#  Statistics Colloquium: Sumit Mukherjee (Columbia University) 

 



####  calendar\_today Date and Time 

 **November 30, 2020** 

 10:30AM - 11:30AM EST 

####  pin\_drop Location 

 **Zoom - please contact emilie_campanelli@fas.harvard.edu for more information**  



 

 



 

###    ![Headshot of Sumit Mukherjee](/sites/g/files/omnuum10116/files/styles/hwp_1_1__360x360_scale/public/statistics-2/files/sumit.jpg?itok=tnkflMU_) 

 

Title:

 Motif Counting via Subgraph sampling

###  Abstract:

 Consider the subgraph sampling model, where we observe a random subgraph of a given (possibly non random) large graph $G\_n$, by choosing vertices of $G\_n$ independently at random with probability $p\_n$. In this setting, we study the question of estimating the number of copies $N(H,G\_n)$ of a fixed motif/small graph (think of $H$ as edges, two stars, triangles) in the big graph $G\_n$. We derive necessary and sufficient conditions for the consistency and the asymptotic normality of a natural Horvitz-Thompson (HT) type estimator.

 As it turns out, the asymptotic normality of the HT estimator exhibits an interesting fourth-moment phenomenon, which asserts that the HT estimator (appropriately centered and rescaled) converges in distribution to the standard normal whenever its fourth-moment converges to 3. We apply our results to several natural graph ensembles, such as sparse graphs with bounded degree, Erdős-Renyi random graphs, random regular graphs, and dense graphons.

 This talk is based on joint work with Bhaswar B. Bhattacharya and Sayan Das



 

 



 

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