Mathematical problems of very large networks Lszl Lovsz Etvs Lornd - PowerPoint PPT Presentation

Mathematical problems of very large networks László Lovász Eötvös Loránd University, Budapest lovasz@cs.elte.hu December 2008 1

Issues on very large graphs The following issues are closely related: • property testing; • parameter estimation; • limit objects for convergent graph sequences; • regularity lemmas; • distance of graphs; • duality of left and right convergence. December 2008 2

Issues on very large graphs The following concepts are cryptomorphic: • a consistent local finite random graph model; • a consistent local countable random graph; • a measurable, symmetric function W : [0,1] 2 → [0,1]; • a multiplicative graph parameter with nonnegative Möbius transform; • a multiplicative, reflection positive graph parameter; • A point in the completion of the set of finite graphs with the cut-distance. December 2008 3

Cut distance of two graphs = (a) ( ) ( ') V G V G − | ( , ) ( , ) | e S T e S T = ' G G ( , ') max d G G � 2 ⊆ n , ( ) S T V G = = (b) | ( ) | | ( ') | V G V G n δ = * ( , ') min ( , ') G G d G G � � ↔ ' G G December 2008 4

Cut distance of two graphs = ≠ = (c) | ( ) | ' | ( ') | V G n n V G blow up nodes, or fractional overlay 1 1 ∑ ∑ ≥ = = ( ) 0 , X X X ∈ ∈ ( ), ( ') ij i V G j V G ij ij ' n n ∈ ∈ ( ) ( ') i V G j V G δ = ( , ') G G � ∑ ∑ = − min ma x ( ' ) X X a a iu jv ij uv ⊆ × , ( ) ( ' ) ( , ) X S T V G V G ∈ ∈ ( , ) i u S j v T December 2008 5

Cut distance of two graphs 1 δ ≈ G Examples: 1 ( , (2 , )) K n � , n n 2 8 = δ � G G ( ) ( o 1 1 ( , ), ( , ) 1) n n 1 2 2 2 1/2 = = δ δ G G ( ) ( ) ( o 1 1 ( , ), ( , ), 1/ 2 1) n n � � 1 1 2 2 December 2008 6

Sampling Lemmas = ': graphs with , ( ) ( ') G G V G V G ⊆ random set of nodes ( ) : S V G k k With large probability, 10 − < ( [ ], '[ ]) ( , ') d G S G S d G G � � k k 1/4 k Alon-Fernandez de la Vega-Kannan-Karpinski+ With large probability, 10 δ < ( , [ ]) G G S � k log k Borgs-Chayes-Lovász-Sós-Vesztergombi December 2008 7

Regularity Lemmas Original Regularity Lemma Szemerédi 1976 “Weak” Regularity Lemma Frieze-Kannan 1999 “Strong” Regularity Lemma Alon – Fischer - Krivelevich - M. Szegedy 2000 December 2008 8

Regularity Lemmas P = partition of { ,..., } ( ) : V V V G 1 k is the complete graph on with edgeweights ( ) G V G P ( , ) ( e V V = ∈ ∈ G i j , ) w u V v V uv i j ⋅ V V i j December 2008 9

Regularity Lemmas “Weak" Regularity Lemma (Frieze-Kannan): ≥ For every graph and 1 G k P there is a partition of with classes such that ( ) V G k 1 δ ≤ ( , ) . G G P � log k ≥ For every graph and 1 G k there is a graph with nodes such that H k 2 δ ≤ ( , ) . G H � log k December 2008 10

“Weak" Regularity Lemma: geometric form = − E E E ( , ) : ( ) ( ) d s t a a a a v u su vu w tw wv 2 v s t u w Fact 1. This is a metric, computable by sampling Fact 2. Weak Szemerédi partition ↔ partition most nodes into sets with small diameter December 2008 11

“Weak" Regularity Lemma: geometric form ⊆ = average ε -net E x [0,1]: ( ) ( , ) S d S d x S P P = partition of [0,1]: ( ) ( , ) r d G G P � regular partition ∀ S ⊆ [0,1] ⇒ Voronoi cells of S form a partition with P < ( ) 8 ( ) r d S ∀ partition P ={ V 1 ,..., V k } of [0,1] ∃ v i ∈ V i with P < ({ ,..., }) 12 ( ) d v v r 1 k LL – B. Szegedy December 2008 12

“Weak” Regularity Lemma: algorithm Algorithm to construct representatives of classes: - Begin with U = ∅ . - Select random nodes v 1 , v 2 , ... - Put v i in U iff d 2 ( v i , u )> ε for all u ∈ U . - Stop if for more than 1/ ε 2 trials, U did not grow. size bounded by O (min # classes) December 2008 13

“Weak” Regularity Lemma: algorithm Algorithm to decide in which class v belongs: Let U ={ u 1 ,..., u k }. Put a node v in V i iff u i is the nearest node to v in U. December 2008 14

Max Cut in huge graphs (Different algorithm implicit by Frieze-Kannan.) Algorithm to construct representation of cut: - Construct U as for the weak Szemerédi partition - Compute p ij = density between classes V i and V j (use sampling) - Compute max cut ( U 1 , U 2 ) in complete graph on U with edge-weights p ij December 2008 15

Max Cut in huge graphs Algorithm to decide in which class does v belong: - Put v ∈ V into V 1 if d 2 ( v , U 1 ) ≤ d 2 ( v , U 2 ) V 2 if d 2 ( v , U 1 ) > d 2 ( v , U 2 ) December 2008 16

Convergent graph sequences = ): # of homomorphisms of into hom( , G H G H hom( , ) F G Probability that random map t F G = ( , ) V ( F ) → V ( G ) is a hom | ( )| V F | ( ) | V G δ � (i) convergent: Cauchy in the -metric. ( G 1 , G 2 , ... ) (ii) ∀ convergent: is convergent ( , ,...) ( , ) G G F t F G 1 2 n distribution of k -samples is convergent for all k (i) and (ii) are equivalent. December 2008 17

Convergent graph sequences Example: random graphs ( ) | ( )| E F ฀ G → 1 1 with probability 1 ( , ( , ) ) t F n 2 2 G G δ → → ∞ ( ) ( , ), 1 ( , ) 1 0 ( , ) n m n m � 2 2 December 2008 18

Convergent graph sequences (i) and (ii) are equivalent. − ≤ δ "Counting lemma": | ( , ) ( , ) | ( ) ( , ) t F G t F H E F G H � 1 − ≤ “Inverse counting lemma": if | ( , ) ( , )| t F G t F H k 10 δ < for all graphs F with k nodes, then ( , ) G H � log k December 2008 19

Limit objects • a consistent local finite random graph model ]: probability distribution on -point graphs [ G S k k consistent has same distribution as ( a) [ ] \{ } [ ] G S v G S − 1 k k local � = ∪ for and areindependent. (b) , [ ] [ ] S S S G S G S 1 2 1 2 Every random graph model with (a) and (b) is the limit of models G [ S ]. December 2008 20

Limit objects • a consistent local finite random graph model • a consistent local countable random graph 1 1/3 2/3 ... 1/24 3/24 3/24 1/24 countable random graph December 2008 21

Limit objects • a consistent local finite random graph model • a consistent local countable random graph • a measurable, symmetric function W : [0,1] 2 → [0,1] December 2008 22

Limit objects 0 0 1 0 0 1 1 0 0 0 1 0 0 1 0 0 1 0 1 0 1 0 0 0 0 0 1 0 1 1 0 1 0 1 1 1 1 0 1 0 1 1 0 0 1 0 1 0 1 0 1 0 1 1 0 0 0 1 0 1 0 1 1 0 0 0 1 0 0 1 1 0 1 0 1 0 1 1 0 1 1 1 0 1 1 1 1 1 1 1 0 1 0 1 1 1 1 0 0 0 1 0 0 1 1 0 1 0 1 0 1 1 0 0 1 1 0 0 0 1 1 1 0 1 0 0 0 0 0 0 0 1 1 0 1 0 1 0 1 0 1 0 1 1 1 1 1 1 0 1 0 1 1 1 0 0 0 1 0 1 1 0 1 0 1 0 1 0 0 1 1 0 0 0 1 1 0 1 1 1 0 1 1 0 1 0 1 1 0 1 0 0 1 0 1 0 December 2008 23

Limit objects A random graph 1/2 with 100 nodes and with 2500 edges December 2008 24

Limit objects Rearranging the rows and columns December 2008 25

Limit objects 1/2 A random graph with 100 nodes and with 2500 edges (no matter how you reorder the nodes) December 2008 26

Limit objects A randomly grown − 1 max( , ) x y uniform attachment graph with 200 nodes December 2008 27

Limit objects = − ( , ) : 1 max( , ) W x y x y → ∫∫∫ ( , ) ( , ) ( , ) ( , ) t K G W x y W y z W x z dx dy dz 3 n December 2008 28

Limit objects { } = → [0,1] symmetric, measurable W 2 : [0,1] W 0 ∏ ∫ = ( , ) ( , ) t F W W x x dx i j ∈ ( ) ij E F V ( F ) [0,1] Adjacency matrix Associated function W G : of graph G: ⎛ ⎞ 0 1 0 1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 1 0 1 1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ 0 1 0 1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎜ ⎟ ⎝ ⎠ 1 1 1 0 = ( , ) ( , ) t F G t F W G December 2008 29

Limit objects Distance of functions ∫ δ = − ( , ') inf sup ( ') W W W W � ⊆ , [0,1] S T × S T δ = δ ( , ') ( , ) G G W W � � ' G G W δ is compact. ( , ) � 0 Equivalent to the Regularity Lemma December 2008 30

Limit objects Converging to a function δ → → (i) ( , ) 0 : W W G W G n n ∀ → (ii) ( ) ( , ) ( , ) F t F G t F W n (i) and (ii) are equivalent. December 2008 31

Limit objects ( , ) W x y G n δ → ( , ) 0 G W ฀ n ∀ → ( , ) ( , ) F t F G t F W n December 2008 32

Limit objects For every convergent graph sequence ( G n ) W ∈ W → there is a such that . G W 0 n → Conversely, ∀ W ∃ ( G n ) such that G W n LL – B. Szegedy W is essentially unique (up to measure-preserving transform). Borgs – Chayes - LL December 2008 33