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A random recursive tree is a rooted tree constructed by successively choosing a vertex uniformly at random and adding a child to the selected vertex. A random preferential attachment tree is grown in a similar manner, but the vertex selection is made proportional to a linear function of the number of children of a vertex. Preferential attachment trees are the tree version of the Barabasi-Albert preferential attachment model.

We consider a red-blue colouring of the vertices of preferential attachment trees, which we call a broadcasting induced colouring: the root is either red or blue with equal probability, while for a fixed value p between 0 and 1, every other vertex is assigned the same colour as its parent with probability p and the other colour otherwise.

In this talk I will discuss properties of preferential attachment trees with broadcasting induced colourings, including limit laws for the number of vertices, clusters (maximal monochromatic subtrees) and leaves of each colour. The main focus of the talk will be on the size of the root cluster, that is, the maximal monochromatic subtree containing the root.

Kontaktperson för denna film
Paul Thévenin, Sannolikhetsteori och kombinatorik

Kanalinformation

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2. Some Experiments with the Coalescent - Raazesh Sainudiin
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5. Fragmentation Process derived from alpha-stable Galton-Watson trees (Gabriel Berzunza-Ojeda, 28/10/2021)
6. How does the chromatic number of a random graph vary? (Annika Heckel, 04/11/2021)
7. Broadcasting induced colourings of preferential attachment trees
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9. Phase transitions of composition schemes: Mittag-Leffler and mixed Poisson distributions (Michael Wallner, 03/02/2022)
10. Asymptotic behaviour of a uniform factorization of a long cycle
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12. On the geometry of biconditioned random trees (Cyril Marzouk, 14/10/2021)