The Mathematical Challenge
of Large Networks
László Lovász
Eötvös Loránd University, Budapest
[email protected]
July 2009
1
Very large graphs
Questions
What properties to study?
- Does it have an even number of nodes?
- How dense is it (average degree)?
- Is it connected?
- What are the connected components?
July 2009
2
Very large graphs
Framework
How to obtain information about them?
- Graph is HUGE.
- Not known explicitly, not even number of nodes.
July 2009
3
Very large graphs
Dense framework
How to obtain information about them?
Dense case: cn2 edges.
- We can sample a uniform random node a
bounded number of times, and see edges
between sampled nodes.
July 2009
4
Very large graphs
Sparse framework
How to obtain information about them?
Sparse case: Bounded degree (d).
- We can sample a uniform random node a
bounded number of times, and explore its
neighborhood to a bounded depth.
July 2009
5
Very large graphs
Alternatives
How to obtain information about them?
- Observing global process locally, for some time.
- Observing global parameters (statistical physics).
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6
Very large graphs
Models
How to model them?
Erdős-Rényi random graphs
Albert-Barabási graphs
Many other randomly growing models
July 2009
7
Very large graphs
Approximations
How to approximate them?
- By smaller graphs
Szemerédi partitions (regularity lemmas)
- By larger graphs
Graph limits
July 2009
8
Very large dense graphs
Distance
How to measure their distance?
cut distance
(a)
V (G ) = V (G ')
| eG ( S , T ) - eG ' ( S , T ) |
dX (G, G ') = max
2
S ,T Н V ( G )
n
(b)
| V (G ) |=| V (G ') |= n
*
X
d (G, G ') = min dX (G, G ')
July 2009
G« G '
9
Very large dense graphs
Distance
How to measure their distance?
(c)
| V (G ) |= n № n ' = | V (G ') |
blow up nodes, or fractional overlay
1
X ij =
,
е
n'
iО V ( G )
( X ij )iОV (G ), jОV (G ') і 0
1
X ij =
е
n
jОV ( G ')
dX (G, G ') =
= min
X
July 2009
max
S Н V (G )ґ V (G ')
е
е
X iu X jv (aij - a 'uv )
i , jОV (G ) u ,vОV (G ')
10
Very large dense graphs
Distance
1
Examples: dX ( K n ,n , G (2n, )) »
8
1
2
dX (G1 (n, 12 ), G 2 (n, 12 )) = o(1)
1/2
dX (G1 (n, ),
1
2
July 2009
) = o(1)
11
Very large dense graphs
Sampling Lemmas
G, G ' : graphs with V (G) = V (G '), S Н V (G), S = k , random
With large probability,
10
dX ( G[ S ], G '[ S ]) - dX ( G, G ') < 1/ 4
k
Alon-Fernandez de la Vega-Kannan-Karpinski+
With large probability,
dX ( G, G[ S ]) <
10
log k
Borgs-Chayes-Lovász-Sós-Vesztergombi
July 2009
12
Approximating by smaller
Regularity Lemmas
Original Regularity Lemma
Szemerédi 1976
“Weak” Regularity Lemma
Frieze-Kannan 1999
“Strong” Regularity Lemma
Alon – Fisher
- Krivelevich
- M. Szegedy 2000
July 2009
13
Approximating by smaller
Regularity Lemmas
The nodes of  graph can be partitioned
into a bounded number
of essentially equal parts
given ε>0, # of parts k
is between 1/ ε and f(ε)
so that
difference at most 1
almost all bipartite graphs between 2 parts
are essentially random
(with different densities).
with εk2 exceptions
for subsets X,Y of the two parts,
# of edges between X and Y
is p|X||Y|  εn2
July 2009
14
Approximating by smaller
Regularity Lemmas
“Weak” Regularity Lemma (Frieze-Kannan):
partition P = {S1 ,..., Sk }
pij: density between Si and Sj
GP: complete graph on V(G) with edge weights pij
July 2009
15
Approximating by smaller
Regularity Lemmas
“Weak" Regularity Lemma (Frieze-Kannan):
For every graph G and k і 1 there is a k -partition P
such that
dX ( G , GP ) Ј
July 2009
2
.
log k
16
“Weak” Regularity Lemma
Similarity distance
d2 (s, t ) := Ev Eu (asu avu ) - Ew (atwawv )
Fact 1. This is a metric, computable by sampling
Fact 2.
Weak Szemerédi partition  partition most nodes
into sets with
small diameter
LL – B. Szegedy
July 2009
17
“Weak” Regularity Lemma
Algorithm
Algorithm to construct representatives of classes:
- Begin with U=.
- Select random nodes v1, v2, ...
- Put vi in U iff d2(vi,u)>ε for all uU.
- Stop if for more than 1/ε2 trials, U did not grow.
size bounded by f(ε)
July 2009
18
“Weak” Regularity Lemma
Algorithm
Algorithm to decide in which class does v belong:
Let U={u1,...,uk}.
Put a node v in Vi iff i is the first index with d2(ui,v)ε.
July 2009
19
Max Cut in huge graphs
Sampling?
cut with many edges
July 2009
20
Approximating by larger
Convergent graph sequences
t(F,G): Probability that random mapV(F)V(G)
preserves edges
(i) (G1, G2,...) convergent:
Cauchy in the dX -metric.
(ii) (G1 , G2 ,...) convergent: " F t ( F , Gn ) is convergent
distribution of k-samples
is convergent for all k
(i) and (ii) are equivalent.
July 2009
21
Approximating by larger
Convergent graph sequences
(i) and (ii) are equivalent.
"Counting lemma": |t ( F , G) - t ( F , H ) |Ј E( F ) dX (G, H )
1
“Inverse counting lemma": if |t ( F , G ) - t ( F , H ) |Ј
k
10
for all graphs F with k nodes, then dX (G, H ) <
log k
July 2009
22
A random graph
with 100 nodes and with 2500 edges
July 2009
1/2
23
Rearranging the rows and columns
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24
Approximation by small: Szemerédi's Regularity Lemma
July 2009
25
Approximation by small: Szemerédi's Regularity Lemma
Nodes can be so labeled
July 2009
essentially random
26
Approximation by infinite
A randomly grown
uniform attachment graph
1- max( x, y )
with 200 nodes
July 2009
27
Approximating by larger
W0 =
{W : [0,1]2 ®
Limit objects
[0,1] symmetric, measurable}
(graphon)
t ( F ,W ) =
т Х
W ( xi , x j ) dx
[0,1]V ( F ) ij О E ( F )
t ( F , G)  t ( F ,WG )
December 2008
28
Approximating by larger
Limit objects
Distance of functions
dX (W ,W ')  sup
 (W  W ')
S ,T [0,1] S T
X (W ,W ')  inf dX (W ,W ')
W W '
dX (G, G ') = dX (WG ,WG ' )
December 2008
29
Approximating by larger
Limit objects
Converging to a function
Gn ® W : (i) d (WGn ,W ) ® 0
(ii) ( " F ) t ( F , Gn ) ® t ( F ,W )
(i) and (ii) are equivalent.
December 2008
30
Approximating by larger
Limit objects
For every convergent graph sequence (Gn)
there is a W О W0 such that Gn ® W .
Conversely, W (Gn) such that Gn ® W
LL – B. Szegedy
W is essentially unique (up to measure-preserving transform).
Borgs – Chayes - LL
December 2008
31
Extension to sparse graphs?
Dense
Bounded degree
Convergence of graph sequences
Erdős-L-Spencer 1979
Borgs-Chayes-L-Sós
-Vesztergombi 2006
Aldous 1989
Benjamini-Schramm 2001
Limit objects
(graphons) L-B.Szegedy 2006
Benjamini-Schramm 2001
(graphings) Elek 2007
Left and right convergence
Borgs-Chayes-L-Sós
-Vesztergombi 2010?
July 2009
Borgs-Chayes-Kahn-L 2010?
32
Extension to sparse graphs?
Dense
Bounded degree
Property testing
Arora-Karger-Karpinski 1995
Goldreich-Goldwasser
-Ron 1998
Goldreich-Ron 1997
Distance
Borgs-Chayes-L-Sós
-Vesztergombi 2008
?
Regularity Lemma
Szemerédi, Frieze-Kannan,
Alon-Fischer-Krivelevich
-Szegedy, Tao, L-Szegedy,…
July 2009
?
33