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The loss surface of deep neural networks has recently attracted interest in the optimization and machine learning communities as a prime example of high-dimensional non-convex problem. Some insights were recently gained using spin glass…
Phylogenetic networks are leaf-labelled directed acyclic graphs that are used to describe non-treelike evolutionary histories and are thus a generalization of phylogenetic trees. The hybridization number of a phylogenetic network is the sum…
Recently there has been considerable interest in the problem of finding a phylogenetic network with a minimum number of reticulation vertices which displays a given set of phylogenetic trees, that is, a network with minimum hybrid number.…
Many real-world complex networks arise as a result of a competition between growth and rewiring processes. Usually the initial part of the evolution is dominated by growth while the later one rather by rewiring. The initial growth allows…
Dynamics and function of neuronal networks are determined by their synaptic connectivity. Current experimental methods to analyze synaptic network structure on the cellular level, however, cover only small fractions of functional neuronal…
Given a rooted, binary phylogenetic network and a rooted, binary phylogenetic tree, can the tree be embedded into the network? This problem, called \textsc{Tree Containment}, arises when validating networks constructed by phylogenetic…
Species networks generalize the notion of species trees to allow for hybridization or other lateral gene transfer. Under the Network Multispecies Coalescent Model, individual gene trees arising from a network can have any topology, but…
Hybridization networks are representations of evolutionary histories that allow for the inclusion of reticulate events like recombinations, hybridizations, or lateral gene transfers. The recent growth in the number of hybridization network…
The network structure (or topology) of a dynamical network is often unavailable or uncertain. Hence, we consider the problem of network reconstruction. Network reconstruction aims at inferring the topology of a dynamical network using…
We introduce and analyze a new technique for model reduction for deep neural networks. While large networks are theoretically capable of learning arbitrarily complex models, overfitting and model redundancy negatively affects the prediction…
The reassembling of a simple connected graph G = (V,E) is an abstraction of a problem arising in earlier studies of network analysis. Its simplest formulation is in two steps: (1) We cut every edge of G into two halves, thus obtaining a…
In mathematical phylogenetics, evolutionary relationships are often represented by trees and networks. The latter are typically used whenever the relationships cannot be adequately described by a tree, which happens when so-called…
In [2] it was proven that the Cass algorithm is a polynomial-time algorithm for constructing level<=2 networks from clusters. Here we demonstrate, for each k>=0, a polynomial-time algorithm for constructing level-k phylogenetic networks…
Phylogenetic networks are a special type of graph which generalize phylogenetic trees and that are used to model non-treelike evolutionary processes such as recombination and hybridization. In this paper, we consider {\em unrooted}…
Most network data are collected from partially observable networks with both missing nodes and missing edges, for example, due to limited resources and privacy settings specified by users on social media. Thus, it stands to reason that…
The characterization of various properties of real-world systems requires the knowledge of the underlying network of connections among the system's components. Unfortunately, in many situations the complete topology of this network is…
How a single fertilized cell gives rise to a complex array of specialized cell types in development is a central question in biology. The cells grow, divide, and acquire differentiated characteristics through poorly understood molecular…
Hybrid evolution and horizontal gene transfer (HGT) are processes where evolutionary relationships may more accurately be described by a reticulated network than by a tree. In such a network, there will often be several paths between any…
Reticulate evolutionary processes result in phylogenetic histories that cannot be modeled using a tree topology. Here, we apply methods from topological data analysis to molecular sequence data with reticulations. Using a simple example, we…
A temporal network is a dynamic graph where every edge is assigned an integer time label that indicates at which discrete time step the edge is available. We consider the problem of hierarchically decomposing the network and introduce an…