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In the effort to capture cultural dynamics, scholars have considered social networks, that is, a graph with people as nodes and their relationships as edges. These social networks are useful; however, to capture dynamics they must be considered over time. In the literature, Time-Varying Graphs (TVGs) have been defined (Aggarwal and Subbian, 2014; Casteigts et al., 2012; Casteigts et al., 2013). In our investigations, we have found benefit in defining TVGs with nodes as societal structures and people as the edges and then considering the dynamics of the societal structures evidenced in the TVGs (Hott et al., 2014; Hott et al., 2015). Here we consider two motivating applications for our extensions to TVGs: early Mormon marital structures and an arXiv.org citation network.

The societal structures represented in the marital and church structures of early Mormons in mid-1800s Nauvoo, Illinois, include binary, polygynous, and polyandrous marriages, as well as child and adult adoptions, and membership of individuals in the church organization hierarchy. In this time period the concept of “marriage” is in flux and part of our research is to consider various conceptualizations of “marriage” to better understand the relationship to the formation of the church structure. Each conceptualization we consider as a different “identity lens,” a term we create to describe these different views.

We therefore define the
identity-lens function that maps one evolving network to another evolving network. More specifically, given a TVG,
identity-lens function
i.

In our marital network research (Hott et al., 2014; Hott et al., 2015), we represent marriages as the nodes, with the individuals connecting the marriages of their parents to their own marriages as adults. Every piece of this network is considered to be evolving, since marital relations change, new children are born, family members are adopted, and individuals change membership in the church organizational structure. Initially, this network

Our second motivating example is the arXiv.org

http://www.arxiv.org

Each of these TIVGs have characteristics that change over time. As we increase the complexity of the nodes through the use of identity lenses, we increase the dynamics of the characteristics, specifically those captured within the nodes. In the Nauvoo dataset, these characteristics include familial relationships among marriage members and church leadership positions held by the members of each marriage. Similarly, in the arXiv dataset, the characteristics include departmental and institutional collaboration. We want and need metrics that are sensitive to these changes within the evolving nodes as well as the overall evolving structure of the network. To capture and analyze these dynamics, we first define sampling methods to produce static graphs depicting the state of the TIVG during a fixed-size interval around each time point, then compute centrality measures over the graph across time for each identity lens. This process creates a distribution of the metric across time, which may then be compared between identity lenses. We conjecture that utilizing different-sized sampling intervals and comparing distributions across identity lenses will provide insights to understanding the TVG and the motivating application it describes.

We therefore define two methods to sample TIVG
t in
T, to a static graph

- The union of all nodes and edges extant at any time during the interval.
and

are the “presence” functions for nodes and edges, respectively, as defined in (Casteigts et al., 2012).

- Only nodes and edges that exist throughout the entire interval.

As a simple example of these sampling methods, consider a correspondence network, where

Using the sampling methods above, we measure characteristics at time points throughout the lifetime of

Allowing
t to range over the entire lifespan of

We have therefore defined a new conceptualization of Time-Varying Graphs, specifically the identity-lens function and resulting Time/Identity-Varying Graphs under each identity mapping. We also defined methods for sampling the TIVGs into series of measurable static graphs and provided metrics over those representations. At the conference, we will present visualizations that represent each of our applications from the various perspectives, as well as the findings of these measures: to better understand the definition of marriage in Nauvoo and its relation to church formation, and to illuminate patterns in author and departmental co-citations.