Systems biology has emerged over the last decade. Driven by the advances in sophisticated measurement technology the research community generated huge molecular biology data sets. These comprise rather static data on the interplay of biological entities, for instance protein-protein interaction network data, as well as quite dynamic data collected for studying the behavior of individual cells or tissues in accordance with changing environmental conditions, such as DNA microarrays or RNA sequencing. Here we bring the two different data types together in order to gain higher level knowledge. We introduce a significantly improved version of the KeyPathwayMiner software framework. Given a biological network modelled as a graph and a set of expression studies, KeyPathwayMiner efficiently finds and visualizes connected sub-networks where most components are expressed in most cases. It finds all maximal connected sub-networks where all nodes but k exceptions are expressed in all experimental studies but at most l exceptions. We demonstrate the power of the new approach by comparing it to similar approaches with gene expression data previously used to study Huntington's disease. In addition, we demonstrate KeyPathwayMiner's flexibility and applicability to non-array data by analyzing genome-scale DNA methylation profiles from colorectal tumor cancer patients. KeyPathwayMiner release 2 is available as a Cytoscape plugin and online at http://keypathwayminer.mpi-inf.mpg.de.
Large real-world networks are typically scale-free. Recent research has shown that such graphs are described best in a geometric space. More precisely, the internet can be mapped to a hyperbolic space such that geometric greedy routing performs close to optimal (Boguná, Papadopoulos, and Krioukov. Nature Communications, 1:62, 2010). This observation pushed the interest in hyperbolic networks as a natural model for scale-free networks. Hyperbolic random graphs follow a powerlaw degree distribution with controllable exponent β and show high clustering (Gugelmann, Panagiotou, and Peter. ICALP, pp. 573-585, 2012).
We investigate the number of maximal cliques, i.e., cliques that are not contained in any larger clique, in three network models: Erdős-Rényi random graphs, inhomogeneous random graphs (also called Chung-Lu graphs), and geometric inhomogeneous random graphs. For sparse and not-too-dense Erdős-Rényi graphs, we give linear and polynomial upper bounds on the number of maximal cliques. For the dense regime, we give super-polynomial and even exponential lower bounds. Although (geometric) inhomogeneous random graphs are sparse, we give superpolynomial lower bounds for these models. This comes form the fact that these graphs have a power-law degree distribution, which leads to a dense subgraph in which we find many maximal cliques. These lower bounds seem to contradict previous empirical evidence that (geometric) inhomogeneous random graphs have only few maximal cliques. We resolve this contradiction by providing experiments indicating that, even for large networks, the linear lower-order terms dominate, before the super-polynomial asymptotic behavior kicks in only for networks of extreme size.
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