Estimating topological properties of weighted networks from limited information

A problem typically encountered when studying complex systems is the limitedness of the information available on their topology, which hinders our understanding of their structure and of the dynamical processes taking place on them. A paramount example is provided by financial networks, whose data are privacy protected: Banks publicly disclose only their aggregate exposure towards other banks, keeping individual exposures towards each single bank secret. Yet, the estimation of systemic risk strongly depends on the detailed structure of the interbank network. The resulting challenge is that of using aggregate information to statistically reconstruct a network and correctly predict its higher-order properties. Standard approaches either generate unrealistically dense networks, or fail to reproduce the observed topology by assigning homogeneous link weights. Here, we develop a reconstruction method, based on statistical mechanics concepts, that makes use of the empirical link density in a highly nontrivial way. Technically, our approach consists in the preliminary estimation of node degrees from empirical node strengths and link density, followed by a maximum-entropy inference based on a combination of empirical strengths and estimated degrees. Our method is successfully tested on the international trade network and the interbank money market, and represents a valuable tool for gaining insights on privacy-protected or partially accessible systems.

Figure 1

Relation between node strengths $\left\{s^{*}\right\}$ and their degree-induced Lagrange multipliers $\left\{x\right\}$ from CM (obtained by knowing the whole degree sequence). The linearity of such a relation is at the basis of the ansatz of our method that $x_i\propto s_i^{*}\forall i$. The left panel refers to WTW, and the right panel to E-mid.

Figure 2

Relation between $k^{*}$ and ${\langle k\rangle }_{{\mathrm{\Omega }}_{\text{ECM}}}$ for WTW (left panel) and E-mid (right panel).

Figure 3

Scatter plots of (a), (e) $s$ vs $s^{nn}$, (b), (f) $k$ vs $k^{nn}$, (c), (g) $s$ vs $c^w$, and (d), (h) $k$ vs $c^k$ for the real quantities $\left[X\left(G_0\right)\right]$, those estimated by our method (${\langle X\rangle }_{{\mathrm{\Omega }}_{\text{ECM}}}$), and those computed by WCM-based reconstruction (${\langle X\rangle }_{{\mathrm{\Omega }}_{\text{WCM}}}$). Insets: Relations $X\left(G_0\right)$ vs ${\langle X\rangle }_{{\mathrm{\Omega }}_{\text{ECM}}}$ and $X\left(G_0\right)$ vs ${\langle X\rangle }_{{\mathrm{\Omega }}_{\text{WCM}}}$ for the same quantities. Upper plots (a)–(d) refer to WTW, and lower plots (e)–(h) to E-mid.