Uncovering the hidden geometry behind metabolic networks

• https://pubs.rsc.org/en/content/articlelanding/2012/mb/c2mb05306c
  • https://arxiv.org/abs/1109.1934
• M. Angeles Serrano, Marian Boguna, Francesc Sagues

Representing metabolites and reactions in the geometric space. A central assumption is that the probability of connection is a function of effective distance (degree-normalized distance in the embedding space).

given a pair metabolite/reaction separated by a geometric distance $d_{mr}$ in the underlying metric space, the probability of existence of a connection between them is here shown to be a decreasing function of the effective distance $d_{eff} = d_{mr} / (k_r k_m)$, where degrees $k_m$ and $k_r$ count the number of their respective neighboring nodes.

The connection probability is defined by $$p\!\left(\frac{d_{mr}}{k_m k_r}\right) = \frac{1}{1 + \left(\dfrac{d_{mr}}{\mu\,k_m k_r}\right)^{\beta}}$$ This is called $\mathbb{S}_1 \times \mathbb{S}_1$ model. 1. Two types of nodes are all distributed on a circle with radius $R$. 2. all nodes are assigned with a degree. 3. Each possible pair is connected based on the probability function.

A more in-depth study in Kitsak2017latent