamic process (Harris coupling): Associate independent uniform [0,1] weights U e to each edge e p as time: Keep edge e, U e p at time p, and then increase p 3

11 Percolation phase transition on finite graphs There exists p c such that for any ε > 0 n := # vertices in the graph (1) p < p c (1 ε): largest component is o(n) subcritical (2) p > p c (1 + ε): largest component is Θ(n) supercritical 4

12 Percolation phase transition on finite graphs There exists p c such that for any ε > 0 n := # vertices in the graph (1) p < p c (1 ε): largest component is o(n) subcritical (2) p > p c (1 + ε): largest component is Θ(n) supercritical Erdős & Rényi (1959), Gilbert (1959) Complete graph Molloy & Reed (1995), Janson (2009) Uniformly chosen graph given degree Bollobás, Borgs, Chayes, Riordan (2010) Dense graph Aldous (2016) General graphs 4

13 Percolation phase transition on finite graphs There exists p c such that for any ε > 0 n := # vertices in the graph (1) p < p c (1 ε): largest component is o(n) subcritical (2) p > p c (1 + ε): largest component is Θ(n) supercritical Erdős & Rényi (1959), Gilbert (1959) Complete graph Molloy & Reed (1995), Janson (2009) Uniformly chosen graph given degree Bollobás, Borgs, Chayes, Riordan (2010) Dense graph Aldous (2016) General graphs (1.5) p = p c (1 ε n ) with ε n 0: Critical behavior is observed 4

14 Critical window: Zoom into the critical value Surplus edges:= # edges to be deleted to turn a graph into tree Features Subcritical p = p c (1 ε) Critical window p = p c (1 n η ) Supercritical p = p c (1 + ε) Surplus edges Component sizes Subcritical Supercritical 5

15 Critical window: Zoom into the critical value Surplus edges:= # edges to be deleted to turn a graph into tree Features Subcritical p = p c (1 ε) Critical window p = p c (1 n η ) Supercritical p = p c (1 + ε) Surplus edges 1 Component sizes Concentrates Concentrates ε > 0 ε > 0 Subcritical Supercritical 5

16 Critical window: Zoom into the critical value Surplus edges:= # edges to be deleted to turn a graph into tree Features Subcritical p = p c (1 ε) Critical window p = p c (1 n η ) Supercritical p = p c (1 + ε) Surplus edges 1 Component sizes Concentrates Concentrates ε > 0 ε n n η ε n n η ε > 0 Subcritical Supercritical 5

17 Critical window: Zoom into the critical value Surplus edges:= # edges to be deleted to turn a graph into tree Features Subcritical p = p c (1 ε) Critical window p = p c (1 n η ) Supercritical p = p c (1 + ε) Surplus edges 1 Poisson Component sizes Concentrates Random Concentrates ε > 0 ε n n η ε n n η ε n n η ε > 0 Subcritical Critical window Supercritical 5

18 Critical window: Zoom into the critical value Surplus edges:= # edges to be deleted to turn a graph into tree Features Subcritical p = p c (1 ε) Critical window p = p c (1 n η ) Supercritical p = p c (1 + ε) Surplus edges 1 Poisson Component sizes Concentrates Random Concentrates ε > 0 ε n n η ε n n η ε n n η ε > 0 Subcritical Critical window Supercritical Mostly trees Components merge Birth of giant 5

19 Critical window: Zoom into the critical value Surplus edges:= # edges to be deleted to turn a graph into tree Features Subcritical p = p c (1 ε) Critical window p = p c (1 n η ) Supercritical p = p c (1 + ε) Surplus edges 1 Poisson Component sizes Concentrates Random Concentrates ε > 0 ε n n η ε n n η ε n n η ε > 0 Subcritical Critical window Supercritical Mostly trees Components merge Birth of giant Critical window: p = p c (1 + λn η ), < λ < 5

20 Highlights of the critical window ε > 0 ε n n η ε n = λn η ε n n η ε > 0 Subcritical Critical window Supercritical Mostly trees Components merge Birth of giant Intermediate behavior is observed: Components are not trees Component sizes do not concentrate Phase transition happens over a whole window of time Race of different components: Components merge over the critical window Unique largest component at the end 6

21 Key questions for percolation over the critical window p = p c (1 + λn η ), < λ < Three fundamental questions: (Q1) Component size and surplus (at each fixed λ) (Q2) Evolution of component size and surplus over the critical window (as λ varies) (Q3) Graph distances within components (at each fixed λ) 7

22 Each question poses novel theoretical challenges (Q1a) Component size: [Aldous & Limic 98], [Aldous & Pittel 00], [Nachmias & Peres 10], [van der Hofstad, Janssen & van Leeuwaarden 10], [Bhamidi, van der Hofstad & van Leeuwaarden 10 12], [van der Hofstad, 13], [Bhamidi, Budhiraja & Wang 14], [Bhamidi, Sen & Wang 14], [Dembo, Levit & Vadlamani 14], [Joseph 14], [van der Hofstad & Nachmias 17] + many more... (Q1) Component size and surplus: [Aldous 97], [Riordan 12], [Bhamidi, Budhiraja & Wang 14] + many more... (Q2) Evolution over the critical window: [Aldous 97], [Aldous & Limic 98], [Bhamidi, van der Hofstad & van Leeuwaarden 12], [Bhamidi, Budhiraja & Wang 14], [Broutin & Marckert 16] + many more... (Q3) Graph distances within components: [Nachmias & Peres 10], [Addario Berry, Broutin & Goldschmidt 12], [Bhamidi, Sen & Wang 14], [Bhamidi, van der Hofstad & Sen 17], [Broutin, Duquesne, & Wang 18] + many more... 8

23 New challenges Strong inhomogeneity (heavy-tailed) in the empirical degree distribution Does heavy-tail lead to fundamentally different behavior? 9

24 New challenges Strong inhomogeneity (heavy-tailed) in the empirical degree distribution Does heavy-tail lead to fundamentally different behavior? YES, with respect to all the above questions 9

25 New challenges Strong inhomogeneity (heavy-tailed) in the empirical degree distribution Does heavy-tail lead to fundamentally different behavior? YES, with respect to all the above questions Three universality classes depending on the degree distribution Finite third moment Infinite third moment Infinite second moment 9

26 Overview of the talk Consider percolation on random graphs with given degrees Finite third moment Erdős-Rényi universality class Insensitivity to the degree distribution 10

27 Overview of the talk Consider percolation on random graphs with given degrees Finite third moment Erdős-Rényi universality class Insensitivity to the degree distribution Infinite third moment Scaling limits are fundamentally different Description depends primarily on vertices of highest degree Deleting the highest degree vertex changes the scaling limits 10

28 Overview of the talk Consider percolation on random graphs with given degrees Finite third moment Erdős-Rényi universality class Insensitivity to the degree distribution Infinite third moment Scaling limits are fundamentally different Description depends primarily on vertices of highest degree Deleting the highest degree vertex changes the scaling limits Infinite second moment p c = 0: Critical behavior is observed for p c = λn η Critical value and component sizes depend on single-edge constraint Configuration model versus erased configuration model 10

29 Preliminaries Random graph: Configuration model/uniform graphs with given degrees Motivation: Power-law degrees (proportion of vertices of degree k Ck τ ) Finite third moment: τ > 4, Infinite third moment: τ (3, 4), Infinite second moment: τ (2, 3) Hubs: Vertices of degree Θ(max degree) 11

30 Preliminaries Random graph: Configuration model/uniform graphs with given degrees Motivation: Power-law degrees (proportion of vertices of degree k Ck τ ) Finite third moment: τ > 4, Infinite third moment: τ (3, 4), Infinite second moment: τ (2, 3) Hubs: Vertices of degree Θ(max degree) Critical window: p = p c (1 + λn η ), < λ < C (i) := the i-th largest component SP(C (i) ):= # surplus edges in C (i) 11

31 Preliminaries Random graph: Configuration model/uniform graphs with given degrees Motivation: Power-law degrees (proportion of vertices of degree k Ck τ ) Finite third moment: τ > 4, Infinite third moment: τ (3, 4), Infinite second moment: τ (2, 3) Hubs: Vertices of degree Θ(max degree) Critical window: p = p c (1 + λn η ), < λ < C (i) := the i-th largest component SP(C (i) ):= # surplus edges in C (i) Topology: U 0 R + N with norm ( i x2 i )1/2 + i x iy i 11

32 Component size and surplus (Q1) Theorem 1 (D, v/d Hofstad, v Leeuwaarden, Sen 16 a,b) For τ > 4 and p = p c (1 + λn 1 3 ) (n 2 3 C (i), SP(C (i) )) i 1 d X 1 in U 0 (Finite third moment) For τ (3, 4) and p = p c (1 + λn τ 3 τ 1 ) (n τ 1 τ 2 d C (i), SP(C (i) )) i 1 X 2 in U 0 (Infinite third moment) X 1 and X 2 are non-degenerate random vectors 12

33 Component size and surplus (Q1) Theorem 1 (D, v/d Hofstad, v Leeuwaarden, Sen 16 a,b) For τ > 4 and p = p c (1 + λn 1 3 ) (n 2 3 C (i), SP(C (i) )) i 1 d X 1 in U 0 (Finite third moment) For τ (3, 4) and p = p c (1 + λn τ 3 τ 1 ) (n τ 1 τ 2 d C (i), SP(C (i) )) i 1 X 2 in U 0 (Infinite third moment) X 1 and X 2 are non-degenerate random vectors X 1 Erdős-Rényi (insensitive to the exact degree distribution) X 2 depends on the precise asymptotics of hubs 12

34 Component size and surplus (Q1) Theorem 1 (D, v/d Hofstad, v Leeuwaarden, Sen 16 a,b) For τ > 4 and p = p c (1 + λn 1 3 ) (n 2 3 C (i), SP(C (i) )) i 1 d X 1 in U 0 (Finite third moment) For τ (3, 4) and p = p c (1 + λn τ 3 τ 1 ) (n τ 1 τ 2 d C (i), SP(C (i) )) i 1 X 2 in U 0 (Infinite third moment) X 1 and X 2 are non-degenerate random vectors X 1 Erdős-Rényi (insensitive to the exact degree distribution) X 2 depends on the precise asymptotics of hubs { = 0, for τ > 4 lim P(two hubs are in the same component) n (0, 1), for τ (3, 4) 12

35 Results: evolution of components (Q2) ( n 2 3 C (i), SP(C (i) ) ) i 1 τ > 4 Z n (λ) := ( n τ 2 τ 1 C(i), SP(C (i) ) ) i 1 τ (3, 4) Z n (λ) is a stochastic process in λ (, ) 13

36 Results: evolution of components (Q2) ( n 2 3 C (i), SP(C (i) ) ) i 1 τ > 4 Z n (λ) := ( n τ 2 τ 1 C(i), SP(C (i) ) ) i 1 τ (3, 4) Z n (λ) is a stochastic process in λ (, ) Theorem 2 (D, van der Hofstad, van Leeuwaarden, Sen 16 a,b) (Z n (λ)) <λ< d (AMC 1 (λ)) <λ< in (U 0 ) k (Finite third moment) (Z n (λ)) <λ< d (AMC 2 (λ)) <λ< in (U 0 ) k (Infinite third moment) AMC 1 and AMC 2 are augmented multiplicative coalescents processes with different entrance boundary conditions Theorems 1 & 2 resolve (Q1)+(Q2) All these results apply to uniform graphs given degree 13

37 Exploration process (Aldous 97): Proof idea for component sizes Explore the graph and encode the component sizes in terms of a walk called exploration process and take scaling limits of the exploration process Recover limits of component sizes from the limiting process 14

38 Exploration process (Aldous 97): Proof idea for component sizes Explore the graph and encode the component sizes in terms of a walk called exploration process and take scaling limits of the exploration process Recover limits of component sizes from the limiting process 14

39 Exploration process (Aldous 97): Proof idea for component sizes Explore the graph and encode the component sizes in terms of a walk called exploration process and take scaling limits of the exploration process Recover limits of component sizes from the limiting process 14

40 Exploration process (Aldous 97): Proof idea for component sizes Explore the graph and encode the component sizes in terms of a walk called exploration process and take scaling limits of the exploration process Recover limits of component sizes from the limiting process 14

41 Exploration process (Aldous 97): Proof idea for component sizes Explore the graph and encode the component sizes in terms of a walk called exploration process and take scaling limits of the exploration process Recover limits of component sizes from the limiting process 14

42 Exploration process (Aldous 97): Proof idea for component sizes Explore the graph and encode the component sizes in terms of a walk called exploration process and take scaling limits of the exploration process Recover limits of component sizes from the limiting process 14

43 Exploration process (Aldous 97): Proof idea for component sizes Explore the graph and encode the component sizes in terms of a walk called exploration process and take scaling limits of the exploration process Recover limits of component sizes from the limiting process 14

44 Exploration process (Aldous 97): Proof idea for component sizes Explore the graph and encode the component sizes in terms of a walk called exploration process and take scaling limits of the exploration process Recover limits of component sizes from the limiting process 14

45 Exploration process (Aldous 97): Proof idea for component sizes Explore the graph and encode the component sizes in terms of a walk called exploration process and take scaling limits of the exploration process Recover limits of component sizes from the limiting process 14

46 Exploration process (Aldous 97): Proof idea for component sizes Explore the graph and encode the component sizes in terms of a walk called exploration process and take scaling limits of the exploration process Recover limits of component sizes from the limiting process 14

47 Exploration process (Aldous 97): Proof idea for component sizes Explore the graph and encode the component sizes in terms of a walk called exploration process and take scaling limits of the exploration process Recover limits of component sizes from the limiting process 14

48 Exploration process (Aldous 97): Proof idea for component sizes Explore the graph and encode the component sizes in terms of a walk called exploration process and take scaling limits of the exploration process Recover limits of component sizes from the limiting process 14

49 Exploration process (Aldous 97): Proof idea for component sizes Explore the graph and encode the component sizes in terms of a walk called exploration process and take scaling limits of the exploration process Recover limits of component sizes from the limiting process 14

50 Exploration process (Aldous 97): Proof idea for component sizes Explore the graph and encode the component sizes in terms of a walk called exploration process and take scaling limits of the exploration process Recover limits of component sizes from the limiting process 14

51 Exploration process (Aldous 97): Proof idea for component sizes Explore the graph and encode the component sizes in terms of a walk called exploration process and take scaling limits of the exploration process Recover limits of component sizes from the limiting process 14

52 Exploration process (Aldous 97): Proof idea for component sizes Explore the graph and encode the component sizes in terms of a walk called exploration process and take scaling limits of the exploration process Recover limits of component sizes from the limiting process 14

53 Exploration process (Aldous 97): Proof idea for component sizes Explore the graph and encode the component sizes in terms of a walk called exploration process and take scaling limits of the exploration process Recover limits of component sizes from the limiting process For ERRG the increment distribution is Binomial but difficult in general... 14

54 Exploration process for percolated CM CM = Configuration model Direct analysis of percolation on CM is difficult... d-regular case was studied by Nachmias & Peres 09 The general case was unknown 15

55 Exploration process for percolated CM CM = Configuration model Direct analysis of percolation on CM is difficult... d-regular case was studied by Nachmias & Peres 09 The general case was unknown A wonderful idea by Fountoulakis 07 and Janson 09 Percolated CM d = CM with a different degree distribution Enough to analyze exploration process on CM 15

56 Exploration process for percolated CM CM = Configuration model Direct analysis of percolation on CM is difficult... d-regular case was studied by Nachmias & Peres 09 The general case was unknown A wonderful idea by Fountoulakis 07 and Janson 09 Percolated CM d = CM with a different degree distribution Enough to analyze exploration process on CM Previous work: Bounded degree: Riordan 12 (convergence in product topology) iid degree: Joseph 14 (component sizes in l 2 topology) 15

57 Analysis of the exploration process: τ > 4 S n (t) = n 1 3 S n ( tn 2 3 ) S n (t) = Fluctuations + Drift. Finite third moment finite variance of increments of S n Fluctuations Brownian motion (Martingale FCLT) Drift c 1 t c 2 t 2 16

58 Analysis of the exploration process: τ > 4 S n (t) = n 1 3 S n ( tn 2 3 ) S n (t) = Fluctuations + Drift. Finite third moment finite variance of increments of S n Fluctuations Brownian motion (Martingale FCLT) Drift c 1 t c 2 t 2 16

59 Analysis of the exploration process: τ > 4 S n (t) = n 1 3 S n ( tn 2 3 ) S n (t) = Fluctuations + Drift. Finite third moment finite variance of increments of S n Fluctuations Brownian motion (Martingale FCLT) Drift c 1 t c 2 t 2 n 2 3 Comp. sizes excursion lengths of S := B(t) + c 1 t c 2 t 2 16

60 Analysis of the exploration process: τ (3, 4) S n (t) = n 1 τ 1 S n ( tn τ 2 τ 1 ) S(t) = i θ i ( 1Exp(θi ) t θ i t ) Limit is a thinned Lévy process: Jumps do not vanish even in limit Standard stochastic process convergence techniques do not work 17

61 Evolution of components as λ increases ( n 2 3 C (i), SP(C (i) ) ) i 1 τ > 4 Z n (λ) := ( n τ 2 τ 1 C (i), SP(C (i) ) ) i 1 τ (3, 4) For ERRG, C (i) and C (j) merge at rate C (i) C (j) SP(C (i) ) increases by 1 at rate ( C (i) ) 2 (multiplicative coalescent) (augmented version) Previous works by [Aldous 97], [Aldous & Limic 98], [Bhamidi, Budhiraja & Wang 14], [Broutin & Marckert 16] 18

62 Challenges for percolation on CM (Z n (λ)) <λ< is not Markov 19

63 Challenges for percolation on CM (Z n (λ)) <λ< is not Markov Approximate by a Markov process 19

64 Challenges for percolation on CM (Z n (λ)) <λ< is not Markov Approximate by a Markov process Even the approx. Markov process is not a multiplicative coalescent Develop techniques to deal with this scenario 19

65 Challenges for percolation on CM (Z n (λ)) <λ< is not Markov Approximate by a Markov process Even the approx. Markov process is not a multiplicative coalescent Develop techniques to deal with this scenario Theorem 2 (D, v/d Hofstad, v Leeuwaarden, Sen 16 a,b) (Z n (λ)) <λ< d (AMC 1 (λ)) <λ< in (U 0 ) k (Finite third moment) (Z n (λ)) <λ< d (AMC 2 (λ)) <λ< in (U 0 ) k (Infinite third moment) Open problem: establish above w.r.t. Skorohod J 1 topology D(R, U 0 ) 19

66 Convergence of distances (Q3) C (i) as a random metric space Elements: vertices in C(i) Metric: graph distance 20

67 Convergence of distances (Q3) C (i) as a random metric space Elements: vertices in C(i) Metric: graph distance Objective: Study the limit of C (i) on the space of metric spaces Outcome: Convergence of global functionals like diameter 20

68 Convergence of distances (Q3) C (i) as a random metric space Elements: vertices in C(i) Metric: graph distance Objective: Study the limit of C (i) on the space of metric spaces Outcome: Convergence of global functionals like diameter Seminal work by Addario-Berry, Broutin, Goldschmidt (2012): Addario-Berry, Bhamidi, Broutin, Conchon-Kerjan, D, Duquesne, Goldschmidt, van der Hofstad, Sen, X. Wang, M. Wang 20

69 Metric structure of critical components Theorem 3 (Bhamidi, Broutin, Sen, Wang 14) (Bhamidi, Sen 16) Re-scale metric by n 1 3. Let τ > 4 and p = p c (1 + λn 1 3 ). Then (C (i) ) i 1 converges in distribution Theorem 4 (Bhamidi, D, v/d Hofstad, Sen 17, 18+) Re-scale metric by n τ 3 τ 1. Let τ (3, 4) and p = pc (1 + λn τ 3 τ 1 ). Then (C (i) ) i 1 converges in distribution The above convergence results hold under S N, where S is the space of measured metric spaces equipped with Gromov Hausdorff Prokhorov topology Scaling limits in Theorem 4 are different for iid degrees Conchon-Kerjan & Goldschmidt (in preparation) 21

70 Limiting object: finite third moment surplus = Poisson 1 distances = n 3 Courtesy: Igor Kortchemski s website 2 size = n 3 22

71 Limiting object: infinite third moment surplus = Poisson τ 3 distances = n τ 1 Courtesy: Igor Kortchemski s website τ 2 size = n τ 1 23

72 Infinite second moment case Degree distribution: Power-law Ck τ with τ (2, 3) p > 0: Always supercritical = p c = 0 24

73 Infinite second moment case Degree distribution: Power-law Ck τ with τ (2, 3) p > 0: Always supercritical = p c = 0 Critical behavior is observed for p 0 Critical window: p c = λn η for λ > 0 24

74 Infinite second moment case Degree distribution: Power-law Ck τ with τ (2, 3) p > 0: Always supercritical = p c = 0 Critical behavior is observed for p 0 Critical window: p c = λn η for λ > 0 Objectives: Identify critical window = Find the exponent η 24

75 Infinite second moment case Degree distribution: Power-law Ck τ with τ (2, 3) p > 0: Always supercritical = p c = 0 Critical behavior is observed for p 0 Critical window: p c = λn η for λ > 0 Objectives: Identify critical window = Find the exponent η Find ρ > 0 and X (nondegenerate scaling limit) such that (n ρ C (i) ) d i 1 X 24

76 Infinite second moment case Degree distribution: Power-law Ck τ with τ (2, 3) p > 0: Always supercritical = p c = 0 Critical behavior is observed for p 0 Critical window: p c = λn η for λ > 0 Objectives: Identify critical window = Find the exponent η Find ρ > 0 and X (nondegenerate scaling limit) such that (n ρ C (i) ) d i 1 X Analyze near critical behavior Component sizes concentrate outside critical window All of these were open questions till date... 24

77 Informal description of the results Models: Configuration model (CM), Erased configuration model (ECM) 25

78 Informal description of the results Models: Configuration model (CM), Erased configuration model (ECM) Properties CM ECM p c λn 3 τ τ 1 λn 3 τ 2 C (i) n τ 1 τ 2 n τ τ 2 25

79 Informal description of the results Models: Configuration model (CM), Erased configuration model (ECM) Properties CM ECM p c λn 3 τ τ 1 λn 3 τ 2 C (i) n τ 1 τ 2 n τ τ 2 Single-edge constraint changes critical value for τ (2, 3)! ECM has larger component sizes and critical value In both cases, critical window is precisely the value when lim P(hubs are in the same component)= ζ (0, 1) n Subcritical regime: P(hubs are in the same component) 0 Supercritical regime: P(hubs are in the same component) 1 25

80 Configuration model results Theorem 5 (D, v/d Hofstad, v Leeuwaarden 18+) For p c = λn 3 τ τ 1 : (n τ 2 d τ 1 C(i), SP(C (i) )) i 1 ( γ i, N(γ i )) i 1 in U 0 topology, where (γ i ) i 1 is the ordered excursions of S (t) = λ i=1 i 1 τ 1 1{Exp(1/i 1 τ 1 µ) t} 2t, N(γ i) = Poisson(area under γ i ) Moreover, diameter(c (i) ) is tight for all i 1 26

81 Configuration model results Theorem 5 (D, v/d Hofstad, v Leeuwaarden 18+) For p c = λn 3 τ τ 1 : (n τ 2 d τ 1 C(i), SP(C (i) )) i 1 ( γ i, N(γ i )) i 1 in U 0 topology, where (γ i ) i 1 is the ordered excursions of S (t) = λ i=1 i 1 τ 1 1{Exp(1/i 1 τ 1 µ) t} 2t, N(γ i) = Poisson(area under γ i ) Moreover, diameter(c (i) ) is tight for all i 1 Theorem 6 (D, v/d Hofstad, v Leeuwaarden 18+) For p p c = λn 3 τ 1 τ 1 : (n τ 1 p) 1 C (i) P ci 1 τ 1 For p p c = λn 3 τ 1 τ 1 : (np 3 τ ) 1 P C (1) c, C (2) C (1) 26

82 Erased configuration model results Theorem 7 (D, v/d Hofstad, v Leeuwaarden 18+) For p = p c = λn 3 τ 2, λ (0, λ 0 ): in l 2 topology ( 1 (n τ 1 pc ) 1 C (i) ) d i 1 (Wi ) i 1 Theorem 8 (D, v/d Hofstad, v Leeuwaarden 18+) For p p c = λn 3 τ 2 : (n 1 τ 1 p) 1 C (i) P ci 1 τ 1 Working on supercritical and λ > λ 0 case Key difficulty: exploration process approach fails Open Problem: find the diameter for ECM 27

83 Summary Critical percolation on configuration model τ = power-law exponent Three fundamental questions: (Q1) Component size and surplus (at each fixed λ) (Q2) Evolution of component size and surplus over critical window (as λ varies) (Q3) Graph distances within components (at each fixed λ) 28

84 Summary Critical percolation on configuration model τ = power-law exponent Three fundamental questions: (Q1) Component size and surplus (at each fixed λ) τ > 4: Erdős-Rényi universality class τ (3, 4): Deleting the highest degree vertex changes scaling limits τ (2, 3): Critical window depends on the single-edge constraint (Q2) Evolution of component size and surplus over critical window (as λ varies) (Q3) Graph distances within components (at each fixed λ) 28

85 Summary Critical percolation on configuration model τ = power-law exponent Three fundamental questions: (Q1) Component size and surplus (at each fixed λ) τ > 4: Erdős-Rényi universality class τ (3, 4): Deleting the highest degree vertex changes scaling limits τ (2, 3): Critical window depends on the single-edge constraint (Q2) Evolution of component size and surplus over critical window (as λ varies) τ > 4 & τ (3, 4): Augmented Multiplicative Coalescent (AMC) τ (2, 3): Open question (Q3) Graph distances within components (at each fixed λ) 28

86 Summary Critical percolation on configuration model τ = power-law exponent Three fundamental questions: (Q1) Component size and surplus (at each fixed λ) τ > 4: Erdős-Rényi universality class τ (3, 4): Deleting the highest degree vertex changes scaling limits τ (2, 3): Critical window depends on the single-edge constraint (Q2) Evolution of component size and surplus over critical window (as λ varies) τ > 4 & τ (3, 4): Augmented Multiplicative Coalescent (AMC) τ (2, 3): Open question (Q3) Graph distances within components (at each fixed λ) 1 τ > 4: Metric structure converges (distance rescaled by n 3 ) τ 3 τ (3, 4): Metric structure converges (distance rescaled by n τ 1 ) τ (2, 3): CM: Finite diameter, ECM: Open question 28

87 References (Q1a) Component size: [Aldous & Limic 98], [Aldous & Pittel 00], [Nachmias & Peres 10], [van der Hofstad, Janssen & van Leeuwaarden 10], [Bhamidi, van der Hofstad & van Leeuwaarden 10 12], [van der Hofstad, 13], [Bhamidi, Budhiraja & Wang 14], [Bhamidi, Sen & Wang 14], [Dembo, Levit & Vadlamani 14], [Joseph 14], [van der Hofstad & Nachmias 17] (Q1) Component size and surplus: [Aldous 97], [Riordan 12], [Bhamidi, Budhiraja & Wang 14] (Q2) Evolution over the critical window: [Aldous 97], [Aldous & Limic 98], [Bhamidi, van der Hofstad & van Leeuwaarden 12], [Bhamidi, Budhiraja & Wang 14], [Broutin & Marckert 16] (Q3) Graph distances within components: [Nachmias & Peres 10], [Addario Berry, Broutin & Goldschmidt 12], [Bhamidi, Sen & Wang 14], [Bhamidi, van der Hofstad & Sen 17], [Broutin, Duquesne, & Wang 18] 29

88 References PhD thesis: Critical percolation on random networks with prescribed degrees Dhara S., van der Hofstad R. & van Leeuwaarden J.S.H., Critical percolation on scale-free random graphs: Effect of the single-edge constraint (2018+), Near completion Bhamidi S., Dhara S., van der Hofstad R. & Sen, S, Global lower mass-bound for critical configuration models in the heavy-tailed regime (2018+), Near completion Bhamidi S., Dhara S., van der Hofstad R. & Sen, S, Universality for critical heavy-tailed network models: Metric structure of maximal components (2017) Dhara S., van der Hofstad R., van Leeuwaarden J.S.H., & Sen, S., Heavy-tailed configuration models at criticality (2016) Dhara S., van der Hofstad R., van Leeuwaarden J.S.H., & Sen, S., Critical window for the configuration model: finite third moment degrees (2016) Thank you 30