Post on 13-Dec-2015
The mathematical challenge
of large networks
László Lovász
Eötvös Loránd University, Budapest
Joint work with Christian Borgs, Jennifer Chayes,Balázs Szegedy, Vera Sós and Katalin Vesztergombi
May 2012 1
Minimize x3-6x over x0
minimum is not attainedin rationals
Minimize 4-cycle density in graphs with edge-density 1/2
minimum is not attainedamong graphs
always >1/16,arbitrarily close for random
graphs
Real numbers are useful
Graph limits are useful
May 2012 2
Graph limits: Why are they needed?
Two extremes:
- dense (cn2 edges)
- bounded degree
well developedgood warm-up case
How dense is the graph?
May 2012 3
Inbetween???Bollobas-RiordanChungConlon-Fox-Zhao
less developed, more difficultmost applications
4May 2012
How is the graph given?
- Graph is HUGE.
- Not known explicitly, not even the number of nodes.
Idealize: define minimum amount of info.
May 2012 5
Dense case: cn2 edges.
- We can sample a uniform random node a bounded number of times, and see edges
between sampled nodes.
Bounded degree (d)
- We can sample a uniform random node a bounded number of times, and explore its neighborhood to a bounded depth.
How is the graph given?
„Property testing”: Arora-Karger-Karpinski,
Goldreich-Goldwasser-Ron, Rubinfeld-Sudan,
Alon-Fischer-Krivelevich-Szegedy, Fischer,
Frieze-Kannan, Alon-Shapira
May 2012 6
Lecture plan
Want to construct completion of the set of graphs.
- What is the distance of two graphs?
- Which graph sequences are convergent?
- How to represent the limit object?
- How does this completion space look like?
- How to approximate graphs? (Regularity Lemmas and sampling)
May 2012 7
Lecture plan
Applications in (dense) extremal graph theory
- Are extremal graph problems decidable?
- Which graphs are extremal?
- Local vs. global extrema
- Is there always an extremal graph?
May 2012 8
Lecture plan
Applications in property testing
- Deterministic and non-deterministic sampling (P=NP)
- Which properties are testable by sampling?
May 2012 9
Lecture plan
The bounded degree case
- Different limit objects: involution-invariant distributions, graphings
- Local algorithms and distributed computing
G
0 0 1 0 0 1 1 0 0 0 1 0 0 10 0 1 0 1 0 1 0 0 0 0 0 1 01 1 0 1 0 1 1 1 1 0 1 0 1 10 0 1 0 1 0 1 0 1 0 1 1 0 00 1 0 1 0 1 1 0 0 0 1 0 0 11 0 1 0 1 0 1 1 0 1 1 1 0 11 1 1 1 1 1 0 1 0 1 1 1 1 00 0 1 0 0 1 1 0 1 0 1 0 1 10 0 1 1 0 0 0 1 1 1 0 1 0 00 0 0 0 0 1 1 0 1 0 1 0 1 01 0 1 1 1 1 1 1 0 1 0 1 1 10 0 0 1 0 1 1 0 1 0 1 0 1 00 1 1 0 0 0 1 1 0 1 1 1 0 11 0 1 0 1 1 0 1 0 0 1 0 1 0
AG
WG
Pixel pictures
May 2012 10
A random graph with 100
nodes and with 2500 edgesMay 2012 11
Pixel pictures
Rearranging rows and columns
May 2012 12
Pixel pictures
May 2012 13
Pixel pictures
A randomly grown uniform
attachment graph on 200 nodes
At step n: - a new node is born; - any two nodes are
joined with probability 1/n
Ignore multiplicity of edges
Approximation by small: Regularity Lemma
Szemerédi1975
May 2012 14
Nodes can be so ordered
essentially random
Approximation by small: Regularity Lemma
May 2012 15
The nodes of any graph can be partitioned
into a small number
of essentially equal parts
so that
the bipartite graphs between 2 parts
are essentially random
(with different densities).with k2 exceptions
for subsets X,Y of parts Vi,Vj# of edges between X and Y
is pij|X||Y| ± (n/k)2
Given >0
22
21 12 }ke e
£ £N
difference at most 1
Approximation by small: Regularity Lemma
May 2012 16
May 2012 17
Original Regularity Lemma Szemerédi 1976
“Weak” Regularity Lemma Frieze-Kannan 1999
“Strong” Regularity Lemma Alon – Fisher- Krivelevich - M. Szegedy 2000
Approximation by small: Regularity Lemma
1 max( , )- x y
May 2012 18
A randomly grown uniform
attachment graph on 200 nodes
Graph limits: Examples
Knowing the limit W
knowing many properties (approximately).
Graph limits: Why are they useful?
3[0,1]
( , ) ( , ) ( , )W x y W y z W z x dx dy dzòtriangle density
May 2012 19
May 2012 20
Limit objects: the math
distribution of k-samples
is convergent for all k
t(F,G): Probability that random map V(F)V(G) preserves edges
(G1,G2,…) convergent: F t(F,Gn) is convergent
May 2012 21
Limit objects: the math
W0 = {W: [0,1]2 [0,1], symmetric, measurable}
( ) ( )[0,1]
( ,( , ) )Î
= ÕòV F
i jij E F
W x x dxt F W
GnW : F: t(F,Gn) t(F,W)
"graphon"
Randomly grown prefix attachment graph
At step n:- a new node is born;- connects to a random previous node and all its predecessors
May 2012 22
Limit objects: an example
Limit objects: an example
A randomly grown prefix attachment graph
with 200 nodes
Is this graph sequenceconvergent at all?
Yes, by computing subgraph densities!
This tends to some shades of gray; is that the limit?
No, by computing triangle densities!
May 2012 23
A randomly grown prefix attachment graph
with 200 nodes (ordered by degrees)
This also tends to some shades of gray; is that the limit?
No…
May 2012 24
Limit objects: an example
The limit of randomly grown prefix attachment graphs
May 2012 25
Limit objects: an example
26
The distance of two graphs
May 2012
'2, ( )
| ( , ) ( , ) |( , ') max G G
S T V G
e S T e S Td G G
nÍ
-=X
( ) ( ')V G V G=(a)
(b) | ( ) | | ( ') |V G V G n= = *
'( , ') min ( , ')
G GG G d G Gd
«=X X
cut distance
| ( ) | ' | ( ') |V G n n V G= ¹ =(c) blow up nodes
*( , ') lim ( ( '), '( ))k
G G G kn G knd d®¥
=X X
finite definitionby fractional overlay
27May 2012
Examples: 1, 2
1( , (2 , ))
8n nK nXd »G
1 11 22 2( , ), ( , ) 1)( ) (n n od =X G G
The distance of two graphs
May 2012 28
{ }1,...,partition = kS SP
pij: density between Si and Sj
GP: complete graph on V(G) with edge weights pij
“Weak” Regularity Lemma
May 2012 29
1
2( , ) .
log
For every graph and there is a -partition
such that
X
G k k
d G Gk
³
£P
P
Frieze-Kannan
“Weak” Regularity Lemma
May 2012 30
| ( , ) ( , ) | ( ) ( , )t F G t F H E F G Hd- £ X
Counting lemma:
1| ( , ) ( , ) |t F G t F H
k- £Inverse counting lemma: If
10( , )
logG H
kd <X
for all graphs F with k nodes, then
Counting Lemmas
May 2012 31
Equivalence of convergence notions
A graph sequence (G1,G2,...) is convergent iff
it is Cauchy in .
December 2008 32
'( , ') inf ( , ')
XX W W
d W WW W
'( , ') ( , )X X G GG G W Wd d=
, [0,1]sup (, ') '( )
XS T S T
W Wd W W
The distance of two graphons
May 2012 33
The semimetric space (W0,) is compact.
“Strong” Regularity Lemma
May 2012 34
Limit objects: the math
For every convergent graph sequence (Gn)
there is a WW0 such that GnW .
W is essentially unique
(up to measure-preserving
transformation).
Conversely, W (Gn) such that GnW .
May 2012 35
Limit objects: the math
For every convergent graph sequence (Gn)
there is a WW0 such that GnW .
W is essentially unique
(up to measure-preserving
transformation).
Conversely, W (Gn) such that GnW . Regularity Lemma + martingales L-SzegedyExchangeable random variables Aldous; Diaconis-JansonUltraproducts Elek-Szegedy
May 2012 36
Limit objects: the math
For every convergent graph sequence (Gn)
there is a WW0 such that GnW .
W is essentially unique
(up to measure-preserving
transformation).
Conversely, W (Gn) such that GnW .
W-random graphs(sampling from a graphon)
May 2012 37
Limit objects: the math
For every convergent graph sequence (Gn)
there is a WW0 such that GnW .
W is essentially unique
(up to measure-preserving
transformation).
Conversely, W (Gn) such that GnW .
Constructing canonical representationBorgs-Chayes-L
Exchangeable random variablesKallenberg; Diaconis-Janson
Inverse Counting Lemma, measure compactnessBollobas-Riordan
May 2012 38
Thanks, that’sall for today!