Transitions from trees to cycles in adaptive flow networks

Transport networks are crucial to the functioning of natural and technological systems. Nature features transport networks that are adaptive over a vast range of parameters, thus providing an impressive level of robustness in supply. Theoretical and experimental studies have found that real-world transport networks exhibit both tree-like motifs and cycles. When the network is subject to load fluctuations, the presence of cyclic motifs may help to reduce flow fluctuations and, thus, render supply in the network more robust. While previous studies considered network topology via optimization principles, here, we take a dynamical systems approach and study a simple model of a flow network with dynamically adapting weights (conductances). We assume a spatially non-uniform distribution of rapidly fluctuating loads in the sinks and investigate what network configurations are dynamically stable. The network converges to a spatially non-uniform stable configuration composed of both cyclic and tree-like structures. Cyclic structures emerge locally in a transcritical bifurcation as the amplitude of the load fluctuations is increased. The resulting adaptive dynamics thus partitions the network into two distinct regions with cyclic and tree-like structures. The location of the boundary between these two regions is determined by the amplitude of the fluctuations. These findings may explain why natural transport networks display cyclic structures in the micro-vascular regions near terminal nodes, but tree-like features in the regions with larger veins

Trinets encode tree-child and level-2 phylogenetic networks

Phylogenetic networks generalize evolutionary trees, and are commonly used to represent evolutionary histories of species that undergo reticulate evolutionary processes such as hybridization, recombination and lateral gene transfer. Recently, there has been great interest in trying to develop methods to construct rooted phylogenetic networks from triplets, that is rooted trees on three species. However, although triplets determine or encode rooted phylogenetic trees, they do not in general encode rooted phylogenetic networks, which is a potential issue for any such method. Motivated by this fact, Huber and Moulton recently introduced trinets as a natural extension of rooted triplets to networks. In particular, they showed that level-1 phylogenetic networks are encoded by their trinets, and also conjectured that all "recoverable" rooted phylogenetic networks are encoded by their trinets. Here we prove that recoverable binary level-2 networks and binary tree-child networks are also encoded by their trinets. To do this we prove two decomposition theorems based on trinets which hold for all recoverable binary rooted phylogenetic networks. Our results provide some additional evidence in support of the conjecture that trinets encode all recoverable rooted phylogenetic networks, and could also lead to new approaches to construct phylogenetic networks from trinets

Trinets encode tree-child and level-2 phylogenetic networks

Phylogenetic networks generalize evolutionary trees, and are commonly used to represent evolutionary histories of species that undergo reticulate evolutionary processes such as hybridization, recombination and lateral gene transfer. Recently, there has been great interest in trying to develop methods to construct rooted phylogenetic networks from triplets, that is rooted trees on three species. However, although triplets determine or encode rooted phylogenetic trees, they do not in general encode rooted phylogenetic networks, which is a potential issue for any such method. Motivated by this fact, Huber and Moulton recently introduced trinets as a natural extension of rooted triplets to networks. In particular, they showed that level-1 level-1 phylogenetic networks are encoded by their trinets, and also conjectured that all “recoverable” rooted phylogenetic networks are encoded by their trinets. Here we prove that recoverable binary level-2 networks and binary tree-child networks are also encoded by their trinets. To do this we prove two decomposition theorems based on trinets which hold for all recoverable binary rooted phylogenetic networks. Our results provide some additional evidence in support of the conjecture that trinets encode all recoverable rooted phylogenetic networks, and could also lead to new approaches to construct phylogenetic networks from trinets

From trees to networks and back

The evolutionary history of a set of species is commonly represented by a phylogenetic tree. Often, however, the data contain conflicting signals, which can be better represented by a more general structure, namely a phylogenetic network. Such networks allow the display of several alternative evolutionary scenarios simultaneously but this can come at the price of complex visual representations. Using so-called circular split networks reduces this complexity, because this type of network can always be visualized in the plane without any crossing edges. These circular split networks form the core of this thesis. We construct them, use them as a search space for minimum evolution trees and explore their properties. More specifically, we present a new method, called SuperQ, to construct a circular split network summarising a collection of phylogenetic trees that have overlapping leaf sets. Then, we explore the set of phylogenetic trees associated with a �fixed circular split network, in particular using it as a search space for optimal trees. This set represents just a tiny fraction of the space of all phylogenetic trees, but we still �find trees within it that compare quite favourably with those obtained by a leading heuristic, which uses tree edit operations for searching the whole tree space. In the last part, we advance our understanding of the set of phylogenetic trees associated with a circular split network. Specifically, we investigate the size of the so-called circular tree neighbourhood for the three tree edit operations, tree bisection and reconnection (tbr), subtree prune and regraft (spr) and nearest neighbour interchange (nni)

The height of random kk-trees and related branching processes

We consider the height of random k-trees and k-Apollonian networks. These random graphs are not really trees, but instead have a tree-like structure. The height will be the maximum distance of a vertex from the root. We show that w.h.p. the height of random k-trees and k-Apollonian networks is asymptotic to clog t, where t is the number of vertices, and c=c(k) is given as the solution to a transcendental equation. The equations are slightly different for the two types of process. In the limit as k-->oo the height of both processes is asymptotic to log t/(k log 2)

An improved connectionist activation function for energy minimization

Symmetric networks that are based on energy minimization, such as Boltzmann machines or Hopfield nets, are used extensively for optimization, constraint satisfaction, and approximation of NP-hard problems. Nevertheless, finding a global minimum for the energy function is not guaranteed, and even a local minimum may take an exponential number of steps. We propose an improvement to the standard activation function used for such networks. The improved algorithm guarantees that a global minimum is found in linear time for tree-like subnetworks. The algorithm is uniform and does not assume that the network is a tree. It performs no worse than the standard algorithms for any network topology. In the case where there are trees growing from a cyclic subnetwork, the new algorithm performs better than the standard algorithms by avoiding local minima along the trees and by optimizing the free energy of these trees in linear time. The algorithm is self-stabilizing for trees (cycle-free undirected graphs) and remains correct under various scheduling demons. However, no uniform protocol exists to optimize trees under a pure distributed demon and no such protocol exists for cyclic networks under central demon