Edge-based analysis of networks: curvatures of graphs and hypergraphs

Abstract

The relations, rather than the elements, constitute the structure of networks. We therefore develop a systematic approach to the analysis of networks, modelled as graphs or hypergraphs, that is based on structural properties of (hyper)edges, instead of vertices. For that purpose, we utilize so-called network curvatures. These curvatures quantify the local structural properties of (hyper)edges, that is, how, and how well, they are connected to others. In the case of directed networks, they assess the input they receive and the output they produce, and relations between them. With those tools, we can investigate biological networks. As examples, we apply our methods here to protein–protein interaction, transcriptional regulatory and metabolic networks.

Fundamental statistics of higher-order networks: a survey

“…Leal等 人 [82] 和 Yadav等 人 [83] 分 别 对 超 图 中 的 Forman-Ricci曲率进行了拓展. 他们根据上述定 义, 认定一条超边e的Forman-Ricci曲率也和成 对交互作用网络的计算思想一致, 并且由于一条超 边e可以包含多个顶点, 那么在无向有权超图中的 公式可以拓展为:…”

What Are Higher-Order Networks?

“…Using higher-order networks to analyze diseases such as cancer [15,226,280,131] offers possibilities for combining data and mathematical models [279,297]. While the structure of some chemical reaction models can be distinguished using persistent homology [298], others are better encoded as a hypergraph and analyzed with discrete Ricci curvature [104].…”

Spaces of mathematical chemistry

“…As discussed in [35,37], only a few mathematical properties of these structures have been studied, for example vertex and hyperedge degrees [35], clustering coefficients [38,39], spectral properties [40], curvatures [37] and more recently the Erdős-Rényi model for the random hypergraph [35]. Nevertheless, other aspects, including different random models, measures of assortativity, and betweenness centrality, among others, remain unexplored, as well as further curvatures and network geometry notions as those pioneered by Jost and collaborators [30,37,[41][42][43][44]. On top of this mathematics to develop, the connection with chemistry is central, that is the interpretation and implications of those mathematical properties for the study of the chemical space.…”