Welcome to the first lesson of your course. Our objective is to build a rigorous framework for designing an experiment and conducting a causal analysis of a polyandrous vee relationship. To do that, we first need a formal language to describe the system itself. This lesson will provide that foundation by introducing the core concepts of network analysis.
Our goal today is to translate the qualitative idea of a "vee relationship" into a precise mathematical object: a network graph. We will define the individuals as nodes, their relationships as edges, the pairs as dyads, and the entire three-person system as a triad. This formalization is the essential first step that will allow us to precisely define concepts like influence, treatment, and satisfaction in our subsequent lessons.
1. From Individuals to Nodes and Edges
At its core, a network is a simplified representation of a system, composed of two fundamental elements: entities and the connections between them. In network analysis, we call the entities nodes (or vertices) and the connections edges (or arcs).
To understand this abstraction, let's watch a short introduction. This video explains how we can model relationships as a graph and why the structure of these connections is as important as the relationships themselves.
Introduction to Social Network Analysis [1/5]: Main Concepts
Please watch the video "Introduction to Social Network Analysis [1/5]: Main Concepts".
Pay close attention to three segments: The initial explanation of how individuals and their letters form a network from the beginning. The discussion of why structure matters, using the network version of Anscombe's quartet, from this section. The formal definition of vertices, edges, and the data modeling process from this segment.
In our scenario, the vee relationship consists of three individuals. Let's label them A (the "pivot," or the woman in the relationship) and B and C (the two men, or "wings"). In our network model:
- Nodes: Individuals A, B, and C are the three nodes of our network.
- Edges: The relationships connecting them are the edges.
However, "relationship" is a vague term. Is it a mutual feeling of love? A one-way flow of support? A sexual connection? The nature of the edge is critical. An edge can be undirected, representing a symmetric relationship (e.g., A and B are married). Or it can be directed, representing an asymmetric relationship (e.g., C feels supported by A, but the reverse may not be equally true).
For analyzing the nuanced dynamics of a personal relationship, directed graphs are far more expressive. They allow us to capture imbalances in feelings, effort, or influence that are often central to relationship satisfaction.
2. Attaching Meaning: Node and Edge Attributes
A simple graph of points and lines is still too abstract. We need to encode the rich details of the people and their connections. We can do this by attaching metadata, or attributes, to our nodes and edges.
- Node Attributes: These are properties of the individuals. For our vee triad, we could define attributes for each node like:
role: 'pivot', 'wing1', 'wing2'age: 45, 47, 46attachment_style: 'secure', 'anxious', 'avoidant'
- Edge Attributes: These are properties of the relationships. For each edge (e.g., the one from A to B), we could define:
type: 'romantic', 'sexual', 'platonic'duration: 25 (years)intensity: A score from 1-10 representing self-reported closeness.
This short clip demonstrates the concept of adding such metadata within a network analysis framework.
Network Analysis Tutorial: Introduction to Networks
Please watch this clip from "Network Analysis Tutorial: Introduction to Networks".
Focus on two key parts: The explanation of associating metadata with nodes and edges from this section. The practical demonstration of storing metadata on a node from this clip.
By adding attributes, our abstract graph becomes a rich data model of the specific relationship system we want to study.
3. The Building Blocks: Dyads and Triads
With nodes and edges defined, we can now describe the structure of the system at different scales.
Dyads: The Pairwise Subsystems
A dyad is simply a pair of nodes. In our three-person system, there are three distinct dyads: (A, B), (A, C), and (B, C). When using directed edges, any given dyad can exist in one of three states:
- Null: No edge exists between the two nodes in either direction. They are unconnected.
- Asymmetric: An edge exists in one direction but not the other (e.g., A → B, but not B → A).
- Mutual: Edges exist in both directions (A ↔ B). This is also called a reciprocated tie.
Understanding the balance of mutual and asymmetric ties is fundamental to analyzing relationship stability and satisfaction. The following reading explains this concept of a "dyad census."
7 Dyads and Triads | Network Analysis
This reading from "Network Analysis with R" formally defines the dyad census. Although it uses R code for its examples, please focus on the conceptual definitions.
In the section "7.2.1 Dyad Census and Reciprocity," read the text from the definition of the dyad census down to the description of the three dyad types. Then, observe how these counts are used to calculate reciprocity, a key network property, in the paragraph starting with this sentence.
The Triad: The Complete System
The triad is the full system of three nodes and their interconnecting edges. It is the fundamental unit of analysis for our vee relationship. While a dyad has only a few possible configurations, a triad of three nodes has a much larger set of possible structures.
With directed edges, there are 16 possible, unique triadic structures. This complexity arises from the combination of the three dyadic states (null, asymmetric, mutual) across the three pairs of nodes. This set of 16 structures is known as the triad census.
The resource below introduces the triad and the idea of a triad census.
11 Transitivity, structural balance, and hierarchy - Bookdown
This text from "Methods for Network Analysis" provides a clear definition of triads and the triad census.
First, read the section "11.3 The Triad" for the definition and the discussion of complexity. Focus on the text from this paragraph. Next, in section "11.4 Calculating a triad census," review the M-A-N classification system, explained in the paragraph here.
This classification system (often labeled by MAN codes: M=Mutual, A=Asymmetric, N=Null relations) gives us a precise vocabulary to describe the exact structure of our vee relationship.
By identifying which of these 16 structures best represents the vee relationship at a given time, we move from a vague description to a formal classification. This allows us to track how the relationship structure changes over time or in response to an intervention.
Conclusion
In this lesson, we have established the foundational vocabulary for analyzing the vee relationship as a formal system. We have seen how to represent individuals as nodes with specific attributes, and their relationships as directed edges, also with attributes. We have defined the pairwise subsystems as dyads, which can be mutual, asymmetric, or null, and the complete three-person system as a triad, which can take one of 16 distinct structural forms.
You have now formalized the vee relationship into a directed, attributed graph. This structured representation is not just an academic exercise; it is the bedrock upon which we will build our entire causal and experimental framework.
In the next lesson, we will use this network model to tackle one of the core challenges in your project: interference. We will define an "exposure mapping" to formally describe how an outcome for one individual (e.g., A's satisfaction) depends not only on treatments applied to them but also on the state of their partners, B and C.