Introduction Edge Expansion & Random Cubic Lattices Open Questions
Modularity in Random Graphs
and on Lattices
Colin McDiarmid, Fiona Skerman
Oxford University
August 9, 2013
Introduction Edge Expansion & Random Cubic Lattices Open Questions
Definition
Let G be a graph on m edges, andE (A) = |{vw 2 E : v ,w 2 A}|.
Modularity qA(G ) :=X
A2A
E (A)
m�✓P
v2C deg(v)
2m
◆2!
Max. Modularity q(G ) := maxA2AV
qA(G )
Notice the sum naturally splits into two components.
Edge contribution Degree tax
qEA(G ) :=X
A2A
|E (C )|m
qDA(G ) :=X
A2A
✓Pv2C deg(v)
2m
◆2
Introduction Edge Expansion & Random Cubic Lattices Open Questions
Definition
Let G be a graph on m edges, andE (A) = |{vw 2 E : v ,w 2 A}|.
Modularity qA(G ) :=X
A2A
E (A)
m�✓P
v2C deg(v)
2m
◆2!
Max. Modularity q(G ) := maxA2AV
qA(G )
Notice the sum naturally splits into two components.
Edge contribution Degree tax
qEA(G ) :=X
A2A
|E (C )|m
qDA(G ) :=X
A2A
✓Pv2C deg(v)
2m
◆2
Introduction Edge Expansion & Random Cubic Lattices Open Questions
Edge contribution Degree tax
qEA(G ) :=X
A2A
|E (C )|m
qDA(G ) :=X
A2A
✓Pv2C deg(v)
2m
◆2
Introduction Edge Expansion & Random Cubic Lattices Open Questions
Edge contribution Degree tax
qEA(G ) :=X
A2A
|E (C )|m
qDA(G ) :=X
A2A
✓Pv2C deg(v)
2m
◆2
Example Graph
1
Introduction Edge Expansion & Random Cubic Lattices Open Questions
Edge contribution Degree tax
qEA(G ) :=X
A2A
|E (C )|m
qDA(G ) :=X
A2A
✓Pv2C deg(v)
2m
◆2
3 Possible Partitions
2 4 3
Introduction Edge Expansion & Random Cubic Lattices Open Questions
Edge contribution Degree tax
qEA(G ) :=X
A2A
|E (C )|m
qDA(G ) :=X
A2A
✓Pv2C 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
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.
Introduction Edge Expansion & Random Cubic Lattices Open Questions
Edge Expansion/ Isoperimetric Number
i(G ) := minV (G)=V1[V2
e(V1,V2)
min{|V1|, |V2|}
q�(G ) := maxA : |A|<�n, 8A2A
qA(G )
Theorem (McDiarmid, S.)
For any " > 0 there exists � > 0 such that the following holds. Letr � 3 and let G be an r-regular graph with at least ��1 vertices.Then
q�(G ) < 1� 2i(G )
r+ ".
Introduction Edge Expansion & Random Cubic Lattices Open Questions
Edge Expansion/ Isoperimetric Number
i(G ) := minV (G)=V1[V2
e(V1,V2)
min{|V1|, |V2|}
q�(G ) := maxA : |A|<�n, 8A2A
qA(G )
Theorem (McDiarmid, S.)
For any " > 0 there exists � > 0 such that the following holds. Letr � 3 and let G be an r-regular graph with at least ��1 vertices.Then
q�(G ) < 1� 2i(G )
r+ ".
Introduction Edge Expansion & Random Cubic Lattices Open Questions
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 ) + ".
Introduction Edge Expansion & Random Cubic Lattices Open Questions
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 ) + ".
Random r-regular graphs.Let Gr be a r -regular graph on n vertices, picked uniformly at ran-dom.
Introduction Edge Expansion & Random Cubic Lattices Open Questions
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 ) + ".
Random r-regular graphs.Let Gr be a r -regular graph on n vertices, picked uniformly at ran-dom.Theorem
Bollobas ’88 i(G3) > 0.18 w .h.p.
Introduction Edge Expansion & Random Cubic Lattices Open Questions
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 ) + ".
Random r-regular graphs.Let Gr be a r -regular graph on n vertices, picked uniformly at ran-dom.Theorem
Bollobas ’88 i(G3) > 0.18 w .h.p.
Kostochka, Melnikov ’92 i(G3) > 0.21 w .h.p.
Introduction Edge Expansion & Random Cubic Lattices Open Questions
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 ) + ".
Random r-regular graphs.Let Gr be a r -regular graph on n vertices, picked uniformly at ran-dom.Theorem
Bollobas ’88 i(G3) > 0.18 w .h.p.
Kostochka, Melnikov ’92 i(G3) > 0.21 w .h.p.
Modularity of random cubic graphs.
Theorem (McDiarmid, S.)
0.66 q�(G3) 0.86 w .h.p.
Introduction Edge Expansion & Random Cubic Lattices Open Questions
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 ) + ".
Random r-regular graphs.Let Gr be a r -regular graph on n vertices, picked uniformly at ran-dom.Theorem
Bollobas ’88 i(G3) > 0.18 w .h.p.
Kostochka, Melnikov ’92 i(G3) > 0.21 w .h.p.
Modularity of random cubic graphs.
Theorem (McDiarmid, S.)
0.66 q�(G3) 0.86 w .h.p.
Introduction Edge Expansion & Random Cubic Lattices Open Questions
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 ) + ".
Observation
q(G ) max{1� i(G )/r , 3/4}.
Consider A = {A1, . . . ,Ak}(a) Suppose that each |Ai | n/2. Then the number of edges be-tween parts Ai is
12
Pi e(Ai , Ai ) � 1
2
Pi |Ai |i(G ) = 1
2ni(G ). HenceqA(G ) 1� i(G )/r .(b) If say |A1| > n/2 than the degree tax is at least (|A1|r/rn)2 >1/4 and so qA(G ) 3/4.
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� � � � � � � �
�
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�R
0� � � � � � � �
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�R
0� � � � � � � �
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�
�
�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� � � � � � � �
�
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�
�R
0� � � � � � � �
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�
�
�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� � � � � � � �
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�
�R
0� � � � � � � �
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�
�
�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� � � � � � � �
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�R
0� � � � � � � �
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0� � � � � � � �
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0� � � � � � � �
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0� � � � � � � �
1
Introduction Edge Expansion & Random Cubic Lattices Open Questions
Proof Outline
G : A1, . . . ,An
1
Fix " and a partition A = A1, . . . ,Ak where �n > |A1| � . . . � |Ak |.
Introduction Edge Expansion & Random Cubic Lattices Open Questions
Proof Outline
G : A1, . . . ,An
(RTP: qA(G ) 1� 2r i(G ) + ").
1
Fix " and a partition A = A1, . . . ,Ak where �n > |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.
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.
These induce pairings on 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 .
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 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.
Choose P↵ 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.
Choose P↵ to minimise edges between paired parts in G .
) E↵PAIRS := # edges between paired parts m/t.
Introduction Edge Expansion & Random Cubic Lattices Open Questions
Proof Outline
G : A1, . . . ,An
. . .
A1
Aj
Ak
(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.
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.
Introduction Edge Expansion & Random Cubic Lattices Open Questions
Proof Outline
G : A1, . . . ,An
. . .
A1
Aj
Ak
(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.
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.
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.
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.
|#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.
|#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.
|#redV �#blueV | t(|A1|� |Aj |)� t|Aj+1| t|A1| by Lemma
E↵R,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.
|#redV �#blueV | t(|A1|� |Aj |)� t|Aj+1| t|A1| by Lemma
E↵R,B := # edges between red and blue parts
) E↵R,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.
|#redV �#blueV | t(|A1|� |Aj |)� t|Aj+1| t|A1| by Lemma
E↵R,B := # edges between red and blue parts
) E↵R,B � i(G )⇥min{#redV , #blueV } � i(G )( n2 � t
2 |A1|)but, E[E↵
R,B ] = E↵PAIRS + 1
2E↵¬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.
|#redV �#blueV | t(|A1|� |Aj |)� t|Aj+1| t|A1| by Lemma
E↵R,B := # edges between red and blue parts
) E↵R,B � i(G )⇥min{#redV , #blueV } � i(G )( n2 � t
2 |A1|)but, E[E↵
R,B ] = E↵PAIRS + 1
2E↵¬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.
) E↵PAIRS := # edges between paired parts m/t.
E↵¬PAIRS := # edges between distinct non-paired parts.
Step 2.
E↵R,B := # edges between red and blue parts
) E↵R,B � i(G )⇥min{#redV , #blueV } � i(G )( n2 � t
2 |A1|)but, E[E↵
R,B ] = E↵PAIRS + 1
2E↵¬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.
) E↵PAIRS := # edges between paired parts m/t.
E↵¬PAIRS := # edges between distinct non-paired parts.
Step 2.
E↵R,B := # edges between red and blue parts
) E↵R,B � i(G )⇥min{#redV , #blueV } � i(G )( n2 � t
2 |A1|)but, E[E↵
R,B ] = E↵PAIRS + 1
2E↵¬PAIRS
Finish. We now have an upper bound for the edge contribution.
qEA(G ) = 1m
PA2A E (A) = 1� 1
m (E↵PAIRS + E↵
¬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.
) E↵PAIRS := # edges between paired parts m/t.
E↵¬PAIRS := # edges between distinct non-paired parts.
Step 2.
E↵R,B := # edges between red and blue parts
) E↵R,B � i(G )⇥min{#redV , #blueV } � i(G )( n2 � t
2 |A1|)but, E[E↵
R,B ] = E↵PAIRS + 1
2E↵¬PAIRS
Finish. We now have an upper bound for the edge contribution.
qEA(G ) = 1m
PA2A E (A) = 1� 1
m (E↵PAIRS + E↵
¬PAIRS)
= 1� 1m (2E[E↵
R,B ]� E↵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.
) E↵PAIRS := # edges between paired parts m/t.
E↵¬PAIRS := # edges between distinct non-paired parts.
Step 2.
E↵R,B := # edges between red and blue parts
) E↵R,B � i(G )⇥min{#redV , #blueV } � i(G )( n2 � t
2 |A1|)but, E[E↵
R,B ] = E↵PAIRS + 1
2E↵¬PAIRS
Finish. We now have an upper bound for the edge contribution.
qEA(G ) = 1m
PA2A E (A) = 1� 1
m (E↵PAIRS + E↵
¬PAIRS)
= 1� 1m (2E[E↵
R,B ]� E↵PAIRS) � 1� 2
rn i(G )(n � t�)� 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.
) E↵PAIRS := # edges between paired parts m/t.
E↵¬PAIRS := # edges between distinct non-paired parts.
Step 2.
E↵R,B := # edges between red and blue parts
) E↵R,B � i(G )⇥min{#redV , #blueV } � i(G )( n2 � t
2 |A1|)but, E[E↵
R,B ] = E↵PAIRS + 1
2E↵¬PAIRS
Finish. We now have an upper bound for the edge contribution.
qEA(G ) = 1m
PA2A E (A) = 1� 1
m (E↵PAIRS + E↵
¬PAIRS)
= 1� 1m (2E[E↵
R,B ]� E↵PAIRS) � 1� 2
r i(G )� 2t�r � 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.
) E↵PAIRS := # edges between paired parts m/t.
E↵¬PAIRS := # edges between distinct non-paired parts.
Step 2.
E↵R,B := # edges between red and blue parts
) E↵R,B � i(G )⇥min{#redV , #blueV } � i(G )( n2 � t
2 |A1|)but, E[E↵
R,B ] = E↵PAIRS + 1
2E↵¬PAIRS
Finish. We now have an upper bound for the edge contribution.
qEA(G ) = 1m
PA2A E (A) = 1� 1
m (E↵PAIRS + E↵
¬PAIRS)
= 1� 1m (2E[E↵
R,B ]� E↵PAIRS) � 1� 2
r i(G )� 2t�r � 1
t
) choose �, t and we are done .
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)).
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 ) + ".
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)).
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 ) + ".
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�
1d+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�
1d+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 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 Edge Expansion & Random Cubic Lattices Open Questions
Open Questions
Modularity =
Edge contribution - Degree tax
qEA(G ) :=X
A2A
|E (C )|m
qDA(G ) :=X
A2A
✓Pv2C deg(v)
2m
◆2
1. Extend upper bound to all partitions
For any graph G and for a random cubic graph G3, 8" > 0, 9� s.t.;
q�(G ) 1� 2r i(G ) + " and also q�(G3) .86 w .h.p.
but currently only for partitions A where 8A 2 A, |A| < �n.
2. Improve lower bound in random cubic
For a random cubic graph G3, 8" > 0; q(G3) > 23 � " w .h.p.
Is this optimal?
Construction based on finding a Hamilton cycle, then cutting intostrips of
pn vertices.
Introduction Edge Expansion & Random Cubic Lattices Open Questions
Open Questions
Modularity =
Edge contribution - Degree tax
qEA(G ) :=X
A2A
|E (C )|m
qDA(G ) :=X
A2A
✓Pv2C deg(v)
2m
◆2
1. Extend upper bound to all partitions
For any graph G and for a random cubic graph G3, 8" > 0, 9� s.t.;
q�(G ) 1� 2r i(G ) + " and also q�(G3) .86 w .h.p.
but currently only for partitions A where 8A 2 A, |A| < �n.
2. Improve lower bound in random cubic
For a random cubic graph G3, 8" > 0; q(G3) > 23 � " w .h.p.
Is this optimal?Construction based on finding a Hamilton cycle, then cutting intostrips of
pn vertices.