• polyamory
• mathematics

Use this interactive exploration widget to map polyamorous relationships, and to examine their characteristics. This is used in my write-up for the Poly Equation.

## Explanations

You'd think the most common features of polyamorous people would be openness and a lack of jealousy. Obsessive time-management (often assisted by PDAs and Google Calendars) and an incredible ability to visualise complex relationship graphs without pen and paper are far more common. For the rest of us, here's a computer-aided way of doing this.

### The Polyamory Side

An explanation of what Polyamory is completely outside the scope of this page. Please read up on it, there are lots of references online. I will, however, explain some useful terms for this page.

A relationship is a romantic involvement between two consenting people.

A polyamorous group is a group of two or more polyamorous people in zero or more relationships (this is purposefully congruent to the definition of a graph).

A vee is a polyamorous group of three people, in two relationships: one person has a relationship with the other two.

A triad is a polyamorous group of three people involved in three relationships: everyone is romantically involved with everyone else. A lot of real-world people identify their relationship configuration as a triad, even if one or more have other relationships too. This graph explorer widget only considers a true triad a triad: exactly three relationships, no people external to the triad.

A quad is a polyamorous group of four people involved in six relationships with each other: everyone is romantically involved with everyone else (count it, six relationships). Like triads, this widget considers quads to be isolated (exactly six relationships, nothing on the side). Many real-world quads define this differently.

A quint is a polyamorous group of five people in ten relationships, much like triads and quads but with an extra person.

### The Graph Theory Side

A vertex (or, sometimes, node) is, in this context, a person.

An edge is a line connecting vertices. In this case, it's a romantic relationship.

A graph is a collection of vertices (people) and edges (relationships between them). This particular type of graph is an undirected one (there are no arrows from one person to another — relationships being equal in nature).

A labelled graph is a graph where the vertices (people) are named and not interchangeable. A graph of Alice, John and Bob, where Alice is in a relationship with both John and Bob is labelled.

An unlabelled graph is a graph where the vertices are anonymous. It's used to study abstract properties of graphs. A vee is an unlabelled graph. The relationship o Alice, John and Bob above becomes the abstract topologial shape of a vee if you ignore their names.

An empty graph has no relationships.

A connected graph is one in which everyone is, directly or indirectly, involved with everyone else. The Alice, John and Bob graph is connected. If we add another two people, Mary and Sean (with a relationship between them), the graph of Alice, John, Bob, Mary and Sean is no longer connected. If you follow the relationships, there's no way to get from Alice, John or Bob to Mary or Sean.

A fully connected graph is a connected graph in which every vertex is directly connected to every other vertex. Polyamorous triads, quads and quints are examples of fully connected graphs.

A subgraph is part of a graph. A graph of Alice and John is a subgraph of the above one. So is Mary and Sean.

An isolated vertex is a vertex unconnected to any other vertices in the graph: a person with no relationships.

The degree sequence of a graph is a list (in decreasing order) of the number of relationships of each person in the graph. In the case of Alice, John, Bob, Mary and Sean, it's `{2,1,1,1,1}`. (Alice has two relationships, everyone else has one). Degree sequences are properties of unlabelled graphs; there's no way to tell who's the person with the two relationships unless you know the labelling of the graph. Graphs with the same degree sequence share various properties.

Two labelled graphs are isomorphic if they represent the same topological relationships. Once the labels are removed, and if you rearrange the vertices (without changing the relationships), you'll end up with identical shapes. The graph Alice, John, Bob (Alice in a relationship with John and Bob) is isomorphic to the graph Steve, Rachel, George (George is in a relationship with Steve and Rachel): they both represent the abstract notion of a vee.

## The Tacit Formula

This was published (among other places) by Tacit in this Livejournal post. The ‘poly formula’, as it's come to be known, supposedly estimates the number of different ways people may form polyamorous groups.

Unfortunately, the formula only counts the total number of mono relationships, triads, quads, quints, and other fully-connected subgraphs. The formula fails to account for vees and any more complicated graphs that are not fully connected. It also doesn't consider mutually isolated graphs (e.g. two triads in a group of six people).

As part of its workings, the widget on this page demonstrates how Tacit's Formula behaves for various graph topologies. A ‘conventionally polyamorous’ explanation is also provided, based on what most people would accept as a polyamorous relationship (one or more people in two or more relationships).

## The Seven Problems (P1 to P7)

In contrast, I suggest seven different counting problems, the answers to which may (or may not) be better than the Tacit formula, depending on people's intent. The main questions are whether or not single people should be allowed in the graph, and whether or not everyone should somehow be connected, or disconnected subgraphs are allowed (e.g. five people, in which three are in a triad, and two in a mono relationship).

### Labelled Graphs

Problem 1. What is the number of ways a group of n specific people may be pairwise related or unrelated such that there are zero or more relationships within the group?

Problem 2. What is the number of ways a group of n specific people may be pairwise related or unrelated such that there are one or more relationships within the group? The answer to this is trivial: it's the answer to Problem 1 minus one. There's exactly one n-person graph in which any number of people may be completely unrelated, after all.

Problem 3. What is the number of ways a group of n specific people may be pairwise related or unrelated such that there is at least one relationship within the group, and no single people?

From a graph theory standpoint, this problem calls for the counting of undirected, labelled graphs of at least one edge, and no isolated vertices.

Problem 4. What is the number of ways a group of n specific people may be pairwise related or unrelated in such a way that every person is related, directly or indirectly, to every other person?

In graph theory terms, what is the number of undirected, labelled, connected n-graphs?

### Unlabelled Graphs

These much harder problems are about unlabelled graphs, and examine abstract ‘relationship configurations’, not relationships between specific people.

Problem 5. How many of the graphs enumerated in Problem 1 are not isomorphic? That is, how many different shapes are there for n-person graphs? For three people, there are eight different graphs listed by Problem 1, but only four of these are different shapes: the empty graph (no relationships), two people in a relationship and a single, a vee, and a triad.

Problem 6. How many of the graphs counted by Problem 5 have at least one relationship? An easy problem: all but one. For three people, there are three such graphs.

Problem 7. How many of the graphs from Problem 5 are connected? There may be no singles, and no isolated polyamorous groups in the graph. There is an answer to this problem, but is quitedifficult to arrive at without advanced discrete maths.