Characterizing Distances of Networks on the Tensor Manifold

At the core of understanding dynamical systems is the ability to maintain and control the systems behavior that includes notions of robustness, heterogeneity, or regime-shift detection. Recently, to explore such functional properties, a convenient representation has been to model such dynamical systems as a weighted graph consisting of a finite, but very large number of interacting agents. This said, there exists very limited relevant statistical theory that is able cope with real-life data, i.e., how does perform analysis and/or statistics over a family of networks as opposed to a specific network or network-to-network variation. Here, we are interested in the analysis of network families whereby each network represents a point on an underlying statistical manifold. To do so, we explore the Riemannian structure of the tensor manifold developed by Pennec previously applied to Diffusion Tensor Imaging (DTI) towards the problem of network analysis. In particular, while this note focuses on Pennec definition of geodesics amongst a family of networks, we show how it lays the foundation for future work for developing measures of network robustness for regime-shift detection. We conclude with experiments highlighting the proposed distance on synthetic networks and an application towards biological (stem-cell) systems.Comment: This paper is accepted at 8th International Conference on Complex Networks 201

Disease-associated pathophysiologic structures in pediatric rheumatic diseases show characteristics of scale-free networks seen in physiologic systems: implications for pathogenesis and treatment

<p>Abstract</p> <p>Background</p> <p>While standard reductionist approaches have provided some insights into specific gene polymorphisms and molecular pathways involved in disease pathogenesis, our understanding of complex traits such as atherosclerosis or type 2 diabetes remains incomplete. Gene expression profiling provides an unprecedented opportunity to understand complex human diseases by providing a global view of the multiple interactions across the genome that are likely to contribute to disease pathogenesis. Thus, the goal of gene expression profiling is not to generate lists of differentially expressed genes, but to identify the physiologic or pathogenic processes and structures represented in the expression profile.</p> <p>Methods</p> <p>RNA was separately extracted from peripheral blood neutrophils and mononuclear leukocytes, labeled, and hybridized to genome level microarrays to generate expression profiles of children with polyarticular juvenile idiopathic arthritis, juvenile dermatomyositis relative to childhood controls. Statistically significantly differentially expressed genes were identified from samples of each disease relative to controls. Functional network analysis identified interactions between products of these differentially expressed genes.</p> <p>Results</p> <p><it>In silico </it>models of both diseases demonstrated similar features with properties of scale-free networks previously described in physiologic systems. These networks were observable in both cells of the innate immune system (neutrophils) and cells of the adaptive immune system (peripheral blood mononuclear cells).</p> <p>Conclusion</p> <p>Genome-level transcriptional profiling from childhood onset rheumatic diseases suggested complex interactions in two arms of the immune system in both diseases. The disease associated networks showed scale-free network patterns similar to those reported in normal physiology. We postulate that these features have important implications for therapy as such networks are relatively resistant to perturbation.</p

Granular matter and the marginal rigidity state

Model experiments are reported on the build-up of granular piles in two dimensions. These show that, as the initial density of falling grains is increased, the resulting pile has decreasing final density and its coordination number approaches the low value predicted for the theoretical marginal rigidity state. This provides the first direct experimental evidence for this state of granular matter. We trace the decrease in the coordination number to the dynamics within an advancing yield front between the consolidated pile and the falling grains. We show that the front's size diverges as the marginal rigidity state is approached, suggesting a critical phenomenon