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