Clustering coefficients of protein-protein interaction networks

The properties of certain networks are determined by hidden variables that are not explicitly measured. The conditional probability (propagator) that a vertex with a given value of the hidden variable is connected to $k$ other vertices determines all measurable properties. We study hidden variable models and find an averaging approximation that enables us to obtain a general analytical result for the propagator. Analytic results showing the validity of the approximation are obtained. We apply hidden variable models to protein-protein interaction networks (PINs) in which the hidden variable is the association free energy, determined by distributions that depend on biochemistry and evolution. We compute degree distributions as well as clustering coefficients of several PINs of different species; good agreement with measured data is obtained. For the human interactome two different parameter sets give the same degree distributions, but the computed clustering coefficients differ by a factor of about 2. This shows that degree distributions are not sufficient to determine the properties of PINs.

Figure 1

(Color online) Average connection probability $\lambda =1$, $\mu =10$. Solid (red), result of Eq. (40); dashed (blue) (containing the step function), result of Eq. (25). The approach to unity is smooth for Eq. (40).

Figure 2

(Color online) Degree distributions $p_k$ and clustering coefficients $\overline{C}\left(k\right)$ of diverse species. Degree distributions $p_k$: The solid (red) curves are derived from the LO theory; the black dots are the results of experimental data as referenced in the text; the small (blue) circles are the results of a numerical simulation using the procedure of [12]. Clustering coefficients $\overline{C}\left(k\right)$: The solid (red) curves are derived from the LO theory; the small (blue) dots are the results of a numerical simulation using the procedure of [12]; the heavy (black) dots represent the measured data.

Figure 3

(Color online) Human degree distribution $p_k$; the solid (red) curve is obtained using both set A, $\lambda =0.67$, $\mu =10.6$, and set B, $\lambda =0.94$, $\mu =8.3$. The black dots represent the experimental data. The data set is that of [21], but nearly identical data are obtained from [22]. Human cluster coefficient $\overline{c}\left(k\right)$: The solid (red) curve is computed using set A, $\lambda =0.67$, $\mu =10.6$; and the dashed (green) using set B, $\lambda =0.94$, $\mu =8.3$. Measured human clustering coefficients are from [21] triangles (blue) and [22] heavy dots (pink).