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]![n](H2)=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