Convergent Dense Graph Sequences

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Convergent Dense Graph Sequences

• Jennifer Tour Chayesjoint work with
• C. Borgs, L. Lovasz, V. Sos, K. Vesztergombi

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Convergent Dense Graph Sequences

• I Metrics, Sampling TestingII Multi-way
  Cuts Statistical Physics

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Outline of I Metrics, Sampling Testing

• Introduction Motivation, Convergence and
  Testing
• Subgraph Densities and Left Convergence
• Graph Metrics
• Convergence in Metric
• Szemeredi Lemma and Sampling
• Parameter Testing
• The Limit Object, Metric Convergence
  Testability

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Introduction Motivation

• Numerous examples of growing graph sequences
  e.g., Internet, WWW, social networks
• Want a succinct but faithful representation for
• Testing properties e.g., clustering
• Testing algorithms e.g., for routing, search
• Here we deal only with dense graphs
• (also have results for bounded-degree graphs)

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Introduction Convergence

• Given
• Sequence Gn of graphs with V(Gn) ! 1
• Questions
• What is the right notion of convergence?
• Is there a useful metric s.t. Gn convergent
• , Gn is Cauchy in the metric?
• What is the limit object?

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Introduction Testing

• Given a simple graph parameter f, i.e. a
  real-valued function on simple graphs, invariant
  under isomorphism
• Question Under what conditions is f testable,
  i.e. 8 e gt 0, 9 k lt 1 such that 8 G with V(G) gt
  k,
• f(G) f(GS) lt e
• with probability at least 1 e, where S ½ V(G)
  is a uniformly random sample of size k?

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Preview

• There is a reasonable notion of convergent graph
  sequences, which turns out to be equivalent to
  convergence in an appropriate metric, and is
  closely related to testability.
• (Some of the) Main Theorems of Part I
• f(Gn) converges 8 convergent graph sequence Gn
• , f is continuous in the metric
• , f is testable
• , f can be extended to a continuous function on
  the limit object

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Outline

• Introduction Motivation, Convergence and
  Testing
• Subgraph Densities and Left Convergence
• Graph Metrics
• Convergence in Metric
• Szemeredi Lemma and Sampling
• The Limit Object and Metric Convergence

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Subgraph Densities

• F, G simple graphs
• (For Part I, think of F as small and G as large)
• Homomorphisms adjacency preserving maps
• Hom(F,G) f V(F) ! V(G) s.t. f(E(F)) ½ E(G)
• Subgraph densities (1979 Erdos, Lovasz, Spencer)
• t(F,G) V(G)-V(F) Hom(F,G)
• E.g., t(K3,G) is the triangle density of G

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Left Convergence

• Our Definition
• Gn is said to be (left) convergent if t(F, Gn)
  converges for all simple F.
• Example Let Gn Gn,p. Then
• t(F, Gn) ! pE(F) .

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Outline

• Introduction Motivation, Convergence and
  Testing
• Subgraph Densities and Left Convergence
• Graph Metrics
• Convergence in Metric
• Szemeredi Lemma and Sampling
• The Limit Object and Metric Convergence

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Outline

• Graph Metrics
• Cut norm on matrices
• Distances between graphs with same number of
  vertices
• Splitting vertices
• Cut metric between arbitrary graphs

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Cut Norm on Matrices

• Definition (1999 Frieze, Kannan)
• Let M be an n n matrix
• The cut norm of M is given by

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Distances Between Graphs on Same Number of
Vertices

• G, G0 weighted graphs on n with
• common vertex weights a1, , an s.t. Siai a
• different edge weights bij and bij0
• adjacency matrices AG with (AG)ij ai bijaj
• and AG0 with (AG0)ij ai bij0aj
• Define the distance between G and G0
• dW(G, G0) a-2 k AG - AG0 kW

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Distances between Graphs on Same Number of
Vertices

• Notice that the distance dW(G, G0) is not
  invariant under isomorphisms of G and G0
• So do an integer overlay and define

where the minimum goes over all relabelings G and
G0
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Different Numbers of Vertices

• Question What do we do if G and G0 have
  different numbers of vertices n ? n0?
• Idea Split each
• vertex of G into n0 new vertices
• vertex of G0 into n new vertices

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Splitting Vertices

• Given Graph G on n,
• with weights ai, bij
• Split i into n0 pieces
• (i,1), , (i,n0)
• Split ai into n0 pieces
• ai Su2n0 Xiu

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Splitting Vertices (continued)

• Given Graph G on n,
• with weights ai, bij
• Replace edge ij by
• complete bipartite graph
• with biu,jv bij
• ) new graph GX on nn0
• Fact t(F,GX) t(F,G)

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Cut Metric between Arbitrary Graphs

• Given G graph on n, weights ai, bij
• G0 graph on n0, weights a0u, b0uv
• with Siai Su a0u
• Define our cut metric

where the minimum goes over all fractional
overlays, i.e. all couplings (or joinings) X of
ai and a0u, with Xiu 0, SuXiu ai and SiXiu
a0u.
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Outline

• Introduction Motivation, Convergence and
  Testing
• Subgraph Densities and Left Convergence
• Graph Metrics
• Convergence in Metric
• Szemeredi Lemma and Sampling
• The Limit Object and Metric Convergence

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Convergence in Metric

• Definition Let (Gn) be a sequence of simple
  graphs. We say that (Gn) is convergent in the
  metric dW if (Gn) is a Cauchy sequence in dW.
• Theorem 1 (Convergence in Metric) A sequence of
  simple graphs (Gn) is left convergent if and only
  if it is convergent in the metric dW.
• I.e., subgraph densities of a sequence of graphs
  converge , the sequence is Cauchy in the cut
  metric.

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Outline

• Introduction Motivation, Convergence and
  Testing
• Subgraph Densities and Left Convergence
• Graph Metrics
• Convergence in Metric
• Szemeredi Lemma and Sampling
• Parameter Testing
• The Limit Object and Metric Convergence

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Szemeredi Lemma

• Given
• simple (unweighted) graph G
• disjoint partition P (V1, ... , Vq) of V(G)
• Define the edge density between classes Vi and
  Vj
• bij Vi-1Vj-1 eG(Vi ,Vj)
• Define the (weighted) average graph GP on V(G)
  with
• nodeweights ax(G) 1
• edgeweights bxy (G) bij if x 2 Vi and y 2 Vj

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Szemeredi Lemma (continued)

• It turns out that Szemerdis Regularity Lemma, in
  the weak form proved by Frieze and Kannan,
  describes precisely the dW-distance between a
  simple graph G and its average over a
  sufficiently large partition.
• Weak Regularity Lemma (1999 Frieze and Kannan)
  For all e gt 0 and all simple graphs G, there
  exists a partition P (V1, ... , Vq) of V(G)
  into q 41/e2 classes such that
• dW(G, GP) lt e.

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Sampling

• Main Technical Lemma Let k be a positive
  integer, and let G, G0 be weighted graphs on at
  least k nodes with nodeweights one and
  edgeweights in 0,1. If S is chosen uniformly
  from all subsets S ½ V of size k, then
• dW(GS,G0S) - dW(G,G0) 10 k-1/4
• with probability at least 1 exp(-k1/2/8).
• related to 2003 Alon, Fernandez de la Vega,
  Kannan and Karpinski, but with different k
  dependence and different proofs

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Sampling (continued)

• Theorem 2 (Closeness of Sample) Let k be a
  positive integer, and let G be a simple graph on
  at least k nodes. If S is chosen uniformly from
  all subsets S ½ V of size k, then
• dW(G,GS) 10 (log2k)-1/2
• with probability at least 1 exp(-k2/2 log2k).

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Sampling (continued)

• Key Elements of the Proof of Theorem 2
• Use an easy sampling argument to prove the
  theorem for the special case in which V(G) can be
  decomposed into only a few large sets V1, , Vq,
  with constant weights for an edge between any
  given pair Vi and Vj
• Use the weak regularity lemma to approximate an
  arbitrary graph by the simple special case above
• Use the previous (main technical) lemma to show
  that this approximation induces only a small error

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Proof of Theorem 1

• Recall Theorem 1 Convergence from the left
  (i.e., subgraph convergence) , convergence in cut
  metric
• Idea of Proof of Theorem 1
• By the triangle inequality, Theorem 2 implies
  that two graphs G and G0 (possibly with n ? n0)
  are close in metric only if their samples are
  close
• But knowledge of all subgraph frequencies is more
  or less equivalent to knowledge of all sampling
  probabilities
• t(F,G) ¼ Prob(GS F)
• where S is uniform among all sets of size V(F)
• thus (lots of work), convergence of subgraph
  frequencies , convergence in metric

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Outline

• Introduction Motivation, Convergence and
  Testing
• Subgraph Densities and Left Convergence
• Graph Metrics
• Convergence in Metric
• Szemeredi Lemma and Sampling
• The Limit Object, Metric Convergence
  Testability

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The Limit Object

• Recall Theorem 1 (Convergence in Metric) A
  sequence of simple graphs (Gn) is left convergent
  if and only if it is convergent in the metric dW.
• ) 9 limit object G1
• Question Is there a useful representation of
  this limit object?
• Answer Yes the graphon.

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Graphons

• Definition A function W 0,12 ! R is called a
  graphon if
• W is measurable
• W(x,y) W(y,x)
• kWk1 lt 1
• Example Step functions
• G graph on n vertices
• WG(x,y) Idxnedyne 2 E(G)

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Graphons as Limit Objects

• For V(F) k, define

• Theorem (2005 Lovasz and B. Szegedy)
• (Gn) is left convergent , 9 graphon W s.t. t(F,
  Gn) ! t(F,W).

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Graphons and Metric Convergence

• Definition (Frieze and Kannan) The cut norm of
  a graphon W is given by

• Theorem 10
• (Gn) is left convergent , 9 graphon W and a
  relabeling of (Gn) s.t.
• kW - WGnkW ! 0.

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Summary of Part I

• There is a reasonable notion of convergent graph
  sequences convergence of subgraph densities,
  which turns out to be equivalent to convergence
  in an appropriate (and useful) metric the cut
  metric, and is closely related to sampling and
  testability.

35
Summary of Part I

• (Some of the) Main Theorems
• f(Gn) converges 8 convergent graph sequence Gn
• , f is continuous in the cut metric
• , f is a testable graph parameter
• , f can be extended to a continuous function on
  the limit object

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Part II Multi-way Cuts Statistical Physics

• In Part I, we learned that a graph sequence Gn
  can be probed from the left by studying the
  densities of subgraphs occurring in it
• In Part II, we will learn that Gn can be probed
  from the right by studying generalized
  colorings (or multi-way cuts or statistical
  mechanical models) on it, giving us a dual notion
  of convergence

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Outline of II Multi-way Cuts Statistical
Physics

• Homomorphisms into (Small) Weighted Graphs
• Naïve Right Convergence and Ground State Energies
• Incomplete Equivalences
• Microcanonical Ensemble and Right Convergence
• Complete Equivalences

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Homomorphisms into (Small) Weighted Graphs

• Recall that in Part I, we considered
• Hom(F,G) f V(F) ! V(G) s.t. f(E(F)) ½ E(G)
• with F small and simple, and G large
• Now instead consider Hom(G,H) with G large and
• H small and weighted, with vertex weights ai
  ai(H) and edge weights bij bij(H), so that

• This is a weighted count of the number of
  colorings

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Example Ising Magnet

• V(H) -1,1
• af ehf , bff eJff

with
Note that this is not the conventional
normalization of the energy for a dense graph,
so. Hom(G,H) expV(G)2 .
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Outline of II Multi-way Cuts Statistical
Physics

• Homomorphisms into (Small) Weighted Graphs
• Naïve Right Convergence and Ground State Energies
• Incomplete Equivalences
• Microcanonical Ensemble and Right Convergence
• Complete Equivalences

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Naïve Right Convergence

• Let r(G,H) V(G)-2 log Hom(G,H)
• Note that r(G,H) is not the free energy
• Our Definition
• Gn is said to be naïvely right convergent if
  r(Gn,H) converges for all soft-core weighted H,
  i.e. all H with
• vertex weights ai ai(H) gt 0 and
• edge weights bij bij(H) gt 0.

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Ground State Energy

• Let bij(H) eJij
• Then the ground state energy of model H on graph
  G is

• Example Maxcut Density

E(G,J) V(G)-2 Maxcut (G)
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Naïve Right Convergence and Ground State Energies

• Lemma If bij(H) eJij , then
• r(G,H) - E(G,J) O(V(G)-1 )
• Again note that r(G,H) is not the free energy
  to leading order, it is just the ground state
  energy. The entropy has been wiped out by the
  V-2 normalization.
• So naïve right convergence of Gn is the same as
  convergence of the ground state energy for all
  soft-core models H on Gn.

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Outline of II Multi-way Cuts Statistical
Physics

• Homomorphisms into (Small) Weighted Graphs
• Naïve Right Convergence and Ground State Energies
• Incomplete Equivalences
• Microcanonical Ensemble and Right Convergence
• Complete Equivalences

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Incomplete Equivalences
Cut densities are testable parameters
OR
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Outline of II Multi-way Cuts Statistical
Physics

• Homomorphisms into (Small) Weighted Graphs
• Naïve Right Convergence and Ground State Energies
• Incomplete Equivalences
• Microcanonical Ensemble and Right Convergence
• Complete Equivalences

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Microcanonical Ensemble

• Given q color classes with fraction ai in color
  class i a (a1, , aq) with ai 0 and Sai
  1
• Define the microcanonical homomorphism number

and microcanonical ground state energy
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Examples

• Max/Min Bisection

• Densest Subgraph

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Right Convergence

• Let ra(G,H) V(G)-2 log Homa (G,H)
• Definition Gn is said to be right convergent if
  ra(Gn,H) converges for all a and all soft-core
  weighted H, i.e. all H with
• vertex weights ai ai(H) gt 0 and
• edge weights bij bij(H) gt 0.
• This is equivalent to convergence of all
  microcanonical ground state energies.

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Outline of II Multi-way Cuts Statistical
Physics

• Homomorphisms into (Small) Weighted Graphs
• Naïve Right Convergence and Ground State Energies
• Incomplete Equivalences
• Microcanonical Ensemble and Right Convergence
• Complete Equivalences

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Complete Equivalences Summary
Cut densities are testable parameters
OR
52
Summary

• There is a reasonable notion of convergent graph
  sequences convergence of subgraph densities
  which turns out to be equivalent to convergence
  in an appropriate (and useful) metric the cut
  metric, and is closely related to sampling and
  testability
• Convergence of subgraph densities is also
  equivalent to the dual notion of convergence of
  all microcanonical ground state energies of all
  soft-core models (and also to convergence of
  quotients, moding out by Szemeredi partitions, in
  the natural Hausdorff metric)

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• THE END

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