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52374 results in Statistics and Probability

Page 108 of 2619

  • On minimum spanning trees for random Euclidean bipartite graphs

  • Part of
  • Mario Correddu, Dario Trevisan
  • Journal:
    Combinatorics, Probability and Computing / Volume 33 / Issue 3 / May 2024
    Published online by Cambridge University Press:
    30 November 2023, pp. 319-350

  • We consider the minimum spanning tree problem on a weighted complete bipartite graph $K_{n_R, n_B}$ whose $n=n_R+n_B$ vertices are random, i.i.d. uniformly distributed points in the unit cube in $d$ dimensions and edge weights are the $p$-th power of their Euclidean distance, with $p\gt 0$. In the large $n$ limit with $n_R/n \to \alpha _R$ and $0\lt \alpha _R\lt 1$, we show that the maximum vertex degree of the tree grows logarithmically, in contrast with the classical, non-bipartite, case, where a uniform bound holds depending on $d$ only. Despite this difference, for $p\lt d$, we are able to prove that the total edge costs normalized by the rate $n^{1-p/d}$ converge to a limiting constant that can be represented as a series of integrals, thus extending a classical result of Avram and Bertsimas to the bipartite case and confirming a conjecture of Riva, Caracciolo and Malatesta.

  • Dynamics of information networks

  • Part of
  • Andrei Sontag, Tim Rogers, Christian A Yates
  • Journal:
    Journal of Applied Probability / Volume 61 / Issue 3 / September 2024
    Published online by Cambridge University Press:
    30 November 2023, pp. 1029-1039
    Print publication:
    September 2024

  • We explore a simple model of network dynamics which has previously been applied to the study of information flow in the context of epidemic spreading. A random rooted network is constructed that evolves according to the following rule: at a constant rate, pairs of nodes (i, j) are randomly chosen to interact, with an edge drawn from i to j (and any other out-edge from i deleted) if j is strictly closer to the root with respect to graph distance. We characterise the dynamics of this random network in the limit of large size, showing that it instantaneously forms a tree with long branches that immediately collapse to depth two, then it slowly rearranges itself to a star-like configuration. This curious behaviour has consequences for the study of the epidemic models in which this information network was first proposed.

Page 108 of 2619