To find out, we measure co-voting similarity networks in the US Senate and trace individual careers over time. Standard network visualization tools fail on dense highly clustered networks, so we used two aggregation strategies to clarify positional mobility over time. First, clusters of Senators who often vote the same way capture coalitions, and allow us to measure polarization quantitatively through modularity (Newman, 2006; Waugh et al., 2009; Poole, 2012). Second, we use role-based blockmodels (White et al., 1976) to identify role positions, identifying sets of Senators with highly similar tie patterns. Our partitioning threshold for roles is stringent, generating many roles occupied by single Senators. This combination allows us to identify movement between positions over time (specifically, we used the Kernighan–Lin improvement of a Louvain method greedy partitioning algorithm for modularity [Blondel et al., 2008], and CONCOR with an internal similarity threshold for roles; see Supplementary materials for details).

Does the human brain have a central connective core, and, if so, how costly is it?

Noninvasive imaging data allow the construction of network maps of the human brain, recording its structural and functional connectivity. A number of studies have reported on various characteristic network attributes, such as a tendency toward local clustering, high global efficiency, the prevalence of specific network motifs, and a pronounced community structure with several anatomically and functionally defined modules and interconnecting hub regions (Bullmore & Sporns, 2009; van den Heuvel & Hulshoff Pol, 2010; Sporns, 2011). Hubs are of particular interest in studies of the brain since they may play crucial roles in integrative processes and global brain communication, thought to be essential for many aspects of higher brain function. Indeed, hubs have been shown to correspond to brain regions that exhibit complex physiological responses and maintain widespread and diverse connection profiles with other parts of the brain. We asked if, in addition to being highly connected, brain hubs would also exhibit a strong tendency to be mutually interconnected, forming what has been called a “rich club” (Colizza et al., 2006). Rich club organization is present in a network if sets of high-degree nodes exhibit denser mutual connections than predicted on the basis of the degree sequence alone. We investigated rich club organization in the human brain in datasets that recorded weighted projections among different anatomical regions of the cerebral cortex, recorded from several cohorts of healthy human volunteers (van den Heuvel & Sporns, 2011; van den Heuvel et al., 2012).

The supplemental online material accompanying this article was not the final version. The final version is now published online. The publisher regrets the error.