In this paper, we develop a formal framework for what a good community should look like and how strong a community is (community strength). One of the key innovations is to incorporate the concept of relative centrality into structural analysis of networks. In our framework, relative centrality is a measure that measures how important a set of nodes in a network is with respect to another set of nodes, and it is a generalization of centrality. Building on top of relative centrality, the community strength for a set of nodes is measured by the difference between its relative centrality with respect to itself and its centrality. A community is then a set of nodes with a nonnegative community strength. We show that our community strength is related to conductance that is commonly used for measuring the strength of a small community. We define the modularity for a partition of a network as the average community strength for a randomly selected node. Such a definition generalizes the original Newman's modularity and recovers the stability in as special cases. For the local community detection problem, we also develop efficient agglomerative algorithms that guarantee the community strength of the detected local community.

Triadic closure has been conceptualized and measured in a variety of ways, most famously the clustering coefficient. Existing extensions to affiliation networks, however, are sensitive to repeat group attendance, which does not reflect common interpersonal interpretations of triadic closure. This paper proposes a measure of triadic closure in affiliation networks designed to control for this factor, which manifests in bipartite models as biclique proliferation. To avoid arbitrariness, the paper introduces a triadic framework for affiliation networks, within which a range of measures can be defined; it then presents a set of basic axioms that suffice to narrow this range to the one measure. An instrumental assessment compares the proposed and two existing measures for reliability, validity, redundancy, and practicality. All three measures then take part in an investigation of three empirical social networks, which illustrates their differences.

Complex networks have recently attracted much interest due to their prevalence in nature and our daily lives (Vespignani, 2009; Newman, 2010). A critical property of a network is its resilience to random breakdown and failure (Albert et al., 2000; Cohen et al., 2000; Callaway et al., 2000; Cohen et al., 2001), typically studied as a percolation problem (Stauffer & Aharony, 1994; Achlioptas et al., 2009; Chen & D'Souza, 2011) or by modeling cascading failures (Motter, 2004; Buldyrev et al., 2010; Brummitt, et al. 2012). Many complex systems, from power grids and the Internet to the brain and society (Colizza et al., 2007; Vespignani, 2011; Balcan & Vespignani, 2011), can be modeled using modular networks comprised of small, densely connected groups of nodes (Girvan & Newman, 2002). These modules often overlap, with network elements belonging to multiple modules (Palla et al. 2005; Ahn et al. 2010). Yet existing work on robustness has not considered the role of overlapping, modular structure. Here we study the robustness of these systems to the failure of elements. We show analytically and empirically that it is possible for the modules themselves to become uncoupled or non-overlapping well before the network disintegrates. If overlapping modular organization plays a role in overall functionality, networks may be far more vulnerable than predicted by conventional percolation theory.

Over the past years, network science has proven invaluable as a means to better understand many of the processes taking place in the brain. Recently, interareal connectivity data of the macaque cortex was made available with great richness of detail. We explore new aspects of this dataset, such as a correlation between connection weights and cortical hierarchy. We also look at the link-community structure that emerges from the data to uncover the major communication pathways in the network, and moreover investigate its reciprocal connections, showing that they share similar properties. A question arising from these analyses is that of determining the role of weak connections in the unfolding of cortical processes. Though we leave this question largely unanswered, we have found that weak connections pervade the entire cortex while giving rise to no community-like structure. We conjecture that whatever function they come to be found to perform will likely involve some form of cortex-wide communication or control.