Important tasks in the study of genomic data include the identification of groups of similar cells (for example by clustering), and visualisation of data summaries (for example by dimensional reduction). In this paper, we develop a novel approach to these tasks in the context of single-cell genomic data. To do so, we propose to model the observed genomic data count matrix X∈Zp×n≥0, by representing these measurements as a bipartite network with multi-edges. Utilising this first-principles network model of the raw data, we cluster single cells in a suitably identified d-dimensional Laplacian Eigenspace (LE) via a Gaussian mixture model (GMM-LE), and employ UMAP to non-linearly project the LE to two dimensions for visualisation (UMAP-LE). This LE representation of the data-points estimates transformed latent positions (of genes and cells), under a latent position statistical model of nodes in a bipartite stochastic network. We demonstrate how transformations of these estimated latent positions can enable fine-grained clustering and visualisation of single-cell genomic data, by application to data from three recent genomics studies in different biological contexts. In each data application, clusters of cells independently learned by our proposed methodology are found to correspond to cells expressing specific marker genes that were independently defined by domain experts. In this validation setting, our proposed clustering methodology outperforms the industry-standard for these data. Furthermore, we validate components of the LE decomposition of the data by contrasting healthy cells from normal and at-risk groups in a machine-learning model, thereby identifying an LE cancer biomarker that significantly predicts long-term patient survival outcome in two independent validation cohorts with data from 1904 and 1091 individuals.

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