Configuration model for correlation matrices preserving the node strength

Correlation matrices are a major type of multivariate data. To examine properties of a given correlation matrix, a common practice is to compare the same quantity between the original correlation matrix and reference correlation matrices, such as those derived from random matrix theory, that partially preserve properties of the original matrix. We propose a model to generate such reference correlation and covariance matrices for the given matrix. Correlation matrices are often analyzed as networks, which are heterogeneous across nodes in terms of the total connectivity to other nodes for each node. Given this background, the present algorithm generates random networks that preserve the expectation of total connectivity of each node to other nodes, akin to configuration models for conventional networks. Our algorithm is derived from the maximum entropy principle. We will apply the proposed algorithm to measurement of clustering coefficients and community detection, both of which require a null model to assess the statistical significance of the obtained results.

Figure 1

Flow of the algorithm for generating random correlation matrices. If the input is a covariance matrix, we first transform it to the correlation matrix and feed it to the configuration model. If the input is a correlation matrix, we directly feed it to the configuration model. Because the output of the configuration model is a random covariance matrix (or samples generated from it), we transform it to the correlation matrix, which is the final output.

Figure 2

Distributions of the degree ($k$, solid lines) and the three types of node strength. The strength of node $i$ is defined as (i) $s_i={\sum }_{j=1;j\ne i}^N{\rho }_{ij}$ (dotted lines), (ii) $s_i^{\mathrm{abs}}={\sum }_{j=1;j\ne i}^N\left|{\rho }_{ij}\right|$ (dashed lines), and (iii) $s_i^{+}={\sum }_{j=1;j\ne i;{\rho }_{ij}>0}^N{\rho }_{ij}$ (dot-dashed lines). In (d) and (e), it holds that $s_i\approx s_i^{\mathrm{abs}}\approx s_i^{+}$ ($1\le i\le N$) because ${\rho }_{ij}^{\mathrm{org}}\ge 0$ for all but three pairs of nodes in (d) and all but one pair of nodes in (e). Therefore, we did not plot $s_i^{\mathrm{abs}}$ or $s_i^{+}$. (a) Motivation. (b) fMRI1. (c) fMRI2. (d) Japanese stocks. (e) US stocks.

Figure 3

Density of the eigenvalues of a random correlation matrix, denoted by ${\rho }^{\mathrm{org}}$, and the corresponding configuration model, denoted by ${\rho }^{\mathrm{con}}$. The black solid lines represent the Marcenko-Pastur distribution. (a) $N=100$ and $L=200$ without community structure. (b) $N=500$ and $L=1000$ without community structure. (c) $N=500$ and $L=1000$ with community structure.

Figure 4

Density of the eigenvalues of the empirical correlation matrix, denoted by ${\rho }^{\mathrm{org}}$, and the corresponding configuration model, denoted by ${\rho }^{\mathrm{con}}$. The black solid lines represent the Marcenko-Pastur distribution. (a) Motivation. (b) fMRI1. (c) fMRI2. (d) Japanese stocks. (e) US stocks. The insets in (d) and (e) are magnifications of the main figures.

Figure 5

Comparison of the node strength between the original correlation matrix, the configuration model, and the correlation matrices constructed from the dominant mode. (a) Motivation. (b) fMRI1. (c) fMRI2. (d) Japanese stocks. (e) US stocks.

Figure 6

Comparison of the node strength between the original correlation matrix, the H-Q-S model, and the models based on random matrix theory. (a) Motivation. (b) fMRI1. (c) fMRI2. (d) Japanese stocks. (e) US stocks.

Figure 7

Survival probability of the values of the off-diagonal elements in the correlation matrix. (a) Motivation. (b) fMRI1. (c) fMRI2. (d) Japanese stocks. (e) US stocks.

Figure 8

Community structure for the fMRI data. (a)–(c) fMRI1. (d)–(f) fMRI2. The null model is $\langle {\rho }^{\mathrm{con}}\rangle$ in (a) and (d), $\langle {\rho }^{\mathrm{MG}2}\rangle$ in (b) and (e), and $\langle {\rho }^{\mathrm{MG}3}\rangle$ in (c) and (f). In each panel, the nodes are vertically stacked. The labels to the left indicate the name of the brain system to which each node belongs. FPN, fronto-parietal network; CON, cingulo-opercular network; SAN, salience network; DAN, dorsal attention network; VAN, ventral attention network; DMN, default mode network. “Uncertain” indicates that the node does not belong to a particular brain system. A bundle to the right in each panel represents a community detected by modularity maximization. For example, in (a), there are four communities, the smallest one of which shown to the bottom mostly consists of the nodes in the motor network.