Grand canonical ensemble of weighted networks

The cornerstone of statistical mechanics of complex networks is the idea that the links, and not the nodes, are the effective particles of the system. Here, we formulate a mapping between weighted networks and lattice gases, making the conceptual step forward of interpreting weighted links as particles with a generalized coordinate. This leads to the definition of the grand canonical ensemble of weighted complex networks. We derive exact expressions for the partition function and thermodynamic quantities, both in the cases of global and local (i.e., node-specific) constraints on the density and mean energy of particles. We further show that, when modeling real cases of networks, the binary and weighted statistics of the ensemble can be disentangled, leading to a simplified framework for a range of practical applications.

Figure 1

Mapping between an undirected graph $G_N$ and a lattice gas on the corresponding triangular graph $T_N$ of $K_N$ (we report the illustrative examples $N=4,5$). Only the existing links of $G_N$ (represented as solid black lines) are placed as particles on the lattice sites (represented as solid dots), and the weight of such links (given by the lines' thickness) corresponds to the generalized coordinates of particles (given by the dots' size). Note that two particles in the lattice gas are neighbors if the corresponding links in $G_N$ have a node in common.

Figure 2

Properties of CECM and SECM ensembles in four real networks: the World Trade Web [13], the eMID interbank network [13], the neural network of C. elegans [21], and the human functional brain network (HFBN) [22]. The upper part of the figure shows the comparison of link probabilities (first row) and of expected weights (second row) obtained by CECM and SECM. The lower part of the figure instead shows how the two ensembles reproduce higher-order statistics of the real networks (defined in Refs. [12, 23]): nearest-neighbor degree $k^{nn}$ (third row), clustering coefficient $c$ (fourth row), nearest-neighbor strength $s^{nn}$ (fifth row), and weighted clustering $c^w$ (sixth row).