variance of the subgraph count for sparse erdős–rényi graphs robert ellis (iit applied math)

Variance of the subgraph count for sparse Erdős–Rényi graphs

Robert Ellis (IIT Applied Math)James Ferry (Metron, Inc.)

AMS Spring Central Section MeetingApril 5, 2008

2

Overview

Definitions– Erdős–Rényi random graph model G(n,p)

– Subgraph H with count XH

Computing the variance of XH

– Encoding in a graph polynomial invariant– Isolating dominating contribution for sparse p = p(n)

– Developing a compact recursive formula

Application– Tight asymptotic variance including two interesting cases

• H a cycle with trees attached• H a tree

3

Subgraph Count XH for G(n,p)

XH = #copies of a fixed graph H in an instance of G(n,p)– Example:

copies ofcopy of

Instance ofG(n,p) forn = 6, p = 0.5

123456788 copies of XH = 8 for this instance

H =

4

[XH]: average #copies of H in an instance of G(n,p)

– From Erdős:

Expected Value of Subgraph Count XH

( )( )![ ]

( ) | Aut( ) |e H

H

n v HX p

v H H

æ ö÷ç= ÷ç ÷÷çè øE

arrange H on v(H)choose v(H) probability of all e(H) edges of H appearing

H#vertices: v(H) = 4

#edges: e(H) = 4

#automorphisms:

|Aut(H)| = 2 :

[ ] =

5

82010

Example: distribution of XH for n = 6, p = 0.5

– Variance:

860

Distribution of Subgraph Count XH

H =

Instance ofG(n,p)

copies of

…0 1 2 3 4 5 6 7 8 9 10 1112 1314 1516 1718 192021 2223 2425 2627

0.025

0.05

0.075

0.1

0.125

0.15

0.175

Pro

bab

ility

XH

[XH] = 180 p2 = 11.25

6

Previous Work on Distribution of XH

Threshold p(n) for H appearing when– H is balanced (Erdős,Rényi `69)– H is unbalanced (Bollobás `81)

H strictly balanced => Poisson distribution at threshold (Bollobás `81; Karoński, Ruciński `83)

Poisson distribution at threshold => H strictly balanced (Ruciński,Vince `85)

Subgraph decomposition approach for distribution of balanced H at threshold (Bollobás,Wierman `89)

7

A Formula for Normalized Variance (XH)

Lemma [Ahearn,Phillips]: For fixed H, and G » G(n,p),

where is all copies with

8

A Formula for Normalized Variance (XH)

Proof: Write . Then

bijection :[n]=H

(symmetric graph process)

reindex

linearity of expectation

9

(n-v(H))k ordered lists

A Formula for Normalized Variance (XH) (II)

Variance Formula:

??

1

r

2

5 6

3

4

s

r

s

Theorem [E,F]:

where the sum is over subgraphs H1,H2 with k ( ) fewer vertices (edges) than H.

10

A Graph Polynomial Invariant

The polynomial invariant for a fixed graph H

11

Normalized Variance (XH) and the Subgraph Plot

Re-express

From: Random Graphs (Janson, Łuczak, & Ruciński)

Subgraph Plot for

1

2

3

4

5

6

7

1 2 3 4 5 6

12

Asymptotic contributors of the Subgraph Plot

Leading variance terms lie on the “roof” Range of p(n) determines contributing terms

From: Random Graphs (Janson, Łuczak, & Ruciński)

Subgraph Plot for

1

2

3

4

5

6

7

1 2 3 4 5 6

13

Restricted Polynomial Invariant

For , contributors contain the “2-core” C(H).

Correspondingly restrict M(H;x,y):

k=2k=1k=0

14

Decomposition of M(H;x)

M(H;x) := mk,k(H) xk expressed as sum over2-core permutations

Breaks M(H;x) into easierrooted tree computations

H

M (H ; x) =X

¼

Y

i2V (C(H ))

B (Ti;T¼(i) ; x)

5

6 3

1 2

4

C(H)T1

T2 T3 T4 T5 T6V (C(H ))

= f 1;2;3;4;5;6g

15

Recursive Computation of M(H;x)

, whereM (H ; x) =X

¼

Y

i2V (C(H ))

B (Ti;T¼(i) ; x)

( ) ( )2 1(0) (0) (0)1 2T T T=

(0)1T (0)

1T (0)2T

( )2(1) (1)1T T=

(1)1T (1)

1Toverlay

16

Concluding Remarks

Compact recursive formula for asymptotic variance for subgraph count of H when when H has nonempty 2-core

Expected value and variance can both be finite when C(H) is a cycle

Case for H a tree uses just B(T(0),T(1);x)

Seems extendable to induced subgraph counts, amenable to bounding variance contribution from elsewhere in the subgraph plot

Preprint: http://math.iit.edu/~rellis/papers/12variance.pdf