This study advances our understanding of inter-and intra-pathways higher order signaling in the cellular system and it leads to new discovery of multiple intracellular structures in signal transduction pathways in yeast Saccharomyces. We present a new tensor decomposition algorithm in reconstructing the pathways based on higher correlations among genes that compose a cellular system. The higher order gene correlation (HOGC) analysis has the power to elucidate gene's higher interaction dependencies which has been barely understood. Recent studies i.e. [24] have experimentally revealed that multiple signaling proteins, yet sometimes infinite, may assemble to meaningful structure to transmit a receptor activation information. In this paper we reveal 3-order genomic correlations among significant component of the cellular system. This is the first time such a systematic and computational model provided for analysis of higher order correlations among genes. We use new fast algorithm to formulate a genes × genes × genes decorrelated rank-1 sub-tensors (complexes) which can be associated with functionally independent pathways. Then we model higher order tensor decomposition T ∈ R d ⊗4 which is constructed by K tensors of genes × genes × genes. Each new tensor is constructed by an orthogonal projection of data signal onto a designated basis signal to keep common sub-tensors in both signals. Our model for decomposing tensor order-4 approximates series of tensors as linear components of deccorelated rank-1 sub-tensors over tensor of order-3 and rank-3 triplings among sub-tensors. The linear components represent intra-pathway in cell signaling and triplings implicate inter-pathways higher order signaling. Through structural studies of inter-and intra-higher order signaling pathways, we uncover different scenario that involves triple formation of signaling proteins into higher order signaling machines for transmission of receptor activation information to cellular responses.

δ-hyperbolic graphs, originally conceived by Gromov in 1987, occur often in many network applications; for fixed δ, such graphs are simply called hyperbolic graphs and include non-trivial interesting classes of "non-expander" graphs. The main motivation of this paper is to investigate the effect of the hyperbolicity measure δ on expansion and cut-size bounds on graphs (here δ need not be a constant), and the asymptotic ranges of δ for which these results may provide improved approximation algorithms for related combinatorial problems. To this effect, we provide constructive bounds on node expansions for δ-hyperbolic graphs as a function of δ, and show that many witnesses (subsets of nodes) for such expansions can be computed efficiently even if the witnesses are required to be nested or sufficiently distinct from each other. To the best of our knowledge, these are the first such constructive bounds proven. We also show how to find a large family of s-t cuts with relatively small number of cut-edges when s and t are sufficiently far apart. We then provide algorithmic consequences of these bounds and their related proof techniques for two problems for δ-hyperbolic graphs (where δ is a function f of the number of nodes, 2 Bhaskar DasGupta et al.the exact nature of growth of f being dependent on the particular problem considered). Mathematics Subject Classification1 IntroductionUseful insights for many complex systems such as the world-wide web, social networks, metabolic networks, and protein-protein interaction networks can often be obtained by representing them as parameterized networks and analyzing them using graph-theoretic tools. Some standard measures used for such investigations include degree based measures (e.g., maximum/minimum/average degree or degree distribution) connectivity based measures (e.g., clustering coefficient, claw-free property, largest cliques or densest sub-graphs), and geodesic based measures (e.g., diameter or betweenness centrality). It is a standard practice in theoretical computer science to investigate and categorize the computational complexities of combinatorial problems in terms of ranges of these parameters. For example:◮ Bounded-degree graphs are known to admit improved approximation as opposed to their arbitrary-degree counter-parts for many graph-theoretic problems. ◮ Claw-free graphs are known to admit improved approximation as opposed to general graphs for graph-theoretic problems such as the maximum independent set problem.In this paper we consider a topological measure called Gromov-hyperbolicity (or, simply hyperbolicity for short) for undirected unweighted graphs that has recently received significant attention from researchers in both the graph theory and the network science community. This hyperbolicity measure δ was originally conceived in a somewhat different group-theoretic context by Gromov [20]. The measure was first defined for infinite continuous metric space via properties of geodesics [10], but was later also adopted for finite graphs. Lately, there have been...

Anomaly detection problems (also called change-point detection problems) have been studied in data mining, statistics and computer science over the last several decades (mostly in non-network context) in applications such as medical condition monitoring, weather change detection and speech recognition. In recent days, however, anomaly detection problems have become increasing more relevant in the context of network science since useful insights for many complex systems in biology, finance and social science are often obtained by representing them via networks. Notions of local and non-local curvatures of higher-dimensional geometric shapes and topological spaces play a fundamental role in physics and mathematics in characterizing anomalous behaviours of these higher dimensional entities. However, using curvature measures to detect anomalies in networks is not yet very common. To this end, a main goal in this paper to formulate and analyze curvature analysis methods to provide the foundations of systematic approaches to find critical components and detect anomalies in networks. For this purpose, we use two measures of network curvatures which depend on non-trivial global properties, such as distributions of geodesics and higher-order correlations among nodes, of the given network. Based on these measures, we precisely formulate several computational problems related to anomaly detection in static or dynamic networks, and provide non-trivial computational complexity results for these problems. This paper must not be viewed as delivering the final word on appropriateness and suitability of specific curvature measures. Instead, it is our hope that this paper will stimulate and motivate further theoretical or empirical research concerning the exciting interplay between notions of curvatures from network and non-network domains, a much desired goal in our opinion.In this paper we seek to address research questions of the following generic nature:"Given a static or dynamic network, identify the critical components of the network that "encode" significant non-trivial global properties of the network".To identify critical components, one first needs to provide details for following four specific items:(i) network model selection,(ii) network evolution rule for dynamic networks, (iii) definition of elementary critical components, and (iv) network property selection (i.e., the global properties of the network to be investigated).The specific details for these items for this paper are as follows:(i) Network model selection: Our network model will be undirected graphs.(ii) Network evolution rule for dynamic networks: Our dynamic networks follow the time series model and are given as a sequence of networks over discrete time steps, where each network is obtained from the previous one in the sequence by adding and/or deleting some nodes and/or edges.(iii) Critical component definition: Individual edges are elementary members of critical components.(iv) Network property selection: The network measure for this paper will be appro...

Background: Since biological systems are complex and often involve multiple types of genomic relationships, tensor analysis methods can be utilized to elucidate these hidden complex relationships. There is a pressing need for this, as the interpretation of the results of high-throughput experiments has advanced at a much slower pace than the accumulation of data. Results: In this review we provide an overview of some tensor analysis methods for biological systems. Conclusions: Tensors are natural and powerful generalizations of vectors and matrices to higher dimensions and play a fundamental role in physics, mathematics and many other areas. Tensor analysis methods can be used to provide the foundations of systematic approaches to distinguish significant higher order correlations among the elements of a complex systems via finding ensembles of a small number of reduced systems that provide a concise and representative summary of these correlations.

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