Total Entropy Fit Index (TEFI)

• TEFI is defined as an entropy-based global fit index that quantifies the internal coherence and complexity of network partitions.
• It uses von Neumann entropy to transform correlation matrices into density-like matrices, comparing overall and within-dimension entropies.
• TEFI is applied in dynamic exploratory graph analysis to optimize embedding depth by balancing entropy reduction with structural accuracy.

Searching arXiv for the specified paper and closely related TEFI/EGA work to ground the article with citations. Total Entropy Fit Index (TEFI) is an entropy-based fit index for dimensional structures in network psychometrics. In "Optimizing the Landscape of LLM Embeddings with Dynamic Exploratory Graph Analysis for Generative Psychometrics: A Monte Carlo Study" (Golino, 14 Jan 2026), TEFI is used to evaluate how well a proposed dimensional partition organizes the correlation structure of networks derived from truncated LLM embeddings. The index is rooted in Von Neumann entropy from quantum information theory, treats a correlation or similarity matrix as a density-like matrix, and quantifies the degree of structural organization in a multivariate system. Within the study, TEFI is not used as a comparative fit index between alternative parametric models; rather, it is an absolute scalar criterion for a given partition, with lower, more negative values indicating better fit.

1. Definition and conceptual scope

TEFI is defined as a global fit index for a proposed dimensional partition. In the study, a partition corresponds to an assignment of items to dimensions or communities produced by Exploratory Graph Analysis (EGA) or its dynamic extension, DynEGA. The index compares the entropy of the full system with the entropies of the dimension-specific submatrices, thereby quantifying whether the partition produces internally coherent dimensions while avoiding unnecessarily complex solutions (Golino, 14 Jan 2026).

The paper characterizes TEFI as an entropy-based measure of how organized and coherent the network’s dimensional structure is. Its interpretation is explicitly tied to a simple structure assumption: each variable belongs to exactly one dimension. Under that assumption, lower within-dimension entropy reflects stronger internal coherence, whereas excessive dimensionality is penalized. This means TEFI operationalizes two simultaneous desiderata: reduction of informational disorder within dimensions and control over overfragmentation of the solution.

A central interpretive point in the study is directional: TEFI is minimized, not maximized. Lower, more negative values indicate better fit because they correspond to partitions that reduce uncertainty without overfitting. The paper therefore treats TEFI as an absolute entropy-based fit index that can be compared across embedding depths or across candidate partitions, but not as a relative index in the sense of nested-model comparison.

2. Mathematical construction

The study builds TEFI from Von Neumann entropy. To compute the index, the correlation matrix $\mathbf{R}$ is first transformed into a density-like matrix $\boldsymbol{\rho}$ by normalizing such that the trace equals one (Golino, 14 Jan 2026). Von Neumann entropy is then defined as

$$\mathcal{S}(\boldsymbol{\rho}) = -\operatorname{tr}(\boldsymbol{\rho}\log \boldsymbol{\rho}),$$

with the equivalent eigenvalue expression

$$\mathcal{S}(\boldsymbol{\rho}) = -\sum_{i=1}^{m}\lambda_i \log \lambda_i,$$

where $\lambda_1, \lambda_2, \ldots, \lambda_m$ are the eigenvalues of $\boldsymbol{\rho}$ and $m$ is the number of variables. In the paper’s terminology, this entropy captures disorder or uncertainty in the correlation structure.

Given a solution with $N_F$ dimensions, TEFI is defined as

$$\text{TEFI} = \left[ \frac{\sum_{k=1}^{N_F}\mathcal{S}(\boldsymbol{\rho}_k)}{N_F} - \mathcal{S}(\boldsymbol{\rho}) \right] + \left[ \left( \mathcal{S}(\boldsymbol{\rho}) - \sum_{k=1}^{N_F}\mathcal{S}(\boldsymbol{\rho}_k) \right)\times \sqrt{N_F} \right].$$

Here, $\boldsymbol{\rho}$ is the density matrix for the full correlation matrix, and $\boldsymbol{\rho}$0 is the density matrix for the submatrix containing only items in dimension $\boldsymbol{\rho}$1. The first bracket is the difference between average within-dimension entropy and total system entropy; according to the authors, it decreases as dimensions become more internally coherent. The second bracket penalizes excessive dimensionality and is scaled by $\boldsymbol{\rho}$2 to control growth of the index as the number of dimensions increases.

The paper explicitly connects this formulation to prior work by Golino (2019) and Golino (2024), and situates its theoretical basis in Von Neumann entropy, quantum information, and entanglement. Within the present study, however, the key operational consequence is that TEFI combines disorder reduction and dimensionality control in a single scalar criterion.

3. Use within DynEGA and embedding-depth search

The study adapts Dynamic Exploratory Graph Analysis to an embedding-search setting by treating the embedding dimension index as a pseudo-temporal ordering analogous to intensive longitudinal trajectories (Golino, 14 Jan 2026). Rather than regarding embeddings as static cross-sectional vectors, the paper treats them as searchable landscapes. At each embedding depth, the first coordinates of the embedding vector are retained, a similarity or correlation matrix among items is constructed, a network is estimated with TMFG, and communities are detected with Walktrap. TEFI is then computed on the resulting partition and the corresponding item correlation matrix.

In this workflow, TEFI is evaluated repeatedly across embedding depths. The investigated depths span from 3 to 1,298 dimensions, in increments of 5, using OpenAI’s text-embedding-3-small model, which produces 1,536-dimensional embeddings. The simulation construct is grandiose narcissism, represented by five dimensions—Authority, Exhibitionism, Superiority, Entitlement, and Exploitativeness—with item-pool sizes varying from 3 to 40 items per dimension.

This design assigns TEFI a specific role: it becomes a trajectory over embedding depth rather than a single summary at one fixed representation. The paper uses that trajectory in three ways. First, TEFI is examined as a stand-alone function of depth. Second, it is contrasted with Normalized Mutual Information (NMI), which measures agreement between detected and known true dimensions. Third, it enters a composite optimization criterion designed to reconcile entropy-based organization and structural accuracy.

4. Relation to NMI and the problem of competing optima

A central finding of the study is that TEFI and NMI define competing optimization trajectories across the embedding landscape (Golino, 14 Jan 2026). NMI is an accuracy measure ranging from 0 to 1, where higher values indicate better correspondence between estimated communities and true dimensions. TEFI, by contrast, measures entropy-based organization, where lower, more negative values indicate better fit. The two criteria do not peak in the same regions of the embedding space.

Criterion	What it evaluates	Direction of improvement
TEFI	Entropy-based organization of the dimensional partition	Lower, more negative
NMI	Agreement between estimated communities and true dimensions	Higher
Composite $\boldsymbol{\rho}$3	Joint balance of NMI and normalized TEFI	Higher

The abstract states that TEFI achieves minima at deep embedding ranges, approximately 900--1,200 dimensions, where entropy-based organization is maximal but structural accuracy degrades. NMI peaks at shallow depths, where dimensional recovery is strongest but entropy-based fit remains suboptimal. The paper therefore argues that single-metric optimization produces structurally incoherent solutions.

The contrast is illustrated with a concrete example for the 4-items-per-dimension condition. The NMI optimum occurs at 93 dimensions, with $\boldsymbol{\rho}$4 and $\boldsymbol{\rho}$5. The TEFI optimum occurs at 983 dimensions, with $\boldsymbol{\rho}$6 and $\boldsymbol{\rho}$7. A weighted composite optimum using 70% NMI and 30% TEFI occurs at 23 dimensions, with $\boldsymbol{\rho}$8 and $\boldsymbol{\rho}$9. The study uses this example to show that optimizing only NMI produces an accurate but entropy-messy solution, whereas optimizing only TEFI produces an organized but inaccurate one.

The vector-field analysis in the paper reinforces the same point. TEFI is plotted on the x-axis and NMI on the y-axis, with arrows showing how both metrics evolve as depth increases. The upper-left region, described as an optimal zone, contains moderate negative TEFI and high NMI at intermediate depths. In deeper regions, TEFI continues to improve while NMI degrades. In shallower regions, NMI can remain high while TEFI stays near zero or positive. The paper interprets these paths as evidence that following the gradient of TEFI alone leads away from the region of strongest structural recovery.

5. Empirical patterns across depth and item-pool size

The Monte Carlo results show systematic TEFI behavior across both embedding depth and item-pool size (Golino, 14 Jan 2026). Across depths, TEFI is not very negative at shallow ranges, roughly 3--183 dimensions. As depth increases, TEFI generally becomes more negative, especially when item pools are larger. The study also reports periodic oscillations with multiple troughs and relative degradations across the landscape.

The principal TEFI troughs are reported at approximately the following depth ranges:

• Initial local optima: about 183--303 dimensions
• Relative degradation: about 323--643 dimensions
• Deeper troughs: about 663--743 dimensions
• Later troughs: about 963--1063 dimensions
• Near maximum depth: around 1163 dimensions

The item-count effect is strong. Sparse pools with 3--8 items per dimension keep TEFI closer to zero and never reach strongly negative values. Moderate pools with 10--20 items per dimension reach approximately $$\mathcal{S}(\boldsymbol{\rho}) = -\operatorname{tr}(\boldsymbol{\rho}\log \boldsymbol{\rho}),$$0 to $$\mathcal{S}(\boldsymbol{\rho}) = -\operatorname{tr}(\boldsymbol{\rho}\log \boldsymbol{\rho}),$$1, which the paper describes as moderate organization. Large pools with 25--40 items per dimension reach below $$\mathcal{S}(\boldsymbol{\rho}) = -\operatorname{tr}(\boldsymbol{\rho}\log \boldsymbol{\rho}),$$2, down to about $$\mathcal{S}(\boldsymbol{\rho}) = -\operatorname{tr}(\boldsymbol{\rho}\log \boldsymbol{\rho}),$$3 at their best depths, which the paper describes as very strong entropy-based organization. This pattern differs from NMI, which follows an inverted-U pattern: 10--20 items per dimension produce the best structural accuracy, whereas too many items, especially 40, degrade NMI.

The interaction of item-pool size and depth is especially important. Large item pools combined with deep embeddings produce extremely negative TEFI but low NMI. Small pools do not permit TEFI to become very negative regardless of depth, because entropy cannot be reduced strongly when dimensions have very few items. The paper does not supply fixed cutoffs such as “TEFI below a given value is good” in an absolute sense; instead, interpretation is comparative within condition. It nevertheless offers qualitative guides: TEFI values near 0 indicate poor organization, values around $$\mathcal{S}(\boldsymbol{\rho}) = -\operatorname{tr}(\boldsymbol{\rho}\log \boldsymbol{\rho}),$$4 to $$\mathcal{S}(\boldsymbol{\rho}) = -\operatorname{tr}(\boldsymbol{\rho}\log \boldsymbol{\rho}),$$5 indicate moderate organization, and values around $$\mathcal{S}(\boldsymbol{\rho}) = -\operatorname{tr}(\boldsymbol{\rho}\log \boldsymbol{\rho}),$$6 to $$\mathcal{S}(\boldsymbol{\rho}) = -\operatorname{tr}(\boldsymbol{\rho}\log \boldsymbol{\rho}),$$7 indicate very strong organization, provided NMI is also considered.

6. Composite optimization, interpretation, and limitations

To reconcile the divergence between accuracy and organization, the study defines a weighted composite metric:

$$\mathcal{S}(\boldsymbol{\rho}) = -\operatorname{tr}(\boldsymbol{\rho}\log \boldsymbol{\rho}),$$8

The paper notes that the printed expression is missing a closing brace, but conceptually the term is $$\mathcal{S}(\boldsymbol{\rho}) = -\operatorname{tr}(\boldsymbol{\rho}\log \boldsymbol{\rho}),$$9 (Golino, 14 Jan 2026). The negative sign aligns TEFI with NMI because lower TEFI is better. Operationally, NMI and TEFI are normalized, $$\mathcal{S}(\boldsymbol{\rho}) = -\sum_{i=1}^{m}\lambda_i \log \lambda_i,$$0 is computed at each embedding depth, and the selected depth is the one that maximizes $$\mathcal{S}(\boldsymbol{\rho}) = -\sum_{i=1}^{m}\lambda_i \log \lambda_i,$$1.

The authors interpret this as a simple multi-objective optimization procedure. In aggregate, DynEGA with NMI-TEFI composite optimization outperforms cross-sectional EGA across all item counts in terms of structural accuracy, with the greatest gains occurring for larger item pools. The study therefore recommends using TEFI in combination with structural accuracy metrics rather than as a stand-alone selector.

Several practical implications follow directly from the reported results. TEFI can function as a diagnostic of internal coherence for partitions produced by EGA or DynEGA. High NMI with poor TEFI indicates a solution that is structurally correct but internally messy. Conversely, highly negative TEFI with poor NMI indicates dimensions that are internally organized but misaligned with the intended construct structure. The paper further argues that embedding spaces are non-uniform semantic spaces and that default full-vector usage is not principled; TEFI helps reveal where the network becomes most orderly, while NMI indicates whether that order corresponds to the target dimensions.

The study also identifies explicit cautions. TEFI is used under a simple structure assumption, so solutions involving substantial cross-loading may be penalized. TEFI is sensitive to the number of items per dimension and to the number of dimensions estimated. The 70/30 weighting scheme is described as arbitrary but justified, and the paper suggests that alternative weights or full Pareto analysis may be preferable depending on the analytic objective. TEFI also depends on the specific network-estimation and community-detection choices; other algorithms could shift the TEFI landscape.

Within the study’s framework, the recommended practice is to embed items, traverse a sequence of embedding depths, construct networks and detect communities at each depth, compute both NMI and TEFI, normalize the two indices, optimize a weighted composite, and report both NMI and TEFI at the selected depth. The authors additionally suggest future work on alternative weighting schemes, Pareto frontiers in the TEFI-NMI space, other constructs and embedding models, and validation against empirical response-data networks.