Modularity in Random Graphs and on Latticesskerman/RSASkermanUpload.pdf · 2015-01-30 ·...

Introduction Edge Expansion & Random Cubic Lattices Open Questions

Modularity in Random Graphs

and on Lattices

Colin McDiarmid, Fiona Skerman

Oxford University

August 9, 2013


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



qEA(G ) :=X

A2A

|E (C )|m

qDA(G ) :=X

A2A

✓Pv2C deg(v)

2m

◆2



qEA(G ) :=X

A2A

|E (C )|m

qDA(G ) :=X

A2A

✓Pv2C deg(v)

2m

◆2

Example Graph

1



qEA(G ) :=X

A2A

|E (C )|m

qDA(G ) :=X

A2A

✓Pv2C deg(v)

2m

◆2

3 Possible Partitions

2 4 3



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

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.


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+ ".



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 ) + ".




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.




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.




q�(G ) < 1� 2r i(G ) + ".


Bollobas ’88 i(G3) > 0.18 w .h.p.

Kostochka, Melnikov ’92 i(G3) > 0.21 w .h.p.




q�(G ) < 1� 2r i(G ) + ".


Bollobas ’88 i(G3) > 0.18 w .h.p.

Kostochka, Melnikov ’92 i(G3) > 0.21 w .h.p.

Modularity of random cubic graphs.


0.66 q�(G3) 0.86 w .h.p.




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.


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)).





��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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0� � � � � � � �

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�

�R

0� � � � � � � �

�

�

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�R

0� � � � � � � �

1





��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

��X

v red

w(v)�X

u blue

w(u)�� = w(1)�w(2)+w(3)� . . .+w(7)�w(8)





��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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0� � � � � � � �

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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)





��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





��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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0� � � � � � � �

1

�

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0� � � � � � � �

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0� � � � � � � �

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0� � � � � � � �

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0� � � � � � � �

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0� � � � � � � �

1


Proof Outline

G : A1, . . . ,An

1

Fix " and a partition A = A1, . . . ,Ak where �n > |A1| � . . . � |Ak |.


Proof Outline

G : A1, . . . ,An

(RTP: qA(G ) 1� 2r i(G ) + ").

1

Fix " and a partition A = A1, . . . ,Ak where �n > |A1| � . . . � |Ak |.


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.


Proof Outline

G : A1, . . . ,An

1

(RTP: qA(G ) 1� 2r i(G ) + ").

1


These induce pairings on parts in G .


Proof Outline

G : A1, . . . ,An

. . .

A1

Aj

Ak

1

(RTP: qA(G ) 1� 2r i(G ) + ").

2


These induce pairings on parts in G .


Proof Outline

G : A1, . . . ,An

. . .

A1

Aj

Ak

1

(RTP: qA(G ) 1� 2r i(G ) + ").

2


Choose P↵ to minimise edges between paired parts in G .


Proof Outline

G : A1, . . . ,An

. . .

A1

Aj

Ak

1

(RTP: qA(G ) 1� 2r i(G ) + ").

2



) E↵PAIRS := # edges between paired parts m/t.


Proof Outline

G : A1, . . . ,An

. . .

A1

Aj

Ak

(RTP: qA(G ) 1� 2r i(G ) + ").

2




E↵¬PAIRS := # edges between distinct non-paired parts.


Proof Outline

G : A1, . . . ,An

. . .

A1

Aj

Ak

(RTP: qA(G ) 1� 2r i(G ) + ").

2





Step 2. For each pair in P↵ randomly colour parts red and blue.

For each part not in P↵ randomly colour red or blue.


Proof Outline

G : A1, . . . ,An

. . .

A1

Aj

Ak

(RTP: qA(G ) 1� 2r i(G ) + ").

1








Proof Outline

G : A1, . . . ,An

. . .

A1

Aj

Ak

(RTP: qA(G ) 1� 2r i(G ) + ").

1







|#redV �#blueV |


Proof Outline

G : A1, . . . ,An

. . .

A1

Aj

Ak

(RTP: qA(G ) 1� 2r i(G ) + ").

1







|#redV �#blueV | t(|A1|� |Aj |)� t|Aj+1| t|A1| by Lemma


Proof Outline

G : A1, . . . ,An

. . .

A1

Aj

Ak

(RTP: qA(G ) 1� 2r i(G ) + ").

1








E↵R,B := # edges between red and blue parts


Proof Outline

G : A1, . . . ,An

. . .

A1

Aj

Ak

(RTP: qA(G ) 1� 2r i(G ) + ").

1









) E↵R,B � i(G )⇥min{#redV , #blueV } � i(G )( n2 � t

2 |A1|)


Proof Outline

G : A1, . . . ,An

. . .

A1

Aj

Ak

(RTP: qA(G ) 1� 2r i(G ) + ").

1










2 |A1|)but, E[E↵

R,B ] = E↵PAIRS + 1

2E↵¬PAIRS


Proof Outline

G : A1, . . . ,An

. . .

A1

Aj

Ak

(RTP: qA(G ) 1� 2r i(G ) + ").

1










2 |A1|)but, E[E↵


2E↵¬PAIRS

Finish. We now have an upper bound for the edge contribution.


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.



Step 2.



2 |A1|)but, E[E↵


2E↵¬PAIRS



Proof Outline

G : A1, . . . ,An

. . .

A1

Aj

Ak

(RTP: qA(G ) 1� 2r i(G ) + ").

1




Step 2.



2 |A1|)but, E[E↵


2E↵¬PAIRS


qEA(G ) = 1m

PA2A E (A) = 1� 1

m (E↵PAIRS + E↵

¬PAIRS)


Proof Outline

G : A1, . . . ,An

. . .

A1

Aj

Ak

(RTP: qA(G ) 1� 2r i(G ) + ").

1




Step 2.



2 |A1|)but, E[E↵


2E↵¬PAIRS


qEA(G ) = 1m

PA2A E (A) = 1� 1

m (E↵PAIRS + E↵

¬PAIRS)

= 1� 1m (2E[E↵

R,B ]� E↵PAIRS)


Proof Outline

G : A1, . . . ,An

. . .

A1

Aj

Ak

(RTP: qA(G ) 1� 2r i(G ) + ").

1




Step 2.



2 |A1|)but, E[E↵


2E↵¬PAIRS


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


Proof Outline

G : A1, . . . ,An

. . .

A1

Aj

Ak

(RTP: qA(G ) 1� 2r i(G ) + ").

1




Step 2.



2 |A1|)but, E[E↵


2E↵¬PAIRS


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


Proof Outline

G : A1, . . . ,An

. . .

A1

Aj

Ak

(RTP: qA(G ) 1� 2r i(G ) + ").

1




Step 2.



2 |A1|)but, E[E↵


2E↵¬PAIRS


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 .



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.


��X

v red

w(v)�X

u blue

w(u)�� t (w(1)� w(n)).



q�(G ) < 1� 2r i(G ) + ".


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.


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.


We extend this result to include any subgraph of the lattice Zdz .


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


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.


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.

Modularity in Random Graphs and on Latticesskerman/RSASkermanUpload.pdf · 2015-01-30 ·...

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