Bakry-Émery Ricci Curvature on Weighted Graphs with Applications to Biological Networks

In recent years, there have been tremendous efforts to elucidate the complex mechanisms of cancer networks by investigating the interactions of different genetic and epigenetic factors. Mathematical tools can help significantly with overcoming many of these challenges and can facilitate better understanding of the complexities of the corresponding networks. This has lead to the emergence of the field of network and systems biology. The formal models employed in biological networks range from graphs as abstract representations of pairwise interactions to complicated systems of partial differential equations that try to capture all details of biological interactions. Therefore, the mathematical methods and tools employed in networks are quite diverse and heterogeneous. We propose an integrative framework to identify genetic features related to cancer networks and to distinguish them from the normal tissue networks by geometrical analysis of the networks provided by The Cancer Genome Atlas (TCGA) data. Our study is based on the analogous notion of fundamental concepts in Riemannian geometry, namely Ricci curvature, on discrete spaces.