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Variance of the subgraph count for sparse Erdős–Rényi graphs Robert Ellis (IIT Applied Math) James Ferry (Metron, Inc.) AMS Spring Central Section Meeting April 5, 2008 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A

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2 Overview Definitions –Erdős–Rényi random graph model G(n,p) –Subgraph H with count X H Computing the variance of X H –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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3 Subgraph Count X H for G(n,p) X H = #copies of a fixed graph H in an instance of G(n,p) –Example: copies of copy of Instance of G(n,p) for n = 6, p = 0.5 123456788 copies of X H = 8 for this instance H =

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4 [X H ] : average #copies of H in an instance of G(n,p) –From Erdős: Expected Value of Subgraph Count X H 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 : [ ] =

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5 82010 Example: distribution of X H for n = 6, p = 0.5 –Variance: 860 Distribution of Subgraph Count X H H = Instance of G(n,p) copies of … Probability XHXH [X H ] = 180 p 2 = 11.25

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6 Previous Work on Distribution of X H 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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7 A Formula for Normalized Variance (X H ) Lemma [Ahearn,Phillips]: For fixed H, and G » G(n,p), where is all copies with

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8 A Formula for Normalized Variance (X H ) Proof: Write. Then bijection :[n] ! [n] (H 2 )=H (symmetric graph process) reindex linearity of expectation

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9 (n-v(H)) k ordered lists A Formula for Normalized Variance (X H ) (II) Variance Formula: ?? 1 r 2 56 3 4 s rs Theorem [E,F]: where the sum is over subgraphs H 1,H 2 with k ( ) fewer vertices (edges) than H.

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10 A Graph Polynomial Invariant The polynomial invariant for a fixed graph H

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11 Normalized Variance (X H ) and the Subgraph Plot Re-express From: Random Graphs (Janson, Łuczak, & Ruciński)

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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)

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

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14 Decomposition of M(H;x) M(H;x) := m k,k (H) x k expressed as sum over 2-core permutations Breaks M(H;x) into easier rooted tree computations H 5 63 12 4 C(H) T1T1 T2T2 T3T3 T4T4 T5T5 T6T6

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

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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

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