Variance of the subgraph count for sparse Erdős–Rényi graphs Robert Ellis (IIT Applied Math)
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Transcript of 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
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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
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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
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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)
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A Formula for Normalized Variance (XH)
Lemma [Ahearn,Phillips]: For fixed H, and G » G(n,p),
where is all copies with
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A Formula for Normalized Variance (XH)
Proof: Write . Then
bijection :[n]![n](H2)=H
(symmetric graph process)
reindex
linearity of expectation
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(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.
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A Graph Polynomial Invariant
The polynomial invariant for a fixed graph H
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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
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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
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Restricted Polynomial Invariant
For , contributors contain the “2-core” C(H).
Correspondingly restrict M(H;x,y):
k=2k=1k=0
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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
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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
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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