Stiffness estimate of information propagation in biological systems modelled as spring networks

We present a mathematical model of the dynamics of a biological network represented as a spring system with n degrees of freedom. The study is inspired by the recent abstraction of a biological network as a system composed of masses (nodes) connected by springs (the arcs between nodes). As a consequence of the mutual interactions between the nodes and/or of stresses deriving from the environmental conditions in which the physical network is immersed, the nodes vibrate around their equilibrium positions. The elastic constant that measures the strength of an ideal spring connecting the vibrating nodes cannot be straightforwardly gauged experimentally, and almost never in the case of large networks whose dynamics has several degrees of freedom. In this study, we show how elastic constants are related to the topology of the network and how they can be derived from the equations of the dynamics at equilibrium. The innovative results of the study are (i) the definition of stiffness vectorial entities as basis of the nullspace of the network incidence matrix, (ii) the introduction of the spring stiffness as an indicator of information propagation between nodes.