Limit Laws for the Number of Groups formed by Social Animals under the Extra Clustering Model (joint with Michael Drmota and Yi-Wen Lee) Michael Fuchs Institute of Applied Mathematics National Chiao Tung University June 19th, 2014 Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 1 / 30
Probabilistic Analysis of a Genealogical Model of Animal Group Patterns Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 2 / 30
Phylogenetic Tree Ordered, binary, rooted tree with leafs representing the animals. Describes the genetic relatedness of animals. Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 3 / 30
Yule-Harding Model (Bottom-Up) Fundamental random model in phylogenetics. Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 4 / 30
Yule-Harding Model (Bottom-Up) Fundamental random model in phylogenetics. Uniformly choose a pair of yellow nodes and let them coalesce. Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 4 / 30
Yule-Harding Model (Bottom-Up) Fundamental random model in phylogenetics. Uniformly choose a pair of yellow nodes and let them coalesce. Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 4 / 30
Yule-Harding Model (Bottom-Up) Fundamental random model in phylogenetics. Uniformly choose a pair of yellow nodes and let them coalesce. Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 4 / 30
Yule-Harding Model (Bottom-Up) Fundamental random model in phylogenetics. Uniformly choose a pair of yellow nodes and let them coalesce. Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 4 / 30
Yule-Harding Model (Bottom-Up) Fundamental random model in phylogenetics. Uniformly choose a pair of yellow nodes and let them coalesce. Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 4 / 30
Yule-Harding Model (Bottom-Up) Fundamental random model in phylogenetics. Uniformly choose a pair of yellow nodes and let them coalesce. Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 4 / 30
Animal Groups under the Yule-Harding Model Durand, Blum and Fran¸ cois (2007) : Groups are formed more likely by animals which are genetically related. Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 5 / 30
Animal Groups under the Yule-Harding Model Durand, Blum and Fran¸ cois (2007) : Groups are formed more likely by animals which are genetically related. − → neutral model. Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 5 / 30
Animal Groups under the Yule-Harding Model Durand, Blum and Fran¸ cois (2007) : Groups are formed more likely by animals which are genetically related. − → neutral model. Clade of a leaf: All leafs of the tree rooted at the parent. Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 5 / 30
Animal Groups under the Yule-Harding Model Durand, Blum and Fran¸ cois (2007) : Groups are formed more likely by animals which are genetically related. − → neutral model. Clade of a leaf: All leafs of the tree rooted at the parent. Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 5 / 30
Animal Groups under the Yule-Harding Model Durand, Blum and Fran¸ cois (2007) : Groups are formed more likely by animals which are genetically related. − → neutral model. # of groups � # of maximal clades Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 6 / 30
Animal Groups under the Yule-Harding Model Durand, Blum and Fran¸ cois (2007) : Groups are formed more likely by animals which are genetically related. − → neutral model. # of groups � # of maximal clades � 2 Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 6 / 30
Yule-Harding Model (Top-Down) Alternative description of Yule-Harding model: Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 7 / 30
Yule-Harding Model (Top-Down) Alternative description of Yule-Harding model: Uniformly choose a yellow node and replace it by a cheery. Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 7 / 30
Yule-Harding Model (Top-Down) Alternative description of Yule-Harding model: Uniformly choose a yellow node and replace it by a cheery. Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 7 / 30
Yule-Harding Model (Top-Down) Alternative description of Yule-Harding model: Uniformly choose a yellow node and replace it by a cheery. Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 7 / 30
Yule-Harding Model (Top-Down) Alternative description of Yule-Harding model: Uniformly choose a yellow node and replace it by a cheery. Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 7 / 30
# of Groups X n = # of groups under the Yule Harding model Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 8 / 30
# of Groups X n = # of groups under the Yule Harding model We have, � 1 , if I n = 1 or I n = n − 1 , d X n = X I n + X ∗ n − I n , otherwise, where I n = Uniform { 1 , . . . , n − 1 } is the # of animals in the left subtree and X ∗ n is an independent copy of X n . Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 8 / 30
# of Groups X n = # of groups under the Yule Harding model We have, � 1 , if I n = 1 or I n = n − 1 , d X n = X I n + X ∗ n − I n , otherwise, where I n = Uniform { 1 , . . . , n − 1 } is the # of animals in the left subtree and X ∗ n is an independent copy of X n . Theorem (Durand and Fran¸ cois; 2010) We have, a := 1 − e − 2 � � E ( X n ) ∼ an . 4 Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 8 / 30
Comparison with Real-life Data Durand, Blum and Fran¸ cois (2007) presented the following data: Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 9 / 30
Extra Clustering Model Durand, Blum and Fran¸ cois (2007): Let p ≥ 0 . We have, � 1 , with probability p d X n = neutral model , otherwise . Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 10 / 30
Extra Clustering Model Durand, Blum and Fran¸ cois (2007): Let p ≥ 0 . We have, � 1 , with probability p d X n = neutral model , otherwise . Remark: p = 0 is neutral model. Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 10 / 30
Extra Clustering Model Durand, Blum and Fran¸ cois (2007): Let p ≥ 0 . We have, � 1 , with probability p d X n = neutral model , otherwise . Remark: p = 0 is neutral model. Introduced to test whether or not genetic relatedness is the sole driving force behind the group formation process. Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 10 / 30
Average Number of Groups Theorem (Durand and Fran¸ cois; 2010) We have, c ( p ) Γ(2(1 − p )) n 1 − 2 p , if p < 1 / 2; log n , if p = 1 / 2; E ( X n ) ∼ 2 p 2 p − 1 , if p > 1 / 2 , where � 1 1 (1 − t ) − 2 p e 2(1 − p ) t � 1 − (1 − p ) t 2 � c ( p ) := d t. e 2(1 − p ) 0 Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 11 / 30
Testing for the Neutral Model Durand, Blum and Fran¸ cois (2007): Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 12 / 30
Yi-Wen’s Thesis (2012) Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 13 / 30
Variance and SLLN Theorem (Lee; 2012) We have, Var( X n ) ∼ (1 − e − 2 ) 2 n log n = 4 a 2 n log n. 4 Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 14 / 30
Variance and SLLN Theorem (Lee; 2012) We have, Var( X n ) ∼ (1 − e − 2 ) 2 n log n = 4 a 2 n log n. 4 Theorem (Lee; 2012) We have, � � � � X n � � P lim E ( X n ) − 1 � = 0 = 1 . � � n →∞ � For SLLN, X n is constructed on the same probability space via the tree evolution process underlying the Yule-Harding model. Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 14 / 30
Higher Moments Theorem (Lee; 2012) For all k ≥ 3 , E ( X n − E ( X n )) k ∼ ( − 1) k 2 k k − 2 a k n k − 1 . Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 15 / 30
Higher Moments Theorem (Lee; 2012) For all k ≥ 3 , E ( X n − E ( X n )) k ∼ ( − 1) k 2 k k − 2 a k n k − 1 . This implies that all moments larger than two of X n − E ( X n ) � Var( X n ) tend to infinity! Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 15 / 30
Higher Moments Theorem (Lee; 2012) For all k ≥ 3 , E ( X n − E ( X n )) k ∼ ( − 1) k 2 k k − 2 a k n k − 1 . This implies that all moments larger than two of X n − E ( X n ) � Var( X n ) tend to infinity! Question: Is there a limit distribution? Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 15 / 30
Random Recursive Trees Unordered, rooted trees. Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 16 / 30
Random Recursive Trees Unordered, rooted trees. Uniformly choose one of the nodes and attach a child. Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 16 / 30
Random Recursive Trees Unordered, rooted trees. Uniformly choose one of the nodes and attach a child. Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 16 / 30
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