Psychometric Equating Techniques

• Psychometric equating techniques are statistical methods used to adjust scores from different test forms, ensuring comparability.
• They employ approaches such as linear equating, equipercentile equating, and item response theory to mitigate bias and form differences.
• These techniques are critical in standardized testing and educational assessments to maintain fairness and consistent measurement.

A multiplex network is a system where a common set of nodes is connected through multiple types of interactions, each encoded as a distinct layer. The complexity and heterogeneity of such systems, prevalent in social, technological, and biological domains, have motivated the development of a rigorous limit theory for multiplex networks—paralleling the "graphon" theory for large, dense single-layer graphs. The theory of "multiplexons" provides canonical limiting objects for sequences of dense multiplex networks and allows the derivation of analytic forms for the limits of structural features such as degree distributions and clustering coefficients.

1. Multiplexons: Definition and Structure

Multiplexons are constructed to capture all statistical dependencies in large multiplex networks as the number of nodes grows while keeping the number of layers fixed. For an $r$-layer multiplex, a multiplexon is a vector $\widebar{\bm{W}} = (\widebar{W}_S)_{S \subset [r], S \neq \emptyset}$, where each

$\widebar{W}_S : [0,1]^2 \to [0,1]$

encodes the limiting density of edges that occur exactly in the subset of layers $S$. The cardinality of the multiplexon vector is $2^r - 1$, representing all possible nonempty combinations of layers in which an edge may appear.

This construction is essential: unlike the single-layer case, where the limiting object is a single symmetric kernel $W$, in the multiplex setting, separate layer-wise graphons are insufficient to capture dependencies and correlations among layers. For example, in correlated Erdős–Rényi multiplexes, joint probabilities for edge co-occurrence in multiple layers determine system-level properties that are invisible in marginal graphons.

2. Convergence Concepts and Limit Theory

The multiplexon framework formalizes convergence for sequences of finite multiplex networks.

Homomorphism densities (left-convergence):

For multiplexes $\bm{H}$ and $\bm{G}$,

$$t(\bm{H}, \bm{G}) = \frac{|\mathrm{hom}(\bm{H}, \bm{G})|}{|V(\bm{G})|^{|V(\bm{H})|}}$$

measures the probability that a random map from $V(\bm{H})$ to $V(\bm{G})$ preserves all layer-specific edge patterns.

Disjoint and cumulative decompositions:

Edges are classified by the specific subset of layers in which they appear (disjoint), or, alternately, by being present in at least a given set of layers (cumulative). The resulting representations are invertible, and both provide a basis for defining limit objects.

Extension of the cut metric:

The classical cut norm for graphons is extended to multiplexons as: $\|\widebar{\bm{W}}\|_\square = \sum_{S} \|\widebar{W}_S\|_\square, \quad \delta_\square(\widebar{\bm{U}}, \widebar{\bm{W}}) = \inf_\sigma \|\widebar{\bm{U}} - \widebar{\bm{W}}^\sigma\|_\square,$ with the infimum taken over measure-preserving bijections of $[0,1]$ applied synchronously across all $\widebar{W}_S$.

Compactness and equivalence:

The space of multiplexons is compact in the cut distance, and convergence of empirical homomorphism densities is equivalent to convergence in cut distance (Theorem 4). A counting lemma shows that convergence in cut distance guarantees convergence of all finite pattern statistics.

3. Limiting Network Features: Degree Distributions and Clustering

The multiplexon formalism yields analytic limits for standard network features, generalized to the multilayer context.

Degrees:

For a subset of layers $S$, the degree of a node in the projected subnetwork present in all layers of $S$ converges to the random variable

$d_{\widebar{W}_S}(\eta) = \int_0^1 \widebar{W}_S(\eta, y) \, dy,$

with $\eta \sim \mathrm{Unif}[0,1]$.

Clustering coefficients:

Local and global clustering coefficients are generalized by accounting for which layers the triangle's edges appear in. For example, the local clustering coefficient involving two different layers is

$$\theta^{(1)}(i) = \frac{\sum_{s_1 \neq s_2} \sum_{j \neq k} a^{(s_1)}_{ij} a^{(s_2)}_{jk} a^{(s_1)}_{ki}} {(r-1) \sum_{s_1} d_{G_{s_1}}(i)(d_{G_{s_1}}(i)-1)},$$

which, in the limit, converges to a functional involving the multiplexon kernels.

4. Illustrative Examples and Applications

The applicability of the multiplexon theory is demonstrated through several classes of models:

Correlated Erdős–Rényi multiplex:

Each edge exists in subset $S$ with probability $p_S$, independently over node pairs but not across layers. The constant multiplexon $\widebar{W}_S(x, y) = p_S$ fully encodes all interlayer dependencies.

Correlated stochastic block models:

Blocks may be defined differently or with correlated probabilities across layers; the multiplexon reflects joint blockwise edge probabilities for all subsets of layers.

Threshold graphs and uniform attachment models:

These are extended to two (or more) layers, resulting in explicit forms for the multiplexon determined by the underlying threshold, attachment, or fitness mechanisms.

Dynamic (temporal) networks:

Viewing a sequence of $T+1$ graphs as a $(T+1)$-layer multiplex, the resulting multiplexon describes the limiting pattern of edge evolution, and allows for the analysis of Markovian or independent edge dynamics within the decorated graphon paradigm.

5. Relationship to Decorated Graphons and Probability Graphons

Multiplex networks can be viewed as edges decorated by subsets of layers. Their limit theory is a specific case of "decorated graphons" or, equivalently, "probability graphons"—measurable functions assigning a probability distribution (here, over layer subsets) to each node pair. Consequently, the topology, convergence results, and analytic formulas for network features in the multiplexon framework are derived as corollaries of the general decorated graphon theory (Ganguly et al., 8 Oct 2025).

The cut distance, sampling convergence, and equivalence of various convergence modes all follow directly from the corresponding results for probability graphons and decorated graphons with a finite decoration space.

6. Theoretical Implications and Scope

The theory of multiplexons unifies and extends the limit-theoretic treatment of large, dense network systems to the multiplex setting. It offers:

• Canonical limiting objects capturing all joint layer-wise dependencies;
• Rigorous equivalence between metric (cut distance) and homomorphism-based notions of convergence;
• Quantitative descriptions for the limiting behavior of degree distributions, clustering, and other structural measures;
• Explicit adaptation and computation of limiting features in wide classes of random, block-structured, thresholded, and evolving networks.

This framework is directly applicable not only to theoretical questions but to empirical analysis and inference in fields where multilayer or dynamic network data are now routinely encountered.

7. Future Directions and Open Problems

The foundation laid by the multiplexon theory suggests multiple lines for further research:

• Extending limit theory to sparse multiplex networks or those with varying numbers of layers;
• Incorporating dynamic node sets, directed multiplexes, or more general interactions (e.g., hyperedges);
• Developing statistical inference and hypothesis testing methods grounded in multiplexon convergence;
• Applying the framework to decorated graph settings arising in real-world complex systems analysis.

The correspondence between multiplexons and probability graphons invites generalization and transfer of tools between classical graph limit theory and decorated network models.

---

Table: Summary of Correspondence Between Convergence Notions

Limiting Object	Edge Decoration	Notions of Convergence
Graphon	None (single-layer)	Cut distance, homomorphism densities
Multiplexon	Subsets of layers	Cut distance (sum over subsets), left-conv.
Decorated graphon	General (finite set $D$)	Probability-graphon cut distance, densities

This underscores that multiplexons are the special case of decorated graphon theory where the decoration space is the power set of layers, directly relating the two frameworks.