Index of Inclusion: Concepts & Applications

• Index of Inclusion is a multifaceted measure capturing accessibility, diversity, equity, and network structure across various disciplines.
• It integrates quantitative methods from operator algebras, combinatorics, and socioeconomic analyses to classify inclusion and drive policy insights.
• Its implementation employs advanced techniques like latent trait models, nerve methods, and network-adjusted indices to improve measurement accuracy.

The Index of Inclusion is a multifaceted concept across mathematics, statistics, economics, social sciences, and organizational analysis, capturing dimensions of accessibility, diversity, equity, and networked structure. The term is not unified to a single formula; rather, its meaning and implementation depend on domain context. It can reference: (i) a formal measure—such as in operator algebras or combinatorics—quantifying substructure inclusion or combinatorial constraints, (ii) an aggregated, often latent, indicator designed to measure inclusivity in education or social services, or (iii) a scalar or vector index combining diversity, category similarity, and network properties within a population. The index often serves as both a quantitative diagnostic and a policy or algorithmic input to reveal, analyze, or optimize inclusionary properties within complex systems.

1. Formal Index of Inclusion in Mathematics

Operator Algebras and Subfactor Theory

In subfactor theory, the index of an inclusion $N \subset M$ of II$_1$ factors is the Jones index $[M:N]$—a numerical invariant quantifying the relative size of $M$ to $N$. For $[M:N] < 4$, allowed index values are quantized as

$$[M:N] = 4 \cos^2\left(\frac{\pi}{n}\right),\quad n=3,4,5,\ldots$$

representing the only possible values for inclusions of II$_1$ factors in this regime. Principal graphs and planar algebras serve as combinatorial invariants, encoding the bimodule structure induced by the inclusion, and are subject to restrictiveness determined by the value of the index. For indices in the interval $(4, 5)$, outside an infinite Temperley–Lieb family, only ten exceptional subfactors exist—each distinguished by unique principal graphs and fusion rules (Jones et al., 2013).

This formal index governs not only the existence of inclusions but also classifies their potential via deep combinatorial, number-theoretic, and planar algebraic constraints. The quantization reflects underlying symmetries and links to tensor category theory, knot invariants (via the Jones polynomial), exactly solvable statistical models, and topological quantum field theory.

Lattice Structure and Angle-based Indices

For inclusions of simple $C^*$-algebras $B\subset A$ with finite Watatani index, the number of intermediate subalgebras is bounded by the minimal index $[A : B]_0$: $\#I(B \subset A) < 9 [A : B]_0$ where rigidity of the angle between intermediate subalgebras (as defined via their Jones projections) is employed to limit the lattice's cardinality (Bakshi et al., 2023).

2. Efficient Implementation in Combinatorial Inclusion-Exclusion

The inclusion–exclusion principle counts elements in unions of sets, typically requiring summation over all $2^h$ subsets for $h$ constraints. For many practical problems, most terms are zero due to incompatible constraints. The nerve method identifies the set ideal of nonzero terms, $\mathcal{N}$, and compacts representation into disjoint unions (e.g., 012n-rows): $$\mathcal{N} = r_1 \uplus r_2 \uplus \cdots \uplus r_R$$ This enables a compressed sum: $N = \sum_{k=0}^{h} (-1)^k f(k) g(k)$ where $f(k)$ is the number of size-$k$ faces in the nerve and $g(k)$ the invariant count for face size $k$ (Wild, 2013). Upgrades A and B collectively allow computational tractability for otherwise intractable inclusion–exclusion calculations and are crucial for enumerative combinatorics, probabilistic models (Bonferroni inequalities), and constraint satisfaction problems.

3. Indices of Inclusion in Socioeconomic and Educational Analysis

Inclusive Development Indices

The Inclusive Development Index (IDI) is constructed as a composite measure, integrating multidimensional economic and social indicators. The REL‑PCANet model computes the IDI by:

• Normalizing raw indicators (e.g., GDP per capita, poverty rates) with direction-sensitive scaling.
• Deriving principal components (via PCA) to form relative attribute vectors.
• Applying k-means clustering to stratify countries and constructing target probability matrices (TRnet) capturing inter-cluster and intra-cluster ranking likelihoods.
• Employing RankNet principles (pairwise learning-to-rank) in a neural network to output final scores:

$\text{rank}_i = \text{weight}^T x_i + \text{bias}$

This approach captures static attributes and dynamic (year-to-year) changes in inclusive development, reflecting cluster movements and interdependencies (Irmatov et al., 2018).

Education: Latent Inclusivity Indices

In education systems, particularly Italian ECEC services, inclusivity is modeled as a latent variable $z$ estimated via a latent trait (Item Response Theory) model: $$\log\left(\frac{\pi_i(z)}{1-\pi_i(z)}\right) = \beta_{0i} + \beta_{1i} z$$ where $\pi_i(z)$ is the probability that facility $i$ exhibits a given inclusivity characteristic. The estimated $z$ is rescaled and then analyzed with a mixed quantile regression model: $Q_{z̃|u_t}(\tau | X, u_t) = X \gamma_\tau + u_t$ to capture heterogeneity and distributional features across service types, regions, and quantiles (Andreella et al., 22 Jul 2024). Small Area Estimation further refines these indices to the NUTS-3 (provincial) level using the empirical best predictor.

Findings consistently show that public ECEC facilities have a higher inclusivity index than private ones, and regional disparities are substantial; bimodal index distributions indicate clustered inclusivity regimes.

4. Network- and Similarity-Adapted Diversity/Inclusion Indices

The DSN index generalizes conventional diversity by incorporating both similarity (matrix $Z$) and network structure (adjacency matrix $E$), with parameter $q$ controlling sensitivity to rare or common categories. The index is defined as:

• For $q\neq1,\infty$:

$$D_{(q)}^{\overline{Z}(p,E)} = \left[ \sum_i p_i \{(L - (\overline{Z} \circ E) p)_i\}^{q-1} \right]^{1/(1-q)}$$

• For $q=1$:

$$D_{(1)}^{\overline{Z}(p,E)} = \prod_i \{(L - (\overline{Z} \circ E) p)_i\}^{-p_i}$$

where $p$ is the category proportion vector, $L$ is an all-ones matrix, $\overline{Z} = L - Z$ (dissimilarity), and $\circ$ indicates the Hadamard product (Kinjo, 8 Oct 2024). Similarity matrices are estimated by optimizing correlation between the DSN index and external outcome variables, using weighted attribute distances. Network relationships can be further expanded to reflect higher-order connections.

This DSN index reduces to classical Hill numbers in appropriate limits, exhibits monotonicity in $q$, and increases with greater similarity or stronger network connectivity among categories. Visualization employs multidimensional scaling and network diagrams to display proportions, similarities, and edges.

5. Empirical Properties, Impact, and Policy Applications

Analysis of inclusion indices across domains has demonstrated:

• In early childhood education, the latent inclusion index exposes both macro (regional) and micro (within-region, public/private) variation, informing targeted policy interventions and performance monitoring.
• For economic development, robust and dynamic scoring models such as REL‑PCANet provide actionable evidence for policy and reform assessment, operationalizing multidimensional inclusion rather than unidimensional growth.
• In organizational and social contexts, DSN-like indices furnished with empirically optimized similarity weights align more closely with subjective or outcome-oriented preference rankings than traditional indices do, supporting tailored diversity and inclusion audits.

A plausible implication is that measurement of inclusion and diversity, when refined to account for latent structure, similarity, and network connectivity, yields indices with stronger explanatory and predictive power, and thus greater relevance both for scientific inference and for policy design.

6. Methodological Advancements and Limitations

Major methodological advances include:

• Nerve-based compression and upgrades for inclusion–exclusion computations, vital for high-dimensional combinatorics.
• Application of latent trait models for unidimensional inclusivity quantification and integration with quantile regression and small area estimation, enhancing resolution and interpretability.
• Introduction of DSN indices coupling similarity and network structure, offering theoretically grounded measures that unify prior diversity concepts and extend their range of application.

Limitations reside in the reliance on accurate modeling of similarity and network structure (critical for DSN indices), assumptions underlying latent variable models (e.g., unidimensionality, logistic link), and sensitivity of cluster-based dynamic updates to the appropriateness of clustering resolution and indicator selection.

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In summary, the Index of Inclusion is a domain-adaptable, mathematically rigorous concept that encapsulates the degree or structure of inclusion—ranging from subfactor index quantization in operator algebra, to computationally efficient counting in combinatorics, composite socioeconomic measures in policy, and modern, network-sensitive diversity in sociological and organizational science. Its implementation, formal constraints, and analytical properties provide not only technical insights but also concrete levers for systemic assessment and improvement in complex structures.