How the latent geometry of a biological network provides information on its dynamics: the case of the gene network of chronic myeloid leukaemia

The concept of the latent geometry of a network that can be represented as a graph has emerged from the classrooms of mathematicians and theoretical physicists to become an indispensable tool for determining the structural and dynamic properties of the network in many application areas, including contact networks, social networks, and especially biological networks. It is precisely latent geometry that we discuss in this article to show how the geometry of the metric space of the graph representing the network can influence its dynamics. We considered the transcriptome network of the Chronic Myeloid Laeukemia K562 cells. We modelled the gene network as a system of springs using a generalization of the Hooke's law to -dimension ( ≥ 1). We embedded the network, described by the matrix of spring's stiffnesses, in Euclidean, hyperbolic, and spherical metric spaces to determine which one of these metric spaces best approximates the network's latent geometry. We found that the gene network has hyperbolic latent geometry, and, based on this result, we proceeded to cluster the nodes according to their radial coordinate, that in this geometry represents the node popularity. Clustering according to radial coordinate in a hyperbolic metric space when the input to network embedding procedure is the matrix of the stiffnesses of the spring representing the edges, allowed to identify the most popular genes that are also centres of effective spreading and passage of information through the entire network and can therefore be considered the drivers of its dynamics. The correct identification of the latent geometry of the network leads to experimentally confirmed clusters of genes drivers of the dynamics, and, because of this, it is a trustable mean to unveil important information on the dynamics of the network. Not considering the latent metric space of the network, or the assumption of a Euclidean space when this metric structure is not proven to be relevant to the network, especially for complex networks with hierarchical or modularised structure can lead to unreliable network analysis results.