Modularity on Random Graphs, Lattices and Embedded Graphsskerman/2015NJCUpload.pdf · Modularity on...

Introduction Random Graphs Lattices & Geometry Treewidth Open Questions

Modularity on Random Graphs,Lattices and Embedded Graphs

Colin McDiarmid, Fiona Skerman

University of Oxford

[email protected]


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


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 )


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 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 )



qEA(G ) :=

X

A2A

e(A)

mqD

A(G ) :=X

A2A

✓degsum(A)

2m

◆2



Defintion


Modularity

Max. Modularity

qA(G ) :=X

A2A

e(A)

m�✓

degsum(A)

2m

◆2!

q(G ) := maxA

qA(G )



qEA(G ) :=

X

A2A

e(A)

mqD

A(G ) :=X

A2A

✓degsum(A)

2m

◆2

Example Graph



qEA(G ) :=

�

A�A

|E (C )|m

qDA(G ) :=

�

A�A

��v�C deg(v)

2m

�2

Example Graph

1



Defintion


Modularity

Max. Modularity

qA(G ) :=X

A2A

e(A)

m�✓

degsum(A)

2m

◆2!

q(G ) := maxA

qA(G )



qEA(G ) :=

X

A2A

e(A)

mqD

A(G ) :=X

A2A

✓degsum(A)

2m

◆2

3 Possible Partitions



qEA(G ) :=

X

A2A

|E (C )|m

qDA(G ) :=

X

A2A

✓�v2C deg(v)

2m

◆2


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



Defintion


Modularity

Max. Modularity

qA(G ) :=X

A2A

e(A)

m�✓

degsum(A)

2m

◆2!

q(G ) := maxA

qA(G )



qEA(G ) :=

X

A2A

e(A)

mqD

A(G ) :=X

A2A

✓degsum(A)

2m

◆2




qEA(G ) :=

X

A2A

|E (C )|m

qDA(G ) :=

X

A2A

✓�v2C deg(v)

2m

◆2


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



qEA(G ) :=

X

A2A

|E (C )|m

qDA(G ) :=

X

A2A

✓�v2C deg(v)

2m

◆2


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


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




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.



Upper Bounds

edge expansion of small setsiu(G ) := min|U|≤un

1|U|e(U,V \U)


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)


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


Simulations


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]


Simulations


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.



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


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.]


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.]


Statistical Physics


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.


Statistical Physics


Statistical Physics



Zdz . Then q(R) � 1 � (d + 1)

�z+12d

� dd+1 n� 1

d+1


� � � � � ��

� � � � � ��

1

� � � � � ��

� � � � � ��

1


3 (right).




Statistical Physics



Zdz . Then q(R) � 1 � (d + 1)

�z+12d

� dd+1 n� 1

d+1 = 1 ��m� 1

d+1

�


� � � � � ��

� � � � � ��

1

� � � � � ��

� � � � � ��

1


3 (right).



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 .


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


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)| ≤ `


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


Idea of Proof

Assumptions: graph G and mapping α : G → Rd such that


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.


Idea of Proof

Assumptions: graph G and mapping α : G → Rd such that


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.


Random Graphs on Surfaces via Treewidth

Trees of bounded degree, Bagrow 2012.

Trees with degree o(n1/5), Montgolfier et. al. 2011.


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→∞.


Random Graphs on Surfaces via Treewidth

Trees of bounded degree, Bagrow 2012.

Trees with degree o(n1/5), Montgolfier et. al. 2011.


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).


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-


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-


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 )


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 ).



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� � � � � � � �

�

�

�

�R

0� � � � � � � �

�

�

�

�R

0� � � � � � � �

�

�

�

�R

0� � � � � � � �

1





��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)





��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)





��X

v red

w(v) �X

u blue

w(u)�� t (w(1) � w(n)).

t = 3

�

�

�

�R

0� � � � � � � �

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0� � � � � � � �

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0� � � � � � � �

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0� � � � � � � �

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0� � � � � � � �

1





��X

v red

w(v) �X

u blue

w(u)�� t (w(1) � w(n)).

t = 3

�

�

�

�R

0� � � � � � � �

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0� � � � � � � �

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0� � � � � � � �

1

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0� � � � � � � �

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0� � � � � � � �

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1


Proof Outline (RTP: qA(G ) ≤ 1− 2r i(G ) + ε).

Fix ε and a partition A = A1, . . . ,Ak where δn > |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.

G : A1, . . . ,Ak



Fix ε and a partition A = A1, . . . ,Ak where δn > |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


G : A1, . . . ,Ak


Proof Outline

G : A1, . . . , An

. . .

A1

Aj

Ak

1

(RTP: qA(G ) � 1 � 2r i(G ) + �).

2


These induce pairings on parts in G .




These induce pairing on parts of G .


Proof Outline

G : A1, . . . , An

1

(RTP: qA(G ) � 1 � 2r i(G ) + �).

1


G : A1, . . . ,Ak


Proof Outline

G : A1, . . . , An

. . .

A1

Aj

Ak

1

(RTP: qA(G ) � 1 � 2r i(G ) + �).

2






Choose Pα 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



G : A1, . . . ,Ak


Proof Outline

G : A1, . . . , An

. . .

A1

Aj

Ak

1

(RTP: qA(G ) � 1 � 2r i(G ) + �).

2







∴ EαPAIRS := # edges between paired parts ≤ m/t.


Proof Outline

G : A1, . . . , An

. . .

A1

Aj

Ak

1

(RTP: qA(G ) � 1 � 2r i(G ) + �).

2



G : A1, . . . ,Ak


Proof Outline

G : A1, . . . , An

. . .

A1

Aj

Ak

1

(RTP: qA(G ) � 1 � 2r i(G ) + �).

2








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


Proof Outline

G : A1, . . . , An

. . .

A1

Aj

Ak

1

(RTP: qA(G ) � 1 � 2r i(G ) + �).

2



G : A1, . . . ,Ak


Proof Outline

G : A1, . . . , An

. . .

A1

Aj

Ak

1

(RTP: qA(G ) � 1 � 2r i(G ) + �).

2









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.


Proof Outline

G : A1, . . . , An

. . .

A1

Aj

Ak

1

(RTP: qA(G ) � 1 � 2r i(G ) + �).

2



G : A1, . . . ,Ak









|#redV −#blueV |


Proof Outline

G : A1, . . . , An

. . .

A1

Aj

Ak

(RTP: qA(G ) � 1 � 2r i(G ) + �).

1


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









|#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







G : A1, . . . ,Ak










EαR,B := # edges between red and blue parts.


Proof Outline

G : A1, . . . , An

. . .

A1

Aj

Ak

(RTP: qA(G ) � 1 � 2r i(G ) + �).

1







G : A1, . . . ,Ak











∴ EαR,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







G : A1, . . . ,Ak












2 |A1|)but, E[EαR,B ] = EαPAIRS + 1

2Eα¬PAIRS


Proof Outline

G : A1, . . . , An

. . .

A1

Aj

Ak

(RTP: qA(G ) � 1 � 2r i(G ) + �).

1







G : A1, . . . ,Ak













2Eα¬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







G : A1, . . . ,Ak



Fix ε and a partition A = A1, . . . ,Ak where δn > |A1| ≥ . . . ≥ |Ak |.Step 1.



Step 2.




2Eα¬PAIRS



Proof Outline

G : A1, . . . , An

. . .

A1

Aj

Ak

(RTP: qA(G ) � 1 � 2r i(G ) + �).

1







G : A1, . . . ,Ak






Step 2.




2Eα¬PAIRS


qEA(G ) = 1m

∑A∈A E (A) = 1− 1

m (EαPAIRS + Eα¬PAIRS)


Proof Outline

G : A1, . . . , An

. . .

A1

Aj

Ak

(RTP: qA(G ) � 1 � 2r i(G ) + �).

1







G : A1, . . . ,Ak






Step 2.




2Eα¬PAIRS


qEA(G ) = 1m

∑A∈A E (A) = 1− 1


= 1− 1m (2E[EαR,B ]− EαPAIRS)


Proof Outline

G : A1, . . . , An

. . .

A1

Aj

Ak

(RTP: qA(G ) � 1 � 2r i(G ) + �).

1







G : A1, . . . ,Ak






Step 2.




2Eα¬PAIRS


qEA(G ) = 1m

∑A∈A E (A) = 1− 1


= 1− 1m (2E[EαR,B ]− EαPAIRS) ≥ 1− 2

rn i(G )(n − tδ)− 1t


Proof Outline

G : A1, . . . , An

. . .

A1

Aj

Ak

(RTP: qA(G ) � 1 � 2r i(G ) + �).

1







G : A1, . . . ,Ak






Step 2.




2Eα¬PAIRS


qEA(G ) = 1m

∑A∈A E (A) = 1− 1


= 1− 1m (2E[EαR,B ]− EαPAIRS) ≤ 1− 2

r i(G )− 2tδr − 1

t


Proof Outline

G : A1, . . . , An

. . .

A1

Aj

Ak

(RTP: qA(G ) � 1 � 2r i(G ) + �).

1







G : A1, . . . ,Ak






Step 2.




2Eα¬PAIRS


qEA(G ) = 1m

∑A∈A E (A) = 1− 1


= 1− 1m (2E[EαR,B ]− EαPAIRS) ≤ 1− 2

r i(G )− 2tδr − 1

t

∴ choose δ, t and we are done. �


Proof Outline

G : A1, . . . , An

. . .

A1

Aj

Ak

(RTP: qA(G ) � 1 � 2r i(G ) + �).

1







G : A1, . . . ,Ak

Introduction Random Graphs Lattices & Geometry Treewidth Open QuestionsIntroduction Edge Expansion & Random Cubic Lattices Open Questions


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)).


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 ) + ".

Modularity on Random Graphs, Lattices and Embedded Graphsskerman/2015NJCUpload.pdf · Modularity on...

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Transcript of Modularity on Random Graphs, Lattices and Embedded Graphsskerman/2015NJCUpload.pdf · Modularity on...