Eigenvalue distribution in random trees Stephan Wagner (joint work with Kenneth Dadedzi) Naples, 13 September 2018
Tree eigenvalues To every graph, and in particular every tree, we can associate an adjacency matrix, a Laplacian matrix, and several other interesting matrices. v 1 0 1 0 0 0 1 − 1 0 0 0 v 2 1 0 1 0 0 − 1 2 − 1 0 0 A = 0 1 0 1 1 L = 0 − 1 3 − 1 − 1 v 3 0 0 1 0 0 0 0 − 1 1 0 0 0 1 0 0 0 0 − 1 0 1 v 4 v 5 Eigenvalue distribution in random trees S. Wagner, Stellenbosch University 2 / 33
Tree eigenvalues We will be interested in the eigenvalues of these matrices. v 1 0 1 0 0 0 1 − 1 0 0 0 v 2 1 0 1 0 0 − 1 2 − 1 0 0 A = 0 1 0 1 1 L = 0 − 1 3 − 1 − 1 v 3 0 0 1 0 0 0 0 − 1 1 0 0 0 1 0 0 0 0 − 1 0 1 v 4 v 5 Eigenvalues of A : − 1 . 84776 , − 0 . 765367 , 0 , 0 . 765367 , 1 . 84776 Eigenvalues of L : 0 , 0 . 518806 , 1 , 2 . 31111 , 4 . 17009 Eigenvalue distribution in random trees S. Wagner, Stellenbosch University 3 / 33
A large random tree A tree with 100 vertices (left) and the distribution of the eigenvalues of its adjacency matrix (right). - 2 - 1 0 1 2 Eigenvalue distribution in random trees S. Wagner, Stellenbosch University 4 / 33
A large random tree A tree with 100 vertices (left) and the distribution of the eigenvalues of its Laplacian matrix (right). 0 1 2 3 4 5 6 Eigenvalue distribution in random trees S. Wagner, Stellenbosch University 5 / 33
The spectral distribution Definition Let T be a fixed tree with n vertices. Picking one of the n eigenvalues (counted with multiplicity) uniformly at random, we obtain a random variable X T : P ( X T = x ) = multiplicity of x as an eigenvalue of T . n The distribution of this random variable is the spectral distribution of T . Eigenvalue distribution in random trees S. Wagner, Stellenbosch University 6 / 33
The spectral distribution Definition Let T be a fixed tree with n vertices. Picking one of the n eigenvalues (counted with multiplicity) uniformly at random, we obtain a random variable X T : P ( X T = x ) = multiplicity of x as an eigenvalue of T . n The distribution of this random variable is the spectral distribution of T . What can we say about the spectral distribution of a large random tree T ? Eigenvalue distribution in random trees S. Wagner, Stellenbosch University 6 / 33
Uniform models The simplest type of model uses the uniform distribution on the set of trees of a given order within a specified family (e.g. the family of all labelled trees, all unlabelled trees or all binary trees). Eigenvalue distribution in random trees S. Wagner, Stellenbosch University 7 / 33
Uniform models The simplest type of model uses the uniform distribution on the set of trees of a given order within a specified family (e.g. the family of all labelled trees, all unlabelled trees or all binary trees). The analysis of such models often involves exact counting and generating functions. Eigenvalue distribution in random trees S. Wagner, Stellenbosch University 7 / 33
Uniform models The simplest type of model uses the uniform distribution on the set of trees of a given order within a specified family (e.g. the family of all labelled trees, all unlabelled trees or all binary trees). The analysis of such models often involves exact counting and generating functions. In particular, this is the case for simply generated families of trees. Eigenvalue distribution in random trees S. Wagner, Stellenbosch University 7 / 33
Simply generated trees On the set of all rooted ordered (plane) trees, we impose a weight function by first specifying a sequence 1 = w 0 , w 1 , w 2 , . . . and then setting w K i ( T ) � w ( T ) = , i i ≥ 0 where K i ( T ) is the number of vertices of outdegree i in T . Then we pick a tree of given order n at random, with probabilities proportional to the weights. For instance, Eigenvalue distribution in random trees S. Wagner, Stellenbosch University 8 / 33
Simply generated trees On the set of all rooted ordered (plane) trees, we impose a weight function by first specifying a sequence 1 = w 0 , w 1 , w 2 , . . . and then setting w K i ( T ) � w ( T ) = , i i ≥ 0 where K i ( T ) is the number of vertices of outdegree i in T . Then we pick a tree of given order n at random, with probabilities proportional to the weights. For instance, w 0 = w 1 = w 2 = · · · = 1 generates random plane trees, Eigenvalue distribution in random trees S. Wagner, Stellenbosch University 8 / 33
Simply generated trees On the set of all rooted ordered (plane) trees, we impose a weight function by first specifying a sequence 1 = w 0 , w 1 , w 2 , . . . and then setting w K i ( T ) � w ( T ) = , i i ≥ 0 where K i ( T ) is the number of vertices of outdegree i in T . Then we pick a tree of given order n at random, with probabilities proportional to the weights. For instance, w 0 = w 1 = w 2 = · · · = 1 generates random plane trees, w 0 = w 2 = 1 (and w i = 0 otherwise) generates random binary trees, Eigenvalue distribution in random trees S. Wagner, Stellenbosch University 8 / 33
Simply generated trees On the set of all rooted ordered (plane) trees, we impose a weight function by first specifying a sequence 1 = w 0 , w 1 , w 2 , . . . and then setting w K i ( T ) � w ( T ) = , i i ≥ 0 where K i ( T ) is the number of vertices of outdegree i in T . Then we pick a tree of given order n at random, with probabilities proportional to the weights. For instance, w 0 = w 1 = w 2 = · · · = 1 generates random plane trees, w 0 = w 2 = 1 (and w i = 0 otherwise) generates random binary trees, w i = 1 i ! generates random rooted labelled trees. Eigenvalue distribution in random trees S. Wagner, Stellenbosch University 8 / 33
Branching processes A classical branching model to generate random trees is the Galton-Watson tree model: fix a probability distribution on the set { 0 , 1 , 2 , . . . } . Eigenvalue distribution in random trees S. Wagner, Stellenbosch University 9 / 33
Branching processes A classical branching model to generate random trees is the Galton-Watson tree model: fix a probability distribution on the set { 0 , 1 , 2 , . . . } . Start with a single vertex, the root. Eigenvalue distribution in random trees S. Wagner, Stellenbosch University 9 / 33
Branching processes A classical branching model to generate random trees is the Galton-Watson tree model: fix a probability distribution on the set { 0 , 1 , 2 , . . . } . Start with a single vertex, the root. At time t , all vertices at level t (i.e., distance t from the root) produce a number of children, independently at random according to the fixed distribution (some of the vertices might therefore not have children at all). Eigenvalue distribution in random trees S. Wagner, Stellenbosch University 9 / 33
Branching processes A classical branching model to generate random trees is the Galton-Watson tree model: fix a probability distribution on the set { 0 , 1 , 2 , . . . } . Start with a single vertex, the root. At time t , all vertices at level t (i.e., distance t from the root) produce a number of children, independently at random according to the fixed distribution (some of the vertices might therefore not have children at all). A random Galton-Watson tree of order n is obtained by conditioning the process. Eigenvalue distribution in random trees S. Wagner, Stellenbosch University 9 / 33
Branching processes A classical branching model to generate random trees is the Galton-Watson tree model: fix a probability distribution on the set { 0 , 1 , 2 , . . . } . Start with a single vertex, the root. At time t , all vertices at level t (i.e., distance t from the root) produce a number of children, independently at random according to the fixed distribution (some of the vertices might therefore not have children at all). A random Galton-Watson tree of order n is obtained by conditioning the process. Simply generated trees and Galton-Watson trees are essentially equivalent. For example, a geometric distribution for branching will result in a random plane tree, a Poisson distribution in a random rooted labelled tree. Eigenvalue distribution in random trees S. Wagner, Stellenbosch University 9 / 33
Simply generated and Galton-Watson trees An example: Consider the Galton-Watson process based on a geometric distribution with P ( X = k ) = pq k ( p = 1 − q ). Eigenvalue distribution in random trees S. Wagner, Stellenbosch University 10 / 33
Simply generated and Galton-Watson trees An example: Consider the Galton-Watson process based on a geometric distribution with P ( X = k ) = pq k ( p = 1 − q ). The tree above has probability p 7 ( pq ) 2 ( pq 2 ) 2 ( pq 3 ) 2 = p 13 q 12 , Eigenvalue distribution in random trees S. Wagner, Stellenbosch University 10 / 33
Simply generated and Galton-Watson trees An example: Consider the Galton-Watson process based on a geometric distribution with P ( X = k ) = pq k ( p = 1 − q ). The tree above has probability p 7 ( pq ) 2 ( pq 2 ) 2 ( pq 3 ) 2 = p 13 q 12 , as does every tree with 13 vertices. Eigenvalue distribution in random trees S. Wagner, Stellenbosch University 10 / 33
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