Core objects

Let \(D\) be a domain, such as names, dates, or identifiers. An \(n\)-ary relation \(R\) is a subset of the Cartesian product \(D_1 \times D_2 \times \cdots \times D_n\), and each tuple in \(R\) is one record in the database.

A database schema specifies the attributes and their domains, while the database instance is the current set of tuples stored at a given time.

Core objects

Let \(D\) be a domain, such as names, dates, or identifiers. An \(n\)-ary relation \(R\) is a subset of the Cartesian product \(D_1 \times D_2 \times \cdots \times D_n\), and each tuple in \(R\) is one record in the database.

A database schema specifies the attributes and their domains, while the database instance is the current set of tuples stored at a given time.

Kelly Miller Network-Organization Database

Metadata as relations

Our metadata records describe the database itself: tables, columns, data types, keys, constraints, and indexes. We use these records in our NER analyses.

NER and metadata interactions

In archival and network research, this metadata often becomes the structured representation of entities and sources, such as person, document, date, collection, and tie type.

If the archive is fragmented, this structure helps preserve provenance while making missingness and partial overlap explicit.

Extracted links

If links are extracted from documents, they can be represented as a binary relation:

\[ L \subseteq E \times E \]

where \(E\) is the set of entities and \(L\) is the set of observed ties.

A more detailed link table can be modeled as a higher-arity relation, for example:

\[ (source, target, type, date) \in L \]

This allows directed, typed, weighted, or time-stamped links to be stored in one normalized structure. Importantly, this normalization helps to solve our issue of fragmentation.

Relational algebra

The basic operations are:

• Selection \(\sigma\): filters rows by a predicate.
• Projection \(\pi\): keeps selected columns.
• Join \(\bowtie\): combines tables on shared keys.
• Union \(\cup\): merges compatible relations.
• Difference \(-\): removes matching tuples.
• Intersection \(\cap\): keeps common tuples.

These operations are closed over relations, meaning the output of each operation is again a relation.

Example

Some of the archival correspondence has information about the sender, recipient, date, and context. A join can connect those records to person authority files, a projection can keep only sender and recipient, and the resulting edge list can be turned into a graph for centrality and broader community analysis.

Letter from Kelly Miller to W. E. B. Du Bois

From tables to networks

A network can be derived from a relational table by constructing an adjacency matrix \(A\), where

\[ A_{ij} = \begin{cases} 1, & \text{if a link exists from } i \text{ to } j \\ 0, & \text{otherwise} \end{cases} \]

Weighted or directed networks replace the 0/1 value with counts, strengths, or direction-specific values. This gives a pipeline:

\[ \text{metadata} \rightarrow \text{relation} \rightarrow \text{relational operations} \]

\[ \text{relational operations} \rightarrow \text{edge list or matrix} \rightarrow \text{network analysis} \]

Interpretation

Mathematically, our databases are not just for storage. They are a formal system for representing entities as sets of tuples and relationships as relations, then using algebraic operations to derive the networks we analyze.

Interpretation

Mathematically, our databases are not just for storage. They are a formal system for representing entities as sets of tuples and relationships as relations, then using algebraic operations to derive the networks we analyze.

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Thank you!

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