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It is nearly 20 years since the concept of a small-world network was first quantitatively defined, by a combination of high clustering and short path length; and about 10 years since this metric of complex network topology began to be widely applied to analysis of neuroimaging and other neuroscience data as part of the rapid growth of the new field of connectomics. Here, we review briefly the foundational concepts of graph theoretical estimation and generation of small-world networks. We take stock of some of the key developments in the field in the past decade and we consider in some detail the implications of recent studies using high-resolution tract-tracing methods to map the anatomical networks of the macaque and the mouse. In doing so, we draw attention to the important methodological distinction between topological analysis of binary or unweighted graphs, which have provided a popular but simple approach to brain network analysis in the past, and the topology of weighted graphs, which retain more biologically relevant information and are more appropriate to the increasingly sophisticated data on brain connectivity emerging from contemporary tract-tracing and other imaging studies. We conclude by highlighting some possible future trends in the further development of weighted small-worldness as part of a deeper and broader understanding of the topology and the functional value of the strong and weak links between areas of mammalian cortex.

Schizophrenia has been conceived as a disorder of brain connectivity, but it is unclear how this network phenotype is related to the underlying genetics. We used morphometric similarity analysis of MRI data as a marker of interareal cortical connectivity in three prior case-control studies of psychosis: in total, n = 185 cases and n = 227 controls. Psychosis was associated with globally reduced morphometric similarity in all three studies. There was also a replicable pattern of case-control differences in regional morphometric similarity, which was significantly reduced in patients in frontal and temporal cortical areas but increased in parietal cortex. Using prior brain-wide gene expression data, we found that the cortical map of case-control differences in morphometric similarity was spatially correlated with cortical expression of a weighted combination of genes enriched for neurobiologically relevant ontology terms and pathways. In addition, genes that were normally overexpressed in cortical areas with reduced morphometric similarity were significantly up-regulated in three prior post mortem studies of schizophrenia. We propose that this combined analysis of neuroimaging and transcriptional data provides insight into how previously implicated genes and proteins as well as a number of unreported genes in their topological vicinity on the protein interaction network may drive structural brain network changes mediating the genetic risk of schizophrenia.