Invariance and attraction properties of Galton-Watson trees [article]

Yevgeniy Kovchegov, Ilya Zaliapin
2020 arXiv   pre-print
We give a description of invariants and attractors of the critical and subcritical Galton-Watson tree measures under the operation of Horton pruning (cutting tree leaves with subsequent series reduction). Under a regularity condition, the class of invariant measures consists of the critical binary Galton-Watson tree and a one-parameter family of critical Galton-Watson trees with offspring distribution {q_k} that has a power tail q_k∼ Ck^-(1+1/q_0), where q_0∈(1/2,1). Each invariant measure has
more » ... non-empty domain of attraction under consecutive Horton pruning, specified by the tail behavior of the initial Galton-Watson offspring distribution. The invariant measures satisfy the Toeplitz property for the Tokunaga coefficients and obey the Horton law with exponent R = (1-q_0)^-1/q_0.
arXiv:1911.08095v3 fatcat:yaqersagxbcfdinr65ewwrfzfi