Neuron-based Multifractal Analysis (NeuroMFA)

• Neuron-based Multifractal Analysis (NeuroMFA) is a framework that uses multifractal theory to quantify and interpret neuron interaction dynamics via graph-based models.
• It generalizes classical multifractal and box-covering analyses by extracting metrics such as fractal dimensions, irregularity exponents, and emergence proxies from neural systems.
• Experimental applications in LLMs, spiking networks, and CNNs demonstrate its capacity to correlate geometric network features with performance, criticality, and self-organization.

Neuron-based Multifractal Analysis (NeuroMFA) is a mathematically-structured framework for quantifying, analyzing, and interpreting the interaction dynamics of neurons in large-scale neural networks. Drawing on multifractal theory and network science, NeuroMFA characterizes emergent abilities and self-organization in systems such as LLMs, stochastic spiking networks, and deep convolutional architectures, translating trained network weights or activity measures into multifractal spectra and structural proxies. The method generalizes classical multifractal and box-covering analysis to neuron interaction graphs, enabling rigorous correlation of geometric network features with performance, criticality, and adaptation across diverse domains (Xiao et al., 2024, Costa et al., 2018, Martins et al., 1 Dec 2025).

1. Network Representation and Measure Construction

NeuroMFA begins by representing a neural system—be it a feed-forward transformer, MLP-style deep network, or a network of stochastic spiking neurons—as a directed, weighted graph termed the Neuron Interaction Network (NIN). For LLMs and similar architectures, neurons are arrayed in layers $L_1, \dots, L_m$, with edge weights $w_{ij}$ connecting inter-layer neurons $i, j$. Edges are retained if $|w_{ij}| > \epsilon$ for threshold $\epsilon>0$, and each is labeled with a distance

$$d_{ij} = f(w_{ij}) = \begin{cases} 1/|w_{ij}|, & |w_{ij}| > \epsilon, \ 0, & \text{otherwise.} \end{cases}$$

yielding a sparse adjacency (distance) matrix. For tractable analysis, subgraphs (“Sampled NINs”) are uniformly sampled per layer, algorithmically subsampled and repeated to ensure robust metric estimation (Xiao et al., 2024); a similar paradigm applies to feature-channel measures in CNNs (Martins et al., 1 Dec 2025) and to membrane-potential time series in spiking networks (Costa et al., 2018).

2. Multifractal Formalism and Key Quantities

NeuroMFA adapts multifractal analysis via the scaling behavior of local measures within this network or time series:

• Mass–radius relation (fractal dimension): For a neuron $v_i$ in layer $l$, the ball of radius $r$ in the next layer is

$$N_{l,i}(r) = \sum_{j \in L_{l+1}} \mathbf{1}\{ d_{ij} \leq r \}$$

satisfying the local fractal law $N_{l,i}(r) \sim r^{D_{l,i}}$.

• Partition function: The $q$th-order network partition function

$$Z_q(r) = \sum_{l=1}^L \sum_{v_i \in L_l} p_{l,i}(r)^q, \qquad p_{l,i}(r) = N_{l,i}(r) / T_{l,i}$$

where $T_{l,i}$ denotes all nonzero edges from $v_i$.

• Mass exponent and spectrum: Empirically, $Z_q(r) \sim (r/d_{\max})^{\tau(q)}$; mass exponent $\tau(q)$ is extracted by log–log regression. Singularity (Hölder) exponent $\alpha(q)$ is obtained as $d\tau/dq$, with the multifractal spectrum given by Legendre transform:

$f(\alpha) = q\,\alpha(q) - \tau(q)$

These quantities are computed per sampled subgraph and pooled over subsamples to yield averaged spectra.

In continuous spaces (e.g., CNN feature channels), local Hölder exponents $\alpha(x)$ are computed by OLS regression on windowed sums, and the spectrum is estimated by soft-histogram or Gaussian approximation (Martins et al., 1 Dec 2025).

3. Computational Workflow and Metric Extraction

The NeuroMFA pipeline consistently follows:

1. Sample neuron subgraphs (or feature/channel regions).
2. Compute shortest-path distances or local measures at multiple radii/scales.
3. Accumulate normalized neighbor counts and measures.
4. Construct partition functions $Z_q(r)$ for prescribed moment orders $q$.
5. Perform log–log regression to extract $\tau(q)$.
6. Compute $\alpha(q) = d\tau/dq$ (finite difference) and $f(\alpha)$.
7. Average results over multiple independent subsamples for robust estimates (Xiao et al., 2024).

Discretization is governed by box radii, chosen $q$ values, and neurons per layer. In CNNs, depthwise convolutions and batch-normalization regularize Hölder maps before attention gating (Martins et al., 1 Dec 2025).

4. Structural Proxies and Emergence Metrics

NeuroMFA provides two central scalar metrics from $f(\alpha)$:

• Irregularity exponent $\alpha_0 = \arg\max_{\alpha} f(\alpha)$, locating the most prevalent singularity strength.
• Heterogeneity width $w = \alpha_{\max} - \alpha_{\min}$, denoting singularity span.

Degree of Emergence is defined as

$$E(t) = \frac{w(t)}{w(0)} \cdot \log\left(\frac{\alpha_0(0)}{\alpha_0(t)}\right)$$

where increased $w$ and leftward $\alpha_0$ shift over training mark self-organization; $E(t)>0$ indicates emergent structure (Xiao et al., 2024). In spiking networks, multifractal spectrum width and tail asymmetry index sensitivity and robustness near critical points (Costa et al., 2018). In CNNs, multifractal recalibration attention layers are shown experimentally to yield consistent segmentation gains, and variability in recalibration correlates with instance-level performance (Martins et al., 1 Dec 2025).

5. Experimental Results and Interpretive Insights

Application to the Pythia model family (14M–2.8B parameters, GPT-NeoX) demonstrates that larger models develop heavier-tailed degree distributions and multifractal spectra during training. Small models show negligible shift in $\alpha_0$, while large models exhibit substantial leftward movement in $f(\alpha)$ and spectrum widening up to plateau epochs. Calculated $E(t)$ tightly tracks performance jumps—correlation coefficients $R^2 > 0.7$ confirm its value as a structure-based proxy for emergent ability, outperforming parameter-count scaling (Xiao et al., 2024). In gain-plasticity spiking models, mid-range $\tau$ yields maximal Hurst exponent and near-critical branching ratio; multifractal spectrum width narrows near criticality, and right-tail asymmetry implies insensitivity to large local fluctuations (robustness), with small fluctuations preserved for adaptability (Costa et al., 2018).

In medical imaging segmentation with U-Nets, multifractal recalibration is the only method to consistently improve Dice scores versus baseline and other channel-attention schemes across three datasets (ISIC18, Kvasir-SEG, BUSI) (Martins et al., 1 Dec 2025). Excitation specificity across encoder depth behaves non-monotonically due to skip connections, and balanced gate-value variability correlates with higher segmentation accuracy.

Application Domain	Key Metric or Effect	Reference
LLMs/GPT-like models	Emergence proxy $E(t)$, degree spectra	(Xiao et al., 2024)
Spiking neuron nets	Criticality, Hurst exponents, spectrum	(Costa et al., 2018)
CNNs/medical images	Dice gain, recalibration variability	(Martins et al., 1 Dec 2025)

6. Generalizations, Limitations, and Open Directions

NeuroMFA extends naturally to models with spatial structure, synaptic plasticity, or dynamic architectures:

• For spiking networks and brain recordings, the DFA→MFDFA analysis pipeline computes multifractal indices on gain series, population activity, or experimental time series, complementing avalanche-based criticality metrics (Costa et al., 2018).
• Fractal priors (monofractal/multifractal recalibration) are fully differentiable and can be inserted into CNNs, FPNs, Vision Transformers, and few-shot learning architectures. Gaussian soft-histogram methods scale favorably with number of bins, and ablation studies suggest $Q\!\geq\!4$ suffices in practice (Martins et al., 1 Dec 2025).

However, several limitations persist:

• Current NeuroMFA implementations are primarily single-layered; cross-layer multifractal analysis remains challenging computationally (Xiao et al., 2024).
• Choice of transfer function $f$ for distance labeling shows robustness but lacks comprehensive theoretical rationale.
• Extension to architectures beyond LLMs (e.g., vision diffusion) gives lower emergence proxy $E$; broader characterization of emergence requires further work.
• Computational tradeoffs between subgraph sampling and accuracy highlight need for scalable graph-coarsening or efficient sampling strategies.

A plausible implication is that NeuroMFA can forecast AGI thresholds and phase transitions in model capabilities by detecting geometric signatures of emergent organization.

7. Significance and Theoretical Foundations

NeuroMFA synthesizes multifractal analysis, network science, and attention mechanisms into a unified framework for probing self-organization and emergence in neural systems. Formally, local fractal laws and multifractal scale invariance (i.e., $$\sum_i [\mu(B_i(\epsilon))]^q \approx C(q)\epsilon^{\tau(q)}$$) underpin the measure construction. Tracking $\alpha_0(t)$ and $w(t)$ through training provides a direct structural link to functional capabilities, offering interpretability for emergent phenomena, guidance for model regularization, and theoretically-founded forecasting of phase transitions in neural architectures (Xiao et al., 2024). The approach complements classical criticality indices and expands the toolkit for both biological and artificial neural dynamics (Costa et al., 2018, Martins et al., 1 Dec 2025).