Patterns of co-membership: Techniques for identifying subgraph composition

Different social processes give rise to network structures with distinctive properties. In this paper our goal is to identify the social processes that give rise to distinct network structures (specifically, subgroups). We examine particular structural meta-relations by identifying the properties of individuals associated with specific subgroups. Clues to the process of group formation and the context in which these groups form and persist may be extracted from the properties of individuals in those groups. Following this intuition, we propose a general technique for identifying systematic patterns of attribute occupancy to determine how individual attributes may drive group formation. To connect the social context in which groups form to their structural signatures, we relate subgroup composition to nodal attributes. We illustrate the utility of comparing subgroup (e.g., clique, n-clique, k-core, etc.) co-membership with nodal co-membership in a variety of attributes. The correlations between these two co-membership matrices illustrate clearly the strength of association between shared attributes and shared subgraph membership. Furthermore, examining these correlations across groups of different sizes indicates where these attributes are most strongly associated with group co-membership. Additionally, these correlations fit well into a QAP framework to determine where shared subgraph membership has a stronger (or weaker) relation to shared attribute membership than we would expect by chance. We demonstrate the technique with a series of large, online friendship networks on the order of thousands of nodes to illustrate how factors such as gender, cohort, residence, and other attributes are associated with co-membership across a range of clique sizes.