No abstract

Neural field theory insights are used to derive effective brain connectivity matrices from the functional connectivity matrix defined by activity covariances. The symmetric case is exactly solved for a resting state system driven by white noise, in which strengths of connections, often termed effective connectivities, are inferred from functional data; these include strengths of connections that are underestimated or not detected by anatomical imaging. Proximity to criticality is calculated and found to be consistent with estimates obtainable from other methods. Links between anatomical, effective, and functional connectivity and resting state activity are quantified, with applicability to other complex networks. Proof-of-principle results are illustrated using published experimental data on anatomical connectivity and resting state functional connectivity. In particular, it is shown that functional connection matrices can be used to uncover the existence and strength of connections that are missed from anatomical connection matrices, including interhemispheric connections that are difficult to track with techniques such as diffusion spectrum imaging.

A spectral algorithm for community detection is presented. The algorithm consists of three stages: (1) matrix factorization of two matrix forms, square signless Laplacian for unipartite graphs and rectangular adjacency matrix for bipartite graphs, using singular value decompostion (SVD); (2) dimensionality reduction using an optimal linear approximation; and (3) clustering vertices using dot products in reduced dimensional space. The algorithm reveals communities in graphs without placing any restriction on the input network type or the output community type. It is applicable on unipartite or bipartite unweighted or weighted networks. It places no requirement on strict community membership and automatically reveals the number of clusters, their respective sizes and overlaps, and hierarchical modular organization. By representing vertices as vectors in real space, expressed as linear combinations of the orthogonal bases described by SVD, orthogonality becomes the metric for classifying vertices into communities. Results on several test and real world networks are presented, including cases where a mix of disjointed, overlapping, or hierarchical communities are known to exist in the network.

Developing a scientific understanding of cities in a fast urbanizing world is essential for planning sustainable urban systems. Recently, it was shown that income and wealth creation follow increasing returns, scaling superlinearly with city size. We study scaling of per capita incomes for separate census defined income categories against population size for the whole of Australia. Across several urban area definitions, we find that lowest incomes grow just linearly or sublinearly (β = 0.94 to 1.00), whereas highest incomes grow superlinearly (β = 1.00 to 1.21), with total income just superlinear (β = 1.03 to 1.05). These findings show that as long as total or aggregate income scaling is considered, the earlier finding is supported: the bigger the city, the richer the city, although the scaling exponents for Australia are lower than those previously reported for other countries. But, we find an emergent scaling behavior with regard to variation in income distribution that sheds light on socio-economic inequality: the larger the population size and densities of a city, while lower incomes grow proportionately or less than proportionately, higher incomes grow more quickly, suggesting a disproportionate agglomeration of incomes in the highest income categories in big cities. Because there are many more people on lower incomes that scale sublinearly as compared to the highest that scale superlinearly, these findings suggest an empirical observation on inequality: the larger the population, the greater the income agglomeration in the highest income categories. The implications of these findings are qualitatively discussed for various income categories, with respect to living costs and access to opportunities and services that big cities provide.