Variance of the subgraph count for sparse Erdős–Rényi graphs Robert Ellis (IIT Applied Math)

1. 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 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAA

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. XH = 8 for this instance Subgraph Count XH for G(n,p) • XH = #copies of a fixed graph H in an instance of G(n,p) • Example: H = Instance ofG(n,p) forn = 6, p = 0.5 4 8 7 6 5 3 2 1 8 copy of copies of copies of

4. #vertices: v(H)= 4 #edges: e(H)= 4 #automorphisms: |Aut(H)|= 2 : E[ ] = choose v(H) arrange H on v(H) probability of all e(H) edges of H appearing Expected Value of Subgraph Count XH • E[XH]: average #copies of H in an instance of G(n,p) • From Erdős: H

5. H = copies of Distribution of Subgraph Count XH • Example: distribution of XH for n = 6, p = 0.5 • Variance: Instance ofG(n,p) Probability E [XH] = 180 p2 = 11.25 20 10 8 8 6 0 … XH

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. bijection:[n]![n] (H2)=H (symmetric graph process) reindex A Formula for Normalized Variance (XH) • Proof: Write . Then linearity of expectation

9. Theorem [E,F]: where the sum is over subgraphs H1,H2 with k ( ) fewer vertices (edges) than H. 1 2 5 6 3 4 ?? r s r s (n-v(H))k ordered lists A Formula for Normalized Variance (XH) (II) • Variance Formula:

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)

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)

13. Restricted Polynomial Invariant For , contributors contain the “2-core” C(H). Correspondingly restrictM(H;x,y): k=0 k=2 k=1

14. 4 5 3 6 2 1 C(H) T1 T2 T3 T4 T5 T6 Decomposition of M(H;x) • M(H;x) := mk,k(H) xkexpressed as sum over2-core permutations • Breaks M(H;x) into easierrooted tree computations H

15. overlay Recursive Computation of M(H;x) , where

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