Phylogenetic Networks that Display a Tree Twice

In the last decade, the use of phylogenetic networks to analyze the evolution of species whose past is likely to include reticulation events, such as horizontal gene transfer or hybridization, has gained popularity among evolutionary biologists. Nevertheless, the evolution of a particular gene can generally be described without reticulation events and therefore be represented by a phylogenetic tree. While this is not in contrast to each other, it places emphasis on the necessity of algorithms that analyze and summarize the tree-like information that is contained in a phylogenetic network. We contribute to the toolbox of such algorithms by investigating the question of whether or not a phylogenetic network embeds a tree twice and give a quadratic-time algorithm to solve this problem for a class of networks that is more general than tree-child networks.     

An efficient algorithm for cable theory,applied to blowfly photoreceptor cells and LMC's

A both simple and efficient algorithm is presented that yields the voltages and currents in an arbitrary cable structure. The algorithm consists of the following steps: 1. The cable structure is divided into homogeneous cable segments; 2. Each cable segment is considered as a two-port, and replaced by an equivalent circuit consisting of discrete elements; 3. The resulting equivalent scheme of the whole cable structure is solved with an algorithm for ladder networks (or, if the structure is not tree-like, with a network analysis program), which yields the input and output voltages and currents of each cable segment; and if required 4. The voltage and current distribution in each segment is determined from the input and output voltages and currents. The algorithm is applied to blowfly photoreceptor cells that are electrically coupled, and to blowfly Large Monopolar Cells. For LMC's it is shown that the loads at the input and output sides of the axon determine whether unidirectional or bidirectional signal transmission occurs.     

Design and Performance Analysis of Divisible Load Scheduling Strategies on Arbitrary Graphs

In this paper, we consider the problem of scheduling divisible loads on arbitrary graphs with the objective to minimize the total processing time of the entire load submitted for processing. We consider an arbitrary graph network comprising heterogeneous processors interconnected via heterogeneous links in an arbitrary fashion. The divisible load is assumed to originate at any processor in the network. We transform the problem into a multi-level unbalanced tree network and schedule the divisible load. We design systematic procedures to identify and eliminate any redundant processor–link pairs (those pairs whose consideration in scheduling will penalize the performance) and derive an optimal tree structure to obtain an optimal processing time, for a fixed sequence of load distribution. Since the algorithm thrives to determine an equivalent number of processors (resources) that can be used for processing the entire load, we refer to this approach as resource-aware optimal load distribution (RAOLD) algorithm. We extend our study by applying the optimal sequencing theorem proposed for single-level tree networks in the literature for multi-level tree for obtaining an optimal solution. We evaluate the performance for a wide range of arbitrary graphs with varying connectivity probabilities and processor densities. We also study the effect of network scalability and connectivity. We demonstrate the time performance when the point of load origination differs in the network and highlight certain key features that may be useful for algorithm and/or network system designers. We evaluate the time performance with rigorous simulation experiments under different system parameters for the ease of a complete understanding.     

HybridFlood: minimizing the effects of redundant messages and maximizing search efficiency of unstructured peer-to-peer networks

Peer-to-peer systems are important Internet applications. A major portion of Internet traffic belongs to such applications. Flooding search is a basic search scheme for unstructured peer-to-peer networks, where a node must send a query message to all its neighbors when seeking a file (in a file sharing situation). Flooding has no knowledge about network topology and files distribution, thus it offers an attractive method for file discovery in dynamic and evolving networks. Although pure flooding can achieve high coverage but it produces exponentially redundant messages in each hop. Consequently, the growth of redundant messages limits system scalability and causes unnecessary traffic in networks. Besides, flooding has no opportunity to get an advantage of node diversity of participating in unstructured P2P networks. To improve this searching scheme and reduce redundant messages, this paper proposes a novel algorithm named HybridFlood. This algorithm is divided into two steps. The first step follows the flooding with a limited number of hops. In the second step, nosey nodes are selected in each searching horizon. The nosey nodes are nodes which have the most links to other nodes. These nodes maintain the data index of all client nodes. We provided analytical studies for flooding and HybridFlood. The analytical results provided the best threshold point of hop for optimum coverage growth rate and redundant messages in flooding. It also proved in HybridFlood broadcasting messages are cut down at least an order of magnitude. Thus, the proposed algorithm extends the search efficiency by reducing redundant messages in each hop. The simulation experiments validated analytical results.     

Testing for treeness: lateral gene transfer,phylogenetic inference,and model selection

A phylogeny that allows for lateral gene transfer (LGT) can be thought of as a strictly branching tree (all of whose branches are vertical) to which lateral branches have been added. Given that the goal of phylogenetics is to depict evolutionary history, we should look for the best supported phylogenetic network and not restrict ourselves to considering trees. However, the obvious extensions of popular tree-based methods such as maximum parsimony and maximum likelihood face a serious problem—if we judge networks by fit to data alone, networks that have lateral branches will always fit the data at least as well as any network that restricts itself to vertical branches. This is analogous to the well-studied problem of overfitting data in the curve-fitting problem. Analogous problems often have analogous solutions and we propose to treat network inference as a case of model selection and use the Akaike Information Criterion (AIC). Strictly tree-like networks are more parsimonious than those that postulate lateral as well as vertical branches. This leads to the conclusion that we should not always infer LGT events whenever it would improve our fit-to-data, but should do so only when the improved fit is larger than the penalty for adding extra lateral branches.     

A thermodynamic view of networks

Networks can be described by the frequency distribution of the number of links associated with each node (the degree of the node). Of particular interest are the power law distributions, which give rise to the so-called scale-free networks, and the distributions of the form of the simplified canonical law (SCL) introduced by Mandelbrot, which give what we shall call the Mandelbrot networks. Many dynamical methods have been obtained for the construction of scale-free networks, but no dynamical construction of Mandelbrot networks has been demonstrated. Here we develop a systematic technique to obtain networks with any given distribution of the degrees of the nodes. This is done using a thermodynamic approach in which we maximise the entropy associated with degree distribution of the nodes of the network subject to certain constraints. These constraints can be chosen systematically to produce the desired network architecture. For large networks we therefore replace a dynamical approach to the stationary state by a thermodynamical viewpoint. We use the method to generate scale-free and Mandelbrot networks with arbitrarily chosen parameters. We emphasise that this approach opens the possibility of insights into a thermodynamics of networks by suggesting thermodynamic relations between macroscopic variables for networks.     

Ongoing Processes in a Fitness Network Model under Restricted Resources

In real networks, the resources that make up the nodes and edges are finite. This constraint poses a serious problem for network modeling, namely, the compatibility between robustness and efficiency. However, these concepts are generally in conflict with each other. In this study, we propose a new fitness-driven network model for finite resources. In our model, each individual has its own fitness, which it tries to increase. The main assumption in fitness-driven networks is that incomplete estimation of fitness results in a dynamical growing network. By taking into account these internal dynamics, nodes and edges emerge as a result of exchanges between finite resources. We show that our network model exhibits exponential distributions in the in- and out-degree distributions and a power law distribution of edge weights. Furthermore, our network model resolves the trade-off relationship between robustness and efficiency. Our result suggests that growing and anti-growing networks are the result of resolving the trade-off problem itself.     

Recurrent fractal neural networks: a strategy for the exchange of local and global information processing in the brain

The regulation of biological networks relies significantly on convergent feedback signaling loops that render a global output locally accessible. Ideally, the recurrent connectivity within these systems is self-organized by a time-dependent phase-locking mechanism. This study analyzes recurrent fractal neural networks (RFNNs), which utilize a self-similar or fractal branching structure of dendrites and downstream networks for phase-locking of reciprocal feedback loops: output from outer branch nodes of the network tree enters inner branch nodes of the dendritic tree in single neurons. This structural organization enables RFNNs to amplify re-entrant input by over-the-threshold signal summation from feedback loops with equivalent signal traveling times. The columnar organization of pyramidal neurons in the neocortical layers V and III is discussed as the structural substrate for this network architecture. RFNNs self-organize spike trains and render the entire neural network output accessible to the dendritic tree of each neuron within this network. As the result of a contraction mapping operation, the local dendritic input pattern contains a downscaled version of the network output coding structure. RFNNs perform robust, fractal data compression, thus coping with a limited number of feedback loops for signal transport in convergent neural networks. This property is discussed as a significant step toward the solution of a fundamental problem in neuroscience: how is neuronal computation in separate neurons and remote brain areas unified as an instance of experience in consciousness? RFNNs are promising candidates for engaging neural networks into a coherent activity and provide a strategy for the exchange of global and local information processing in the human brain, thereby ensuring the completeness of a transformation from neuronal computation into conscious experience.     

Flux balance analysis: A geometric perspective

Advances in the field of bioinformatics have led to reconstruction of genome-scale networks for a number of key organisms. The application of physicochemical constraints to these stoichiometric networks allows researchers, through methods such as flux balance analysis, to highlight key sets of reactions necessary to achieve particular objectives. The key benefits of constraint-based analysis lie in the minimal knowledge required to infer systemic properties. However, network degeneracy leads to a large number of flux distributions that satisfy any objective; moreover, these distributions may be dominated by biologically irrelevant internal cycles. By examining the geometry underlying the problem, we define two methods for finding a unique solution within the space of all possible flux distributions; such a solution contains no internal cycles, and is representative of the space as a whole. The first method draws on typical geometric knowledge, but cannot be applied to large networks because of the high computational complexity of the problem. Thus a second method, an iteration of linear programs which scales easily to the genome scale, is defined. The algorithm is run on four recent genome-scale models, and unique flux solutions are found. The algorithm set out here will allow researchers in flux balance analysis to exchange typical solutions to their models in a reproducible format. Moreover, having found a single solution, statistical analyses such as correlations may be performed.     

Stability analysis of higher-order neural networks for combinatorial optimization

Recurrent neural networks with higher order connections, from here on referred to as higher-order neural networks (HONNs), may be used for the solution of combinatorial optimization problems. In Ref. 5 a mapping of the traveling salesman problem (TSP) onto a HONN of arbitrary order was developed, thereby creating a family of related networks that can be used to solve the TSP. In this paper, we explore the trade-off between network complexity and quality of solution that is made available by the HONN mapping of the TSP. The trade-off is investigated by undertaking an analysis of the stability of valid solutions to the TSP in a HONN of arbitrary order. The techniques used to perform the stability analysis are not new, but have been widely used elsewhere in the literature. The original contribution in this paper is the application of these techniques to a HONN of arbitrary order used to solve the TSP. The results of the stability analysis show that the quality of solution is improved by increasing the network complexity, as measured by the order of the network. Furthermore, it is shown that the Hopfield network, as the simplest network in the family of higher-order networks, is expected to produce the poorest quality of solution.     

Efficient and Exact Sampling of Simple Graphs with Given Arbitrary Degree Sequence

Uniform sampling from graphical realizations of a given degree sequence is a fundamental component in simulation-based measurements of network observables, with applications ranging from epidemics, through social networks to Internet modeling. Existing graph sampling methods are either link-swap based (Markov-Chain Monte Carlo algorithms) or stub-matching based (the Configuration Model). Both types are ill-controlled, with typically unknown mixing times for link-swap methods and uncontrolled rejections for the Configuration Model. Here we propose an efficient, polynomial time algorithm that generates statistically independent graph samples with a given, arbitrary, degree sequence. The algorithm provides a weight associated with each sample, allowing the observable to be measured either uniformly over the graph ensemble, or, alternatively, with a desired distribution. Unlike other algorithms, this method always produces a sample, without back-tracking or rejections. Using a central limit theorem-based reasoning, we argue, that for large , and for degree sequences admitting many realizations, the sample weights are expected to have a lognormal distribution. As examples, we apply our algorithm to generate networks with degree sequences drawn from power-law distributions and from binomial distributions.     

Contact Trees: Network Visualization beyond Nodes and Edges

Node-Link diagrams make it possible to take a quick glance at how nodes (or actors) in a network are connected by edges (or ties). A conventional network diagram of a “contact tree” maps out a root and branches that represent the structure of nodes and edges, often without further specifying leaves or fruits that would have grown from small branches. By furnishing such a network structure with leaves and fruits, we reveal details about “contacts” in our ContactTrees upon which ties and relationships are constructed. Our elegant design employs a bottom-up approach that resembles a recent attempt to understand subjective well-being by means of a series of emotions. Such a bottom-up approach to social-network studies decomposes each tie into a series of interactions or contacts, which can help deepen our understanding of the complexity embedded in a network structure. Unlike previous network visualizations, ContactTrees highlight how relationships form and change based upon interactions among actors, as well as how relationships and networks vary by contact attributes. Based on a botanical tree metaphor, the design is easy to construct and the resulting tree-like visualization can display many properties at both tie and contact levels, thus recapturing a key ingredient missing from conventional techniques of network visualization. We demonstrate ContactTrees using data sets consisting of up to three waves of 3-month contact diaries over the 2004-2012 period, and discuss how this design can be applied to other types of datasets.     

The circular topology of rhythm in asynchronous random Boolean networks

The analysis of previously evolved rhythmic asynchronous random Boolean networks [Biosystems 59 (2001) 185] reveals common topological characteristics indicating that rhythm originates from a circular functional structure. The rhythm generating core of the network has the form of a closed ring which operates as a synchronisation substrate by supporting a travelling wave of state change; the size of the ring corresponds well with the period of oscillation. The remaining nodes in the network are either stationary or follow the activity of the ring without feeding back into it so as to form a coherent whole. Rings are typically formed early on in the evolutionary search process. Alternatively, long chains of nodes are favoured before they close upon themselves to stabilize. Analysis of asynchronous networks with de-correlated (non-rhythmic, non-stationary) attractors reveals no such common topological characteristics. These results have been obtained using statistical analysis and a specifically developed bottom-up pruning algorithm. This algorithm works from local interactions to global configuration by eliminating redundant links. The suitability of the algorithm has been confirmed by both numerical and single lesion analysis. The ring topology solution for the generation of rhythm implies that it will be harder to evolve rhythmic networks for big sizes and small periods and for bigger number of connections per node. These trends are confirmed empirically. Finally, the identified mechanisms are utilised to handcraft rhythmic networks of different periods showing that a low number of connections suffices for a large variety of rhythms. Random asynchronous update forces the evolved solutions to be highly robust maintaining their performance in the presence of intrinsic noise. The biological implications of such robust designs for molecular clocks are discussed.     

Polytomies and Bayesian phylogenetic inference

Bayesian phylogenetic analyses are now very popular in systematics and molecular evolution because they allow the use of much more realistic models than currently possible with maximum likelihood methods. There are, however, a growing number of examples in which large Bayesian posterior clade probabilities are associated with very short branch lengths and low values for non-Bayesian measures of support such as nonparametric bootstrapping. For the four-taxon case when the true tree is the star phylogeny, Bayesian analyses become increasingly unpredictable in their preference for one of the three possible resolved tree topologies as data set size increases. This leads to the prediction that hard (or near-hard) polytomies in nature will cause unpredictable behavior in Bayesian analyses, with arbitrary resolutions of the polytomy receiving very high posterior probabilities in some cases. We present a simple solution to this problem involving a reversible-jump Markov chain Monte Carlo (MCMC) algorithm that allows exploration of all of tree space, including unresolved tree topologies with one or more polytomies. The reversible-jump MCMC approach allows prior distributions to place some weight on less-resolved tree topologies, which eliminates misleadingly high posteriors associated with arbitrary resolutions of hard polytomies. Fortunately, assigning some prior probability to polytomous tree topologies does not appear to come with a significant cost in terms of the ability to assess the level of support for edges that do exist in the true tree. Methods are discussed for applying arbitrary prior distributions to tree topologies of varying resolution, and an empirical example showing evidence of polytomies is analyzed and discussed.     

Self-sustained Oscillations in Blood Flow Through a Honeycomb Capillary Network

Numerical simulations of unsteady blood flow through a honeycomb network originating at multiple inlets and terminating at multiple outlets are presented and discussed under the assumption that blood behaves as a continuum with variable constitution. Unlike a tree network, the honeycomb network exhibits both diverging and converging bifurcations between branching capillary segments. Numerical results based on a finite difference method demonstrate that as in the case of tree networks considered in previous studies, the cell partitioning law at diverging bifurcations is an important parameter in both steady and unsteady flow. Specifically, a steady flow may spontaneously develop self-sustained oscillations at critical conditions by way of a Hopf bifurcation. Contrary to tree-like networks comprised entirely of diverging bifurcations, the critical parameters for instability in honeycomb networks depend weakly on the system size. The blockage of one or more network segments due to the presence of large cells or the occurrence of capillary constriction may cause flow reversal or trigger a transition to unsteady flow.     

Optimal, efficient reconstruction of phylogenetic networks with constrained recombination

A phylogenetic network is a generalization of a phylogenetic tree, allowing structural properties that are not tree-like. In a seminal paper, Wang et al.(1) studied the problem of constructing a phylogenetic network, allowing recombination between sequences, with the constraint that the resulting cycles must be disjoint. We call such a phylogenetic network a "galled-tree". They gave a polynomial-time algorithm that was intended to determine whether or not a set of sequences could be generated on galled-tree. Unfortunately, the algorithm by Wang et al.(1) is incomplete and does not constitute a necessary test for the existence of a galled-tree for the data. In this paper, we completely solve the problem. Moreover, we prove that if there is a galled-tree, then the one produced by our algorithm minimizes the number of recombinations over all phylogenetic networks for the data, even allowing multiple-crossover recombinations. We also prove that when there is a galled-tree for the data, the galled-tree minimizing the number of recombinations is "essentially unique". We also note two additional results: first, any set of sequences that can be derived on a galled tree can be derived on a true tree (without recombination cycles), where at most one back mutation per site is allowed; second, the site compatibility problem (which is NP-hard in general) can be solved in polynomial time for any set of sequences that can be derived on a galled tree. Perhaps more important than the specific results about galled-trees, we introduce an approach that can be used to study recombination in general phylogenetic networks. This paper greatly extends the conference version that appears in an earlier work.(8) PowerPoint slides of the conference talk can be found at our website.(7).     

Tree-based networks: characterisations,metrics, and support trees

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. One way to obtain such a network is by starting with a (rooted) phylogenetic tree T, called a base tree, and adding arcs between arcs of T. The class of phylogenetic networks that can be obtained in this way is called tree-based networks and includes the prominent classes of tree-child and reticulation-visible networks. Initially defined for binary phylogenetic networks, tree-based networks naturally extend to arbitrary phylogenetic networks. In this paper, we generalise recent tree-based characterisations and associated proximity measures for binary phylogenetic networks to arbitrary phylogenetic networks. These characterisations are in terms of matchings in bipartite graphs, path partitions, and antichains. Some of the generalisations are straightforward to establish using the original approach, while others require a very different approach. Furthermore, for an arbitrary tree-based network N, we characterise the support trees of N, that is, the tree-based embeddings of N. We use this characterisation to give an explicit formula for the number of support trees of N when N is binary. This formula is written in terms of the components of a bipartite graph.

     

Reconstructing an ultrametric galled phylogenetic network from a distance matrix

Given a distance matrix M that specifies the pairwise evolutionary distances between n species, the phylogenetic tree reconstruction problem asks for an edge-weighted phylogenetic tree that satisfies M, if one exists. We study some extensions of this problem to rooted phylogenetic networks. Our main result is an O(n(2) log n)-time algorithm for determining whether there is an ultrametric galled network that satisfies M, and if so, constructing one. In fact, if such an ultrametric galled network exists, our algorithm is guaranteed to construct one containing the minimum possible number of nodes with more than one parent (hybrid nodes). We also prove that finding a largest possible submatrix M' of M such that there exists an ultrametric galled network that satisfies M' is NP-hard. Furthermore, we show that given an incomplete distance matrix (i.e. where some matrix entries are missing), it is also NP-hard to determine whether there exists an ultrametric galled network which satisfies it.     

Scaling of differentiation in networks: nervous systems,organisms, ant colonies,ecosystems, businesses,universities, cities,electronic circuits,and Legos

Nodes in networks are often of different types, and in this sense networks are differentiated. Here we examine the relationship between network differentiation and network size in networks under economic or natural selective pressure, such as electronic circuits (networks of electronic components), Legos (networks of Lego pieces), businesses (networks of employees), universities (networks of faculty), organisms (networks of cells), ant colonies (networks of ants), and nervous systems (networks of neurons). For each of these we find that (i) differentiation increases with network size, and (ii) the relationship is consistent with a power law. These results are explained by a hypothesis that, because nodes are costly to build and maintain in such "selected networks", network size is optimized, and from this the power-law relationship may be derived. The scaling exponent depends on the particular kind of network, and is determined by the degree to which nodes are used in a combinatorial fashion to carry out network-level functions. We find that networks under natural selection (organisms, ant colonies, and nervous systems) have much higher combinatorial abilities than the networks for which human ingenuity is involved (electronic circuits, Legos, businesses, and universities). A distinct but related optimization hypothesis may be used to explain scaling of differentiation in competitive networks (networks where the nodes themselves, rather than the entire network, are under selective pressure) such as ecosystems (networks of organisms).     

Phylogenomic networks

Phylogenomics is aimed at studying functional and evolutionary aspects of genome biology using phylogenetic analysis of whole genomes. Current approaches to genome phylogenies are commonly founded in terms of phylogenetic trees. However, several evolutionary processes are non tree-like in nature, including recombination and lateral gene transfer (LGT). Phylogenomic networks are a special type of phylogenetic network reconstructed from fully sequenced genomes. The network model, comprising genomes connected by pairwise evolutionary relations, enables the reconstruction of both vertical and LGT events. Modeling genome evolution in the form of a network enables the use of an extensive toolbox developed for network research. The structural properties of phylogenomic networks open up fundamentally new insights into genome evolution.