Modeling Multiplex Networks: Different Relationship Dimensions

Beyond the Simple Graph: Introducing Multiplexity

So far, we have been working with what are called monoplex or single-layer networks. In these models, a single type of edge represents "a relationship." This is useful but limiting. A connection between a hinge and a partner is not a monolith; it is a bundle of different strands. Are they emotionally intimate? Sexually active? Do they share a home or finances? These dimensions do not always coexist, and their unique combination defines the character of the relationship.

To capture this richness, we turn to multilayer networks. A particularly useful form of multilayer network for our purposes is the multiplex network.

In essence, a multiplex network consists of the same set of individuals (nodes) arranged across multiple "layers," where each layer represents a distinct type of tie. For our vee (Hinge, Partner 1, Partner 2), we can visualize this as a stack of three separate networks built on the same three people:

• Layer 1: Emotional Intimacy. Edges on this layer represent a strong, supportive emotional bond.
• Layer 2: Sexual Connection. Edges represent an active sexual relationship.
• Layer 3: Logistical Interdependence. Edges represent shared responsibilities like cohabitation, finances, or co-parenting.

In this framework, the entity we previously called a "node" (e.g., P1) is now represented by a "node-tuple"—a collection of its presence across all layers (P1 on the emotional layer, P1 on the sexual layer, etc.).

Constructing a Multiplex Model of a Vee

Let's imagine a specific vee scenario. The Hinge (H) has an emotionally intimate and sexually active relationship with both P1 and P2. However, H only lives with and shares finances with P1. P1 and P2 are not connected in any of these dimensions.

We can represent this with three adjacency matrices, one for each layer. Let's use 1 for the presence of a tie and 0 for its absence.

Layer 1: Emotional Intimacy

Layer 2: Sexual Connection

Layer 3: Logistical Interdependence

This representation immediately reveals crucial structural details:

• The emotional and sexual layers are congruent; they have the identical structure.
• The logistical layer is incongruent with the others. The relationship between H and P2 does not exist in this dimension.
• The overall brokerage role of the hinge is maintained on the emotional and sexual layers, but on the logistical layer, P2 is an isolate.

This method allows for a much more precise description of the relationships. We can even describe the dyadic bond between any two individuals as a vector of their connections across all layers.

Using the N-gram concept, we can summarize our example:

• H-P1 Relationship: [Emotional: 1, Sexual: 1, Logistical: 1]
• H-P2 Relationship: [Emotional: 1, Sexual: 1, Logistical: 0]
• P1-P2 Relationship: [Emotional: 0, Sexual: 0, Logistical: 0]

This vector representation provides a rich, quantitative fingerprint for each dyadic bond, moving far beyond a simple "connected" or "not connected" label.

Combining and Analyzing Multiplex Data

A stack of matrices is descriptive, but for many analytical tasks, we need to synthesize this information. Your background in statistics involves reducing complex datasets to meaningful summaries, and the same principle applies here. There are two general approaches to combining multiplex data into a single, analyzable network.

Let's apply these methods to our example vee.

1. Quantitative Combination: Creating a "Strength of Tie" Index

This approach reduces the multiple layers into a single, weighted network that represents the overall intensity or multiplicity of the connection.

• Sum: We can create a new matrix where the edge weight is the sum of weights across all layers.

  • The H-P1 tie strength would be $1+1+1=3$.
  • The H-P2 tie strength would be $1+1+0=2$.
  • This immediately quantifies that the H-P1 bond is more "multiplex" or multifaceted than the H-P2 bond.

• Intersection (Logical AND): We could create a network that only shows ties that exist on all layers. In our example, only the H-P1 tie would remain. This isolates the most robust, fully-encompassing relationships.

• Union (Logical OR): We could create a network showing any tie that exists on at least one layer. This would produce a network identical in structure to our emotional/sexual layers.

2. Qualitative Combination: Creating a Relationship Typology

Instead of measuring strength, this method classifies the type of relationship. The values in the resulting matrix are categorical, not quantitative.

Following the logic from the textbook, we could define a new matrix with a code for each unique N-gram:

• 0 = No connection of any kind (P1-P2)
• 1 = Emotional & Sexual connection only (Hypothetical)
• 2 = Emotional, Sexual, & Logistical connection (H-P1)
• 3 = Emotional & Sexual, but not Logistical connection (H-P2)

The resulting matrix would look like this:

This approach preserves the qualitative differences between the relationships. We can now clearly see that the hinge has two different types of relationships with their partners, a distinction that was invisible in a simple graph.

Conclusion

Today we have moved from a one-dimensional view of relationships to a much richer, multidimensional one. By adopting a multiplex network framework, we can formally model the distinct emotional, sexual, and logistical strands that constitute relationships in a polyamorous vee.

• Key Takeaways:
  • Relationships are multifaceted; modeling them with single-layer (monoplex) networks is an oversimplification.
  • Multiplex networks allow us to represent different relationship dimensions (e.g., emotional, sexual, logistical) as distinct layers built upon the same set of individuals.
  • This approach enables us to analyze the congruence (similarity) between different layers and to describe dyadic bonds with a detailed vector called an N-gram.
  • To analyze a multiplex network, we can combine its layers into a single summary matrix, either quantitatively (e.g., by summing ties to get a "strength" score) or qualitatively (e.g., by creating a typology of relationship types).
  • This richer model allows for a more precise analysis of roles and structures. For instance, a hinge's power and influence may not be uniform across all dimensions of their relationships.

Preview of the next lesson:
Now that we have a more sophisticated model for the vee's structure, we can revisit the concept of power with greater precision. In our next lesson, we will apply betweenness centrality to quantify the hinge partner's brokerage power in a vee. We will explore how a hinge's centrality—and therefore their structural power—can differ across the various layers of a multiplex network, leading to a more nuanced understanding of their role.