Exposing a component [ B REATH -F IRST -S EARCH : K ARP 90 ] v For a given vertex v we want to determine the order of the component C ( v ) that contains v . • First we expose the neighbours ( „children”) of v • Then we expose the neighbours of each neighbour of v • We continue this procedure, until there are no more vertices contained in C ( v ) .
Exposing a component [ B REATH -F IRST -S EARCH : K ARP 90 ] v For a given vertex v we want to determine the order of the component C ( v ) that contains v . • First we expose the neighbours ( „children”) of v • Then we expose the neighbours of each neighbour of v • We continue this procedure, until there are no more vertices contained in C ( v ) .
Exposing a component [ B REATH -F IRST -S EARCH : K ARP 90 ] v For a given vertex v we want to determine the order of the component C ( v ) that contains v . • First we expose the neighbours ( „children”) of v • Then we expose the neighbours of each neighbour of v • We continue this procedure, until there are no more vertices contained in C ( v ) .
Exposing a component [ B REATH -F IRST -S EARCH : K ARP 90 ] v For a given vertex v we want to determine the order of the component C ( v ) that contains v . • First we expose the neighbours ( „children”) of v • Then we expose the neighbours of each neighbour of v • We continue this procedure, until there are no more vertices contained in C ( v ) .
Exposing a component [ B REATH -F IRST -S EARCH : K ARP 90 ] k When k ≪ n vertices are exposed, • the number of new neighbours (“children”) of a vertex: Bi( n − k, p ) • the expected number of children: ( n − k ) p = ( n − k ) c n − 1 ∼ c.
Exposing a component [ B REATH -F IRST -S EARCH : K ARP 90 ] k Bi( n − k, p ) When k ≪ n vertices are exposed, • the number of new neighbours (“children”) of a vertex: Bi( n − k, p ) • the expected number of children: ( n − k ) p = ( n − k ) c n − 1 ∼ c.
Exposing a component [ B REATH -F IRST -S EARCH : K ARP 90 ] k Bi( n − k, p ) ∼ Po( c ) When k ≪ n vertices are exposed, • the number of new neighbours (“children”) of a vertex: Bi( n − k, p ) • the expected number of children: ( n − k ) p = ( n − k ) c n − 1 ∼ c. � n − k � p i (1 − p ) n − k − i n →∞ P (Bi( n − k, p ) = i ) lim = lim i n →∞ = c i i ! e − c = P (Po( c ) = i )
Branching process • It starts with a unisexual individual • The number of children: i.i.d. random variable Y ∼ Po( c ) . • If c < 1 , the process dies out with probability 1. • If c > 1 , with positive probability the process continues forever. It corresponds to a „small” component in G ( n, p )
Branching process • It starts with a unisexual individual • The number of children: i.i.d. random variable Y ∼ Po( c ) . • If c < 1 , the process dies out with probability 1. • If c > 1 , with positive probability the process continues forever. Po( c ) Po( c ) It corresponds to a „small” component in G ( n, p )
Branching process • It starts with a unisexual individual • The number of children: i.i.d. random variable Y ∼ Po( c ) . • If c < 1 , the process dies out with probability 1. • If c > 1 , with positive probability the process continues forever. It corresponds to a „small” component in G ( n, p )
Branching process • It starts with a unisexual individual • The number of children: i.i.d. random variable Y ∼ Po( c ) . • If c < 1 , the process dies out with probability 1. • If c > 1 , with positive probability the process continues forever. It corresponds to the „giant” component of order Θ( n ) in G ( n, p )
Branching process • It starts with a unisexual individual • The number of children: i.i.d. random variable Y ∼ Po( c ) . • If c < 1 , the process dies out with probability 1. • If c > 1 , with positive probability the process continues forever. Survival probability ρ : 1 − ρ = e − cρ It corresponds to the „giant” component of order Θ( n ) in G ( n, p )
Branching process • It starts with a unisexual individual • The number of children: i.i.d. random variable Y ∼ Po( c ) . • If c < 1 , the process dies out with probability 1. • If c > 1 , with positive probability the process continues forever. Let T be the total number of organisms. The prob. generating function � i< ∞ P [ T = i ] z i q ( z ) := k P [Po( c ) = k ] q ( z ) k = z � k ! q ( z ) k = ze c ( q ( z ) − 1) . k e − c c k satisfies q ( z ) = z �
Branching process • It starts with a unisexual individual • The number of children: i.i.d. random variable Y ∼ Po( c ) . • If c < 1 , the process dies out with probability 1. • If c > 1 , with positive probability the process continues forever. Let T be the total number of organisms. The prob. generating function � i< ∞ P [ T = i ] z i q ( z ) := k P [Po( c ) = k ] q ( z ) k = z � k ! q ( z ) k = ze c ( q ( z ) − 1) . k e − c c k satisfies q ( z ) = z � Po( c )
Branching process • It starts with a unisexual individual • The number of children: i.i.d. random variable Y ∼ Po( c ) . • If c < 1 , the process dies out with probability 1. • If c > 1 , with positive probability the process continues forever. Let T be the total number of organisms. The prob. generating function � i< ∞ P [ T = i ] z i q ( z ) := k P [Po( c ) = k ] q ( z ) k = z � k ! q ( z ) k = ze c ( q ( z ) − 1) . k e − c c k satisfies q ( z ) = z � The extinction probability 1 − ρ := � i< ∞ P [ T = i ] = q (1) satisfies 1 − ρ = q (1) = e c ( q (1) − 1) = e c (1 − ρ − 1) = e − cρ .
Giant component Let N p be the order of the giant component after the phase transition. Then E ( N p ) = ρn and N p = ρn + o ( n ) whp where 1 − ρ = e − cρ , ρ � = 0 .
Giant component Let N p be the order of the giant component after the phase transition. Then E ( N p ) = ρn and N p = ρn + o ( n ) whp where 1 − ρ = e − cρ , ρ � = 0 .
Giant component Let N p be the order of the giant component after the phase transition. Then E ( N p ) = ρn and N p = ρn + o ( n ) whp where 1 − ρ = e − cρ , ρ � = 0 . Central Limit Theorem [ P ITTEL 90; B ARREZ –B OUCHERON – DE LA V EGA 00 ] The variance N p satisfies σ 2 := Var( N p ) = ρ − ρ 2 (1 − c (1 − ρ )) 2 n . For any fixed numbers a < b � b − x 2 � � 1 P [ ρn + a ≤ N p ≤ ρn + b ] ∼ √ exp dx. 2 σ 2 σ 2 π a
Giant component Let N p be the order of the giant component after the phase transition. Then E ( N p ) = ρn and N p = ρn + o ( n ) whp where 1 − ρ = e − cρ , ρ � = 0 . Central Limit Theorem [ P ITTEL 90; B ARREZ –B OUCHERON – DE LA V EGA 00 ] The variance N p satisfies σ 2 := Var( N p ) = ρ − ρ 2 (1 − c (1 − ρ )) 2 n . For any fixed numbers a < b � b − x 2 � � 1 P [ ρn + a ≤ N p ≤ ρn + b ] ∼ √ exp dx. 2 σ 2 σ 2 π a Locally?? Gaussian distribution ρn + a ρn ρn + b
Giant component Let N p be the order of the giant component after the phase transition. Then E ( N p ) = ρn and N p = ρn + o ( n ) whp where 1 − ρ = e − cρ , ρ � = 0 . Central Limit Theorem [ P ITTEL 90; B ARREZ –B OUCHERON – DE LA V EGA 00 ] The variance N p satisfies σ 2 := Var( N p ) = ρ − ρ 2 (1 − c (1 − ρ )) 2 n . For any fixed numbers a < b � b − x 2 � � 1 P [ ρn + a ≤ N p ≤ ρn + b ] ∼ √ exp dx. 2 σ 2 σ 2 π a Locally?? Gaussian distribution c √ n c √ n ⌊ ρn − c √ n ⌋ ⌊ ρn + c √ n ⌋ ⌊ ρn ⌋
Giant component Let N p be the order of the giant component after the phase transition. Then E ( N p ) = ρn and N p = ρn + o ( n ) whp where 1 − ρ = e − cρ , ρ � = 0 . Central Limit Theorem [ P ITTEL 90; B ARREZ –B OUCHERON – DE LA V EGA 00 ] The variance N p satisfies σ 2 := Var( N p ) = ρ − ρ 2 (1 − c (1 − ρ )) 2 n . For any fixed numbers a < b � b − x 2 � � 1 P [ ρn + a ≤ N p ≤ ρn + b ] ∼ √ exp dx. 2 σ 2 σ 2 π a Local Limit Theorem [ B EHRISCH –C OJA -O GHLAN –K. 07+ ] For any integer k with k = ρn + x and x = O ( √ n ) = O ( σ ) , − x 2 � � 1 √ P [ N p = k ] ∼ 2 π exp . 2 σ 2 σ
Joint distribution • Let M p denote # edges in the giant component in G ( n, p ) . y 2 x 2 σ 2 − 2 σ N M xy + 1 σ 2 σ 2 σ 2 P [ N p = k ∧ M p = l ] ∼ − M M exp � 1 − σ 2 � � 1 − σ 2 2 N M 2 πσσ M N M σ 2 σ 2 σ 2 σ 2 M M
Joint distribution • Let M p denote # edges in the giant component in G ( n, p ) . y 2 x 2 σ 2 − 2 σ N M xy + 1 σ 2 σ 2 σ 2 P [ N p = k ∧ M p = l ] ∼ − M M exp � 1 − σ 2 � � 1 − σ 2 2 N M 2 πσσ M N M σ 2 σ 2 σ 2 σ 2 M M • # C ( k, l ) of connected graphs with k vertices and l edges satisfies � − 1 � n n − k p − l (1 − p ) − ( n 2 ) + ( 2 ) + l C ( k, l ) ∼ P [ N p = k ∧ M p = l ] k
Joint distribution • Let M p denote # edges in the giant component in G ( n, p ) . y 2 x 2 σ 2 − 2 σ N M xy + 1 σ 2 σ 2 σ 2 P [ N p = k ∧ M p = l ] ∼ − M M exp � 1 − σ 2 � � 1 − σ 2 2 N M 2 πσσ M N M σ 2 σ 2 σ 2 σ 2 M M • # C ( k, l ) of connected graphs with k vertices and l edges satisfies � − 1 � n n − k p − l (1 − p ) − ( n 2 ) + ( 2 ) + l C ( k, l ) ∼ P [ N p = k ∧ M p = l ] k ⇒ its asymptotic formula via probabilistic analysis [ B EHRISCH –C OJA -O GHLAN –K. 07+ ] =
Joint distribution • Let M p denote # edges in the giant component in G ( n, p ) . y 2 x 2 σ 2 − 2 σ N M xy + 1 σ 2 σ 2 σ 2 P [ N p = k ∧ M p = l ] ∼ − M M exp � 1 − σ 2 � � 1 − σ 2 2 N M 2 πσσ M N M σ 2 σ 2 σ 2 σ 2 M M • # C ( k, l ) of connected graphs with k vertices and l edges satisfies � − 1 � n n − k p − l (1 − p ) − ( n 2 ) + ( 2 ) + l C ( k, l ) ∼ P [ N p = k ∧ M p = l ] k ⇒ its asymptotic formula via probabilistic analysis [ B EHRISCH –C OJA -O GHLAN –K. 07+ ] = Cf. asymptotic formula for C ( k, l ) via – – enumerative method [ B ENDER –C ANFIELD –M C K AY 90 ] – – saddle-point method [ F LAJOLET –S ALVY –S CHAEFFER 04 ]
The critical phase What about the order of the largest component of G ( n, p ) with p = c n n − 1 when the expected degree c n → 1 , in the so-called critical phase?
The critical phase What about the order of the largest component of G ( n, p ) with p = c n n − 1 when the expected degree c n → 1 , in the so-called critical phase? [ B OLLOBÁS 84; Ł UCZAK 90; Ł UCZAK –P ITTEL –W IERMAN 94 ] Let c n = 1 + λ n n − 1 / 3 log n where λ n n − 1 / 3 log n → 0 as n → ∞ . • If λ n → −∞ , whp all components have ≪ n 2 / 3 vertices. • If λ n → λ , whp the largest component has Θ( n 2 / 3 ) vertices. • If λ n → + ∞ , whp there is exactly one component with ≫ n 2 / 3 ver- tices, while all other components have ≪ n 2 / 3 vertices.
The critical phase What about the order of the largest component of G ( n, p ) with p = c n n − 1 when the expected degree c n → 1 , in the so-called critical phase? [ B OLLOBÁS 84; Ł UCZAK 90; Ł UCZAK –P ITTEL –W IERMAN 94 ] Let c n = 1 + λ n n − 1 / 3 log n where λ n n − 1 / 3 log n → 0 as n → ∞ . � � � � � � � � • If λ n → −∞ , whp all components have ≪ n 2 / 3 vertices. • If λ n → λ , whp the largest component has Θ( n 2 / 3 ) vertices. • If λ n → + ∞ , whp there is exactly one component with ≫ n 2 / 3 ver- tices, while all other components have ≪ n 2 / 3 vertices. c n = 1 + λ n n − 1 / 3 Scaling window of Mean-Field width of n − 1 / 3 (Percolation Theory)
The critical phase What about the order of the largest component of G ( n, p ) with p = c n n − 1 when the expected degree c n → 1 , in the so-called critical phase? [ B OLLOBÁS 84; Ł UCZAK 90; Ł UCZAK –P ITTEL –W IERMAN 94 ] Let c n = 1 + λ n n − 1 / 3 log n where λ n n − 1 / 3 log n → 0 as n → ∞ . � � � � � � � � • If λ n → −∞ , whp all components have ≪ n 2 / 3 vertices. • If λ n → λ , whp the largest component has Θ( n 2 / 3 ) vertices. • If λ n → + ∞ , whp there is exactly one component with ≫ n 2 / 3 ver- tices, while all other components have ≪ n 2 / 3 vertices. c n = 1 + λ n n − 1 / 3 Scaling window of Mean-Field width of n − 1 / 3 (Percolation Theory)
Outline I. Phase transition • Introduction to phase transition • Erd˝ os–Rényi random graph – Phase transition – Limit theorems for the giant component – Critical phase • Random graphs with given degree sequence II. Enumeration and random sampling • Recursive decomposition • Singularity analysis, Boltzmann sampler, probabilistic analysis • Planar structures, minors and genus
Degree distribution G ( n, p ) as a stochastic model for large complex systems?
Degree distribution G ( n, p ) as a stochastic model for large complex systems? • In G ( n, p ) , degree of each vertex ∼ ( n − 1) p : homogeneous
Degree distribution G ( n, p ) as a stochastic model for large complex systems? • In G ( n, p ) , degree of each vertex ∼ ( n − 1) p : homogeneous • In some complex systems/networks, e.g. www, epidemic networks, • some vertices are of high degree, while most vertices are of low • degree: non-homogeneous
Random graph models Random graph processes • To model and analyse dynamic nature of complex systems/networks arising from the real world • Random „internet” graph [ B OLLOBÁS –R IORDAN ; C OOPER –F RIEZE 03 ] • Degree constraints [ W ORMALD ; K.–S EIERSTAD 07; C OJA -O GHLAN –K. 08+ ]
Random graph models Random graph processes • To model and analyse dynamic nature of complex systems/networks arising from the real world • Random „internet” graph [ B OLLOBÁS –R IORDAN ; C OOPER –F RIEZE 03 ] • Degree constraints [ W ORMALD ; K.–S EIERSTAD 07; C OJA -O GHLAN –K. 08+ ] Inhomogenous random graphs [ B OLLOBÁS –J ANSON –R IORDAN 07 ] • Vertices come in different types
Random graph models Random graph processes • To model and analyse dynamic nature of complex systems/networks arising from the real world • Random „internet” graph [ B OLLOBÁS –R IORDAN ; C OOPER –F RIEZE 03 ] • Degree constraints [ W ORMALD ; K.–S EIERSTAD 07; C OJA -O GHLAN –K. 08+ ] Inhomogenous random graphs [ B OLLOBÁS –J ANSON –R IORDAN 07 ] • Vertices come in different types Random graphs with given degree sequence [ M OLLOY –R EED 95, 98; J ANSON –M. L UCZACK 07+; K.–S EIERSTAD 08 ]
Asymptotic degree sequence [ M OLLOY –R EED 95, 98 ] Let G n ( d 0 ( n ) , d 1 ( n ) , . . . ) be a uniform random graph on n vertices, d i ( n ) of which are of degree i .
Asymptotic degree sequence [ M OLLOY –R EED 95, 98 ] Let G n ( d 0 ( n ) , d 1 ( n ) , . . . ) be a uniform random graph on n vertices, d i ( n ) of which are of degree i . The asymptotic degree sequence D = { d 0 ( n ) , d 1 ( n ) , . . . } satisfies: • � i ≥ 0 d i ( n ) = n and d i ( n ) = 0 for i ≥ n • δ i ( n ) = d i ( n ) i as n → ∞ → δ ∗ n 4 − ε for some ε > 0 • "well behaves" and d i ( n ) = 0 whenever i > n 1
Asymptotic degree sequence [ M OLLOY –R EED 95, 98 ] Let G n ( d 0 ( n ) , d 1 ( n ) , . . . ) be a uniform random graph on n vertices, d i ( n ) of which are of degree i . The asymptotic degree sequence D = { d 0 ( n ) , d 1 ( n ) , . . . } satisfies: • � i ≥ 0 d i ( n ) = n and d i ( n ) = 0 for i ≥ n • δ i ( n ) = d i ( n ) i as n → ∞ → δ ∗ n 4 − ε for some ε > 0 • "well behaves" and d i ( n ) = 0 whenever i > n 1 The phase transition in G n ( D ) occurs when � Q ( D ) := ( i − 2) iδ i ( n ) = 0 . i (We will come back to this later)
Phase transition [ M OLLOY –R EED 95, 98 ] If Q ( D ) < 0 , whp all components have O (log n ) vertices. If Q ( D ) > 0 , whp there is a unique component of order Θ( n ) , while all other components have O (log n ) vertices. To study the critical phase Q ( D ) = � i ( i − 2) iδ i ( n ) → 0 let τ n be s.t. � n = 0 and τ n → 1 i ( i − 2) iδ i ( n ) τ i
Phase transition [ M OLLOY –R EED 95, 98 ] If Q ( D ) < 0 , whp all components have O (log n ) vertices. If Q ( D ) > 0 , whp there is a unique component of order Θ( n ) , while all other components have O (log n ) vertices. To study the critical phase Q ( D ) = � i ( i − 2) iδ i ( n ) → 0 let τ n be s.t. � n = 0 and τ n → 1 i ( i − 2) iδ i ( n ) τ i
Phase transition [ M OLLOY –R EED 95, 98 ] If Q ( D ) < 0 , whp all components have O (log n ) vertices. If Q ( D ) > 0 , whp there is a unique component of order Θ( n ) , while all other components have O (log n ) vertices. To study the critical phase Q ( D ) = � i ( i − 2) iδ i ( n ) → 0 let τ n be s.t. � n = 0 and τ n → 1 . i ( i − 2) iδ i ( n ) τ i
Phase transition [ M OLLOY –R EED 95, 98 ] If Q ( D ) < 0 , whp all components have O (log n ) vertices. If Q ( D ) > 0 , whp there is a unique component of order Θ( n ) , while all other components have O (log n ) vertices. To study the critical phase Q ( D ) = � i ( i − 2) iδ i ( n ) → 0 let τ n be s.t. � n = 0 and τ n → 1 . i ( i − 2) iδ i ( n ) τ i [ K.–S EIERSTAD 08; J ANSON –M. L UCZAK 07+ ] Let λ n = (1 − τ n ) n 1 / 3 . • If λ n → −∞ , whp all the components have ≪ n 2 / 3 vertices. • If λ n → + ∞ and λ n ≥ c log n , whp there is a unique component of order ≫ n 2 / 3 , while all other components have ≪ n 2 / 3 vertices. 1 − τ n plays the same role as λ n n − 1 / 3 for G ( n, p ) with p = 1+ λ n n − 1 / 3 . n − 1
Phase transition [ M OLLOY –R EED 95, 98 ] If Q ( D ) < 0 , whp all components have O (log n ) vertices. If Q ( D ) > 0 , whp there is a unique component of order Θ( n ) , while all other components have O (log n ) vertices. To study the critical phase Q ( D ) = � i ( i − 2) iδ i ( n ) → 0 let τ n be s.t. � n = 0 and τ n → 1 . i ( i − 2) iδ i ( n ) τ i [ K.–S EIERSTAD 08; J ANSON –M. L UCZAK 07+ ] Let λ n = (1 − τ n ) n 1 / 3 . • If λ n → −∞ , whp all the components have ≪ n 2 / 3 vertices. • If λ n → + ∞ and λ n ≥ c log n , whp there is a unique component of order ≫ n 2 / 3 , while all other components have ≪ n 2 / 3 vertices. 1 − τ n plays the same role as λ n n − 1 / 3 for G ( n, p ) with p = 1+ λ n n − 1 / 3 . n − 1
Phase transition [ M OLLOY –R EED 95, 98 ] If Q ( D ) < 0 , whp all components have O (log n ) vertices. If Q ( D ) > 0 , whp there is a unique component of order Θ( n ) , while all other components have O (log n ) vertices. To study the critical phase Q ( D ) = � i ( i − 2) iδ i ( n ) → 0 let τ n be s.t. � n = 0 and τ n → 1 . i ( i − 2) iδ i ( n ) τ i [ K.–S EIERSTAD 08; J ANSON –M. L UCZAK 07+ ] Let λ n = (1 − τ n ) n 1 / 3 . • If λ n → −∞ , whp all the components have ≪ n 2 / 3 vertices. • If λ n → + ∞ and λ n ≥ c log n , whp there is a unique component of � � � � order ≫ n 2 / 3 , while all other components have ≪ n 2 / 3 vertices. 1 − τ n plays the same role as λ n n − 1 / 3 for G ( n, p ) with p = 1+ λ n n − 1 / 3 . n − 1
Random configuration Why does the phase transition in G n ( D ) occur when � i ( i − 2) iδ i ( n ) = 0 ?
Random configuration Why does the phase transition in G n ( D ) occur when � i ( i − 2) iδ i ( n ) = 0 ? R ANDOM CONFIGURATION [ B ENDER –C ANFIELD ; B OLLOBÁS ; W ORMALD ] Given a degree sequence D n = { a 1 , . . . , a n } of V = { v 1 , . . . , v n } s.t. for a i = deg( v i ) 1 ≤ i ≤ n, • L n = { a i distinct copies of v i , called half-edges, for 1 ≤ i ≤ n } • M n = perfect matching of L n , chosen uniformly at random. Then a random configuration C n = L n + M n . v n v n − 1 v 1 v 2 v 3 . . .
Random configuration Why does the phase transition in G n ( D ) occur when � i ( i − 2) iδ i ( n ) = 0 ? R ANDOM CONFIGURATION [ B ENDER –C ANFIELD ; B OLLOBÁS ; W ORMALD ] Given a degree sequence D n = { a 1 , . . . , a n } of V = { v 1 , . . . , v n } s.t. for a i = deg( v i ) 1 ≤ i ≤ n, • L n = { a i distinct copies of v i , called half-edges, for 1 ≤ i ≤ n } • M n = a perfect matching of L n , chosen uniformly at random. Then a random configuration C n = L n + M n . v n v n − 1 v 1 v 2 v 3 . . .
Random configuration Why does the phase transition in G n ( D ) occur when � i ( i − 2) iδ i ( n ) = 0 ? R ANDOM CONFIGURATION [ B ENDER –C ANFIELD ; B OLLOBÁS ; W ORMALD ] Given a degree sequence D n = { a 1 , . . . , a n } of V = { v 1 , . . . , v n } s.t. for a i = deg( v i ) 1 ≤ i ≤ n, • L n = { a i distinct copies of v i , called half-edges, for 1 ≤ i ≤ n } • M n = a perfect matching of L n , chosen uniformly at random. Then a random configuration C n = L n + M n . v n v n − 1 v 1 v 2 v 3 . . .
Random configuration Given a configuration C n , let G ∗ n be the multigraph obtained by • identifying all a i copies of v i for every i = 1 , . . . , n , and • letting the pairs of the perfect matching in C n become edges. v n v n − 1 v 1 v 2 v 3 . . . v n v n − 1 v 1 v 2 v 3 . . .
Random configuration Given a configuration C n , let G ∗ n be the multigraph obtained by • identifying all a i copies of v i for every i = 1 , . . . , n , and • letting the pairs of the perfect matching in C n become edges. v n v n − 1 v 1 v 2 v 3 . . . v n v n − 1 v 1 v 2 v 3 . . .
Branching process e · · · · · · v
Branching process e · · · · · · v Let X be the number of children of the vertex v to which the edge e belongs. Its distribution is given by = |{ k : deg( v k ) = i }| iδ i ( n ) δ i ( n ) = d i ( n ) P [ X = i − 1] = i iδ i ( n ) , . � n n = P [ v is one of the d i vertices of degree i ] = P [ v is one of the d i vertices of degree i ]
Branching process e · · · · · · v Let X be the number of children of the vertex v to which the edge e belongs. Its distribution is given by = |{ k : deg( v k ) = i }| iδ i ( n ) δ i ( n ) = d i ( n ) P [ X = i − 1] = i iδ i ( n ) , . � n n = P [ v is one of the d i vertices of degree i ] = P [ v is one of the d i vertices of degree i ]
Branching process e · · · · · · v Let X be the number of children of the vertex v to which the edge e belongs. Its distribution is given by = |{ k : deg( v k ) = i }| iδ i ( n ) δ i ( n ) = d i ( n ) P [ X = i − 1] = i iδ i ( n ) , . � n n = P [ v is one of the d i vertices of degree i ] = P [ v is one of the d i vertices of degree i ] = P [ e is matched to one of the i clones of a vertex of degree i ]
Branching process e · · · · · · v Let X be the number of children of the vertex v to which the edge e belongs. Its distribution is given by = |{ k : deg( v k ) = i }| iδ i ( n ) δ i ( n ) = d i ( n ) P [ X = i − 1] = i iδ i ( n ) , . � n n = P [ v is one of the d i vertices of degree i ] = P [ v is one of the d i vertices of degree i ] = P [ e is matched to one of the i clones of a vertex of degree i ]
Branching process e · · · · · · v Let X be the number of children of the vertex v to which the edge e belongs. Its distribution is given by = |{ k : deg( v k ) = i }| iδ i ( n ) δ i ( n ) = d i ( n ) P [ X = i − 1] = i iδ i ( n ) , . � n n iδ i ( n ) � � E [ X ] = ( i − 1) P [ X = i − 1] = ( i − 1) i iδ i ( n ) . � i i
Branching process e · · · · · · v Let X be the number of children of the vertex v to which the edge e belongs. Its distribution is given by = |{ k : deg( v k ) = i }| iδ i ( n ) δ i ( n ) = d i ( n ) P [ X = i − 1] = i iδ i ( n ) , . � n n iδ i ( n ) � � E [ X ] = ( i − 1) P [ X = i − 1] = ( i − 1) i iδ i ( n ) . � i i The critical point of the branching process is when E [ X ] = 1 , that is, � Q ( D ) := i ( i − 2) iδ i ( n ) = 0 .
Outline I. Phase transition • Introduction to phase transition • Erd˝ os–Rényi random graph – Phase transition – Limit theorems for the giant component – Critical phase • Random graphs with given degree sequence II. Enumeration and random sampling • Recursive decomposition • Singularity analysis, Boltzmann sampler, probabilistic analysis • Planar structures, minors and genus
Planar graphs Graphs that can be embedded in the plane without crossing edges
Planar graphs Graphs that can be embedded in the plane without crossing edges [ D ENISE –V ASCONCELLOS –W ELSH 96 ] • How many labeled planar graphs are there on n vertices? • How can we sample a random planar graph uniformly at random?
Planar graphs Graphs that can be embedded in the plane without crossing edges [ D ENISE –V ASCONCELLOS –W ELSH 96 ] • How many labeled planar graphs are there on n vertices? • How can we sample a random planar graph uniformly at random? Markov chain (addition-deletion) – upper bound on # planar graphs [ D ENISE –V ASCONCELLOS –W ELSH 96 ] – typical properties [ M C D IARMID –S TEGER –W ELSH 05 ] – unknown mixing time
Planar graphs Graphs that can be embedded in the plane without crossing edges [ D ENISE –V ASCONCELLOS –W ELSH 96 ] • How many labeled planar graphs are there on n vertices? • How can we sample a random planar graph uniformly at random? Markov chain (addition-deletion) – upper bound on # planar graphs [ D ENISE –V ASCONCELLOS –W ELSH 96 ] – typical properties [ M C D IARMID –S TEGER –W ELSH 05 ] – unknown mixing time
Planar graphs Graphs that can be embedded in the plane without crossing edges [ D ENISE –V ASCONCELLOS –W ELSH 96 ] • How many labeled planar graphs are there on n vertices? • How can we sample a random planar graph uniformly at random? Markov chain (addition-deletion) – upper bound on # planar graphs [ D ENISE –V ASCONCELLOS –W ELSH 96 ] – typical properties [ M C D IARMID –S TEGER –W ELSH 05 ] – unknown mixing time ∃ alternative methods?
An alternative method? “A nonstandard method of counting trees: Put a cat into each tree, walk your dog, and count how often he barks.” [ Proofs from THE BOOK , M. A IGNER AND G. Z IEGLER ]
Recursive decomposition [ N IJENHUIS –W ILF 79; F LAJOLET –Z IMMERMAN –V AN C UTSEM 94; F LAJOLET –S EDGEWICK 08+ ]
Recursive decomposition [ N IJENHUIS –W ILF 79; F LAJOLET –Z IMMERMAN –V AN C UTSEM 94; F LAJOLET –S EDGEWICK 08+ ] Decomposition Recursive Counting Formulas
Recursive decomposition [ N IJENHUIS –W ILF 79; F LAJOLET –Z IMMERMAN –V AN C UTSEM 94; F LAJOLET –S EDGEWICK 08+ ] Decomposition Recursive Counting Formulas Recursive Method Uniform Generation
Recursive decomposition [ N IJENHUIS –W ILF 79; F LAJOLET –Z IMMERMAN –V AN C UTSEM 94; F LAJOLET –S EDGEWICK 08+ ] Decomposition Recursive Counting Formulas Equations of Generating Functions Recursive Method Uniform Generation
Recursive decomposition [ N IJENHUIS –W ILF 79; F LAJOLET –Z IMMERMAN –V AN C UTSEM 94; F LAJOLET –S EDGEWICK 08+ ] Decomposition Recursive Counting Formulas Equations of Generating Functions Singularity Analysis Recursive Method Uniform Generation Asymptotic Number
Recursive decomposition [ N IJENHUIS –W ILF 79; F LAJOLET –Z IMMERMAN –V AN C UTSEM 94; F LAJOLET –S EDGEWICK 08+ ] Decomposition Recursive Counting Formulas Equations of Generating Functions Singularity Analysis Recursive Method Uniform Generation Asymptotic Number Probabilistic Analysis Bolzmann Sampler Typical Properties
Recursive decomposition [ N IJENHUIS –W ILF 79; F LAJOLET –Z IMMERMAN –V AN C UTSEM 94; F LAJOLET –S EDGEWICK 08+ ] Decomposition Recursive Counting Formulas Equations of Generating Functions Singularity Analysis Recursive Method Uniform Generation Asymptotic Number Probabilistic Analysis Bolzmann Sampler Typical Properties
Recursive decomposition [ N IJENHUIS –W ILF 79; F LAJOLET –Z IMMERMAN –V AN C UTSEM 94; F LAJOLET –S EDGEWICK 08+ ] Decomposition Recursive Counting Formulas Equations of Generating Functions Singularity Analysis Recursive Method Uniform Generation Asymptotic Number Probabilistic Analysis Bolzmann Sampler Typical Properties
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