Post on 11-Mar-2018
Introduction Random Graphs Lattices & Geometry Treewidth Open Questions
Modularity on Random Graphs,Lattices and Embedded Graphs
Colin McDiarmid, Fiona Skerman
University of Oxford
skerman@stats.ox.ac.uk
Introduction Random Graphs Lattices & Geometry Treewidth Open Questions
Introduction
Introduction Edge Expansion & Random Cubic Lattices Open QuestionsIntroduction Edge Expansion & Random Cubic Lattices Open Questions
Introduction
Figure 1: Modularity used to study e�ect ofschizophrenia on brain cell interaction2
First introduced by Newman andGirvan 2004 as a measure of how wella network is clustered intocommunities.
Many clustering algorithms; based onoptimising modularity; includingprotein discovery and social networks
Finding the optimal partition of a
graph shown to be NP-hard by
Brandes. et. al. 2007
Disrupted modularity and local connectivity of brain functionalnetworks in childhood-onset schizophrenia.Alexander-Bloch A.F., Gogtay N., Meunier D., Birn R., Clasen L.,Lalonde F., Lenroot R., Giedd J., Bullmore E.T.
Modularity
Introduction Random Graphs Lattices & Geometry Treewidth Open Questions
Defintion
Let G be a graph on m edges and A a vertex partition of V (G )
Modularity
Max. Modularity
qA(G ) :=∑
A∈A
(e(A)
m−(degsum(A)
2m
)2)
q(G ) := maxA
qA(G )
Introduction Random Graphs Lattices & Geometry Treewidth Open Questions
Defintion
Let G be a graph on m edges and A a vertex partition of V (G )
Modularity
Max. Modularity
qA(G ) :=∑
A∈A
(e(A)
m−(degsum(A)
2m
)2)
q(G ) := maxA
qA(G )
Notice the sum naturally splits into two components.
Edge contribution Degree tax
qEA(G ) :=∑
A∈A
e(A)
mqDA(G ) :=
∑
A∈A
(degsum(A)
2m
)2
Introduction Random Graphs Lattices & Geometry Treewidth Open Questions
Introduction Edge Expansion & Random Cubic Lattices Open Questions
Defintion
Let G be a graph on m edges,
Modularity
Max. Modularity
qA(G ) :=X
A2A
e(A)
m�✓
degsum(A)
2m
◆2!
q(G ) := maxA
qA(G )
Notice the sum naturally splits into two components.
Edge contribution Degree tax
qEA(G ) :=
X
A2A
e(A)
mqD
A(G ) :=X
A2A
✓degsum(A)
2m
◆2
Introduction Random Graphs Lattices & Geometry Treewidth Open Questions
Introduction Edge Expansion & Random Cubic Lattices Open Questions
Defintion
Let G be a graph on m edges,
Modularity
Max. Modularity
qA(G ) :=X
A2A
e(A)
m�✓
degsum(A)
2m
◆2!
q(G ) := maxA
qA(G )
Notice the sum naturally splits into two components.
Edge contribution Degree tax
qEA(G ) :=
X
A2A
e(A)
mqD
A(G ) :=X
A2A
✓degsum(A)
2m
◆2
Example Graph
Introduction Edge Expansion & Random Cubic Lattices Open Questions
Edge contribution Degree tax
qEA(G ) :=
�
A�A
|E (C )|m
qDA(G ) :=
�
A�A
��v�C deg(v)
2m
�2
Example Graph
1
Introduction Random Graphs Lattices & Geometry Treewidth Open Questions
Introduction Edge Expansion & Random Cubic Lattices Open Questions
Defintion
Let G be a graph on m edges,
Modularity
Max. Modularity
qA(G ) :=X
A2A
e(A)
m�✓
degsum(A)
2m
◆2!
q(G ) := maxA
qA(G )
Notice the sum naturally splits into two components.
Edge contribution Degree tax
qEA(G ) :=
X
A2A
e(A)
mqD
A(G ) :=X
A2A
✓degsum(A)
2m
◆2
3 Possible Partitions
Introduction Edge Expansion & Random Cubic Lattices Open Questions
Edge contribution Degree tax
qEA(G ) :=
X
A2A
|E (C )|m
qDA(G ) :=
X
A2A
✓�v2C deg(v)
2m
◆2
3 Possible Partitions
2 4 3
qEA1
= 0.96, qDA1
= 0.56 qEA2
= 0.94, qDA2
= 0.50 qEA3
= 0.59, qDA3
= 0.29
qA1= 0.40 qA2
= 0.44 qA3= 0.30
Introduction Random Graphs Lattices & Geometry Treewidth Open Questions
Introduction Edge Expansion & Random Cubic Lattices Open Questions
Defintion
Let G be a graph on m edges,
Modularity
Max. Modularity
qA(G ) :=X
A2A
e(A)
m�✓
degsum(A)
2m
◆2!
q(G ) := maxA
qA(G )
Notice the sum naturally splits into two components.
Edge contribution Degree tax
qEA(G ) :=
X
A2A
e(A)
mqD
A(G ) :=X
A2A
✓degsum(A)
2m
◆2
3 Possible Partitions
Introduction Edge Expansion & Random Cubic Lattices Open Questions
Edge contribution Degree tax
qEA(G ) :=
X
A2A
|E (C )|m
qDA(G ) :=
X
A2A
✓�v2C deg(v)
2m
◆2
3 Possible Partitions
2 4 3
qEA1
= 0.96, qDA1
= 0.56 qEA2
= 0.94, qDA2
= 0.50 qEA3
= 0.59, qDA3
= 0.29
qA1= 0.40 qA2
= 0.44 qA3= 0.30
Introduction Edge Expansion & Random Cubic Lattices Open Questions
Edge contribution Degree tax
qEA(G ) :=
X
A2A
|E (C )|m
qDA(G ) :=
X
A2A
✓�v2C deg(v)
2m
◆2
3 Possible Partitions
2 4 3
qEA1
= 0.96, qDA1
= 0.56 qEA2
= 0.94, qDA2
= 0.50 qEA3
= 0.59, qDA3
= 0.29
qA1= 0.40 qA2
= 0.44 qA3= 0.30
Introduction Random Graphs Lattices & Geometry Treewidth Open Questions
Random r-Regular Graphs
Theorem (McDiarmid, S.)
Let Gr be an r -regular random graph. Then with high probability -
r = 3 4 5 6 7 8 9 10
q(Gr ) > 0.66 0.50 0.40 0.33 0.28 0.25 0.22 0.21q(Gr ) < 0.80 0.69 0.61 0.55 0.51 0.48 0.45 0.41
Introduction Random Graphs Lattices & Geometry Treewidth Open Questions
Random r-Regular Graphs
Theorem (McDiarmid, S.)
Let Gr be an r -regular random graph. Then with high probability -
r = 3 4 5 6 7 8 9 10
q(Gr ) > 0.66 0.50 0.40 0.33 0.28 0.25 0.22 0.21q(Gr ) < 0.80 0.69 0.61 0.55 0.51 0.48 0.45 0.41
Lower Bounds r = 3, . . . , 8Hamilton cycle construction,
√n parts.
Lower Bounds r = 9, 10Two equal sized parts.
Introduction Random Graphs Lattices & Geometry Treewidth Open Questions
Random r-Regular Graphs
Upper Bounds
edge expansion of small setsiu(G ) := min|U|≤un
1|U|e(U,V \U)
Theorem (McDiarmid, S.)
Let G be an r -regular graph. Suppose for all u ≤ 1/2 thatu + iu(G )/r ≥ α. Then,
q(G ) ≤ max{1− α, 3/4}.
Results of Kolesnik and Wormald1 give numerical bounds on edgeexpansion of small sets in random regular graphs whp.
1B. Kolesnik and N. Wormald, Lower bounds for the isoperimetric numbers of random regular graphs,
SIAM Journal on Discrete Mathematics 28, 553 (2014)
Introduction Random Graphs Lattices & Geometry Treewidth Open Questions
Simulations
Modularity of Random r-Regular Graphs (whp)
r = 3 4 5 6 7 8 9 10
q(Gr ) > 0.66 0.50 0.40 0.33 0.28 0.25 0.22 0.21s(G ∗r ) = 0.68 0.53 0.44 0.38 0.34 0.31 0.28 0.26q(Gr ) < 0.80 0.69 0.61 0.55 0.51 0.48 0.45 0.41
Introduction Random Graphs Lattices & Geometry Treewidth Open Questions
Simulations
Modularity of Random r-Regular Graphs (whp)
r = 3 4 5 6 7 8 9 10
q(Gr ) > 0.66 0.50 0.40 0.33 0.28 0.25 0.22 0.21s(G ∗r ) = 0.68 0.53 0.44 0.38 0.34 0.31 0.28 0.26q(Gr ) < 0.80 0.69 0.61 0.55 0.51 0.48 0.45 0.41
1. Graphs generated via configuration model
MATLAB, 10 000 nodes, reject if not simple graph.
[Image Credit: D. Nykamp, Univ. Minnesota]
Introduction Random Graphs Lattices & Geometry Treewidth Open Questions
Simulations
Modularity of Random r-Regular Graphs (whp)
r = 3 4 5 6 7 8 9 10
q(Gr ) > 0.66 0.50 0.40 0.33 0.28 0.25 0.22 0.21s(G ∗r ) = 0.68 0.53 0.44 0.38 0.34 0.31 0.28 0.26q(Gr ) < 0.80 0.69 0.61 0.55 0.51 0.48 0.45 0.41
1. Graphs generated via configuration model
MATLAB, 10 000 nodes, reject if not simple graph.
2. Modularity estimated via Louvain method
Etienne Lefebvre 2007,Vincent Blondel, Jean-Loup Guillaume and Renaud Lambiotte 2008.MATLAB implementation by Antoine Scherrer ENS Lyon.(available from Vincent Blondel’s website)results averaged over 10 trials.
Introduction Random Graphs Lattices & Geometry Treewidth Open Questions
Modularity of Random r-Regular Graphs (whp)
X - results of simulations6
-
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
3 4 5 6 7 8 9 10
q(Gr)
s(G⇤r)
r
Figure 1: Simulation results for n = 10, 000 nodes and degrees r = 3, . . . , 10. Each crossindicates the optimal modularity returned averaged over ten sampled graphs. The almost surerange proven in Theorem 1.2 for large n is shown in blue on the same graph. [The upper boundfor degrees 7,8,9 is missing and still needs to be calculated.]
References
[1] Aaron F Alexander-Bloch, Nitin Gogtay, David Meunier, Rasmus Birn, Liv Clasen, FrancoisLalonde, Rhoshel Lenroot, Jay Giedd, and Edward T Bullmore. Disrupted modularity andlocal connectivity of brain functional networks in childhood-onset schizophrenia. Frontiersin systems neuroscience, 4, 2010.
[2] James P Bagrow. Communities and bottlenecks: Trees and treelike networks have highmodularity. Physical Review E, 85(6):066118, 2012.
[3] Vincent D Blondel, Jean-Loup Guillaume, Renaud Lambiotte, and Etienne Lefebvre. Fastunfolding of communities in large networks. Journal of Statistical Mechanics: Theory andExperiment, 2008(10):P10008, 2008.
[4] Hans L Bodlaender. A partial k-arboretum of graphs with bounded treewidth. Theoreticalcomputer science, 209(1):1–45, 1998.
[5] Fabien De Montgolfier, Mauricio Soto, and Laurent Viennot. Asymptotic modularity ofsome graph classes. In Algorithms and Computation, pages 435–444. Springer, 2011.
[6] Zdenek Dvorak and Sergey Norin. Treewidth of graphs with balanced separations. 08 2014.
[7] Brett Kolesnik and Nick Wormald. Lower bounds for the isoperimetric numbers of randomregular graphs. SIAM Journal on Discrete Mathematics, 28(1):553–575, 2014.
[8] AV Kostochka and LS Melnikov. On a lower bound for the isoperimetric number of cubicgraphs. Probabilistic methods in discrete mathematics, 1:251–265, 1993.
5
Introduction Random Graphs Lattices & Geometry Treewidth Open Questions
Phase Transition in Erdos-Renyi
Connected components in the random graph Gn,c/n.
c < 1 c = 1 c > 1
Critical Phase
How big is the largest component in G(n, p), when pn = 1 + " for " = o(1) ?
[ BOLLOBÁS 84; ŁUCZAK 90; JANSON–KNUTH–ŁUCZAK–PITTEL 93; BOLLOBÁS–RIORDAN 13+]
If " n1/3 ! �1, whp L(n) = o(n2/3).
If " n1/3 ! �, a constant, whp L(n) = ⇥(n2/3).
If " n1/3 ! 1, whp L(n) = (1 + o(1)) 2"n.
2/3<< 2/3 2/3~ >>n nn
B Uniform random graph G(n, m): m = n/2 + s, s n�2/3 = " n1/3
Mihyun Kang Phase Transitions in Random Discrete Structures
O(log n) ∼ n2/3 ∼ n
[Image Credit: M. Kang, TU Graz.]
Introduction Random Graphs Lattices & Geometry Treewidth Open Questions
Phase Transition in Erdos-Renyi
Connected components in the random graph Gn,c/n.
c < 1 c = 1 c > 1
Critical Phase
How big is the largest component in G(n, p), when pn = 1 + " for " = o(1) ?
[ BOLLOBÁS 84; ŁUCZAK 90; JANSON–KNUTH–ŁUCZAK–PITTEL 93; BOLLOBÁS–RIORDAN 13+]
If " n1/3 ! �1, whp L(n) = o(n2/3).
If " n1/3 ! �, a constant, whp L(n) = ⇥(n2/3).
If " n1/3 ! 1, whp L(n) = (1 + o(1)) 2"n.
2/3<< 2/3 2/3~ >>n nn
B Uniform random graph G(n, m): m = n/2 + s, s n�2/3 = " n1/3
Mihyun Kang Phase Transitions in Random Discrete Structures
O(log n) ∼ n2/3 ∼ n
q(Gn,c/n)→ 1 q(Gn,c/n) 6→ 1
[Image Credit: M. Kang, TU Graz.]
Introduction Random Graphs Lattices & Geometry Treewidth Open Questions
Statistical Physics
Introduction Edge Expansion & Random Cubic Lattices Open QuestionsIntroduction Edge Expansion & Random Cubic Lattices Open Questions
Statistical Physics
Theorem (R. Guimera et. al.3)
Fix d , z 2 N+ and let R be an n-vertex complete rectangular section of
Zdz . Then q(R) � 1 � (d + 1)
�z+12d
� dd+1 n� 1
d+1
Definition Zdz : d-dim lattice with axis-parallel edges lengths 1, ..., z .
� � � � � �� � � � � �� � � � � �� � � � � �
� � � � � �� � � � � �� � � � � �� � � � � �
1
� � � � � �� � � � � �� � � � � �� � � � � �
� � � � � �� � � � � �� � � � � �� � � � � �
1
Figure 2: Rectangular sections of Z22 (left) and Z2
3 (right).
3R. Guimera, M. Sales-Pardo and L.A. Amaral, Modularity from fluctuations in random graphs and complex
networks, Phys. Rev. E 70 (2) (2004) 025101.
Introduction Random Graphs Lattices & Geometry Treewidth Open Questions
Statistical Physics
Introduction Edge Expansion & Random Cubic Lattices Open QuestionsIntroduction Edge Expansion & Random Cubic Lattices Open Questions
Statistical Physics
Theorem (R. Guimera et. al.3)
Fix d , z 2 N+ and let R be an n-vertex complete rectangular section of
Zdz . Then q(R) � 1 � (d + 1)
�z+12d
� dd+1 n� 1
d+1
Definition Zdz : d-dim lattice with axis-parallel edges lengths 1, ..., z .
� � � � � �� � � � � �� � � � � �� � � � � �
� � � � � �� � � � � �� � � � � �� � � � � �
1
� � � � � �� � � � � �� � � � � �� � � � � �
� � � � � �� � � � � �� � � � � �� � � � � �
1
Figure 2: Rectangular sections of Z22 (left) and Z2
3 (right).
3R. Guimera, M. Sales-Pardo and L.A. Amaral, Modularity from fluctuations in random graphs and complex
networks, Phys. Rev. E 70 (2) (2004) 025101.
Introduction Edge Expansion & Random Cubic Lattices Open Questions
Statistical Physics
Theorem (R. Guimera et. al.3)
Fix d , z 2 N+ and let R be an n-vertex complete rectangular section of
Zdz . Then q(R) � 1 � (d + 1)
�z+12d
� dd+1 n� 1
d+1 = 1 ���m� 1
d+1
�
Definition Zdz : d-dim lattice with axis-parallel edges lengths 1, ..., z .
� � � � � �� � � � � �� � � � � �� � � � � �
� � � � � �� � � � � �� � � � � �� � � � � �
1
� � � � � �� � � � � �� � � � � �� � � � � �
� � � � � �� � � � � �� � � � � �� � � � � �
1
Figure 2: Rectangular sections of Z22 (left) and Z2
3 (right).
3R. Guimera, M. Sales-Pardo and L.A. Amaral, Modularity from fluctuations in random graphs and complex
networks, Phys. Rev. E 70 (2) (2004) 025101.
Introduction Random Graphs Lattices & Geometry Treewidth Open QuestionsIntroduction Edge Expansion & Random Cubic Lattices Open QuestionsIntroduction Edge Expansion & Random Cubic Lattices Open Questions
We extend this result to include any subgraph of the lattice Zdz .
Theorem (McDiarmid, S.)
Fix d , z 2 N+, and let L be an m-edge subgraph of Zdz . Then
q(L) = 1 � O�m� 1
d+1
�as m ! 1.
� � � � � � � � �� � � � � � � � �� � � � � � � � �� � � � � � � � �� � � � � � � � �� � � � � � � � �� � � � � � � � �
e(S1) = 2 e(�S1) = 5 w(S1) = 4.5 vol(S1) = 4 �(S1) = 1.125
e(S2) = 11 e(�S2) = 12 w(S2) = 17 vol(S2) = 20 �(S2) = 0.85
e(S3) = 7 e(�S3) = 4 w(S3) = 9 vol(S3) = 12 �(S3) = 0.75
Figure 2.9: Some example near-squares with their weights, w, volume, vol, and density,
�, shown.
36
Figure 3: A subgraph of Z21
Introduction Random Graphs Lattices & Geometry Treewidth Open Questions
Graph Geometry
An embedding α of a graph G into Rd is said to have warp ` if
∀x , y ∈ V (G ), 1 ≤ |α(x)− α(y)|∀uv ∈ E (G ), |α(u)− α(v)| ≤ `
Theorem (McDiarmid, S.)
Let G be a graph, d ≥ 2. Suppose α : V (G )→ Rd embeds G
with warp `. Then q(G ) ≥ 1− O(`d−1d+1m
−1d+1 ).
↵�!
1
0 1
R2
6
-
For d dimensions the result will extend to;
q(G) � 1 �p
d⌧m� 1d+1 � �m� 1
d+1 (1 + O(m� 1d+1 )).
Definition 1 (distortion). An embedding ↵ of a graph G into metric space (X, �) is said to have distortion
` if for all x, y 2 V (G),
dG(x, y) �(↵(x),↵(y)) `dG(x, y).
We show that a bounding distortion bounds the maximum degree of our graph. notation
denote by � the frac-
tion of space covered
by balls in the pack-
ing and � be the cen-
tre density, i.e. the
number of spheres per
unit volume.
good to cite [?]
As the following example shows, having small distortion is not enough to guarantee a high modularity.
Example 6 (Distortion 1, modularity 0.). Fix a bipartite graph G = Km,n on vertex parts U, V . Then
define our embedding ↵ : U [ V ! Rn, by ↵(u) = (0, 0, . . . , 0), 8u 2 U and ↵(v) = (1, 0, . . . , 0), 8v 2 V .
Observe this embedding has distortion 1; as all edges in G are between U and V . cite the lemma that
says a bipartite graph
has modularity zero.
?
If we have both small distortion and a minimum vertex separation then we can achieve modularity bounds.
Theorem 7. Let G be a graph and suppose ↵ : V (G) ! R3 is an embedding with distortion ` and min
vertex separation of �. Then
q(G) � 1 � m� 14�2⌧ + 2
p2
3 �⌧3�.
We pause to establish some graph properties implied by the distortion and vertex separation properties
of our embedding which will help to prove our theorem.
Lemma 8. Suppose embedding ↵ : G ! R3 has distortion `. Then �(G) 13p
2�(l + 1
2 )3.technically � maps
V (G) � X not
all of G � X -
but we want to know
where the edges are
- otherwise distortion
defn makes no sense.
Think about this!
Proof. Fix u 2 V (G). As the distortion is bounded, 8v 2 �(u), d(↵(u),↵(v)) `. Also observe that any
two neighbours v, w 2 �(u) have graph distance at most two following the path vuw. Possibly, v, w are
neighbours and so d(↵(v),↵(w)) � 1.
Construct a set of open balls of radius a half about each neighbour of u. Let;
B = {Bv( 12 ) : v 2 �(u)}.
Note 8v, w 2 �(u); Bv( 12 ) \ Bw( 1
2 ) = ? and Bv( 12 ) ⇢ Bu(` + 1
2 ). Hence by the sphere packing bound of
Hales [2].
6
Figure 3: Example: The bipartite graph K3,5 embeds into R2 such that all edges are of unit length.
Example 7 (Ratio of min/max edge lengths 1, modularity 0.). Fix a bipartite graph G = Km,n on
vertex parts U, V . Then define our embedding ↵ : U [ V ! Rd, by ↵(u) = (0, 0, . . . , 0), 8u 2 U and
↵(v) = (1, 0, . . . , 0), 8v 2 V . Observe that in this embedding all edges have unit length.
We pause to establish Lemmas 8 and 9 which show graph properties are implied by the warp and vertex
separation of our embedding. These will enable us to prove Theorem 4.
Definition 3 (unit packing). For X ⇢ Rd define U(X) maximum number of non-overlapping unit balls
each of whose centres lies within X.
Lemma 8. Let G be a graph. Suppose ↵ : G ! Rd embeds G such that
8x, y 2 V (G), 1 |↵(x) � ↵(y)| and 8uv 2 E(G), |↵(u) � ↵(v)| `.
Then �(G) U(B2`) � 1.
Proof. Define ↵0 by stretching the embedding along each axis by a factor of 2. The maximum edge length
is now at most 2` and minimum vertex separation at least 2.
Fix u 2 V (G). Construct a set B of open unit balls about the embedding of each neighbour of u and
about u itself,
B := {B1(↵0(v)) : v 2 �(u) [ {u}}.
All centres are at least distance two apart so will not intersect i.e. 8v, w 2 �(u); B1(↵0(v))\B1(↵
0(w)) = ?and each ball has centre within B2`(↵
0(u)). Now, deg(u) + 1 = |�(u) [ {u}| = |B| U(B2`) and we are
done.
Lemma 9. Let Bdr be a ball of radius r and Ad
r be a hypercube of side length r in Rd. Then,
U(Bdr ) (r + 1)d and U(Ad
r) rd
vol(Bd1 )
Proof. Note the maximum number of vertices intersecting Br must be less than the number of unit spheres
that can pack into Br+1. Thus, |U(Br)|vol(B1) vol(Br+1) and so |U(Br)| (r+1)d. The second bound
follows similarly.
Proof. of Theorem 4. Define ↵0 by stretching the embedding along each axis by a factor of 2. The
maximum edge length is now at most 2` and minimum vertex separation at least 2. Thus by Lemma 8
5
Introduction Random Graphs Lattices & Geometry Treewidth Open Questions
Idea of Proof
Assumptions: graph G and mapping α : G → Rd such that
∀x , y ∈ V (G ), 1 ≤ |α(x)− α(y)|∀uv ∈ E (G ), |α(u)− α(v)| ≤ `
min vertex separation
max edge length
These imply: ∆(G ) ≤ U(B2`) #unit spheres in ball of radius 2`.
Let Hs be a hypercube of side length s. Then the max sum of degreesof vertices embedded inside of any hypercube Hs is..
degsumα(G)(Hs) ≤ ∆(G )U(Hs) #unit spheres in hypercube.
Introduction Random Graphs Lattices & Geometry Treewidth Open Questions
Idea of Proof
Assumptions: graph G and mapping α : G → Rd such that
∀x , y ∈ V (G ), 1 ≤ |α(x)− α(y)|∀uv ∈ E (G ), |α(u)− α(v)| ≤ `
min vertex separation
max edge length
These imply: ∆(G ) ≤ U(B2`) #unit spheres in ball of radius 2`.
degsumα(G)(Hs) ≤ ∆(G )U(Hs) #unit spheres in hypercube.
Lemma (McDiarmid, S.)
Let G be a graph. Suppose α : V (G )→ Rd embeds G with maxedge length `. Then for s � `;
q(G ) ≥ 1− `√d
s− degsum(Hs)
2m.
Introduction Random Graphs Lattices & Geometry Treewidth Open Questions
Random Graphs on Surfaces via Treewidth
Trees of bounded degree, Bagrow 2012.
Trees with degree o(n1/5), Montgolfier et. al. 2011.
Theorem (McDiarmid, S.)
Let G be a graph with m edges, treewidth tw(G ) = t andmaximum degree ∆ = ∆(G ). Then the modularity q(G ) satisfies
q(G ) ≥ 1− 2((t + 1)∆/m)1/2.
For m = 1, 2, . . . let Gm be a graph with m edges. Iftw(Gm) ·∆(Gm) = o(m) then q(Gm)→ 1 as m→∞.
Introduction Random Graphs Lattices & Geometry Treewidth Open Questions
Random Graphs on Surfaces via Treewidth
Trees of bounded degree, Bagrow 2012.
Trees with degree o(n1/5), Montgolfier et. al. 2011.
Theorem (McDiarmid, S.)
Let G be a graph with m edges, treewidth tw(G ) = t andmaximum degree ∆ = ∆(G ). Then the modularity q(G ) satisfies
q(G ) ≥ 1− 2((t + 1)∆/m)1/2.
For m = 1, 2, . . . let Gm be a graph with m edges. Iftw(Gm) ·∆(Gm) = o(m) then q(Gm)→ 1 as m→∞.
Corollary
Fix a surface S and let GS(n) be chosen uniformly from all graphson n vertices which embed into S with no crossing edges. Thenwith high probability q(GS(n)) ≥ 1− O(ln n/
√n).
Introduction Random Graphs Lattices & Geometry Treewidth Open Questions
Open Questions
qEA(G ) :=∑
A∈A
e(A)
mqDA(G ) :=
∑
A∈A
(degsum(A)
2m
)2
1. Edge expansion of small sets
Is there a cubic graph G for which iu(G ) ≥ 1, ∀u?
October 12, 2014
sc sc sc sc sc sc sc sc sc
1
Modularity =Edge contribution Degree tax-
Introduction Random Graphs Lattices & Geometry Treewidth Open Questions
Open Questions
qEA(G ) :=∑
A∈A
e(A)
mqDA(G ) :=
∑
A∈A
(degsum(A)
2m
)2
1. Edge expansion of small sets
Is there a cubic graph G for which iu(G ) ≥ 1, ∀u?
2. Improve bounds in random cubic.
Let G3 be a random cubic graph. Then whp,
0.66 ≤ q(G3) ≤ 0.8
Is the lower bound optimal? i.e. q(G3) = 2/3 whp?Construction based on finding a Hamilton cycle, then cutting intostrips of
√n vertices.
Modularity =Edge contribution Degree tax-
Introduction Random Graphs Lattices & Geometry Treewidth Open Questions
bonus: a nice proof
Edge expansion Small sets modularity
i(G ) := min|U|≤n/2
1|U|e(U,V \U) qδ(G ) := max
A : |A|<δn,∀A∈AqA(G )
Theorem (McDiarmid, S.)
For any ε > 0 there exists δ > 0 such that the following holds. Letr ≥ 3 and let G be an r -regular graph with at least δ−1 vertices.Then,
qδ(G ) < 1− 2r i(G ) + ε.
Observation q(G ) ≤ max{1− 1r i(G ), 34}.
Why? Fix A = {A1, . . . ,Ak}.(a) If some |Ai | > n/2 then degree tax is at least (|Ai |r/rn)2 > 1
4 .(b) If all |Ai | ≤ n/2 use edge expansion. The number of edgesbetween parts is 1
2
∑i e(Ai ,V \Ai ) ≥ 1
2 |Ai |i(G ) = 12 i(G )n.
So the edge contribution is less than 1− 2rn
12 i(G )n = 1− 1
r i(G ).
Introduction Edge Expansion & Random Cubic Lattices Open Questions
Lemma (McDiarmid, S.)
Fix vertices [n] with weights w(1) � . . . � w(n) � 0.Let M be a perfect matching such that |a � b| t 8edges ab 2 M.
For any red-blue colouring of M,
���X
v red
w(v) �X
u blue
w(u)��� t (w(1) � w(n)).
Introduction Edge Expansion & Random Cubic Lattices Open Questions
Lemma (McDiarmid, S.)
Fix vertices [n] with weights w(1) � . . . � w(n) � 0.Let M be a perfect matching such that |a � b| t 8edges ab 2 M.
For any red-blue colouring of M,
���X
v red
w(v) �X
u blue
w(u)��� t (w(1) � w(n)).
t = 1
�
�
�
�R
0� � � � � � � �
�
�
�
�R
0� � � � � � � �
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�
�R
0� � � � � � � �
�
�
�
�R
0� � � � � � � �
�
�
�
�R
0� � � � � � � �
1
Introduction Edge Expansion & Random Cubic Lattices Open Questions
Lemma (McDiarmid, S.)
Fix vertices [n] with weights w(1) � . . . � w(n) � 0.Let M be a perfect matching such that |a � b| t 8edges ab 2 M.
For any red-blue colouring of M,
���X
v red
w(v) �X
u blue
w(u)��� t (w(1) � w(n)).
t = 1
�
�
�
�R
0� � � � � � � �
�
�
�
�R
0� � � � � � � �
�
�
�
�R
0� � � � � � � �
�
�
�
�R
0� � � � � � � �
�
�
�
�R
0� � � � � � � �
1
Introduction Edge Expansion & Random Cubic Lattices Open Questions
Lemma (McDiarmid, S.)
Fix vertices [n] with weights w(1) � . . . � w(n) � 0.Let M be a perfect matching such that |a � b| t 8edges ab 2 M.
For any red-blue colouring of M,
���X
v red
w(v) �X
u blue
w(u)��� t (w(1) � w(n)).
t = 1
�
�
�
�R
0� � � � � � � �
�
�
�
�R
0� � � � � � � �
�
�
�
�R
0� � � � � � � �
�
�
�
�R
0� � � � � � � �
�
�
�
�R
0� � � � � � � �
1
���X
v red
w(v)�X
u blue
w(u)��� = w(1)�w(2)+w(3)� . . .+w(7)�w(8)
Introduction Edge Expansion & Random Cubic Lattices Open Questions
Lemma (McDiarmid, S.)
Fix vertices [n] with weights w(1) � . . . � w(n) � 0.Let M be a perfect matching such that |a � b| t 8edges ab 2 M.
For any red-blue colouring of M,
���X
v red
w(v) �X
u blue
w(u)��� t (w(1) � w(n)).
t = 1
�
�
�
�R
0� � � � � � � �
�
�
�
�R
0� � � � � � � �
�
�
�
�R
0� � � � � � � �
�
�
�
�R
0� � � � � � � �
�
�
�
�R
0� � � � � � � �
1
���X
v red
w(v)�X
u blue
w(u)��� = w(1)�w(2)+w(3)� . . .+w(7)�w(8)
w(1) � w(8)
Introduction Edge Expansion & Random Cubic Lattices Open Questions
Lemma (McDiarmid, S.)
Fix vertices [n] with weights w(1) � . . . � w(n) � 0.Let M be a perfect matching such that |a � b| t 8edges ab 2 M.
For any red-blue colouring of M,
���X
v red
w(v) �X
u blue
w(u)��� t (w(1) � w(n)).
t = 3
�
�
�
�R
0� � � � � � � �
�
�
�
�R
0� � � � � � � �
�
�
�
�R
0� � � � � � � �
�
�
�
�R
0� � � � � � � �
�
�
�
�R
0� � � � � � � �
1
Introduction Edge Expansion & Random Cubic Lattices Open Questions
Lemma (McDiarmid, S.)
Fix vertices [n] with weights w(1) � . . . � w(n) � 0.Let M be a perfect matching such that |a � b| t 8edges ab 2 M.
For any red-blue colouring of M,
���X
v red
w(v) �X
u blue
w(u)��� t (w(1) � w(n)).
t = 3
�
�
�
�R
0� � � � � � � �
�
�
�
�R
0� � � � � � � �
�
�
�
�R
0� � � � � � � �
�
�
�
�R
0� � � � � � � �
�
�
�
�R
0� � � � � � � �
1
�
�
�
�R
0� � � � � � � �
�
�
�
�R
0� � � � � � � �
�
�
�
�R
0� � � � � � � �
�
�
�
�R
0� � � � � � � �
�
�
�
�R
0� � � � � � � �
1
Introduction Random Graphs Lattices & Geometry Treewidth Open Questions
Proof Outline (RTP: qA(G ) ≤ 1− 2r i(G ) + ε).
Fix ε and a partition A = A1, . . . ,Ak where δn > |A1| ≥ . . . ≥ |Ak |.
Introduction Edge Expansion & Random Cubic Lattices Open Questions
Proof Outline
G : A1, . . . , An
1
(RTP: qA(G ) � 1 � 2r i(G ) + �).
1
Fix � and a partition A = A1, . . . , Ak where �n > |A1| � . . . � |Ak |.Step 1. Choose t, and factor Kt+1 into perfect matchings.
G : A1, . . . ,Ak
Introduction Random Graphs Lattices & Geometry Treewidth Open Questions
Proof Outline (RTP: qA(G ) ≤ 1− 2r i(G ) + ε).
Fix ε and a partition A = A1, . . . ,Ak where δn > |A1| ≥ . . . ≥ |Ak |.Step 1. Choose t, and factor Kt+1 into perfect matchings.
Introduction Edge Expansion & Random Cubic Lattices Open Questions
Proof Outline
G : A1, . . . , An
1
(RTP: qA(G ) � 1 � 2r i(G ) + �).
1
Fix � and a partition A = A1, . . . , Ak where �n > |A1| � . . . � |Ak |.Step 1. Choose t, and factor Kt+1 into perfect matchings.
G : A1, . . . ,Ak
Introduction Edge Expansion & Random Cubic Lattices Open Questions
Proof Outline
G : A1, . . . , An
. . .
A1
Aj
Ak
1
(RTP: qA(G ) � 1 � 2r i(G ) + �).
2
Fix � and a partition A = A1, . . . , Ak where �n > |A1| � . . . � |Ak |.Step 1. Choose t, and factor Kt+1 into perfect matchings.
These induce pairings on parts in G .
Introduction Random Graphs Lattices & Geometry Treewidth Open Questions
Proof Outline (RTP: qA(G ) ≤ 1− 2r i(G ) + ε).
Fix ε and a partition A = A1, . . . ,Ak where δn > |A1| ≥ . . . ≥ |Ak |.Step 1. Choose t, and factor Kt+1 into perfect matchings.
These induce pairing on parts of G .
Introduction Edge Expansion & Random Cubic Lattices Open Questions
Proof Outline
G : A1, . . . , An
1
(RTP: qA(G ) � 1 � 2r i(G ) + �).
1
Fix � and a partition A = A1, . . . , Ak where �n > |A1| � . . . � |Ak |.Step 1. Choose t, and factor Kt+1 into perfect matchings.
G : A1, . . . ,Ak
Introduction Edge Expansion & Random Cubic Lattices Open Questions
Proof Outline
G : A1, . . . , An
. . .
A1
Aj
Ak
1
(RTP: qA(G ) � 1 � 2r i(G ) + �).
2
Fix � and a partition A = A1, . . . , Ak where �n > |A1| � . . . � |Ak |.Step 1. Choose t, and factor Kt+1 into perfect matchings.
These induce pairings on parts in G .
Introduction Random Graphs Lattices & Geometry Treewidth Open Questions
Proof Outline (RTP: qA(G ) ≤ 1− 2r i(G ) + ε).
Fix ε and a partition A = A1, . . . ,Ak where δn > |A1| ≥ . . . ≥ |Ak |.Step 1. Choose t, and factor Kt+1 into perfect matchings.
These induce pairing on parts of G .
Introduction Edge Expansion & Random Cubic Lattices Open Questions
Proof Outline
G : A1, . . . , An
1
(RTP: qA(G ) � 1 � 2r i(G ) + �).
1
Fix � and a partition A = A1, . . . , Ak where �n > |A1| � . . . � |Ak |.Step 1. Choose t, and factor Kt+1 into perfect matchings.
G : A1, . . . ,Ak
Introduction Edge Expansion & Random Cubic Lattices Open Questions
Proof Outline
G : A1, . . . , An
. . .
A1
Aj
Ak
1
(RTP: qA(G ) � 1 � 2r i(G ) + �).
2
Fix � and a partition A = A1, . . . , Ak where �n > |A1| � . . . � |Ak |.Step 1. Choose t, and factor Kt+1 into perfect matchings.
These induce pairings on parts in G .
Introduction Random Graphs Lattices & Geometry Treewidth Open Questions
Proof Outline (RTP: qA(G ) ≤ 1− 2r i(G ) + ε).
Fix ε and a partition A = A1, . . . ,Ak where δn > |A1| ≥ . . . ≥ |Ak |.Step 1. Choose t, and factor Kt+1 into perfect matchings.
Choose Pα to minimise edges between paired parts in G .
Introduction Edge Expansion & Random Cubic Lattices Open Questions
Proof Outline
G : A1, . . . , An
. . .
A1
Aj
Ak
1
(RTP: qA(G ) � 1 � 2r i(G ) + �).
2
Fix � and a partition A = A1, . . . , Ak where �n > |A1| � . . . � |Ak |.Step 1. Choose t, and factor Kt+1 into perfect matchings.
These induce pairings on parts in G .
G : A1, . . . ,Ak
Introduction Edge Expansion & Random Cubic Lattices Open Questions
Proof Outline
G : A1, . . . , An
. . .
A1
Aj
Ak
1
(RTP: qA(G ) � 1 � 2r i(G ) + �).
2
Fix � and a partition A = A1, . . . , Ak where �n > |A1| � . . . � |Ak |.Step 1. Choose t, and factor Kt+1 into perfect matchings.
These induce pairings on parts in G .
Introduction Random Graphs Lattices & Geometry Treewidth Open Questions
Proof Outline (RTP: qA(G ) ≤ 1− 2r i(G ) + ε).
Fix ε and a partition A = A1, . . . ,Ak where δn > |A1| ≥ . . . ≥ |Ak |.Step 1. Choose t, and factor Kt+1 into perfect matchings.
Choose Pα to minimise edges between paired parts in G .
∴ EαPAIRS := # edges between paired parts ≤ m/t.
Introduction Edge Expansion & Random Cubic Lattices Open Questions
Proof Outline
G : A1, . . . , An
. . .
A1
Aj
Ak
1
(RTP: qA(G ) � 1 � 2r i(G ) + �).
2
Fix � and a partition A = A1, . . . , Ak where �n > |A1| � . . . � |Ak |.Step 1. Choose t, and factor Kt+1 into perfect matchings.
These induce pairings on parts in G .
G : A1, . . . ,Ak
Introduction Edge Expansion & Random Cubic Lattices Open Questions
Proof Outline
G : A1, . . . , An
. . .
A1
Aj
Ak
1
(RTP: qA(G ) � 1 � 2r i(G ) + �).
2
Fix � and a partition A = A1, . . . , Ak where �n > |A1| � . . . � |Ak |.Step 1. Choose t, and factor Kt+1 into perfect matchings.
These induce pairings on parts in G .
Introduction Random Graphs Lattices & Geometry Treewidth Open Questions
Proof Outline (RTP: qA(G ) ≤ 1− 2r i(G ) + ε).
Fix ε and a partition A = A1, . . . ,Ak where δn > |A1| ≥ . . . ≥ |Ak |.Step 1. Choose t, and factor Kt+1 into perfect matchings.
Choose Pα to minimise edges between paired parts in G .
∴ EαPAIRS := # edges between paired parts ≤ m/t.
Eα¬PAIRS := # edges between distinct non-paired parts.
Introduction Edge Expansion & Random Cubic Lattices Open Questions
Proof Outline
G : A1, . . . , An
. . .
A1
Aj
Ak
1
(RTP: qA(G ) � 1 � 2r i(G ) + �).
2
Fix � and a partition A = A1, . . . , Ak where �n > |A1| � . . . � |Ak |.Step 1. Choose t, and factor Kt+1 into perfect matchings.
These induce pairings on parts in G .
G : A1, . . . ,Ak
Introduction Edge Expansion & Random Cubic Lattices Open Questions
Proof Outline
G : A1, . . . , An
. . .
A1
Aj
Ak
1
(RTP: qA(G ) � 1 � 2r i(G ) + �).
2
Fix � and a partition A = A1, . . . , Ak where �n > |A1| � . . . � |Ak |.Step 1. Choose t, and factor Kt+1 into perfect matchings.
These induce pairings on parts in G .
Introduction Random Graphs Lattices & Geometry Treewidth Open Questions
Proof Outline (RTP: qA(G ) ≤ 1− 2r i(G ) + ε).
Fix ε and a partition A = A1, . . . ,Ak where δn > |A1| ≥ . . . ≥ |Ak |.Step 1. Choose t, and factor Kt+1 into perfect matchings.
Choose Pα to minimise edges between paired parts in G .
∴ EαPAIRS := # edges between paired parts ≤ m/t.
Eα¬PAIRS := # edges between distinct non-paired parts.
Step 2. For each pair in Pα randomly colour parts red and blue.
For each part not in Pα randomly colour it red or blue.
Introduction Edge Expansion & Random Cubic Lattices Open Questions
Proof Outline
G : A1, . . . , An
. . .
A1
Aj
Ak
1
(RTP: qA(G ) � 1 � 2r i(G ) + �).
2
Fix � and a partition A = A1, . . . , Ak where �n > |A1| � . . . � |Ak |.Step 1. Choose t, and factor Kt+1 into perfect matchings.
These induce pairings on parts in G .
G : A1, . . . ,Ak
Introduction Random Graphs Lattices & Geometry Treewidth Open Questions
Proof Outline (RTP: qA(G ) ≤ 1− 2r i(G ) + ε).
Fix ε and a partition A = A1, . . . ,Ak where δn > |A1| ≥ . . . ≥ |Ak |.Step 1. Choose t, and factor Kt+1 into perfect matchings.
Choose Pα to minimise edges between paired parts in G .
∴ EαPAIRS := # edges between paired parts ≤ m/t.
Eα¬PAIRS := # edges between distinct non-paired parts.
Step 2. For each pair in Pα randomly colour parts red and blue.
For each part not in Pα randomly colour it red or blue.
|#redV −#blueV |
Introduction Edge Expansion & Random Cubic Lattices Open Questions
Proof Outline
G : A1, . . . , An
. . .
A1
Aj
Ak
(RTP: qA(G ) � 1 � 2r i(G ) + �).
1
Fix � and a partition A = A1, . . . , Ak where �n > |A1| � . . . � |Ak |.Step 1. Choose t, and factor Kt+1 into perfect matchings.
Choose P� to minimise edges between paired parts in G .
� E�PAIRS := # edges between paired parts � m/t.
E�¬PAIRS := # edges between distinct non-paired parts.
Step 2. For each pair in P� randomly colour parts red and blue.
For each part not in P� randomly colour red or blue.
G : A1, . . . ,Ak
Introduction Random Graphs Lattices & Geometry Treewidth Open Questions
Proof Outline (RTP: qA(G ) ≤ 1− 2r i(G ) + ε).
Fix ε and a partition A = A1, . . . ,Ak where δn > |A1| ≥ . . . ≥ |Ak |.Step 1. Choose t, and factor Kt+1 into perfect matchings.
Choose Pα to minimise edges between paired parts in G .
∴ EαPAIRS := # edges between paired parts ≤ m/t.
Eα¬PAIRS := # edges between distinct non-paired parts.
Step 2. For each pair in Pα randomly colour parts red and blue.
For each part not in Pα randomly colour it red or blue.
|#redV −#blueV | ≤ t(|A1| − |Aj |)− t|Aj+1| ≤ t|A1| by Lemma
Introduction Edge Expansion & Random Cubic Lattices Open Questions
Proof Outline
G : A1, . . . , An
. . .
A1
Aj
Ak
(RTP: qA(G ) � 1 � 2r i(G ) + �).
1
Fix � and a partition A = A1, . . . , Ak where �n > |A1| � . . . � |Ak |.Step 1. Choose t, and factor Kt+1 into perfect matchings.
Choose P� to minimise edges between paired parts in G .
� E�PAIRS := # edges between paired parts � m/t.
E�¬PAIRS := # edges between distinct non-paired parts.
Step 2. For each pair in P� randomly colour parts red and blue.
For each part not in P� randomly colour red or blue.
G : A1, . . . ,Ak
Introduction Random Graphs Lattices & Geometry Treewidth Open Questions
Proof Outline (RTP: qA(G ) ≤ 1− 2r i(G ) + ε).
Fix ε and a partition A = A1, . . . ,Ak where δn > |A1| ≥ . . . ≥ |Ak |.Step 1. Choose t, and factor Kt+1 into perfect matchings.
Choose Pα to minimise edges between paired parts in G .
∴ EαPAIRS := # edges between paired parts ≤ m/t.
Eα¬PAIRS := # edges between distinct non-paired parts.
Step 2. For each pair in Pα randomly colour parts red and blue.
For each part not in Pα randomly colour it red or blue.
|#redV −#blueV | ≤ t(|A1| − |Aj |)− t|Aj+1| ≤ t|A1| by Lemma
EαR,B := # edges between red and blue parts.
Introduction Edge Expansion & Random Cubic Lattices Open Questions
Proof Outline
G : A1, . . . , An
. . .
A1
Aj
Ak
(RTP: qA(G ) � 1 � 2r i(G ) + �).
1
Fix � and a partition A = A1, . . . , Ak where �n > |A1| � . . . � |Ak |.Step 1. Choose t, and factor Kt+1 into perfect matchings.
Choose P� to minimise edges between paired parts in G .
� E�PAIRS := # edges between paired parts � m/t.
E�¬PAIRS := # edges between distinct non-paired parts.
Step 2. For each pair in P� randomly colour parts red and blue.
For each part not in P� randomly colour red or blue.
G : A1, . . . ,Ak
Introduction Random Graphs Lattices & Geometry Treewidth Open Questions
Proof Outline (RTP: qA(G ) ≤ 1− 2r i(G ) + ε).
Fix ε and a partition A = A1, . . . ,Ak where δn > |A1| ≥ . . . ≥ |Ak |.Step 1. Choose t, and factor Kt+1 into perfect matchings.
Choose Pα to minimise edges between paired parts in G .
∴ EαPAIRS := # edges between paired parts ≤ m/t.
Eα¬PAIRS := # edges between distinct non-paired parts.
Step 2. For each pair in Pα randomly colour parts red and blue.
For each part not in Pα randomly colour it red or blue.
|#redV −#blueV | ≤ t(|A1| − |Aj |)− t|Aj+1| ≤ t|A1| by Lemma
EαR,B := # edges between red and blue parts.
∴ EαR,B ≥ i(G )×min{#redV , #blueV } ≥ i(G )( n2 − t
2 |A1|)
Introduction Edge Expansion & Random Cubic Lattices Open Questions
Proof Outline
G : A1, . . . , An
. . .
A1
Aj
Ak
(RTP: qA(G ) � 1 � 2r i(G ) + �).
1
Fix � and a partition A = A1, . . . , Ak where �n > |A1| � . . . � |Ak |.Step 1. Choose t, and factor Kt+1 into perfect matchings.
Choose P� to minimise edges between paired parts in G .
� E�PAIRS := # edges between paired parts � m/t.
E�¬PAIRS := # edges between distinct non-paired parts.
Step 2. For each pair in P� randomly colour parts red and blue.
For each part not in P� randomly colour red or blue.
G : A1, . . . ,Ak
Introduction Random Graphs Lattices & Geometry Treewidth Open Questions
Proof Outline (RTP: qA(G ) ≤ 1− 2r i(G ) + ε).
Fix ε and a partition A = A1, . . . ,Ak where δn > |A1| ≥ . . . ≥ |Ak |.Step 1. Choose t, and factor Kt+1 into perfect matchings.
Choose Pα to minimise edges between paired parts in G .
∴ EαPAIRS := # edges between paired parts ≤ m/t.
Eα¬PAIRS := # edges between distinct non-paired parts.
Step 2. For each pair in Pα randomly colour parts red and blue.
For each part not in Pα randomly colour it red or blue.
|#redV −#blueV | ≤ t(|A1| − |Aj |)− t|Aj+1| ≤ t|A1| by Lemma
EαR,B := # edges between red and blue parts.
∴ EαR,B ≥ i(G )×min{#redV , #blueV } ≥ i(G )( n2 − t
2 |A1|)but, E[EαR,B ] = EαPAIRS + 1
2Eα¬PAIRS
Introduction Edge Expansion & Random Cubic Lattices Open Questions
Proof Outline
G : A1, . . . , An
. . .
A1
Aj
Ak
(RTP: qA(G ) � 1 � 2r i(G ) + �).
1
Fix � and a partition A = A1, . . . , Ak where �n > |A1| � . . . � |Ak |.Step 1. Choose t, and factor Kt+1 into perfect matchings.
Choose P� to minimise edges between paired parts in G .
� E�PAIRS := # edges between paired parts � m/t.
E�¬PAIRS := # edges between distinct non-paired parts.
Step 2. For each pair in P� randomly colour parts red and blue.
For each part not in P� randomly colour red or blue.
G : A1, . . . ,Ak
Introduction Random Graphs Lattices & Geometry Treewidth Open Questions
Proof Outline (RTP: qA(G ) ≤ 1− 2r i(G ) + ε).
Fix ε and a partition A = A1, . . . ,Ak where δn > |A1| ≥ . . . ≥ |Ak |.Step 1. Choose t, and factor Kt+1 into perfect matchings.
Choose Pα to minimise edges between paired parts in G .
∴ EαPAIRS := # edges between paired parts ≤ m/t.
Eα¬PAIRS := # edges between distinct non-paired parts.
Step 2. For each pair in Pα randomly colour parts red and blue.
For each part not in Pα randomly colour it red or blue.
|#redV −#blueV | ≤ t(|A1| − |Aj |)− t|Aj+1| ≤ t|A1| by Lemma
EαR,B := # edges between red and blue parts.
∴ EαR,B ≥ i(G )×min{#redV , #blueV } ≥ i(G )( n2 − t
2 |A1|)but, E[EαR,B ] = EαPAIRS + 1
2Eα¬PAIRS
Finish. We now have an upper bound for the edge contribution.
Introduction Edge Expansion & Random Cubic Lattices Open Questions
Proof Outline
G : A1, . . . , An
. . .
A1
Aj
Ak
(RTP: qA(G ) � 1 � 2r i(G ) + �).
1
Fix � and a partition A = A1, . . . , Ak where �n > |A1| � . . . � |Ak |.Step 1. Choose t, and factor Kt+1 into perfect matchings.
Choose P� to minimise edges between paired parts in G .
� E�PAIRS := # edges between paired parts � m/t.
E�¬PAIRS := # edges between distinct non-paired parts.
Step 2. For each pair in P� randomly colour parts red and blue.
For each part not in P� randomly colour red or blue.
G : A1, . . . ,Ak
Introduction Random Graphs Lattices & Geometry Treewidth Open Questions
Proof Outline (RTP: qA(G ) ≤ 1− 2r i(G ) + ε).
Fix ε and a partition A = A1, . . . ,Ak where δn > |A1| ≥ . . . ≥ |Ak |.Step 1.
∴ EαPAIRS := # edges between paired parts ≤ m/t.
Eα¬PAIRS := # edges between distinct non-paired parts.
Step 2.
EαR,B := # edges between red and blue parts.
∴ EαR,B ≥ i(G )×min{#redV , #blueV } ≥ i(G )( n2 − t
2 |A1|)but, E[EαR,B ] = EαPAIRS + 1
2Eα¬PAIRS
Finish. We now have an upper bound for the edge contribution.
Introduction Edge Expansion & Random Cubic Lattices Open Questions
Proof Outline
G : A1, . . . , An
. . .
A1
Aj
Ak
(RTP: qA(G ) � 1 � 2r i(G ) + �).
1
Fix � and a partition A = A1, . . . , Ak where �n > |A1| � . . . � |Ak |.Step 1. Choose t, and factor Kt+1 into perfect matchings.
Choose P� to minimise edges between paired parts in G .
� E�PAIRS := # edges between paired parts � m/t.
E�¬PAIRS := # edges between distinct non-paired parts.
Step 2. For each pair in P� randomly colour parts red and blue.
For each part not in P� randomly colour red or blue.
G : A1, . . . ,Ak
Introduction Random Graphs Lattices & Geometry Treewidth Open Questions
Proof Outline (RTP: qA(G ) ≤ 1− 2r i(G ) + ε).
Fix ε and a partition A = A1, . . . ,Ak where δn > |A1| ≥ . . . ≥ |Ak |.Step 1.
∴ EαPAIRS := # edges between paired parts ≤ m/t.
Eα¬PAIRS := # edges between distinct non-paired parts.
Step 2.
EαR,B := # edges between red and blue parts.
∴ EαR,B ≥ i(G )×min{#redV , #blueV } ≥ i(G )( n2 − t
2 |A1|)but, E[EαR,B ] = EαPAIRS + 1
2Eα¬PAIRS
Finish. We now have an upper bound for the edge contribution.
qEA(G ) = 1m
∑A∈A E (A) = 1− 1
m (EαPAIRS + Eα¬PAIRS)
Introduction Edge Expansion & Random Cubic Lattices Open Questions
Proof Outline
G : A1, . . . , An
. . .
A1
Aj
Ak
(RTP: qA(G ) � 1 � 2r i(G ) + �).
1
Fix � and a partition A = A1, . . . , Ak where �n > |A1| � . . . � |Ak |.Step 1. Choose t, and factor Kt+1 into perfect matchings.
Choose P� to minimise edges between paired parts in G .
� E�PAIRS := # edges between paired parts � m/t.
E�¬PAIRS := # edges between distinct non-paired parts.
Step 2. For each pair in P� randomly colour parts red and blue.
For each part not in P� randomly colour red or blue.
G : A1, . . . ,Ak
Introduction Random Graphs Lattices & Geometry Treewidth Open Questions
Proof Outline (RTP: qA(G ) ≤ 1− 2r i(G ) + ε).
Fix ε and a partition A = A1, . . . ,Ak where δn > |A1| ≥ . . . ≥ |Ak |.Step 1.
∴ EαPAIRS := # edges between paired parts ≤ m/t.
Eα¬PAIRS := # edges between distinct non-paired parts.
Step 2.
EαR,B := # edges between red and blue parts.
∴ EαR,B ≥ i(G )×min{#redV , #blueV } ≥ i(G )( n2 − t
2 |A1|)but, E[EαR,B ] = EαPAIRS + 1
2Eα¬PAIRS
Finish. We now have an upper bound for the edge contribution.
qEA(G ) = 1m
∑A∈A E (A) = 1− 1
m (EαPAIRS + Eα¬PAIRS)
= 1− 1m (2E[EαR,B ]− EαPAIRS)
Introduction Edge Expansion & Random Cubic Lattices Open Questions
Proof Outline
G : A1, . . . , An
. . .
A1
Aj
Ak
(RTP: qA(G ) � 1 � 2r i(G ) + �).
1
Fix � and a partition A = A1, . . . , Ak where �n > |A1| � . . . � |Ak |.Step 1. Choose t, and factor Kt+1 into perfect matchings.
Choose P� to minimise edges between paired parts in G .
� E�PAIRS := # edges between paired parts � m/t.
E�¬PAIRS := # edges between distinct non-paired parts.
Step 2. For each pair in P� randomly colour parts red and blue.
For each part not in P� randomly colour red or blue.
G : A1, . . . ,Ak
Introduction Random Graphs Lattices & Geometry Treewidth Open Questions
Proof Outline (RTP: qA(G ) ≤ 1− 2r i(G ) + ε).
Fix ε and a partition A = A1, . . . ,Ak where δn > |A1| ≥ . . . ≥ |Ak |.Step 1.
∴ EαPAIRS := # edges between paired parts ≤ m/t.
Eα¬PAIRS := # edges between distinct non-paired parts.
Step 2.
EαR,B := # edges between red and blue parts.
∴ EαR,B ≥ i(G )×min{#redV , #blueV } ≥ i(G )( n2 − t
2 |A1|)but, E[EαR,B ] = EαPAIRS + 1
2Eα¬PAIRS
Finish. We now have an upper bound for the edge contribution.
qEA(G ) = 1m
∑A∈A E (A) = 1− 1
m (EαPAIRS + Eα¬PAIRS)
= 1− 1m (2E[EαR,B ]− EαPAIRS) ≥ 1− 2
rn i(G )(n − tδ)− 1t
Introduction Edge Expansion & Random Cubic Lattices Open Questions
Proof Outline
G : A1, . . . , An
. . .
A1
Aj
Ak
(RTP: qA(G ) � 1 � 2r i(G ) + �).
1
Fix � and a partition A = A1, . . . , Ak where �n > |A1| � . . . � |Ak |.Step 1. Choose t, and factor Kt+1 into perfect matchings.
Choose P� to minimise edges between paired parts in G .
� E�PAIRS := # edges between paired parts � m/t.
E�¬PAIRS := # edges between distinct non-paired parts.
Step 2. For each pair in P� randomly colour parts red and blue.
For each part not in P� randomly colour red or blue.
G : A1, . . . ,Ak
Introduction Random Graphs Lattices & Geometry Treewidth Open Questions
Proof Outline (RTP: qA(G ) ≤ 1− 2r i(G ) + ε).
Fix ε and a partition A = A1, . . . ,Ak where δn > |A1| ≥ . . . ≥ |Ak |.Step 1.
∴ EαPAIRS := # edges between paired parts ≤ m/t.
Eα¬PAIRS := # edges between distinct non-paired parts.
Step 2.
EαR,B := # edges between red and blue parts.
∴ EαR,B ≥ i(G )×min{#redV , #blueV } ≥ i(G )( n2 − t
2 |A1|)but, E[EαR,B ] = EαPAIRS + 1
2Eα¬PAIRS
Finish. We now have an upper bound for the edge contribution.
qEA(G ) = 1m
∑A∈A E (A) = 1− 1
m (EαPAIRS + Eα¬PAIRS)
= 1− 1m (2E[EαR,B ]− EαPAIRS) ≤ 1− 2
r i(G )− 2tδr − 1
t
Introduction Edge Expansion & Random Cubic Lattices Open Questions
Proof Outline
G : A1, . . . , An
. . .
A1
Aj
Ak
(RTP: qA(G ) � 1 � 2r i(G ) + �).
1
Fix � and a partition A = A1, . . . , Ak where �n > |A1| � . . . � |Ak |.Step 1. Choose t, and factor Kt+1 into perfect matchings.
Choose P� to minimise edges between paired parts in G .
� E�PAIRS := # edges between paired parts � m/t.
E�¬PAIRS := # edges between distinct non-paired parts.
Step 2. For each pair in P� randomly colour parts red and blue.
For each part not in P� randomly colour red or blue.
G : A1, . . . ,Ak
Introduction Random Graphs Lattices & Geometry Treewidth Open Questions
Proof Outline (RTP: qA(G ) ≤ 1− 2r i(G ) + ε).
Fix ε and a partition A = A1, . . . ,Ak where δn > |A1| ≥ . . . ≥ |Ak |.Step 1.
∴ EαPAIRS := # edges between paired parts ≤ m/t.
Eα¬PAIRS := # edges between distinct non-paired parts.
Step 2.
EαR,B := # edges between red and blue parts.
∴ EαR,B ≥ i(G )×min{#redV , #blueV } ≥ i(G )( n2 − t
2 |A1|)but, E[EαR,B ] = EαPAIRS + 1
2Eα¬PAIRS
Finish. We now have an upper bound for the edge contribution.
qEA(G ) = 1m
∑A∈A E (A) = 1− 1
m (EαPAIRS + Eα¬PAIRS)
= 1− 1m (2E[EαR,B ]− EαPAIRS) ≤ 1− 2
r i(G )− 2tδr − 1
t
∴ choose δ, t and we are done. �
Introduction Edge Expansion & Random Cubic Lattices Open Questions
Proof Outline
G : A1, . . . , An
. . .
A1
Aj
Ak
(RTP: qA(G ) � 1 � 2r i(G ) + �).
1
Fix � and a partition A = A1, . . . , Ak where �n > |A1| � . . . � |Ak |.Step 1. Choose t, and factor Kt+1 into perfect matchings.
Choose P� to minimise edges between paired parts in G .
� E�PAIRS := # edges between paired parts � m/t.
E�¬PAIRS := # edges between distinct non-paired parts.
Step 2. For each pair in P� randomly colour parts red and blue.
For each part not in P� randomly colour red or blue.
G : A1, . . . ,Ak
Introduction Random Graphs Lattices & Geometry Treewidth Open QuestionsIntroduction Edge Expansion & Random Cubic Lattices Open Questions
Lemma (McDiarmid, S.)
Fix vertices [n] with weights w(1) � . . . � w(n) � 0.Let M be a perfect matching such that |a � b| t, 8edges ab 2 M.
For any red-blue colouring of M,
���X
v red
w(v) �X
u blue
w(u)��� t (w(1) � w(n)).
Theorem (McDiarmid, S.)
For any " > 0 there exists � > 0 such that the following holds. Let r � 3and let G be an r-regular graph with at least ��1 vertices. Then
q�(G ) < 1 � 2r i(G ) + ".