Isotopy and Energy of Physical Networks

Physical exclusion imposes severe limitations on most real systems, from granular media to networks. For example, if the links of a spatial network are physical objects and are unable to cross one another, in two dimensions only planar networks can exist. While the structural characteristics of a network are uniquely determined by its adjacency matrix, in physical networks, such as the brain or the vascular system, the network’s three-dimensional layout also affects the system’s structure and function.

In this project, we introduce the concept of network isotopy, representing different network layouts that can be transformed into one another without link crossings, as a tool to distinguish physical networks with identical wiring but different geometrical layouts. To determine whether two network embeddings are non-isotopic, we define the graph linking number (GLN), a single quantity that captures the entangledness of a layout, defining distinct isotopy classes.

We further find that a network’s elastic energy (defined as the sum of the elastic energy of all links) depends linearly on the graph linking number, indicating that each local tangle offers an independent contribution to the total energy. This finding allows us to formulate a statistical model for the formation of tangles in physical networks. We apply the developed framework to a diverse set of real physical networks, finding that the mouse connectome is more entangled than expected based on optimal wiring.