the topology of random - pennsylvania state university · them from a continuous graph. these...
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The topology of randomgraphons
Jan Reimann, Penn State April 29, 2015
joint work with Cameron Freer (MIT)
Homogeneous structuresA countable (relational) structure is homogeneous ifevery isomorphism between finite substructures of extends to an automorphism of .
Fraissé: Any homogeneous structure arises as aamalgamation process of finite structures over the samelanguage (Fraissé limits).
Examples:
,the Rado (random) graphthe universal ‑free graphs, (Henson)
(ℚ, <)
Kn n ~ 3
Randomnized constructionsMany universal homogeneous structures can obtained(almost surely) by adding new points according to arandomized process.
: add the ‑th point between (or at the ends) ofany existing point with uniform probability .
Rado graph: add the ‑th vertex and connect to everyprevious vertex with probability (uniformly andindependently).
Vershik: Urysohn space, Droste and Kuske: universalposet
Henson graph: ???
(ℚ, <) n1/n
np
Constructions ʺfrom belowʺA naive approach to ʺrandomizeʺ the construction of theHenson graph would be as follows:
In the ‑th step of the construction, pick a one‑vertexextension uniformly among all possible extensions thatpreserve ‑freeness.
However: Erdös, Kleitman, and Rothschild showed thatthis asymptotically almost surely yields a bipartite graph(in fact, the random countable bipartite graph)
The Henson graph(s), in contrast, has to contain everyfinite ‑free graph as an induced subgraph, inparticular, and hence cannot be bipartite.
n
Kn
KnC5
Constructions ʺfrom aboveʺPetrov and Vershik (2010) showed how to constructuniversal ‑free graphs probabilistically by samplingthem from a continuous graph.
These continuous graphs, known as graphons, have beenstudied extensively over the past decade.
See, for example the recent book by Lovasz, Largenetworks and graph limits (2012).
Kn
GraphonsOne basic motivation behind graphons is to capture theasymtotic behavior of growing sequences of densegraphs, e.g. with respect to subgraph densities.
While the Rado graph can be seen as the limit object of asequence of finite random graphs, it does notdistinguish between the distributions with which theedges are produced.
For any , converges almost surely to(an isomorphic copy of) the Rado graph.
However, , will exhibit verydifferent subgraph densities than
( )Gn
0 < p < 1 �(n, p)
�p1 p2 �(n, )p1�(n, )p2
Graphons and graph limitsBasic idea: ʺpixel picturesʺ
from Lovasz (2012), Large networks and graph limits
Graphons and graph limitsConvergence of pixel pictures
from Lovasz (2012), Large networks and graph limits
Graphons and graph limitsConvergence of pixel pictures
from Lovasz (2012), Large networks and graph limits
ConvergenceLet be a graph sequence with .
We say converges if
( )Gn |V( )| → 7Gn
( )Gn
for every finite graph , the relative number of embeddings of into
converges.
F(F, )ti Gn F Gn
Graphons measurable, and for all ,
and .
Think: is the probability there is an edge between and .
Subgraph densities:
edges: triangles: this can be generalized to define .
W : [0, 1 → [0, 1]]2 x, y
W(x, x) = 0 W(x, y) = W(y, x)
W(x, y)x y
D W(x, y) dx dyD W(x, y)W(y, z)W(z, x) dx dy dz
(F, W)ti
The limit graphonTHM: For every convergent graph sequence thereexists (up to weak isomorphim) exactly one graphon
such that for all finite :
( )GnW
F
.(F, ) ⟶ (F, W)ti Gn ti
A compatible metricEdit distance: . Cut distance: , where isthe cut norm
can be extended to graphs of different order... ... and to graphons:
(F, G) => +d1 AF AG>1
(F, G) => +d◻ AF AG>◻ >. >◻
>A = .>◻1n2 max
S,T�[n]<< ∑i!S,j!T
Aij <<
d◻
>W = W(x, y) dx dy.>◻ supS,T�[0,1] ∫S×T
A compatible metric
A sequence converges iff it is a Cauchy sequencewith respect to .
iff
( )Gnd◻
→ WGn ( , W) → 0d◻ Gn
Sampling from graphonsWe can obtain a finite graph from by(independently) sampling points from and filling edges according to probabilities .
almost surely, we get a sequence with .
If we sample ‑many points from , wealmost surely get the random graph.
�(n, W) Wn ,… ,x1 xn W
W( , )xi xj
�(n, W) → W
ω W(x, y) z 1/2
The Petrov‑Vershik graphonPetrov and Vershik (2010) constructed, for each , agraphon such that we almost surely sample auniversal ‑free graph.
The graphons are (necessarily) ‑valued.
Such graphons are called random‑free.
The constructions resembles a finite extensionconstruction with simple geometric forms, where eachstep satisfies a new type requiring attention.
The method can also be used to construct random‑freegraphons universal for all finite graphs.
n ~ 3WKn
0, 1
Universal graphonsA random‑free graphon is countably universal if for everyset of distinct points from , ,
, the intersection
has non‑empty interior. Here is the neighborhood of .
For countably ‑free universal graphs, we require this tohold only for such tuples where the induced subgraph bythe has no induced ‑subgraph,additionally, require that there are no n‑tuples in Xwhich induce a .
[0, 1] , ,… ,x1 x2 xn,… ,y1 ym
B⋂i,j
Exi ECyj
= {y: W(x, y) = 1}Ex x
Kn
xi Kn+1
Kn
which induce a .Kn
The topology of graphonsNeighborhood distance:
and mod out by .
Example: is a singleton space.
THM: (Freer & R.) (informal) If is a random‑free universalgraphon obtained via a ʺtameʺ extension method, then is notcompact in the topology.
(x, y) = > W(x, . ) + W(y, . ) = ∫ |W(x, z) + W(y, z)rW >1
(x, y) = 0rW
W(x, y) z p
WW
rW
ʺTameʺ extensionsDEF: A random‑free graphon has continuous realizationof extensions if there exists a function
that is continuous a.e. such that for all ,
Here is the neighborhood of . The Petrov‑Vershik graphons have uniformly continuousrealization of extensions.
W
f : ( ,… , ), ( ,… , ) ↦ (l, r)x1 xn y1 ym,x ⃗ y ⃗
[l, r] � B .⋂i,j
Exi ECyj
= {y: W(x, y) = 1}Ex x
Non‑compactness
THM: If a countably ( ‑free) universalgraphon has uniformly continuous realizationof extensions, then it is not compact in the
‑topology.
Kn
rW
Features of the proofBuilding a ʺCantor sequenceʺ in .
Apply the Szemeredi regularity lemma to pass to asequence of stepfunctions that approximate the graphonuniformly.
Use universality to find the next splitting.
Uniform continuity guarantees that the Szemerediʺsquaresʺ are filled with the right measure.
W
Regularity lemmaFor every there is an such that everygraph with at least vertices has an equitablepartition of V into pieces (1/ε ≤ k ≤ S(ε)) such that for allbut pairs of indices , the bipartite graph is ‑regular.
For every graphon and there is stepfunction with steps such that
ϵ > 0 S(ϵ) ! ℕG S(ϵ)
kϵk2 i, j G[ , ]Vi Vj
ϵW k ~ 1 U
k
(W, U) < > Wd◻2
log k‾ ‾‾‾‾3>2