Scale-free percolation Remco van der Hofstad Simons Conference on Random Graph Processes, May 9–12, 2016, UT Austin Joint work with: ⊲ Mia Deijfen (Stockholm) ⊲ Gerard Hooghiemstra (TU Delft)
Complex networks Yeast protein interaction network Internet topology in 2001 Attention focussing on unexpected commonality.
Scale-free paradigm 10 0 10 − 1 10 − 1 10 − 2 proportion proportion 10 − 3 10 − 3 10 − 4 10 − 5 10 − 5 10 − 6 10 − 7 10 − 7 10 0 10 1 10 2 10 3 10 4 10 5 10 0 10 1 10 2 10 3 10 4 degree degree Loglog plot degree sequences Internet Movie Database and Internet ⊲ Straight line: proportion p k of vertices with degree k satisfies p k = ck − τ .
Small-world paradigm 0 . 6 proportion of pairs proportion of pairs 2003 0 . 4 0 . 4 0 . 2 0 . 2 0 0 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 distance distance Distances in SCC WWW and IMDb in 2003.
Random graphs for complex networks ⊲ Inhomogeneous random graph: Vertex set [ n ] = { 1 , . . . , n } , edge ij independently present w.p. p ij . Example: Erd˝ os-Rényi model, for which p = λ/n for some λ > 0 . ⊲ Configuration model: Vertices in [ n ] have prescribed degree, graph constructed by pairing half-edges. ⊲ Preferential attachment model: Growing network, new vertices more likely to attach to old vertices having high degree. Models typically are non-spatial and have small clustering. AIM: construct simple spatial scale-free random graph model.
Inhomogeneous rgs Norros-Reittu model: Equip each vertex i ∈ [ n ] = { 1 , . . . , n } with random weight W i , where ( W i ) i ∈ [ n ] are i.i.d. random variables. Attach edge with probability p ij between vertices i and j, where p ij = 1 − e − λW i W j /n . Different edges are conditionally independent given weights, and os-Rényi RG with p = 1 − e − λ/n λ > 0 is parameter. Retrieve Erd˝ when W i ≡ 1 . ⊲ Related models: Chung-Lu model: p ij = ( W i W j /n ) ∧ 1; Generalized random graph: p ij = W i W j / ( n + W i W j ); Janson (2010): Conditions for asymptotic equivalence. Bollobás-Janson-Riordan (2007): General set-up inhomogeneous random graphs.
Long-range percolation Consider model on Z d where we attach edge between x, y ∈ Z d independently with probability p x,y = 1 − e − λ/ | x − y | α . Degree distribution: � D x = I x,y , y ∈ Z d with I x,y independent Bernoulli variables with success prob. p xy . Properties: ⊲ Percolation function continuous when α ∈ ( d, 2 d ) (Berger 02); ⊲ Graph distances polylogarithmic when α ∈ ( d, 2 d ) (Biskup 04); ⊲ Model has high clustering, i.e., many triangles; ⊲ Model never scale-free, i.e., either degrees are infinite a.s., or have thin tails; ⊲ Instantaneous percolation only when degrees are infinite a.s.
Percolation in random environment ⊲ Equip each vertex x ∈ Z d with random weight W x , where ( W x ) x ∈ Z d are i.i.d. random variables. ⊲ Conditionally on weights, edges in graph are independent, and probability that edge between x and y is present equals p xy = 1 − e − λW x W y / | x − y | α . ⊲ Special attention to weights with power-law distribution: P ( W x ≥ w ) = w − ( τ − 1) L ( w ) , where τ > 1 , w �→ L ( w ) is slowly varying. (Often take L ( w ) ≡ c. ) ⊲ Long-range nature determined by parameter α > 0 . ⊲ Percolative properties determined by parameter λ > 0 . ⊲ Inhomogeneity determined by distribution of ( W x ) .
Questions and remarks Model interpolates between ⊲ long-range percolation, obtained when W x ≡ 1; ⊲ inhomogeneous random graphs, more precisely, Poissonian random graph or Norros-Reittu model (06). ⊲ small-world model (Strogatz-Watts) which has torus as vertex set, and rare macroscopic connections. We have connections on all length scales. Investigate: ⊲ Degree structure: How many neighbors do vertices have? ⊲ Percolation: For which λ > 0 is there infinite component? ⊲ Distances: What is graph distance x and y as | x − y | → ∞ ?
Inhomogeneous RG τ = 1 . 95 (Joost Jorritsma)
Long-range percolation d = 2 , α = 3 . 9 , λ = 0 . 1 (Joost Jorritsma)
Scale-free percolation d = 2 , α = 3 . 9 , τ = 1 . 95 , λ = 0 . 1 (Joost Jorritsma)
Scale-free percolation d = 1 , α = 2 , τ = 1 . 95 , λ = 0 . 1 (Joost Jorritsma)
Degrees Special attention to weights with power-law distribution: P ( W x ≥ w ) = w − ( τ − 1) L ( w ) , where τ > 1 , w �→ L ( w ) is slowly varying. (Often take L ( w ) ≡ c. ) Theorem 1 (Infinite degrees). P ( D 0 = ∞ | W 0 > 0) = 1 when either α ≤ d, or α > d for power-law weights with γ = α ( τ − 1) /d < 1 . Theorem 2 (Power-law degrees). For power-law weights, when α > d and γ = α ( τ − 1) /d > 1 , there exists a function s �→ ℓ ( s ) that is slowly varying at infinity s.t. P ( D 0 > s ) = s − γ ℓ ( s ) . Power-law degrees in percolation model: Scale-free percolation.
Degrees: Proof Theorem 1 W.l.o.g. take λ = 1 . First take α > d, so that γ = α ( τ − 1) /d ≤ 1 im- plies τ ∈ (1 , 2) . For power-law weight distributions with τ ∈ (1 , 2) , E [ W y ✶ { W y ≤ s } ] = Θ( s 2 − τ ) . Thus, when γ = α ( τ − 1) /d ≤ 1 , using 1 − e − x ≥ x ✶ [0 , 1] ( x ) / 2 , � 1 − e − wW y / | y | α � � � P ((0 , y ) occupied | W 0 = w )= E y � =0 y � =0 ≥ 1 � � � wW y / | y | α ✶ { W y ≤| y | α /w } E 2 y � =0 1 ≥ Cw − (2 − τ ) � | y | α ( τ − 1) = ∞ . y � =0 By Borel-Cantelli, implies that P ( D 0 = ∞| W 0 = w ) = 1 when w > 0 . Similar (and easier) when α ≤ d.
Degrees: Proof Theorem 2 Crucially use that, for α > d, as a → ∞ , (1 − e − a/ | y | α ) = v d,α a d/α (1 + o (1)) . � y � =0 Thus, when w > 1 is large, and with ξ = v d,α E [ W d/α ] < ∞ , � � e − wW y / | y | α �� � ≈ ξw d/α , E [ D 0 | W 0 = w ] = 1 − E y � =0 Conditionally on W 0 = w, D 0 is sum independent indicators, and thus highly concentrated when mean is large, i.e., P ( D 0 ≥ s ) ≈ P ( W 0 ≥ ( s/ξ ) α/d ) ≈ ℓ ( s ) s − α ( τ − 1) /d = ℓ ( s ) s − γ . γ > 1 : finite-mean degrees; γ > 2 : finite-variance degrees.
Percolation critical value From now on, assume that long-range parameter α > d and power- law exponent γ = α ( τ − 1) /d > 1 . Write x ← → y when there is path of occupied bonds connecting x and y. Let C ( x ) = { y : x ← → y } be cluster of x. ⊲ Percolation probability: θ ( λ ) = P ( |C (0) | = ∞ ) . ⊲ Critical percolation value: λ c = inf { λ : θ ( λ ) > 0 } . Theorem 3 (Finiteness critical value). (a) λ c < ∞ in d ≥ 2 if P ( W = 0) < 1 . (b) λ c < ∞ in d = 1 if α ∈ (1 , 2] , P ( W = 0) < 1 . (c) λ c = ∞ in d = 1 if α > 2 , γ = α ( τ − 1) /d > 2 .
Positivity threshold Theorem 4 (Positivity critical value). λ c > 0 when γ = α ( τ − 1) /d > 2 . Theorem 5 (Zero critical value). λ c = 0 when γ = α ( τ − 1) /d ∈ (1 , 2) , i.e., θ ( λ ) > 0 for every λ > 0 . Robustness of phase transition (Jacob, Mörters) Identical to Norros-Reittu model, novel for percolation models: Norros-Reittu model: G = K n , p ij = 1 − e − λW i W j /n . Giant component exists for every λ > 0 when variance degrees is infinite. NR-model: degrees have same number of moments as weights W.
Proof Theorem 4 We first assume that for E [ W 2 ] < ∞ . When |C (0) | = ∞ , there exists paths of arbitrary length from origin: n � � � � � θ ( λ ) ≤ P (( x i − 1 , x i ) occupied ) = p x i − 1 ,x i , E x 1 ,...,x n x 1 ,...,x n i =1 where sum is over distinct vertices, with x 0 = 0 . Bound p x,y = 1 − e − λW x W y | x − y | − α ≤ λW x W y | x − y | − α : n � W x i − 1 W x i | x i − 1 − x i | − α � θ ( λ ) ≤ λ n � � E x 1 ,...,x n i =1 n | x | − α � n | x i − 1 − x i | − α ≤ � = λ n � E [ W ] 2 E [ W 2 ] n − 1 � λ E [ W 2 ] � . x 1 ,...,x n i =1 x � =0
Proof Theorem 4 When E [ W 2 ] = ∞ , instead use Cauchy-Schwarz and bound p x,y = 1 − e − λW x W y | x − y | − α ≤ � λW x W y | x − y | − α ∧ 1 � : n � � λW x i − 1 W x i | x i − 1 − x i | − α ∧ 1 �� � � θ ( λ ) ≤ E x 1 ,...,x n i =1 � 2 � 1 / 2 � n � � �� λW 0 W 1 | x | − α ∧ 1 ≤ . E x � =0 Key estimate: if P ( W ≥ w ) ≤ cw − ( τ − 1) with τ ∈ (1 , 3) , then � 2 � �� ≤ C (1 + log u ) u − ( τ − 1) . g ( u ) ≡ E W 1 W 2 /u ∧ 1 α ( τ − 1) / 2 > d when γ = α ( τ − 1) /d > 2 , so above sum finite.
Proof Theorem 5 We use renormalization argument for γ ∈ (1 , 2) . Prove θ ( λ ) > 0 for any λ > 0 small. Take r λ large. By extreme value theory, W x = Θ P ( r d/ ( τ − 1) max ) . λ | x | <r λ For x ∈ Z d , let x ( λ ) be maximal weight vertex in { y : | y − r λ x | ≤ r λ } . Say ( x, y ) occupied when ( x ( λ ) , y ( λ )) occupied. For nearest-neighbor x, y, and with high probability, ≈ 1 − e − λr 2 d/ ( τ − 1) − α P (( x, y ) occ. | ( W x ) x ∈ Z d ) ≈ 1 − e − λW x ( λ ) W y ( λ ) r − α . λ λ Note 2 d/ ( τ − 1) − α > 0 precisely when γ = α ( τ − 1) /d < 2 . Take r λ so large that λr 2 d/ ( τ − 1) − α ≫ 1 . Then nearest-neighbor per- λ colation model supercritical for small λ > 0 . Implies that θ ( λ ) > 0 .
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