Moment-Based Spectral Analysis of Random Graphs with Given Expected Degrees

In this paper, we analyze the limiting spectral distribution of the adjacency matrix of a random graph ensemble, proposed by Chung and Lu, in which a given expected degree sequence $\overline{w}_n^{^{T}} = (w^{(n)}_1,\ldots,w^{(n)}_n)$ is prescribed on the ensemble. Let $\mathbf{a}_{i,j} =1$ if there is an edge between the nodes $\{i,j\}$ and zero otherwise, and consider the normalized random adjacency matrix of the graph ensemble: $\mathbf{A}_n$ $=$ $[\mathbf{a}_{i,j}/\sqrt{n}]_{i,j=1}^{n}$. The empirical spectral distribution of $\mathbf{A}_n$ denoted by $\mathbf{F}_n(\mathord{\cdot})$ is the empirical measure putting a mass $1/n$ at each of the $n$ real eigenvalues of the symmetric matrix $\mathbf{A}_n$. Under some technical conditions on the expected degree sequence, we show that with probability one, $\mathbf{F}_n(\mathord{\cdot})$ converges weakly to a deterministic distribution $F(\mathord{\cdot})$. Furthermore, we fully characterize this distribution by providing explicit expressions for the moments of $F(\mathord{\cdot})$. We apply our results to well-known degree distributions, such as power-law and exponential. The asymptotic expressions of the spectral moments in each case provide significant insights about the bulk behavior of the eigenvalue spectrum

Test-retest reliability of structural brain networks from diffusion MRI

Structural brain networks constructed from diffusion MRI (dMRI) and tractography have been demonstrated in healthy volunteers and more recently in various disorders affecting brain connectivity. However, few studies have addressed the reproducibility of the resulting networks. We measured the test–retest properties of such networks by varying several factors affecting network construction using ten healthy volunteers who underwent a dMRI protocol at 1.5 T on two separate occasions. Each T1-weighted brain was parcellated into 84 regions-of-interest and network connections were identified using dMRI and two alternative tractography algorithms, two alternative seeding strategies, a white matter waypoint constraint and three alternative network weightings. In each case, four common graph-theoretic measures were obtained. Network properties were assessed both node-wise and per network in terms of the intraclass correlation coefficient (ICC) and by comparing within- and between-subject differences. Our findings suggest that test–retest performance was improved when: 1) seeding from white matter, rather than grey; and 2) using probabilistic tractography with a two-fibre model and sufficient streamlines, rather than deterministic tensor tractography. In terms of network weighting, a measure of streamline density produced better test–retest performance than tract-averaged diffusion anisotropy, although it remains unclear which is a more accurate representation of the underlying connectivity. For the best performing configuration, the global within-subject differences were between 3.2% and 11.9% with ICCs between 0.62 and 0.76. The mean nodal within-subject differences were between 5.2% and 24.2% with mean ICCs between 0.46 and 0.62. For 83.3% (70/84) of nodes, the within-subject differences were smaller than between-subject differences. Overall, these findings suggest that whilst current techniques produce networks capable of characterising the genuine between-subject differences in connectivity, future work must be undertaken to improve network reliability