Graphing Polyamorous Relationships

From Social Networks to Mathematical Graphs

At its core, network analysis is about abstraction. We take a complex, real-world system—in our case, a network of human relationships—and model it as a mathematical object called a graph. This allows us to apply powerful analytical tools to uncover properties of the system that are not immediately obvious.

A graph consists of two basic components: vertices (also called nodes) and edges (also called links or ties). In social network analysis, vertices almost always represent the actors in the network (e.g., people, organizations), and edges represent the relationships or connections between them.

The following video provides an excellent introduction to these core concepts and explains why this structural perspective is so valuable. It also makes an important distinction between a "network" (the real-world phenomenon) and a "graph" (its mathematical representation).

As the video notes, the modeling process requires simplification. For our purposes, we will typically model relationships as undirected edges, signifying a mutual connection. However, it's worth remembering that edges can also have properties like direction (e.g., unrequited affection) or weight (e.g., the intensity or frequency of contact), which allows for more complex models.

Defining Polyamorous Structures as Graphs

Now, let's apply this framework directly to polyamorous relationships. A person becomes a vertex, and a romantic relationship between two people becomes an edge connecting them. This simple mapping allows us to precisely define common polyamorous configurations.

The following resource provides clear, graph-based definitions for several key structures.

To summarize and visualize these definitions:

• Vee: A graph with three vertices and two edges, where one central vertex is connected to the other two. The two non-central vertices are not connected to each other.
• Triad (or Triangle): A graph with three vertices and three edges. Every vertex is connected to every other vertex. This is also known as a complete graph on three vertices, denoted $K_3$.
• Quad: A group of four people. If all six possible relationships exist, it forms a complete graph on four vertices ($K_4$). However, many other configurations are possible with four people.

The image below illustrates these common structures, along with a few more complex examples. Notice how each is simply a collection of vertices (dots) and edges (lines).

Describing Graphs with Formal Properties

Representing relationships as graphs gives us access to a rich mathematical vocabulary for describing them. This allows us to move beyond simple pictures and characterize structures with quantitative precision. Your background in statistics and algorithms will make these concepts straightforward.

The "Polyamory Graph Explorer" page introduces a few of the most important properties.

Let's quickly apply these terms to our main example, the polyandrous vee:

• Vertices and Edges: The vee has 3 vertices ($n=3$) and 2 edges ($m=2$).
• Labelling: If we are talking about a specific vee—say, "Hera (hinge), with Zeus and Ares"—it is a labelled graph. If we are discussing the general structure, it is an unlabelled graph. Two different vees with different people are isomorphic because they have the same abstract structure.
• Degree: The degree of a vertex is the number of edges connected to it, which corresponds to the number of relationships a person has within that specific group.
• Degree Sequence: The degree sequence of the vee is $\{2,1,1\}$. The hinge has a degree of 2, and the two non-hinge partners each have a degree of 1. This sequence is a powerful signature of the vee structure. In contrast, a triad has a degree sequence of $\{2,2,2\}$.
• Connectivity: A single vee is a connected graph, as you can get from any vertex to any other by following the edges.

This graph representation is not just a theoretical exercise; it's actively used in tools and research. The image below shows the interface of the "Polyamory Graph Explorer," which visualizes a more complex network and automatically calculates various network statistics, including the degree sequence and the corresponding adjacency matrix.

Furthermore, this graph-based approach is the standard in academic research on the topic. For example, the paper "Polyamorous Scheduling" by Gąsieniec, Smith, and Wild (2024), which you can find in the resources (LINK), formally defines a "polycule" as an undirected graph $(P,R)$, where $P$ is the set of persons and $R$ is the set of pairwise relationships. This establishes graph theory as the formal language for analyzing these social structures.

Conclusion

In this lesson, we established the fundamental methodology for the rest of this course's section on network theory. By abstracting people into vertices and relationships into edges, we can represent complex polyamorous configurations as mathematical graphs.

• Key Takeaways:
  • Social networks can be modeled as graphs, with people as vertices and relationships as edges.
  • This framework allows for precise, unambiguous definitions of structures like the vee (3 vertices, 2 edges, degree sequence {2,1,1}) and the triad (a complete graph on 3 vertices, degree sequence {2,2,2}).
  • Formal graph properties like degree, degree sequence, and connectivity provide a powerful vocabulary to describe and compare different relationship networks.
  • This representation is the standard foundation for both practical tools and academic research in the field.

Preview of the next lesson:
Now that we know how to represent these structures, our next task is to analyze them. In the next lesson, we will begin exploring network metrics. You will learn to compute and interpret three of the most fundamental measures of centrality—degree, betweenness, and closeness—to quantify the structural importance and role of each partner within a network.