random graphs, finite - pennsylvania state universitya random‑free graphon is countably universal...
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Random graphs, finiteextension constructions, and
complexityJan Reimann, Penn State
October 4, 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
Randomized constructionsMany homogeneous structures can obtained (almostsurely) by adding new points according to a randomizedprocess.
: 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 (2004): Urysohn space, Droste and Kuske (2003): universal poset
Henson graph: ??? (until 2008)
(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 (1976)showed that this asymptotically almost surely yields abipartite graph (in fact, the universal countable bipartitegraph).
The Henson graph(s), in contrast, has to contain and hence cannot be bipartite.
n
Kn
C5
Symmetric constructionsOn the other hand, one could (degenerately) ensure thatevery triangle‑free subgraph appears, and indeed witnessall extension axioms, by deterministically building theHenson graph.
But this violates symmetry: we would like the jointdistribution of any distinct k‑tuple to be the same as anyother (i.e., exchangeability).
from Lloyd–Orbanz–Ghahramani–Roy (2012), Random function priors for exchangeable arrays
from Lloyd–Orbanz–Ghahramani–Roy (2012), Random function priors for exchangeable arrays
Symmetric constructionsIs there an exchangeable construction of the Hensongraph?
Equivalently, is there a probability measure on graphswith vertex set that is concentrated on the isomorphismclass of the Henson graph, and is invariant under thelogic action of the symmetric group on the underlyingset of vertices?
ω
S7ω
Constructions ʺfrom aboveʺPetrov and Vershik (2010) showed how to constructuniversal ‑free graphs probabilistically by samplingthem from a continuous graph.
Indeed every exchangeable structure in a countablelanguage must arise in essentially this way, as shown byAldous (1981) and Hoover (1979).
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 , ʺconvergesʺ almost surely to(an isomorphic copy of) the Rado graph.
However, if , will exhibit subgraphdensities very different from
( )Gn
0 < p < 1 �(n, p)
�p1 p2 �(n, )p1
�(n, )p2
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
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
The limit graphonTHM: For every convergent graph sequence thereexists (up to weak isomorphim) exactly one graphon
such that for all finite :
( )Gn
WF
.(F, ) ⟶ (F, W)ti Gn ti
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 [0, 1]
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 a Hensongraph for .
The graphons are (necessarily) {0,1}‑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 from which we sample the Rado graph.
n ~ 3Wn
Invariant measuresThe Petrov‑Vershik graphon also yields a measure on theset of countable infinite graphs concentrating on the set ofuniversal, homogeneous ‑free graphs.
This measure will be invariant under the ʺlogic actionʺ,the natural action of on the space of countable(relational) structures with universe .
This method was generalized by Ackerman, Freer, and Patel(2014) to other homogeneous structures.
It can be used to define algorithmic randomness for suchstructures (as suggested by Nies and Fouché).
Kn
S7ℕ
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,also require that no n‑tuples 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
The topology of graphonsNeighborhood distance:
and mod out by .
Example: is a singleton space.
THM: (Freer & R.) (informal) If is a random‑freeuniversal graphon obtained via a ʺtameʺ extensionmethod, then is not compact in the topology.
(x, y) = > W(x, . ) + W(y, . ) = ∫ |W(x, z) + W(y, z)| dzrW >1
(x, y) = 0rW
W(x, y) z p
W
W 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
Compactness of graphonsThis contrasts the following result due to Lovasz andSzegedy.
THM: If a pure graphon misses some signedbipartite graph , then
(i) is compact, and (ii) has Minkowski dimension at most .
(J, W)F
(J, )rW
10v(F)
Complexity of universal graphonsconstruction: fully
randomtamedeterministic
generaldeterministic
complexityof graphon
low
(singleton)
high
(not compact,infiniteMinkowskidimension)
?