Low-rank network decomposition reveals structural characteristics of small-world networks

Small-world networks occur naturally throughout biological, technological, and social systems. With their prevalence, it is particularly important to prudently identify small-world networks and further characterize their unique connection structure with respect to network function. In this work we develop a formalism for classifying networks and identifying small-world structure using a decomposition of network connectivity matrices into low-rank and sparse components, corresponding to connections within clusters of highly connected nodes and sparse interconnections between clusters, respectively. We show that the network decomposition is independent of node indexing and define associated bounded measures of connectivity structure, which provide insight into the clustering and regularity of network connections. While many existing network characterizations rely on constructing benchmark networks for comparison or fail to describe the structural properties of relatively densely connected networks, our classification relies only on the intrinsic network structure and is quite robust with respect to changes in connection density, producing stable results across network realizations. Using this framework, we analyze several real-world networks and reveal new structural properties, which are often indiscernible by previously established characterizations of network connectivity.

Figure 1

Small-world network characteristics. (a) Average path length (red dots, lower curve) and clustering coefficient (blue dots, upper curve) as a function of rewiring probability, $p$, for WS networks of 300 nodes with mean degree 30. (b) Small-world characterization $\sigma$ as a function of $p$ for networks of sizes $n=300$, 500, and 700 nodes with average degree $0.1n$. (c) Small-world characterization $\omega$ as a function of $p$ for the same networks as in (b).

Figure 2

Dependence of component structures on the network rewiring probability. (a) Normalized rank of low-rank component $\nu \left(L\right)$ as a function of rewiring probability, $p$, for WS networks of 300 nodes with mean degree 30. (b) Density of sparse component, $\mathrm{\Sigma }\left(S\right)$, for the same networks as in (a). The inset in (a) depicts the density of low-rank component, $\mathrm{\Sigma }\left(L\right)$, and the inset in (b) depicts the normalized rank of the sparse component, $\nu \left(S\right)$. The mean of these statistics over 20 realizations of the WS network for a given $p$ is plotted with the corresponding standard deviation depicted by the error bars.

Figure 3

Example network and low-rank decomposition. (a) Adjacency matrix $A$ for a small-world network with $n=300$ nodes, mean degree 30, and rewiring probability $p=0.1$. (b) Low rank component $L$ in low-rank network decomposition of $A$ followed by a thresholding process as in Eq. (5). (c) Sparse component $S$ in low-rank network decomposition of $A$ followed by a thresholding process as in Eq. (5). (d) Recovered adjacency matrix $A_r=L+S$. In each plot, black pixels mark connections between nodes of the indicated indices. The relative Frobenius-norm error in the recovered network adjacency matrix in (d) is $0.00089$.

Figure 4

Scaling of low-rank network decomposition. (a) Normalized rank of the thresholded low-rank component $L$ as a function of rewiring probability, $p$, for WS networks of size $n=500$, 700, and 900 nodes with mean degree $n/10$. (b) Density of the thresholded sparse component $S$ for the same networks as in (a). (c) Normalized rank of the thresholded low-rank component $L$ as a function of rewiring probability, $p$, for WS networks of size $n=500$ nodes with mean degree $0.05n, 0.1n$, and $0.15n$. (d) Density of the thresholded sparse component $S$ for the same networks as in (c). The mean of these statistics over 20 realizations of the WS network for a given $p$ is plotted with the corresponding standard deviation depicted by the error bars.

Figure 5

Example network decomposition in two dimensions. (a) Adjacency matrix $A$ for a small-world network with $n=600$ nodes, mean degree 60, and rewiring probability $p=0.1$. In this case, the network is defined over a two-dimensional lattice with periodic boundary conditions and thickness of 3 nodes. (b) Low rank component $L$ in low-rank network decomposition of $A$ followed by a thresholding process as in Eq. (5). (c) Sparse component $S$ in low-rank network decomposition of $A$ followed by a thresholding process as in Eq. (5). In each plot, black pixels mark connections between nodes of the indicated indices.