The Quasi-Randomness
of
Hypergraph Cut Properties
Asaf Shapira & Raphael Yuster
Background
Chung, Graham, and Wilson ’89, Thomason ‘87: 1. Defined the notion of a p-quasi-random graph = A graph
that “behaves” like a typical graph generated by G(n,p).
2. Proved that several “natural” properties that hold in G(n,p)
whp “force” a graph to be p-quasi-random.
Abstract Question: When can we say that a single graph
behaves like a random graph?
“Concrete” problem: Which graph properties “force” a graph
to behave like a “truly” random one.
The CGW Theorem
Theorem [CGW ‘89]: Fix any 0<p<1, and let G=(V,E) be a
graph on n vertices. The following are equivalent:
1. Any set U V spans ½p|U|2 edges
2. Any set U V of size ½n spans ½p|U|2 edges
3. G contains ½pn2 edges and p4n4 copies of C4
4. Most pairs u,v have co-degree p2n
Definition: A graph that satisfies any (and therefore all) theabove properties is p-quasi-random, or just quasi-random.
Note: All the above hold whp in G(n,p).
Quasi-Random Properties
Definition: Say that a graph property is quasi-random if it isequivalent to the properties in the CGW theorem.
“The” Question: Which graph properties are quasi-random?
Any (reasonable) property that holds in G(n,p) whp?
Example 1: Having ½pn2 edges and p3n3 copies of K3
is not a quasi-random property.
Recall that if we replace K3
with C4 we do get a quasi-random property.
No!
Example 2: Having degrees pn is not a quasi-random prop.
…but having co-degrees p2n is a quasi-random property.
The Chung-Graham Theorem
Theorem [Chung-Graham ’89]:
1. Having ¼ pn2 edges crossing all cuts of size (½n,½n) is not a quasi-random property.
2. For any 0<<½ , having (1-)pn2 edges crossing all cuts of size (n,(1-)n) is a quasi-random property.
[CG ‘89] Gave two proofs of (2). One using a counting argument, and another algebraic proof based on the rank of certain intersection matrices.
To get (1), take an Independent set on n/2 vertices, a cliqueon the rest, and connect them with a random graph.
[Janson ‘09] Gave another proofs of (2), using graph limits.
Quasi-Random Hypergraphs
What is a quasi-random hypergraph?
Answer 1: The “obvious” generalization of quasi-random
graphs. Every set of vertices has the “correct” edge density.
Definition: This is called “weak” quasi-randomness.
Why? Because it does not imply certain things that are
implied by quasi-randomness in graphs.
Answer 2: “Strong” quasi-randomness.
Fact: Strong Quasi-Randomness Weak Quasi-Randomness
Notation: “P is Quasi-Random” means P Weak Quasi-Rand
Our Main Results
Theorem 1 [S-Yuster ’09]:
1. If = (1/k,…,1/k) then P is not quasi-random.
Definition: Let =(1,…,k) satisfy 0<i<1 and I =1.
Let P be the following property of k-uniform hypergraphs:
Any (1n,…,kn)-cut has the correct number of edges
crossing it.
2. If (1/k,…,1/k) then P is quasi-random.
Theorem 2 [S-Yuster ’09]:1. When = (½,½) the only way a non-quasi random graph can satisfy P is the “trivial” one.2. Same result conditionally holds in hypergraphs.
Proof Overview
Theorem: If = (1/k,…,1/k) then P is not quasi-random.
Definition: Let =(1,…,k) satisfy 0<i<1 and I =1. We let P be the
following property of k-uniform hypergraphs: Any (1n,…,kn)-cut has the
correct number of edges crossing it.
0 2pp
[CG‘89] For k=2:
For arbitrary k2:
i verticesk-i vertices2ip/k
Proof OverviewTheorem: If (1/k,…,1/k) then P is not quasi-random.
Proof (of k=2): It is enough to show that every set of verticesof size n has the correct edge density. Let A be such a set.
p2p1 p3
Let 0c, and “re-shuffle” the partition by randomly picking cn vertices from A and (-c)n vertices from V-A.
1. We know the expected number of edges in the new cut.
2. This expectation is a linear function in p1 , p2 , p3.
3. Using c{0,,/2} we get 3 linear equations, which have a unique solution p1=p2=p3=p when 1/2.
A |A|=n
V-A |V-A|=(1-)n
Proof OverviewTheorem 2: When = (½,½) the only way a non-quasi random graph can satisfy P is the “trivial” one.
Instead of thinking about graphs, let’s consider the problemof assigning weights to the edges of the complete graph,s.t. for any (n/2,n/2)-cut, the total weight crossing it is p.
p
The random graph
0 2pp
The example showing that P is not quasi random
What is “trivial”? Two ways a graph can satisfy P
There are many such solutions
Proof Overview
What is “trivial”? Two ways a graph can satisfy P
p
The random graph
(1)
0 2pp
The example showing that P is not quasi random
(2)Satisfying P is equivalent to satisfying a set of linear equations:
1. Unknowns are the weights of the edges.
2. We have one linear equation for any (n/2,n/2)-cut
Definition: A trivial solution is any affine combination of solution (1) and (a collection of) solution (2).
Proof Overview
p
The random graph
(1)
0 2pp
The example showing that P is not quasi random
(2)
Definition: A trivial solution is any affine combination of solution (1) and (a collection of) solution (2).
Theorem 2: When = (½,½) the only solutions satisfying P are the trivial ones.
Proof Overview
Definition: A trivial solution is any affine combination of solution (1) and (2).
Theorem 2: When = (½,½) the only solutions satisfying P are the trivial ones.
Proof: Any solution is a solution of the linear system Ax=p.
Recall Satisfying P is equivalent to satisfying a set of linear equations: 1. Unknowns are the weights of the edges. 2. We have one equation for any (n/2,n/2)-cut
p The random graph (1)
0 2pp The example showing that
P is not quasi random
(2)
Step 2: span[solutions (2) – (1)] has dimension n-1.
Step 1: rank(A)
Definition: Let us write this as Ax=p.
Note: A is an matrix.
Proof Overview
Definition: A trivial solution is any affine combination of solution (1) and (2).
p The random graph (1)
0 2pp The example showing that
P is not quasi random
(2)
Step 2: trivial solutions have dimension n-1.
Proof: For every solution (2) consider the vector of pairs (v1,vi). 2p0
p
n/2-1 of the entries are 2p, the other are p.
After subtracting (1) from these vectors, we get, for everysubset S [n-1] of size n/2, a vector vS, satisfying:
1. vS(i) = 0 if i S.2. vS(i) = p if i S .
This collection spans Rn-1
Proof Overview
Note: A is an matrix
n/2 n/2n/2 n/2-1
Proof: Take the vector vS corresponding to some cut.
vS,t = vector of cut obtained by moving t from S to V-S.
We first prove that matrix of (n/2,n/2-1)-cuts has full rank.
(vS,t – vS) 1-1 t
(vS,t – vS) tvs-
S V-S
C-c
Step 1: rank(A)
Proof Overview
Conclusion: A spans the rows of the matrix I(2,n/2,n-1)
n/2-element subsets of [n-1]
2-element subsets of [n-1]
IS,T = 1 iif ST
[Gottlieb ‘66]: rank(I(2,h,k)) = .
Step 1: rank(A)
Concluding Remarks
Coro: If (1/3,1/3,1/3) then P is quasi-random
Definition: Let =(1,2,3) satisfy 0<i<1 and i =1.
Let P be the following graph property:
Any (1n,2n,3n)-cut is crossed by the “correct” number of K3.
Open Problem: What happens when = (1/3,1/3,1/3)?
Proof: Replace every K3 with a 3-hyper edge. We geta hypergraph satisfying P, which must be quasi-random byTheorem 1. This means that in the graph, any set of verticeshas the “correct” number of K3. A theorem of Simonovits-Sosimplies that the graph must be quasi-random.
Thank You
Background
Relation to (theoretical) computer science:
1. Conditions of randomness that are verifiable in polynomial
time. For example, using number of C4, or using 2(G).
2. Algorithmic version of Szemeredi’s regularity-lemma:
[Alon et al. ’95] Uses equivalence between quasi-randomenss
and co-degrees.
Background
Relation to Extremal Combinatorics:
1. Central in the strong hypergraph generalizations of
Szemeredi’s regularity-lemma [RSSN’04, Gowers’06, Tao’06].
Quasi-Random Groups [Gowers ‘07]
Generalized Quasi-Random Graphs [Lovasz-Sos ‘06]
Quasi-Random Set Systems [Chung-Graham ‘91]
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