Related papers: Link Patterns and the Catalan Tree
Dynamical patterns in complex networks of coupled oscillators are both of theoretical and practical interest, yet to fully reveal and understand the interplay between pattern emergence and network structure remains to be an outstanding…
We address the problem of link reciprocity, the non-random presence of two mutual links between pairs of vertices. We propose a new measure of reciprocity that allows the ordering of networks according to their actual degree of correlation…
We investigate the tree gonality of a genus-$g$ metric graph, defined as the minimum degree of a tropical morphism from any tropical modification of the metric graph to a metric tree. We give a combinatorial constructive proof that this…
Neural Networks and Decision Trees: two popular techniques for supervised learning that are seemingly disconnected in their formulation and optimization method, have recently been combined in a single construct. The connection pivots on…
This paper provides a relationship between a geometric structure of a suspended tree and the number of link components of the associated link diagram.
A result about spanning forests for graphs yields a short proof of Krebes's theorem concerning embedded tangles in links.
We present an explicit connected spanning structure that appears in a random graph just above the connectivity threshold with high probability.
We are surrounded by spatio-temporal patterns resulting from the interaction of the numerous basic units constituting natural or human-made systems. In presence of diffusive-like coupling, Turing theory has been largely applied to explain…
We develop a canonical pairing between trees and graphs, which passes to their quotients by Jacobi identities. This pairing is an effective and simple tool for understanding the Lie and Poisson operads, providing canonical duals. In the…
New applications of data mining, such as in biology, bioinformatics, or sociology, are faced with large datasetsstructured as graphs. We introduce a novel class of tree-shapedpatterns called tree queries, and present algorithms for…
Phylogenetic networks are a generalisation of phylogenetic trees that allow for more complex evolutionary histories that include hybridisation-like processes. It is of considerable interest whether a network can be considered `tree-like' or…
We introduce a random graph model based on k-trees, which can be generated by applying a probabilistic preferential attachment rule, but which also has a simple combinatorial description. We carry out a precise distributional analysis of…
We develop basic cluster theory from an elementary point of view using a variation of binary trees which we call mixed cobinary trees. We show that the number of isomorphism classes of such trees is given by the Catalan number Cn where n is…
We study the set of networks, which consist of sources, sinks and neutral points, bijective to the permutations. The set of directed edges, which characterizes a network, is constructed from a polyomino or a Rothe diagram of a permutation…
We consider an infinite sequence of rooted trees naturally emerging in a number-theoretical context. We advance some ideas on its structure by discussing some elementary properties. Some of those properties are shown to be related to…
Many physical, biological, and social phenomena can be described by cascades taking place on a network. Often, the activity can be empirically observed, but not the underlying network of interactions. In this paper we offer three…
Understanding the origins of complexity is a fundamental challenge with implications for biological and technological systems. Network theory emerges as a powerful tool to model complex systems. Networks are an intuitive framework to…
Building taxonomies is often a significant part of building an ontology, and many attempts have been made to automate the creation of such taxonomies from relevant data. The idea in such approaches is either that relevant definitions of the…
Terms are a concise representation of tree structures. Since they can be naturally defined by an inductive type, they offer data structures in functional programming and mechanised reasoning with useful principles such as structural…
Phylogenetic networks generalise phylogenetic trees and allow for the accurate representation of the evolutionary history of a set of present-day species whose past includes reticulate events such as hybridisation and lateral gene transfer.…