Neuron-based Multifractal Analysis of Neuron Interaction Dynamics in Large Models
Authors: Xiongye Xiao
Heng Ping
Chenyu Zhou
Defu Cao
Yaxing Li
Yi-Zhuo Zhou
Shixuan Li
Nikos Kanakaris
Paul Bogdan
Venue: ICLR 2025 Poster (2025)
Keywords: Emergent Ability
Large Language Models (LLMs)
Multifractal Analysis
Primary area: interpretability and explainable AI
TL;DR: Our study primarily lays a theoretical foundation for exploring the multifractal structure of neural networks in LLMs, supplemented by initial empirical evidence suggesting practical implications.

Abstract: In recent years, there has been increasing attention on the capabilities of large-scale models, particularly in handling complex tasks that small-scale models are unable to perform. Notably, large language models (LLMs) have demonstrated ``intelligent'' abilities such as complex reasoning and abstract language comprehension, reflecting cognitive-like behaviors. However, current research on emergent abilities in large models predominantly focuses on the relationship between model performance and size, leaving a significant gap in the systematic quantitative analysis of the internal structures and mechanisms driving these emergent abilities. Drawing inspiration from neuroscience research on brain network structure and self-organization, we propose (i) a general network representation of large models, (ii) a new analytical framework — *Neuron-based Multifractal Analysis (NeuroMFA)* - for structural analysis, and (iii) a novel structure-based metric as a proxy for emergent abilities of large models. By linking structural features to the capabilities of large models, *NeuroMFA* provides a quantitative framework for analyzing emergent phenomena in large models. Our experiments show that the proposed method yields a comprehensive measure of the network's evolving heterogeneity and organization, offering theoretical foundations and a new perspective for investigating emergence in large models.

Published as a conference paper at ICLR 2025
NEURON-BASEL MULTIFRACTAL ANALYSIS OF NEU- RON INTERACTION DYNAMICS IN LARGE MODELS
Xiongye Xiao 'f
Heng Ping"
Chenyu Zhou'
Defu Cao
Yaxing Li'
Yi-Zhuo Zhou²
Shixuan Li'
Nikos Kanakaris'
Paul Bogdan"
"University of Southern California, CA, USA University of California, Riverside, CA, USA
ABSTRACT
In recent years, there has been increasing attention the capabilities of large-scale models, particularly in handling complex tasks that small-scale models are unable perform. Notably, large language models (LLMs) have demonstrated "intelli- gent" abilities such as complex reasoning and abstract language comprehension, reflecting cognitive-like behaviors. However, current research on emergent abil- ities in large models predominantly focuses on the relationship between model performance and size, leaving a significant gap in the systematic quantitative anal- ysis of the internal structures and mechanisms driving these emergent abilities. Drawing inspiration from neuroscience research on brain network structure and self- organization, propose (i) general network representation of large models, (ii) a new analytical Paamwork-Neuue -Neurnn-uuasss basee Multifractal Analysis (NeuroMFA) J-for structural analysis, and (iii) novel structure- aruuttrr-based metric as proxy for emergent abilities of large models. By linking structural features to the capabilities of large models, NewroMFA provides quantitative framework for analyzing emergent phe- nomena in large models. Our experiments show that the proposed method yields a comprehensive measure of the network's evolving heterogeneity and organization, offering theoretical foundations and a new perspective for investigating emergent abilities in large models.
INTRODUCTION
The advent of foundation models triggered paradigm shift in artificial intelligence (AI), demon- strating unparalleled capabilities across many domains, particularly natural language processing (Chang et al. [2024) Brown et al.) [2020] Chowdhery et al.) (0223
( He et al. 2020: Duan et al.) (024). Recent studies observed that, compared to traditional small-scale neural networks (NNs), large language models (LLMs) exhibit advanced cognitive and reasoning abilities and higher level of comprehension (Brown et al. 2020 BBubeck et al. 2023 Achiam et al. 2023). This behavior, where larger models develop qualitatively
abilities absent smaller ones, is referred as "emergent abilities" (Srivastava et al. 2022; Wei et al., 2022a). Recent works have focused on using scaling laws and other analytical methods to study the relationship between model size and the development of these new capabilities, suggesting that emergent abilities may arise predictably as models scale
up (Kaplan et al. 2020 Hu al. 2023a b Michaud al. 2023 Schaeffer et al. 2024).
Existing research, limitations, and motivation. As depicted in Fig. [ previous research on the "emergence" in large models and particularly LLMs (Wei et al. 2022a| Fu et al. 2023 Zhong et al..
2024; Schaeffer et al. 2024) focuses on the observation-based analysis and the model size (e.g.. number of parameters, model depth). Traditional metrics (Paperno al. 2016: Sakaguchi al.
2021 Bisk et al. 2020) have largely assessed LLMs' emergent abilitiees by capturing the relationship between model's 's size and output performance. However, these approachs have several limitations: (i) observation-basee methods often require exhaustively testing various outputs, making it challenging
Correspondence: Xiongye Xiao <xiongyex @ uuuc,edu>, Paul Bogdan <phogdan @usc.edu>, 'Fqual
Contribution.

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Low Transparency Exhaustive Tesling Needed Limited Interpretabillty
Degroe Emtergence
Observation- based Analysis
Performance metrics
Large Models
Model Size
LLMs, diffusion models, etc.
Emergent Abilities
Degree Emergence
Neuron Interaction
Structure- Analysis
Networks
Weighted directed networks
Training
Multifractal analysis
SettOrrgeiizttinn
Modill
Figure 1: (a) The proposed paradigm shift for assessing the degree of emergence in large models. (b) Comparison of traditional methods (top) and our approach (down); we study the emergent abilities of large models by analyzing structural dynamics during training.
to establish unified standard for evaluating the full scope of large model's emergent abilities; (ii) the approaches treat the model as black box, limiting the understanding of the internal mechanisms that drive the model's behavior and emergent abilities; (iii) relying solely on model size to measure performance overlooks other critical factors that significantly influence the model's capabilities, such as architecture design, data quality, and training dynamics. To address these limitations, approach emphasizes structure- analysis by examining the dynamic changes in a model's Is internal structure during the training process. The iterative training process is crucial for the improvement and development of capabilities in large models, as enables the model to transition from having no abilities to exhibiting emergent abilities, as illustrated in Fig. 2 Current work has demonstrated that large models undergo significant structural changes in their internal mechanisms and dynamic characteristics throughout training (Ding et al. 2023) Moreover, inspired by neuroscience research
on how the brain's 's network microstructure supports its functions (Bullmore & Sporns 2009 Bassett & Sporns 2017). we aim to develop similar framework for the structural analysis large models. This framework seeks to link the internal architecture and interaction patterns of large models with their emergent capabilities, providing deeper insights into the mechanisms that drive their complex behaviors. By adopting structure- based analysis approach, we can move beyond treating models as black boxes and gain a deeper understanding how internal structural dynamics connect to the development of advanced functionalities in large models.
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Our approach and its connection to neuroscience. To address the above limitations, our research bridges the existing gap by analyzing the internal network structures of large models and linking the structure-based properties to the models' emergent capabilities. In the field of neuroscience, extensive research has established that the brain's 's network microstructure is fundamental its functionality, with model organisms such as Drosophila and mice serving as key examples (Mizutani et al.) 2013 Shih et al. 2020 Xiao et al. 2021 Yin al. 2020 Coletta et al. 2020). These studies emphasize the critical role of network structure in supporting and determining functional capabilities, inspiring us to explore the relationship between the structure and the emergent abilities of large artificial models. In the study of human intelligence, the brain is considered a self-organizing complex system (Kelso) 1995; Singer 1986; Pribram 2018 Antonello et al., (022).. The manifestation of intelligence and human behavior from neurons to mind is governed by the generic process of ceff-organization. A key aspect of studying intelligence involves quantitatively analyzing and investigating the complex interactions among numerous neurons (Sporns 2004 Bassett & Sporns 2017). Self-organization
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phenomena, widely observed in natural systems, involve numerous micro-level interactions leading to complex macro-level behaviors, and the emergence can be understood by modeling and analyzing the self-organizing micro-structure (Turing 1990) (see Appendix A.3). Inspired by this, aim to explore whether the emergent behaviors ex exhibited by large models can be understood and connected through structural self-organization. By abstracting large models network representations, can analyze their interaction structures and correlate them with capabilities. Drawing an analogy to structural sell-organization, we propose relevant structure-based metrics that enable us quantify the degree of emergence (this is structure-baut metric that can be viewed as proxy for the emergent abilities of large models) large models.
Our contributions. In summary, the contributions of our work are three-fold: (i) We propose unified network representation for large models, where large models are represented as Neuron Interaction Networks (NINs), enabling the structural analysis of artificial neuron interactions. (ii) We introduce nturon-based multifractal analysis (NeuroMFA), a framework that quantifies the regularity and heterogeneity of NINs by statistically analyzing the fractal properties of neuron interactions. This framework measures the neuron interaction dynamics at the micro-level and is the first flexible framework that seamlessly integrates techniques for seff-organization, fractal, and structural analysis to estimate the degree of emergence of large models. (iii) We propose a novel metric based on MeuroMFA to quantify the degree of structural emergence in large models by analyzing the interaction dynamics of neurons during the training process, opening a new direction for the study of "emergence" in large models.
2 RELATED WORK
The concept of emergent ability has gained prominence in LLM research, initially through the chain-of-thouuhh technique for complex mathematical problem-solving (Wei et al., (0022b). The interpretability and quantification of emergent ability were advanced with the PASSUNTIL evaluation strategy (Fu et al. 2223), utilizing massive sampling for high high-resolution analysis (Hu al. 2023b). Michaud et al. (2023) proposed quantization model to elucidate neural scaling laws, addressing both the power-law decrease in loss with increasing model and dataset sizes, and the development of new abilities as models scale up. However, [Schaeffer et al. (2024)qqestione the notion of emergent abilities in LLMs, arguing that the appearance of these abilities may be more a consequence of the evaluation metrics chosen, rather than fundamental changes in the model's 's behavior. This critique has sparked further inquiry into how evaluation strategies influence our understanding of emergent abilities in large models.
Following this, Chen et al. (0024) proposed a quantitative approach to measuring "emergence" by comparing entropy reductions at semantic and token levels. Gurnee al. (2024) explored the concept of "universal neurons" " in GPT-2 models, discovering that a small subset of neurons exhibits consistent behavior across models trained with different random seeds, suggesting neuron-level mechanism that may drive emergent phenomena. Most recently, [uu et al. (2024) introduced a new perspective, emphasizing pre-training loss critical driver of emergent ability in large models. Their study showed that models with lower pre- tr-iniiiiig losses exhibit emergent abilities, independent of model size or dataset scale.
3 METHODOLOGY
In this section, we introduce the motivation behind our approach and outline its three main components. Briefly, we first represent a large model as a neuron interaction network. Then, we propose the neuron-based multifractal analysis (NeuroMFA). Finally, based on NeuroMFA, we propose new metric as proxy for the emergent abilities large models by quantifying and analyzing the dynamic changes in their structure.
3.1 MOTIVATION: CONNECTING SELF-ORGANIIATION EMERGENCE
Self-organization is foundational concept for understanding emergent phenomena in natural sys-
tems (Goldstein) 1999; Crutchfield 1994). It describes how systems gradually evolve through
local interactions to develop more diverse interaction patterns and achieve overall order, ultimately
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leading emergence (Correia 2006: Ay et al. 2007). This process involves two key aspects: the development of new interaction patterns (reflecting increased heterogeneity in interactions) and the formation of orderly structures (representing enhanced regularity in global structure). Emer- gence, in this context, refers to the novel macro-level properties formed during the process of ceaniiaation-ccaaaatteeettee
that arise through collective interactions and cannot be fully ex- plained by the dynamics of individual components. Inspired by these principles, we extend the concept of self-organization study the degree of emergence large artificial models (please refer to Appendix A.3.3) and design setwork-based metrics to quantify the two key aspects of self-organization: heterogeneity (capturing the diversity of neuron interactions) and regularity (re- flecting the order in global network structure). By introducing neuron-based multifractal analysis (NeuroMFA), we provide a unified framework investigate these dimensions, enabling a quantitative analysis of the relationship between structure and emergent capabilities in LLMs..
3.2 NEURON INTERACTION NETWORK
To illustrate approach, we begin by defining the structure of neural network (NN) within large models. A large model is usually com- posed of multiple NN layers, represented as L1, L2..... La, where k N+ denotes the num- ber of layers. Within each layer Ly. the artificial neurons specified as n/. nz.. ..- m23, where P, N denotes the number of neurons in layer L.j. Neurons between layers are interconnected by weights Was E W, where a and denote the indices of the neurons. For instance, could rep- resent ni and could represent n/+1. Notably, neurons in one layer connect only to neurons in the subsequent layer.
13
0367
025
Figure 3: The transformation from partial neu- ral network (NN) large models into a neuron interaction network (NIN).
The structure of an NN contains key information about its architecture and neuron connections. However, applying network and structure analysis to NNs presents challenge: (i) the presence of negative weights can create inconsistencies in distance-based metrics, complicating the interpretation of interactions; (ii) defining measures of distance and shortest path crucial for studying network
structures, where stronger connections typically imply shorter distances (Newman) 2001). To address these issues, we first transform NNs into the Neuron Interaction Networks (NINs), defined follows.
Definition 3.1 (Neuron Interaction Network (NIN)). NIN is weighted directed graph G (V, E), where:
[1] Node set: V denotes the of neurons (nodes) in the NIN, directly inherited from the NN;
[2] Edge set: E' C V x V represents the directed edges between nodes, where each edge ea, E E' connects nodes a and b. Each edge has non-negative weight wab = |w l", where wab is the original weight in the NN, |wab| reflects the interaction strength between neurons, and p -1 is parameter that inversely maps the original weight the edge distance (Newman 200T Liu al.) 2017).
the distance di E I between two nodes i and j is defined as the shortest path distance, calculated as the minimum sum edge weights over all possible paths between the nodes. A path jii,j) = (e4, U1, ... ... connecting i to j. spanning multiple layers. The formal expression for the distance given by:
[3] Shortest path distance:
..)) consists of a valid sequence of neurons and edges
d:j min P(i.)
Σ
Whr + (P(,g))))
(a.))EP(L))
The term [P(, is defined as the number edges in the path P(i.j) connecting nodes and j. representing the length of the shortest path. This distance metric accumulates the edge weights way along each valid path and incorporates a weighted term based on the path length. The parameters IR and y IR balance the influence of edge weights and path lengths, respectively, allowing for
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comprehensive measure of distance that accounts for both interaction strengths and the complexity of the connectivity paths.
[4] Neighbors (u;): To facilitate the proposed analysis of neuron-based local interaction patterss in the NIN, we define the neighbors of a neuron e; E as the set of neurons whose shortest path distance from Ut is less than or equal to predefined threshold distance diwrehold- Formally, the neighbor set (wz) is defined as:
((ve)) = (uy e V dr S J
(2)
where dıj denotes the shortest path distance between neurons v; and vi. This definition identifies all neurons that are sufficiently close v; in terms of network proximity, enabling to efficiently and systematically explore local node-based interaction dynamics.
In the subsequent NeuroMFA framework, these neighborhood relationships are leveraged to conduct multifractal analysis, revealing multi-scale structures and complex interaction patterns within the network. Since LLMs contain a huge number of neurons, it is inefficient to apply the fractal analysis the whole NIN. To this end, define a Sampled Neural Interaction Network improve the analysis efficiency and the generalization capabilities of our method.
Definition 3.2 (Sampled Neural Interaction Network (SNIN)). An SNIN represents sampled subgraph G' (V', E') of the original NIN. In each layer Li, we sample subset of nodes V/ e L; and add them to SNIN with weights E' {Eable E, a,a,h V').
To enhance the precision and efficiency of the NIN analysis, we adopt strategy where we average the outcomes from the study of 10 randomly selected SNIN sets in NIN. substantiated Ap- pendix I.1 this sampling methodology does not alter the estimation of the fractal dimensions. For clarity and simplicity, we consistently use the notation = (V, E') and refer to the NIN instead of SNIN in subsequent sections.
3.3 NEURON-BE NEURON-BASED MULTIFRACTAL ANALYSIS
To perform multifractal analysis of LLMs, we extend the box- cov-covening and box- tox-growing methods (Song et al. 2005 Evertsz & Mandelbrot 1992 Salat et al. 2017; Xiao al. (02I)) to capture the us consider a neuron v; as box with an initial radius of zero at layer I, indexed by i. We then increase the radius of the box and record the mass distribution (r) E N (i.c., the number of nodes covered by the box), as shown in Fig. (a). That is to say, (r) denotes the number of neighbors of neuron UI within radius
local neuron- neuron-based fractal properties NINs. Specifically,
For a specific neuron o at layer I and radius r, (r) calculated as Nt,(r) Ee [ddy where 1 (d.,5r) Sr) is an indicator function that takes the value when the distance diş between neighbor neurons 0/ and Vy less than or equal box radius until all the neighbors of the neuron are covered. Based on this approach, we capture the log-log relationship between the box radius r and the mass distribution (r,,(r) in the box as shown in Fig. (b), indicating the fractal properties of the neuron interactions. The log-log relationship between (r) and through the whole NIN presented in Appendix Table 9 leads to the following observation.
and 0 otherwise. The process continues increase the
Observation 1 (Fractal Properties). In NIN, the relation between the mass distribution N()) and box radius can be expressed as:
N r
where E IR denotes the fractal dimension characterizing the fractal properties of the interactions among neuron and its neighbors.
To capture and distinguish the neuron-based multifractality of the NIN, we develop the Neuron-based Multifractal Analysis (NeuroMFA) framework. This method integrates the fractal features of different neurons in the NIN. The distortion factor q E R is introduced to differentiate the nuances of fractal structural features. For a given neuron v; at layer / in the neural network G = (V, E), the mass where Nr,(r) is the number
Nr)(r) probability measure (r) at radius is defined as (T) Tus
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Figure 4: (a) Examples of weighted box growing method, illustrating two neurons randomly selected from the NIN of pythia IB model the 143,000t epoch. The radius of each box, is determined by the distance to its neighboring neurons, with N representing the number of neurons contained within the box. (b) The power-law relationship between N and r.
of neurons within the radius from neuron U; and Ti,; is the total number of neighbors of neuron vs- To characterize the fractal properties at different scales, the partition function defined as follows:
Partition function. The partition function (r,(r) for the neuron-based multifractal analysis is defined as the sum of the q-th power of the probability measures across all neurons in the network, given by:
Zg(r) Σ Σ PL,(r) 1=1 NE
(4)
where g is the distortion factor designed to distinguish different fractal features. We provide the algorithm to obtain the partition function in Appendix Algorithm 2 By capturing the power-law relationship between the partition function (r) and the scale J. we have the following observation:
Lemma 3.1 (Multifractal Scale Invariance). Let μ be a mass probability measure defined on the network. For any positive number and scale e, if there exists constant C(q) such that
Σlu(B.(O) (()))
(5)
then the network said possess multifractal scale invariance (Falconer, 2014). Here, B,(e) represents the box of scale and T(g) is the mass exponent.
Observation 2 (Multifractal Properties). A log-log relationship exists between (r)r) and the obser- vational scale r/d, validated in Appendix Table
g
(9) 7(g)
(r(r)
(6)
where d is the maximum box radius and r(g) is the mass exponent.
T(4) is estimated as the slope obtained from the linear regression of log (rk) against log (7)):
TT
log Zy(rk) log Zy(r)) (1og (7k)
log (7)
[())
log (5) log
Here, T& represents the distance from neuron its k-th nearest neighbor within the threshold distance d, with & ranging from to m, where m is the total number of neighbors within the box, and rm < is the maximum box radius. The terms log (r) and log ("x) denote the mean values of log (rx) and log (*), respectively. The mass exponent 7(q) quantifies the scaling behavior of the partition function Zr(r) across different observational scales thereby characterizing the multifractal nature of the neural interactions within the network. The algorithm for computing is provided in Appendix Algorithm

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Multifractal spectrum. The Legendre transform enables the computation of the the Lipschitz- exponent a(q) and multifractal spectrum f(a) characterizing the multifractal structural features of the network:
Lipschitz-Hölder Hölder
a(q) dr(g)
(8)
dq
f(a) (a 7 (4))
(9)
The detailed algorithm to calculate the multifractal spectrum is provided in Appendix Algorithm 4
3.4 DEGREE OF EMERGENCE
While the Lipschitz-Hölder exponent a provides information about the nature of regularity/order and singularity, the multifractal spectrum f(xx) offers insight into the frequency (commonly or rarely occurring pattern) and thus can help us quantify the complex and heterogeneous nature of weighted network topology. Here extract measures from the multifractal spectrum to quantify the irregularity and heterogeneity of network structures, and use these to construct our measure for the degree of emergence.
Irregularity metric ogt The exponent ay is defined as the Lipschitz- Hölder exponent value a at which the multifractal spectrum ((x) attains its maximum, i.e.,
0g arg max f(a).
(10)
The exponent ag represents the most prevalent degree of singularity within the network, and higher value og indicates a greater level irregularity and complexity the network's 's structure. Therefore, we employ og as metric for assessing the NIN's regularity. A more detailed explanation is provided in Appendix A.2.I
Heterogeneity metric w: The width w of the multifractal spectrum is defined the difference between the maximum and minimum values of the Lipschitz-Hölder exponent a, represented by
O'mxx
O'min-
(11)
This width measures the heterogeneity of the network, with larger value indicating broader range of fractal structures within the network. Further details are provided in Appendix [A2.22
Proposed metric. To evaluate the degree of emergence of large models based on structural dynamics of NIN, we estimate two key metrics at a given epoch I: the order/regularity on (l) and heterogeneity w().. These metrics are derived from the NIN's evolving network representation, which provides insights into the model's 's internal structure as it progresses through training.
Definition 3.3 (Degree of Emergence). The degree emergence can be calculated through the following formula:
mo(t) log (O)(O) (0) avu(t)
E
(12)
This formula integrates the changes in heterogeneity and regularity over time, providing comprehen-
sive measure of the system's 's structural evolution. Specifically, w[0] signifies the relative change in
=*t)
heterogeneity, where values greater than 1 indicate increase in the diversity of interaction patterns
within the network. Similarly, log co(0) quantifies the shift in regularity, where values greater than
represent the process of the network structure becoming more organized. Together, the product of these terms offers normalized view of the structural shifts in the network. illustrating the degree of self-organization. In this context, self-organization refers to the spontaneous development of new interaction patterns and the increasing regularity within the system, which ultimately leads to the development of skills in large models. complete explanation is provided in Appendix A.2
4 EXPERIMENTS
In this section, we describe the experiments conducted to analyze the NINs to observe the structural dynamics process during training and evaluate our proposed metrics. Our experiments conducted
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on Pythia from 14M to 2.8B (Biderman al. 2023) models (except for specific requirements like different architectures). Pythia models provide checkpoints during training phases and different model scales, which is helpful for us reveal the rules behind these scales of training steps and model size. A detailed description of all the benchmarks used for comparison is provided in Appendix B.6 In the following experiments, we sample 10 networks with 64 nodes per layer and average the results to obtain precise, model-independent calculations. We set parameter À to I and to 5 in Eq. T to achieve an appropriate distance between neurons. The variance of the calculations shown in Appendix D.1 demonstrates the reliability of our sampling method.
4.1 SELF-ORGANIZATION PROCESS OF THE NIN IN LLMS
We analyze the dynamic LLM networks during their training process. The self-organization process, characterized by the emergence of new patterns and increasingly organized structures, be quantitatively assessed using the w and og metrics from Section 3.4 For each model with different sizes, NeuroMFA is applied to networks at various epochs. The curves showing the changes in the values of and w with epochs are provided in Appendix Fig. [18 The spectra, which vary with the epochs, exhibit distinct behaviors that reveal certain trends, as discussed below.
Increasing heterogeneity. The spectrum width wo measures the degree heterogeneity of the fractal interaction patterns within LLMss. Figure 3 shows that w tends to broaden with more training epochs. This widening, evident from the initial spectrum to later stages, sig- nals increasing heterogeneity in the net- work's ' structure. This phenomenon is largely due to the emergence of varied fractal patterns, which are intricate structures formed through neuronal interactions during training. These patterns contribute to the spectrum width's 's ex-
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Decreasing irregularity. The leftward shift of the multifractal spectrum indicates self- organization. This shift suggests increase in the network's regularity, as marked by a decrease in the a parameter (metric of irreg- ularity). However, Fig. S (a) shows for the smaller model (e.g.. 14M) an increasing trend of og. As training progresses through epochs, spectrum tends to shift rightward slightly instead of leftward. This indicates that, in the 14M model, the network's 's regularity does not increase. Similarly. the 31M model displays subtle leftward shift at the 5,000th epoch, but this trend does not continue significantly thereafter.
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Figure 5: Multifractal spectra of Neuronal Interac- tion Networks (NINs) for models with different sizes (14M-2.8B) throughout the training process. The gradient of spectral lines, shifting from blue red, represents the multifractal spectra of the same model at different epochs, with blue indicating early-stage training and red signifying later stages.
In contrast, for larger models (exceeding 100M parameters). consistent leftward shift of the network spectra is observed from the Oth to the 35,000th epoch. This shift suggests an increase in regularity and corresponding enhancement in self-organization training progresses. After reaching the 3,5000th
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Figure 6: Comparison of the proposed emergence metric E' (blue line) with two established metrics
(red line). The strong correlation between our proposed metric and existing metrics demonstrates the validity of our approach in quantifying emergent properties.
epoch, this leftward shift the spectrum stabilizes across these larger models. This stabilization implies that further increases in training epochs do not significantly affect the network's degree of regularity. This could indicate that the model training has reached a certain threshold, beyond which additional orfa-organzzttion is observed. Notably, since the degree of emergence is fundamentally tied to self-organization phenomena, we did not compute this metric for the 14M and 31M models in the following experiments.
4.2 QUANTITATIVE ANALYSIS OF EMERGENCE
In this section, we assess the degree of emergence for LLMs during the training process. Figure shows the emergence analysis across models of varying scales (160M, 1.4B, 2.8B) and their perfor- mance for different benchmarks. We observe a positive correlation, i.e., higher levels of emergence are associated with enhanced expected model performance, evidenced by increases in testing accuracy; of note, we acknowledge that this is restricted to these datasets and a more comprehensive evaluation should be done across more datasets (see more results across different metries in Appendix D.2).
Data from Lambada and PIQA a tests reveal a significant increase in model accuracy during the ini- tial training phase, up to approxi- mately the 15,000th epoch. This trend was paralleled by swift increase in E. From around the 15,000th to the 40,000th epochs, we note gradual enhancement in model performance, aligning with the trajectory of the de- gree of emergence E. Post this phase, models tend to stabi- lize, exhibiting fluctuating accu- racies. These fluctuations mirror the emergence levels, further val- idating our metric.
b
Degree of Emergence
Degree of Emergence
335
235
143%
Figure 7: Radar charts showing the performance of (a) IB and (b) 1.4B models across cognitive and reasoning benchmarks over different training epochs. A: LAMBADA dataset challenges, B: science exam questions, C: Physical Interaction QA, D: A12 Reasoning Challenge Easy.
Moreover, we observe significant
performance oscillations in two other datasets (WSC and LogiQA, see Appendix D.2) suggesting their limited utility in assessing model capabilities. This underscores the limitations of relying solely on performance metrics for evaluating the model's emergent abilities. Therefore, our approach to studying emergence focuses on analyzing the model itself, rather than merely its output or performance. This methodology provides a more comprehensive view of the emergence phenomenon in LLMs, encapsulating both the intricacies of the training process and the nuanced evolution of the model. Information about the benchmarks can be found in Appendix B.6
Degree of emergence and model capabilities. Fig. compares degree of emergence metric with established LLM benchmarks: Lambada, SciQ. PIQA, and ARC-easy. These benchmarks assess different aspects of LLM emergent abilities: Lambada evaluates contextual comprehension, testing

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the model's ability to understand long-range dependencies in text. SciQ measures scientific reasoning, challenging models to apply scientific concepts beyond mere fact recall. PIQA assesses physical commonsense, requiring models to reason about everyday physical interactions. ARC-easy tests basic science knowledge and fundamental reasoning across various disciplines. Together, these metrics provide comprehensive assessment of LLM capabilities, covering language understanding, scientific reasoning, commonsense knowledge, and basic problem-solving skills. We analyze Pythia IB and Pythia 1.4B models across different training epochs. The radar charts demonstrate that our proposed degree of emergence aligns closely with these established metrics as training progresses.
Our metric reflects the gradual enhancement of LLM's overall capabilities as the training epoch increases, mirroring the trends observed in other L.L benchmarks. This correlation sug- gests that our degree of emergence, derived from the NeuroMFA method, effectively captures the comprehensive development LLMs over time. The consistent behavior between structure- based metric and the diverse set established performance benchmarks throughout the train- ing process validates the NeuroMFA approach. also demonstrates the feasibility of assessing LLM capabilities from structural perspective, offering new pathway to evaluate the compre- hensive abilities of language models based on their internal organization rather than just output performance.
0.8-
Normlizze Score
0.6
Tox (bwch 823)
"ou
AMME 515
0.4
Chilkeg
Chillgg Phecc (I0I))
0.0-
18.5
18.0
21.
In(Model Size)
Figure 8: Degree of Emergence Model Size. Avg. Performance means the average score of all the metries. All metric scores have been normal- ized between 0 and I for comparison.
Degree of emergence vs. model size. We eval- uate the degree of emergence across models of varying sizes by comparing their performance on different benchmarks, specifically PIQA, LAMBADA, and ARC Challenge, along with the av- erage score across all evaluation metrics. As illustrated in Fig. g at an early stage of training (512 epochs), all models, regardless of size, exhibit low degree of emergence, reflecting limited emergent abilities across the benchmarks. However, at a later stage of training (53,000 epochs), the degree of emergence increases logarithmically with model size, suggesting that larger models demonstrate more pronounced emergent abilities across a range of tasks.
4.3 IMPACT EVALUATIONS
We applied NeuroMFA to analyze varying architectures (e.g., diffusion models, CNNs) and training data (e.g., contaminated vs. clean datasets, Pythia-standard vs. Pythia-deduped). The results, detailed in Appendix C highlight differences in emergent abilitics across architectures and the impact of training data on self-organization and emergence.
5 DISCUSSION AND CONCLUSION
Our Neuron-basee Multifractal Analysis (NeuroMFA) framework addresses critical gap in quantify- ing the internal dynamics of large language models (LLMs). By analyzing the multifractal properties of neuron interactions, NeuroMFA offers unique insights into emergent abilities during training. Metrics such as the regularity metric and heterogeneity metric track the evolving complexity and organization within neural networks, providing cohesive explanation for learning evolution in LLMs. The experimental results demonstrate that emergence in LLMs can be explained by modeling and analyzing the structural and self-organizatton properties among neurons, aligning with theories of brain function. While accessing human brain neuronal parameters remains challenging. LLMs provide a transparent and accessible platform for studying self-organization phenomena, making this research particularly promising. In future applications, NeuroMFA holds potential not only for enhancing interpretability and assessment but also for forecasting LLMs' behavior and guiding their development. Moreover, by linking structure to the capability of large artificial models, NeuroMFA enables a deeper exploration of potential connections between artificial and biological intelligence.
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ACKNOWLEDGEMENTS
The authors want to thank Ziming Liu for meaningful discussions. The authors acknowledge the support by the U.S. Army Research Office (ARO) under Grant No. W911NF-23-1-0111, the National Science Foundation (NSF) under the Career Award CPS-1453860, CCF-1837131, MCB-1936775, NNS-1192620 and the NSF award No. 2243 104 under the Center for Complex Particle Systems (COMPASS), the Defense Advanced Research Projects Agency (DARPA) Young Faculty Award and DARPA Director Fellowship Award under Grant Number N66001-17-1-4044 an Intel faculty award and a Northrop Grumman grant. P.B. is also grateful to National Institute of Health (NIH) for the grants R01 AG 079957 "Interpretable machine learning to synergize brain age estimation and neuroimaging genetics" and RF1 AG 082201 "Neurovascular calcification and ADRD in two nonindustrial Native American populations", It was a wonderful experience designing and writing the grant application entitled "Neurovascular calcification and ADRD in two nonindustrial Native American populations" and awarded under RF1 AG 082201. The views, opinions, and/or findings in this article are those of the authors and should not be interpreted as official views policies of the Department of Defense, the National Institute of Health or the National Science Foundation.
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Appendix
Theories and Methodology Details
17
A.1
Multifractal analysis
17
A.2
Elaboration for NeuroMFA
19
A.3
Discussion on self-organization
23
A.4
Comparison with traditional metrics
26
implementation Details
29
B.
Code availability
29
Evaluation LLMS
29
B.3
Algorithms for NeuroMFA
30
4
Time complexity of NeuroMFA
31
B 5
Network construction
31
B.6
Benchmarks description
33
Impact Study
35
Varying architecture
35
Varying training data
35
Varying training directions
36
D Additional Results
37
D.I NeuroMFA spectra with variance
37
D.2 Assessment of the degree of emergence across various datasets
40
D.3 Log-log relationship evaluation
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D.4 Average weighted degree distribution
41
Evaluation Metrics
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Kurtosis (32)
47
Coefficient of determination (R²2
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A THEORIES AND METHODOLOGY DETAILS
A.1 MULTIFRACTAL ANALYSIS
A.1.1 HAUSDORFF MEASURE
The Hausdorff measure is key concept in fractal geometry and measure theory, extending the notion of Lebesgue measure imegular sets, and is particularly useful for characterizing the size of fractals. It's named after Felix Hausdorff, a pioneer in set theory and topology.
Formally, consider a A within R" For a metric space (X, p), the diameter of U is defined as:
diamU == sup(p(z, v) x. E U}, diames := O,
(13)
where U is any subset U X.For 8 >> define the ô-approximate d-dimensional Hausdorff measure as:
U U, 2 A, diam( < 8 ((U)) is a 5-cover of mn) (14)
H((A) inf Σ (diam
where diam(U;) the diameter of the set U₁ and a 5-cover countable collection of sets {U+ covering A with dU; < 8 for all The d-dimensional Hausdorff measure of A is then defined as the limit of (A) as approaches zero:
HA = sup Hg (A)
lim (A) 8-0
(15)
6>0
The Hausdorff dimension dim (A) of set A is defined as the infimum over all d such that the d-dimensional Hausdorff measure of A is zero:
dim (A) = inf(d2 (A) 0}.
(16)
Equivalently, it can be described as the supremum over all such that the d-dimensional Hausdorff measure of A is infinite:
dim sup 0: (A) = oo}.
(17)
This dimension is critical value that separates the scales at which the set A appears "large" from those at which it appears "small" in terms of its dddimenniioal Hausdorff measure. It provides a precise mathematical way quantify the notion of dimension for irregular sets, like multifractals, which do not fit neatly into the traditional framework of integer dimensions.
A.1.2 MULTIFRACTAL ANALYSIS BY BOX-COUNTING METHOO
In the domain of network analysis, multifractal analysis (Sog et al. 2011) employs the box-counting method to elucidate the intricate, multifractal structures inherent in complex networks. This analytical approach aligns with the renormalization procedure, essential for understanding multifractality within these systems. The process commences with covering the network using a series of boxes of variable sizes, denoted as The count of boxes, (()). required to encompass the network at each scale is meticulously recorded. This methodology resonates with the principles of renormalization. highlighting the dynamic interplay between different scales within the network. This approach is akin to assessing the Hausdorff measure a foundational concept in fractal geometry that quantifies the size of fractal object. Here provide the process of the calculation of the generalized fractal dimension and the multifractal spectrum.
2005 2007 Furuya & Yakubo
The fractal structure is divided into boxes (or elements) of size Each box covers a part of the fractal, with a total (e)) boxes. For each box, we calculate the proportion of the mass probability measure within that box, denoted as Pt, where represents the i-th box. By applying q-th power weighting
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to the probability p, in each box, obtain the partition function Σ N(t) Li=1 p. The distortion factor
can adjust the relative importance of different box probabilities. Considering the scale factor as approaches capture the fractal, the generalized fractal dimension is calculated
log Σ -i=1 P
Dg
lim c00
(18)
log E
Subsequently, through Legendre transform, the Lipschitz- Hölder exponent a(g) is determined through the differential equation:
di a()) dg (11 -1-- (1g-1)D,))
(19)
This exponent a(g) provides insights into the local regularity of the network structure various scales. Advancing this analytical framework, the multifractal spectrum f(a) is derived from the relationship:
qua
(20)
The multifractal spectrum f(a), in conjunction with a(g). offers a comprehensive understanding of network's ' scaling behaviors, revealing the multifractal and heterogeneous nature of structure. This spectrum delineates the distribution of singularities across the network, thus encapsulating the essence of multifractality. The utilization of these calculations computational; they provide profound insights into the network's 's heterogeneity, unraveling the multifractal core of its structure and the dynamics that shape its evolution.
network analysis is not merely
A.1.3 POWER LAW IN FRACTAL DIMENSION
The definition of fractal dimension is based on unconventional views of scaling and dimension. In conventional geometry, dimension can be defined by the intuitive space scaling law. For example, a line with finite length can contain 3 lines with 1/3 size. But for square, with a box of side length 1/3 the size of square, one will find 9 times many squares as with the original. In general, mathematical definition of such a scaling law can be given by
-
(21)
where N stands for the number of measurement units, is the scaling factor. Thus, this gives the definition of dimension D:
D = -= N N
(22)
Ing
For lines, we can see N = 3 when = 3- Then we get D = 1. In the case of a square D = 2
because N = 9 when 3'
Then the definition can be generalized into fractal geometry. For example, the Koch snowflake shown in Fig 9 N = 4 when
3. Thus the fractal dimension D = log, 4 &s 1.261 15251859,, which is not integer. Even though it is not intuitive any more, the scaling law N E =D still exists. Then for fractal objects embedded in the Euclidean space we can use box-covering method to define fractal dimensions, which is useful for fractal networks:
ND(rR) ~
(23)
where (B(rg) is the minimum number of boxes needed to tile given fractal network with the box radius TB. Then we can define the fractal dimension in another way. The mass distribution
defined by the average number of vertices (N(rc) within box with radius rc- The mass-radius
relation can be given when the relation M ~ (N((B)) (NV holds for ra = rc = r. we can then characterize the power law between the mass distribution N (r) and the box radius
N()~r² ~
(24)
where D is the fractal dimension.
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Order
Ontar
Odd
Figure 9: Fractal pattern.
Lemma A.1 (Scale Invariance). Let μ a mass measure defined on the network. For any radius around specific node, if there exists constant C and real number D) such that the following relationship holds:
(B((c) = CC.
(25)
where B(c) is the set of nodes within radius of the specified node, then the network exhibits scale invariance at the local scale around the node. D represents the local fractal dimension.
Definition A.1 (Neuron-based Fractal Dimension). Consider neuron v; layer of the NIN. The neuron-based fractal dimension Di,; is defined to capture the fractal characteristic based on each neuron. Given the set of distances (ri, r2. I'm from ny, to its neighbors in the next layer I + the fractal dimension Dr, is calculated as follows:
(log(rk) log(r)) log (r))))
D1.5
(26)
Σ (log(rk) log(r))2
where 7x represents each distances in the set, (rs. Tk. and log(r), log( (r)))) are the mean values of log(rk) and (rk)), respectively.
(rx) is the number of neurons within the distance
In practice, calculate the fractal dimension of large network we do need count all nodes. practical method is to randomly sample the nodes in the box. Note that, when we plot a In In graph, the slope is the fractal dimension D:
In = D rl.:
(27)
When randomly sample the nodes with constant ratio kt it only generates constant intercept compared to the original:
In N' = In KN = In N + In Dlnr + In k.
(28)
where In k is constant. Then we can still calculate the fractal dimension from its slope.
A.2 ELABORATION FOR NEUROMFA
A.2.1
HIPSCHITZ-HOLDER EXPONENT
In the context of multifractal analysis, the Lipschitz-Hööldr exponent, denoted a, is erucial for characterizing the local scaling properties dataset. Mathematically, a is defined as the derivative
dr(g) of the mass exponent 7 r(y) with respect to g, expressed as a(g) -dy Here, T(y) represents the mass exponent that characterizes the scaling behavior of the data at different moments (g).
The value a provides insights into the degree of irregularity at various scales within the dataset. Lower values of a suggest higher level of regularity, where the data exhibits smoother transitions and less variability in its local structure. In contrast, higher values of a are representative of regions with more irregular and disordered scaling behavior. This distinction is fundamental to multifractal analysis, offering a detailed perspective on the heterogeneous nature of the data and its scaling properties across various scales.
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Therefore, the distribution and range of a values within the multifractal spectrum offer valuable insights into the underlying scaling dynamics of the dataset, revealing the intricate interplay between uniformity and complexity that defines its multifractal nature.
A.2.2 SPECTRUM WIDTH
The spectrum width, denoted as w, in multifractal analysis plays pivotal role in quantifying the heterogeneity of a network's structure. It is defined as the difference between the maximum and minimum values of the Lipschitz-Hölder exponent a, mathematically expressed
Omax
Qmin-
This width w captures the range of scaling behaviors exhibited by the network across different regions. larger indicates broader spectrum of a values, signifying a network with wide variety of scaling behaviors. Such diversity in scaling properties points to a network with a rich mixture of regular and irregular structures, highlighting the complexity and variability in its composition.
In essence, the spectrum width w serves as crucial metric for assessing the structural heterogeneity of network. It reflects the degree which the network deviates from uniform scaling behavior, offering insights into the multifaceted and complex nature of the network's 's fractal characteristics. The broader the spectrum, the more pronounced the heterogeneity, revealing a network structure that is rich in its diversity.
A.2.3 DEGREE OF EMERGENCE
In the study of self- self-organize offa-irgai networks, the phenomenon of emergence is a key concept, representing the evolution from simpler initial states to more complex and ordered structural patterns. The measure of emergence, denoted as E, aims to quantify this transition from disorder to order, capturing the key characteristics of the network's self-organization process.
The measure of emergence E considers two principal aspects: the increase in network regularity and the growth heterogeneity. Regularity, indicated by the parameter dgo, reflects the uniformity and predictability of the network's s structure. As the process of self-organization progresses, changes in ag suggest transition the network structure towards greater regularity and consistency. This transition is typically marked by increase in uniform patterns, indicating move from more irregular to a more regular state.
Heterogeneity, represented by the parameter w, illustrates the diversity of the network's fractal structure. A larger w value points to a network exhibiting a variety of structural characteristics across different regions. The emergence of new patterns and structures during eeff-orgaizzatinn leads increase in w, reflecting a rise in the diversity and heterogeneity of the network's structure.
Thus, the emergence measure E' is defined as:
wc()) log
axp( (t)
(29)
This metric accounts for the relative changes in regularity and heterogeneity over time, offering prehensive perspective to quantify the phenomenon emergence in the network's self-organization. In this formula, wn(t)
(oo(0)
captures the relative change in regularity. The utilization of the logarithm function is crucial, as it allows the emergence measure to have positive or negative values, reflecting whether the multifractal spectrum shifts left or right, critical indicator of emergence in the model. By integrating these dimensions, E effectively describes the network's evolution from its initial state to a more advanced level of organization, revealing the complex dynamics and characteristics of emergence in the self-organizing process.
w(() represents the relative change in heterogeneity, while log ao(t))
A.2.4 ILLUSTRATION EXAMPLES
We provide concrete examples to illustrate the multifractality and monofractality in Fig. 10 Cantor sets classical fractal structures created by iteratively removing the middle part of line segment. Fig. 10 (a) shows the patterns with self-similar statistical properties characterized by a single fractal dimension across all scales while Fig. (b) shows the patterns characterized by different fractal
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dimensions across multiple scales. Here, the mass probability p; is defined the ratio of the number of segments in each subinterval to the total number of segments. By applying the multifractal analysis, as shown in Fig. TO (c), the spectrum of monofractal Cantor set results in single point, indicating uniform scale characteristics. In contrast, the multifractal Cantor set's spectrum forms a bell-shaped curve, reflecting the diversity of scale characteristics.
In our NeuroMFA, the mass probability is represented as the ratio of nodes in a box to the total number of nodes, with the observation scale being the radius of the box (see Figs. I0 (f-g)). By measuring and distorting the mass distribution in different boxes, we can capture the node-based multifractal characteristics in the network. In Figs. (d-e). we provide an example of Watts- Stroggtt (WS) network model transitioning from regular network to a completely random network. A completely regular network exhibits monofractal properties, with the multifractal spectrum being single point. As edges are randomly rewired, the network structure's ' diversity increases, leading to an increased spectrum width. Simultaneously, the network structure becomes more irregular, indicated by right-shifted spectrum.
n)
Nonotactel-(1..0
bj
Nuttfaas -1.0 [1.0.2)
0.2
0.6G
c)
Horofractel
Mondtwrri
vit
Tio in ex00
Mustiructal
DU- 11
(G)
Upechib-Haadee exponent,
Raguirr
Smalnworts
Random
@)
Nonotiactal
-0.35
-0.39
-0.20
-025
2-0.02
Rewiing probability
20
23
Unwetightn MnweiggoodMMwwr
8)
Divocted
wrigh
Figure 10: Concrete examples of monofractal and multifractal. Fig. (a) illustrates self-similar patterns with single fractal dimension across scales, while Fig- (b) depicts patterns with varying fractal dimensions across multiple scales. Fig. (c) shows the monofractal Cantor set yielding single-point spectrum, contrasted by the bell-shaped multifractal spectrum for multifractal Cantor set. In Figs. (d-e), a Watts-Strogatz (WS) network model transitions from regular random network, where rewiring edges increases spectrum width and irregularity, shifting the multifractal spectrum. Figs. (f-g) demonstrate the node-based multifractal analysis in network, using box-counting techniques.
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b)
9)
0.0
Nutt-dracial spectrum, (la)
www.--rrcciil--o flet
-0.01
-6.1-
-0.02
-0.04
-03
-0.66
Reg jarty
Hetengenelly
-0.
-0.07
naltc
-0.5
0.500
0.575
0.009
9.255
Hopochiz-Hotlrr exponent,
Upschiz-Hotder exponent,
Figure 11: Illustration of the proposed metrics. (a-c) Experiment illustrates the transformation from homogeneous to more heterogeneous network, as shown by (g) broadened spectral width. (d-f) Experiment 2 demonstrates increasing regularity, starting from random configurations and progressing through multiple Weighted Preferential Attachment Mechanism (wPAM) iterations, reflected in (h) the shift and decrease in Org-
To intuitively illustrate the metrics, we apply NeuroMFA toy examples. These metrics provide quantitative insights into the heterogeneity and regularity of neuronal networks. Using simplified models consisting of three layers with 32 nodes each, we demonstrate how network transformations influence these metrics through two experiments:
Experiment Focused on the effects increasing network heterogeneity. Starting with homogeneous network, systematically varied node characteristies simulate increased complexity.
Experiment 2: Explored the introduction of regularity into network using Algorithm I We began with a randomly structured network and used the wPAM to gradually increase regularity.
The results are visualized Fig. II showing distinct phases the experiments and how NeuroMFA metrics quantify changes in network heterogeneity and regularity.
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Algorithm 1 Multiple Rounds of Weight Adjustment
Require: N (number of iterations)
1: for n - 1 to N do
2: for i = 1 to 128 do
3:
for j = 1 to 32 do
4:
if W median (W) then
5:
Wij t Way+
>. Decrease weight
6:
else
7:
Wi t Wij- -
Increase weight
8:
end if
9: end for
end for
11: end for
A.3 DISCUSSION ON SELF-ORGANIZATION
A.3.1 EMERGENCE AND SELF-ORGANNZATION
Emergence and self- orgaaniaaazznnnn are fundamental concepts that span across multiple disciplines, including physics, biology, computer science, and sociology, among others. They describe the way complex systems and patterns arise out of multiplicity of relatively simple interactions.
Emergence. Emergence refers the phenomenon where larger entities, patterns, or regularities arise through interactions among smaller simpler entities that themselves do not exhibit such properties. The essential point about emergent properties that they are not properties any component the system but of the system whole.
Self-organization. Self-organization closely related to emergence and is often seen as a process leading emergent properties. It is the process by which system, without external guidance, spontaneously forms a coherent and ordered structure.
Emergence and seff-organization underpin complex systems across physics, biology, computer science, and sociology. Emergence captures how complex systems and patterns spring from simple interactions. Self-organization, integral to emergence, deseribes systems spontaneously forming ordered structures without external direction. In Physics, emergence is seen in phenomena like thermodynamies, where macroscopic properties such temperature and pressure emerge from the collective behavior of particles(Epster) et al.) (2006). oeff-orgaanzzation is observed in ee-euiiiiiiium processes such as the formation of crystal structures the spontaneous formation of cyclones in the atmosphere. In Biology, emergence explains how complex biological phenomena, such as life, arise from the interactions of non-living molecules the right Anaditions(Kaufmann 1993). The behavior of ant colonies, where complex organization and problem-solvee abilities emerge from simple rules followed by individual ants, another example. The development of an organism from fertilized egg, where highly ordered structures and functions emerge, is a classic example of orgaaiiaaazz.oon.. These concepts highlight the transition from simple interactions to complex behaviors, also exemplified by Turing pattern.. Turing patterns serve model, illustrating how basic reaction-diffusion processes can create intricate patterns, mirroring the self-organization leading emergence.
A.3.2 TURING PATTERN
Turing pattern is well-known example of how self-organization occurs in various systems. It was first proposed by Alan Turing when he tried to explain the patterns in morphogenesis. His daccion-ddifussie theory has provided insights for understanding pattern formation of various complex objects ranging from chemical molecules sandpiles.
In general, we examine the reaction ceeetion-diffusion equations
ac:
Øt F.((C))) D +
(30)
Here, D, is the diffusion coefficient for object species (which could be totally different things in different systems, for example, molecular in chemical process) with concentration Cs, and reactions
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described by local non-linear terms included in {C.). A stochastic term n7( t) is also included to describe the general case with stochastic noise: In the microscopic system it could result from thermal fluctuations, whereas for macroscopic ones it could stand for variations in the environment.
Now we demonstrate the simplest example. The key idea is that simple chemical processes, when subject to diffusion and reaction, can give rise to complex and diverse patterns. The necessary conditions for the formation of Turing patterns in the simplest case are:
Morphogens: There must be least two components Activator and Inhibitor. For example, in biological systems, they can be signaling molecules.
For the activator it must be autocatalytic (positive feedback), while the inhibitor could suppress the activator. In short, the reactions (auto-catalysis and cross-catalysis) must occur between them.
Reaction:
Diffussin:
Both activator and inhibitor diffuse spatially. And the inhibitor diffuse much faster than the activator.
The simplest form is like
Za Da v a + fa(a, 0r D, 2 fo(a,b),
(31)
where C1 a is the concentration of activator, C2 b is the concentration of inhibitor, Da,b are diffusion coefficients, fa,b(a, b) are the reaction functions.
In Turing's 's mathematical analysis, such reaction-diffusion system could yield six potential steady states, depending on the dynamies of reaction term and wavelength of the pattern(Kondo & Miura 2010). When the diffusion of the inhibitor is much faster than that of the activator, that is, D. < Dr- the steady state becomes a Turing pattern, kind nonlinear wave maintained by the dynamic equilibrium of the system.
Figure 12: Turing pattern Turing 2022
The process of pattern formation is symmetry-breakinn one: at the beginning all spices diffuse in an isotropic way but finally the patterns become like the one shown Fig In principle, Turing patterns possess an intrinsic wavelength, which is determined by interactions between molecules and their rates of diffusion. More specifically, it depends only on the ratio of the parameters in the equation, which means it is scale-independent property.
To demonstrate this mathematically, we try to examine the Eq 30
We need to find a stable fixed point {CF} as solution. First, we apply the linear expansion on the cnaction-diffsiinn equations around the fixed point
Ci(FF = C7 + cF(F.))
dc:
Σ Miy + D.V&& + Mj =
0F; 0C; C+
(32)
Øt
The stability of the solution requires all eigenvalues of the matrix Mij have negative real parts. We introduce Fourier transforms
(FFFt) / dketkFFaal t).
(33)
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then Eq becomes
do
Σ (My - 6eyDih tk, ((( t(k,t)
(34)
dl
To get the Turing instability, the eigenvalues question after the transformation becomes whether the matrix Miy(k) Mij- dig D,h² can have positive eigenvalue finite wave-vector k. Then come back to the simplest case. Let us examine the 2 x linear-stability matrix
(k) (M11 M21
D2
M12
(35)
M22 - D2k²
Here we denote two eigenvalues of the matrix by E (k), and we can get their sum
E+ (k) + (k) Tr) (M11 + M22) (Di + D2)1²
(36)
and the product of them E+ (k) (k) det M(k)
(37)
(M1 M22 M22 (M1.22 M22nD22+2222M1 (o..Dii>++DDDapp D2
(38)
When & = 0 it is obviously stable state (uniform state), so we have E+ (0) < 0-)-(0) < 0. Thus the first term det (0) (0)=-(0) is positive
0. And the last term D22 is always
positive. Within the unstable state (e+ (0) > 0, (0) < 0) we must have e (k)s (k) < 0, thus the second term is the only route to instability, that is
(M1 + M1) O, while (M11 + M22)
(39)
where (Mn M22) 0 because of the stability at k 0. The requires e + +0( Tr)(()) = (M1 + M22) < O. Then we can give curve to describe Eq 38
E (K)= (k)
A:
1:2
km
It crosses zero at two points kn Ai and there is minimum km requiring
M11 D2 + M22
k 12 = 22
(40)
D2
The band unstable modes spans wave numbers from h to kz. Now recall the necessary conditions for instability in [9] it is clear that when M11, M22 are all negative the conditions can not be satisfied. Without loss of generality, we can choose M11 > O, M22 < which means C₁ stands for the activator and C2 stands for the inhibitor. The requirement for instability now becomes
D1
M2a
<
M1 < [Mzel
D2
(41)
D2
M2a
1.
D1
M11
Also, we must require
6+ (km) + 8- +-(mm det M(0) (M11 + M12
4D, < 0
(42)
(Mi M12 2 D2 D2 det M()),
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which gives more restrictive condition the ratio D2/D1. For example, in the case of the matrix
M(k) -1- 11-1111 2
(43)
-1 D
the condition in Eq requires D2/ D1 > 1, while Eq 42
requires D2/ D1 > + 4/5 17.9.
Let us come back to talk about the stochastic term in Eq 32 For chemical reactions, fluctuations obey typical VN rules for number of N molecules. We assume the stochastic fluctuations are described. by white noise with covariance,
= 2NO (2) F.)
(44)
After the Fourier transformation, it becomes
2N
t)(k, +)) = (2.)) 6(t 17)8 (K))
(45)
For many variable cases in Eq 32 the power spectrum of fluctuations in steady state is proportional 1/ det (k) Eq When D2/ D2 |Mzal/M11, the peak in the power spectrum for noisy fluctuations shifts to km-
A.3.3 EMERGENCE AND SELF-ORGANIZATION IN LARGE MODELS
"Emergence refers to the arising of novel and coherent structures, patterns, and properties during the process of reff-organization" (Goldstein) 1999). Emergence can be described as the formation of new properties "something new appears"
the evolution of a collective system that cannot be fully captured by its dynamical equations. These emergent properties arise from local interactions and can be harnessed by the system to develop additional functionalities or capabilities (Crutchfield 1994). In the context of large models, emergence is the complex behavior through which new learning capabilities are acquired by the LLM from the collective interactions among numerous simple elements (i.c., the neurons following consistent rules within the neural network).
"Self-organization is the process of forming patterns and achieving overall order through internal local interactions" (Correia 2006 Ay et al., 2007). In the human brain, intelligence and complex behaviors arise from the collective behavior of neurons, with the brain operating as a self-organizing system (Kelso 1995 Singer 1986; Pribram 2018). In the context of large models, self-organization refers to the process where the interactions of artificial neurons during training evolve from initial randomness, developing new patterns and orderly structures, thus leading emergent abilities in large models.
A.4 COMPARISON WITH TRADITIONAL METRICS
In this section, further discuss why traditional network analysis methods fail to effectively meet the requirements of an analytical framework for large models, while also highlighting how our proposed NeuroMFA method enables efficient analysis of neural networks in large models, representing key contribution of our work.
Here some key challenges in applying network analysis to large-scale models:
Unique Structural Characteristics of Neural Networks: Neural networks exhibit **di- rected, weighted, and hierarchical** structures, which make traditional metrics designed for unweighted or undirected networks less applicable.
2. Scalability to Large Models: Large models have billions of parameters, creating networks of enormous size. Many traditional metrics become computationally infeasible for networks exceeding 10 million nodes or edges.
3. Need for Multiscale Analysis: Our objective is to bridge local (microscopic) structural changes in neural networks with their emergent macroscopic capabilities. This requires metrics capable of capturing both local interactions and global multiscale properties.
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Based on these points, we analyzed the potential applications of traditional network metries in neuronal networks and neural networks, as well as their limitations when applied to large-scale models as below.
1. Centrality Metrics: Centrality measures such as degree, closeness, and betweenness centrality quantify the importance of nodes within a network.
Potential Applications: Useful for identifying key neurons in neuronal networks that are critical for information propagation or specific task processing.
Limitations: Centrality metrics focus individual node-level properties, making them less suitable for capturing global network dynamics or multiscale interactions.
2. Small Worldness: Small worldness quantifies the balance between local clustering and global efficiency using metrics such as local clustering coefficients and average shortest path lengths.
Potential Applications: In neuronal networks, helps understand the trade-off be- tween local processing and global information integration, as well as network efficiency. Limitations:
Primarily designed for undirected and unweighted networks. small worldness (especially for the calculation of average shortest path) is computationally expensive for large, weighted networks like those in LLMs.
3. Modularity: Modularity measures the degree to which a network can be divided into distinct modules communitics.
Potential Applications: In neuronal networks, modularity can be used to identify functionally distinct sub-networks or quantify modular brain region connections.
Limitations: Modularity identifies large-scale structures but fails to capture node- level interactions or multiscale properties. Additionally, it was initially designed for undirected and unweighted networks. Although adaptations exist, they are often computationally expensive to analyze the directed and weighted structures in neural networks.
4. Entropy-Based Metrics: Entropy metrics quantify the diversity or randomness of network structures, often focusing on degree distributions or edge weights.
Potential Applications: Useful for measuring the diversity of synaptic connections or activity patterns in neuronal networks.
Limitations: Entropy metrics analyze distributions rather than interactions, making them unsuitable for multiscale analysis. Additionally, entropy calculations for large, directed, and weighted networks are resource interceei
5. Ricei Curvature:
Ricci curvature provides geometric insights into connectivity and robust- ness by measuring edge-based properties.
Potential Applications: Applied in neural networks to identify critical pathways for signal transmission or bottlenecks in artificial neural networks.
Limitations: Ricci curvature is computationally prohibitive for large-scale, weighted networks and focuses on edge properties, offering limited insights into global network structure.
Why NeuroMFA? Compared to the metrics above, NeuroMFA offers several distinct advantages:
1. Node-Based Design: NeuroMFA analyzes nodes directly, making it well-suited for studying local interactions and scaling up to multiscale analyses.
2. Efficiency: In NeuroMFA, the analysis starts from node-based local properties and leverages multiscale characteristics to derive the overall network's multifractal features. This method, which relies on local properties rather than direct computation over the entire network, sigmificantly improves efficiency. Additionally, by leveraging scaling laws, NeuroMFA allows for network sampling without compromising the accuracy of fractal dimension estimation (see Appendix A.1.3, revised version, page 19), making NeuroMFA computationally efficient.
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3. Multiscale Analysis:
NeuroMFA captures both local and global properties, enabling us to capture the heterogeneity (diversity local interactions) and irregularity (global structure) of the network structure across multiple scales, aligning with natural systems' self-organization and emergence rationale.
In summary, although traditional metrics can be useful for analyzing smaller neural networks, they often limited by computational inefficiency lack of the ability for multiscale analysis when applied to large-scale neural networks. NeuroMFA overcomes these challenges by providing an efficient, scalable, and multiscale framework tailored the unique characteristics of large neural models.
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B IMPLEMENTATION DETAILS
B.1
CODE AVAILABILITY
The source code available btttss://gtthub com/:oshaarioo08//eeee... we provide the hyperrparamettess we used to produce our experiments in Table
LIM
Here
B.2 EVALUATION LLMS
We experiment on Pythia Biderman et al. (2023) dataset and models. The detailed information can be found in the table
Model Size
Non-Emmeddding Params
Layers
Model Dim
Heads
Learning Rate
Equivalent Models
70 M
18.915.328
512
10.0 10-1
160 M
$5,006,000
12
768
12
6.0 x 10
GPT-Neo 125M, OPT-125M OPT-350M
410 M
302.311 .424
24
1024
16
3.0 x 10-4
1.0 B
805,736,44h,
16
2048
3.0 x 10
B B
1.208.602.62.
24
2048
16
2.0 x 10-4
GPT-Nco 1.3B, OPT-1.3B
2.8
2,517,652,480
32
2560
32
1.6 x 10
GPT-Neo 2.7B, OPT-2.7B
Table
Models in the Pythia suite and select hyperparameters from Biderman al. (0023).
And the model being used is based on GPT-NeoXAndoniaa et al. (0023). It is an open source library GPT-NeoX developed by EleutherAI. The model checkpoints are saved at initialization, at the first 2, 4, 8, 16, 32, 64, 128, 256, 512 iterations, and then every 1000 iterations, making it total of 154 checkpoints.
The design of GPT-NeoX is based on currently most wide-spread design of open source GPT models. To be more specific, it has the follow features:
The model uses sparse and dense attention layers in alternation introduced by Brown et It is used in the fully dense layers of the model.
The model uses Flash Attentior et al. (2022) during training for improved device throughput.
The model uses rotary embeddings introduced by Su et al. Now it is widely used as positional embedding type of choice.
The model uses parallelized attention and feed-forward technique.
The model's 's initialization methods are introduced by Wang & Komatsuzaki and adopted by Black al, and Chowdhery et al. because they improve training efficiency without losing performance.
Aside from rotary embeddings, the model uses untied embedding / unembedding matrices Belrose et al. (2023). This makes interpretability research easier, which is very important for our research.
Sample1 (x 10)
Sample2 (x 10)
Sample3 x 10)
Name
Blocks
Model Dim
Nodes Per Layer
Total Nodes
Nodes
Total Nodes
Nodes Per Layer
Total Nodes
Per Layer
Pythia-14M
128
128
5504
2752
1376
Pythia-31M
6
256
128
5504
64
2752
32
1376
Pythia-70M-ddduped
6
512
128
5504
64
2752
32
1376
Pythia-160M-den
768
128
10880
64
5440
2720
4100-a-400M
24
1024
128
21632
64
5440
32
2720
Pythia- 1B -edeueed
16
2048
128
14464
64
7232
32
3616
Pythia- LB--deduped
24
2048
128
21632
64
10816
5408
Pythia 2.8B-BBBB. deddupped
2560
128
28800
14400
7200
Table SNIN sample parameters
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B.3 ALGORTHTM FOR NEUROMFA
The detailed algorithms of the calculation of the mass exponent (()) and the multifractal spectrum f(a) shown in Algorithms 3 and
Algorithm 2 Calculation of the Partition Function Z(g)
Require: S (a dictionary with nodes keys and lists of distances from the node its neighbors as values), Qust (a list of distortion factors)
Mist [
2: R <- ()
3: for all S do
4: num <- empty map
5: for all S[s] do
6: rdist +- rownd(d)
7: num[rdist]] + nummrdist + 1
8: end for
9: R 6 RU keys(nam)
10: append numn Misa
11: end for
rsoned sort(R)
13: lens + length of Mis
len, length of ssoted
15: W +- matrix of ones with dimensions lenas X len
16: for = 0
len - do
17: for 0 len4 1 do
18:
Wi + Es Li] Mist [k]
19: end for
20: end for
21: Zynaa +
22: for all q Mist do
23: Zq Elenv
Wh. WK.
24: append Zq to Zqua
25: end for
26: return (Fsorted: Z Zqist)
Algorithm 3 Calculation of the Mass Exponent (g)) in Eq.
Require:
Tsorted (a list sorted box radius, output from Algorithm 2 Zgnist (a list of Partition functions, output from Algorithm 2 d (maximum of box radius)
1: for all Zq in Zqust do
2: Calculate the slope of the linear regression of log(Zq) on log [muaal
3: Save the slopes to an empty list: Tia
4: end for
5: return Thit
Algorithm 4 Calculation of the Multifractal Spectrum ((a) in Eq, 9
Require: Tist (a list of mass exponents, output from Algorithm Qist (a list of distortion factors)
1: Calculate the Oriiaa using Eq
dris
Olist +- dgs
2: Calculate the f(a)sis using Eg
((a)tis + Qist Oja Tis
3: &- Otia [index of maximum value in fa)
4: width <- (aaa) min(ana)
5: return (Co, width, Onis I (a()is))
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B.4 TIME COMPLEXITY OF NEUROMFA
Here are the full expressions for each algorithm's complexity:
Algorithm 2: O[|5] [S[+]] + [R]] log [R] + +S| | + Iquisl - [S])
Algorithm O(12glsel) (
Algorithm O(
We simplify these expressions based on certain considerations:
Tiadl and Zgliet| equal Iqueel which is predefined constant.
ssorted] is equal to RR. Since R represents positive integers and we set an upper limit for computational convenience, so |R] becomes a constant.
After applying these simplifications:
Algorithm 2: O([S) [[[s]]
Algorithm 3: 0(1)
Algorithm 4: O(1)
To connect Algorithm 2's complexity neural network parameters, consider:
L: number of layers
D: average neurons per layer
N: number of neurons of the whole network
a: sampling ratio
The number of sampled nodes |S| = L D and each sampled node considers only nodes in the next layer, so [S[$]| a D. Therefore, Algorithm 2's complexity is expressed as:
O(S (S[$]) = O(L D) (a D)) = 0(a² - L. L- D²)
Since is a constant, we simplify to:
OL. D(-- D)
This final expression represents the overall time complexity of NeuroMFA, as it is dominated by Algorithm 2. This analysis shows that NeuroMFA's complexity scales linearly with the number of layers (L) and quadratically with the average number of neurons per layer (D).
B.5 NETWORK CONSTRUCTION
Shortest Path Calculation. The time complexity of the algorithm of calculating the shortest paths among each node in different layers is (kn?), here denotes how many layers we crossed, and denotes the number of nodes in each layer. This algorithm calculates the passing closure with a limitation of steps to calculate the shortest paths with an allowance of layers. Here since the network is a leverage graph, can reduce the number of nodes we need to compute in each iteration to the number nodes in each layer. The algorithm is as follows:
However, the time cost of the calculation of shortest paths in neural networks is unacceptable, because in large language models, the total number of nodes and edges is extremely large. So in real-world situation we need to sample some nodes from each layer build manageable network for analysis. We will explain the sample and shortest path estimation in the Appendix.
Even though have already sampled nodes in each layer, calculating the shortest paths across multiple layers is still very expensive. The complexity what we mentioned in Section 4 Okn??). Here the is the number of layers cross and is the number of nodes in each layer. This is
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Algorithm 5 Calculation of the Precise Shortest-Path in Neural Networks
Require: Dis (A map from node pairs (u, v) to distance value w)) Nodes No.ees2...NV (An array contain- ing nodes in each layer),NV (Number of layers)
1: for all in N Nodes [0] do
2: for = to N do
3:
for all in [odes] do
4:
for all i in Nodes|! - 1] do
5:
Dis , y] & Dis x nl, Disla, Disli,
6:
end for 7: end for
8: end for
9: end for
10: return Dis
modified version of Floyd- Warshall algorithm caleulating the passing closure of the whole graph. However, since the number of each layer is still very large, it is still too expensive to calculate the distance. Here cannot calculate the distance the sampled subgraph it will be too inaccurate.
To address this problem, we use a binary search based method to estimate the shortest paths among nodes. We will give budget to sample passing node and calculate the distance passing by this node at every binary checking. The historical biggest value is also recorded to accumulate historical trials. The detailed process of the algorithm can be found at Algorithm
Weight Matrices in Transformers. In our network construction of transformer attention mechanisms, we focus specifically on the static weight matrices (Wq. WK. and Wv) rather than the dynamic attention weights that are generated during inference. While the interaction between query and key matrices produces dynamic attention weights that vary with input data, these static weight matrices represent the model's fundamental learned parameters and therefore better reflect the inherent structural properties of the trained model.
Algorithm 6 Calculation of the Estimated Shortest- Path in Neural Networks
Require:
Dis (A map from node pairs (u, v) to distance value w), Nodes N2....V (An array contain- ing nodes in each layer), N (Number of layers) Nodes t- Sample( Nodes)
2: for all x in Nodes [0] do
3: for = N do
4:
for all in Nodes[[] do
5:
1 1-0
6:
r inf
7:
hisest +-
8:
mid
9:
while I < do
mid +
11:
IntN odes + Sample 1)) est + ShortestAmong( (Int odes)
13:
hisest, est t- min(est, hisest) if est S mid then + mid
15:
else
17:
1 mid
18:
end ir
19:
end while
20:
dis[æ, y] < mid
21:
end for
22:
end for
23:
end for
24:
return Dis
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B.6 BENCHMARKS DESCRIPTION
In this subsection we introduce the metrics used to evaluate the Pythia model's performance. Here the traditional metries all testing-based metrics, each focusing on different aspects.
Lambada-OpenA.. LAMBADA Paperro et al. (2016) is specialized metric developed by OpenAl to evaluate the capabilities of Large Language Models (L.LMs). It stands for "Language Modeling Broadened to Account for Discourse Aspects." This metric designed to assess an LLM's ability to understand and generate text within the context of broader discourse, going beyond simple sentence-level understanding. LAMBADA specifically focuses on testing the model's s proficiency in handling long-range dependencies and context in text. This is achieved through a set of challenging tasks that require the model to make predictions or generate responses based on extended passages of text, rather than just individual sentences or short snippets. By employing LAMBADA, researchers can gain deeper insights into LLM's understanding of complex linguistic structures and its capacity to maintain coherence over longer stretches of discourse, which are critical for more advanced natural language processing applications.
PIQA. PIQA (Physical Interaction: Question Answering Bissk et al. 2020) is dataset created for assessing commonsense reasoning in natural language processing (NLP) models, particularly focusing on their understanding of physical knowledge. PIQA challenges these systems with questions that require physical commonsense, such as choosing the most sensible physical action among various options. While humans exhibit high accuracy (95%) in answering these questions, pretrained models like BERT struggle, achieving around 77% accuracy. This dataset exposes the gap in Al systems' ability to reliably answer physical commonsense questions and provides a vital benchmark for advancing research in natural language understanding, especially in the realm of physical knowledge.
WinGrande. WinoGrandeSakaguchh et al. (022)) is a large-scale dataset designed to evaluate neural language models' commonsense reasoning capabilities. Comprising 44,000 problems and inspired by the original Winograd Schema Challenge, WinoGrande developed through a meticulous crowdsourcing process and systematic bias reduction using the AfLite algorithm. This dataset addresses the overestimation of machine commonsense in previous benchmarks by reducing dataset-specific biases and providing a more challenging set of problems. While state-of-the-art methods on WinoGrande achieve accuracy rates between 59.4% and 79.1% %, they fall short of human performance at 94%, highlighting the gap in true commonsense reasoning capabilities of AI models. WinoGrande not only serves as crucial benchmark for transfer learning but also underscores the importance of algorithmic bias reduction in evaluating machine commonsense.
WSC. The Winograd Schema Challenge (WSC]Kocijan et al. (2020) is a test of artificial intelligence that focuses on evaluating system's 's ability to perform commonsense reasoning. WSC consists of a series of questions based on Winograd schemes, pairs of sentences that differ in only one or two words and contain a highly ambiguous pronoun. The challenge requires deep understanding of text content and the situations described to resolve these pronouns correctly. Initially containing 100 examples constructed manually by Al experts, the dataset has since expanded to 285 examples, with the WSC273 variant often used for consistency in model evaluations. Despite its initial design to be difficult for machines, recent advances AI, such as the BERT and GPT-3 models, have achieved high levels of accuracy on the WSC, raising questions about the true extent of their commonsense reasoning capabilities. The challenge highlights the importance of knowledge and commonsense reasoning in AI and serves benchmark for evaluating progress in natural language understanding.
ARC.The A12 Reasoning Challenge (ARC) et al. (2018) dataset, introduced by Clark et al., is multiple-choice anestion-anssering. dataset featuring questions sourced from science exams spanning grades to 9. ARC is divided into two sections: the Easy Set and the Challenge Set, with the latter containing more difficult questions that necessitate higher levels of reasoning. The dataset primarily consists of questions with four answer choices, although small percentage have either three five options. Accompanying ARC is a supporting knowledge base (KB) of 14.3 million unstructured text passages. This dataset is used to assess the reasoning capabilities of language models, particularly in the
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context of science and general knowledge. Models like GPT-4 have shown remarkable performance on the ARC dataset, reflecting advancements in Al's ability to handle complex crestion-answering tasks that require blend of knowledge retrieval and reasoning skills.
SciQ. The SeiQ Welbi et al. (017)) dataset, developed by the Allen Institute for Artificial In- telligence (A12), consists of 13,679 crowdsourced science exam questions covering subjects such as Physics, Chemistry, and Biology. These questions are formatted as multiple-choiee queries, each offering four answer options. A distinctive feature of the SciQ datasct is that for the majority of its questions, evidence for the correct answer. This dataset a valuable tool for evaluating language mod- els' ability to perform in reading comprehension, question generation, and understanding complex scientific content. It challenges models to not only select the correct answer from multiple choices but also utilize supporting evidence effectively, thereby testing their comprehension and reasoning skills in the scientific domain.
additional paragraph is provided, offering supporting
LogiQA. LogiQALiu et al. (2020) is a dataset consisting of 8,678 QA instances that focus evaluating machine reading comprehension with emphasis logical reasoning. It is derived from the National Civil Servants Examination of China and covers various types of deductive reasoning. This dataset presents a significant challenge for
neural models, which perform notably worse than humans in these tasks. LogiQA serves as a unique benchmark for testing logical Al under deep learning NLP environments, requiring models to demonstrate a blend of language understanding and complex logical reasoning. It includes different types of logical problems, such as categorical reasoning, sufficient and necessary conditional reasoning, disjunctive reasoning. and conjunctive reasoning, all key to deductive reasoning. This dataset provides a rigorous test of Al's logical reasoning capabilities and its ability to handle problems similar to those faced by human experts.
Hendrycks Test. The SeaddycckTTesHHenyyckk. et al. (2020), also known as the Measuring Massive Multitask Language Understanding (MMIU)) test, a massive multitask test that includes multiple-choice questions from wide range of knowledge domains, covering 57 tasks in areas such as elementary mathematics, US history, computer science, and law. This test aims measure the multitask accuracy of text models, requiring them demonstrate extensive world knowledge and problem-sole solving ability. The results show that while most recent models have near random- aedom--chance accuracy, larger models like GPT-3 have shown improvement, but still fall short of expert-level accuracy across all tasks. The HendrycksTest serves comprehensive tool for evaluating the breadth and depth of models" academic and professional understanding, identifying significant shortcomings, and highlighting areas needing substantial improvement, especially in socially important subjects like morality and law.
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C IMPACT STUDY
C.I VARYING ARCHITECTURE
Although in the area of computer vision (CV), several generative models (Rombach et 2022 providing personalized image generation exhibit emergent abilities, no analysis of their emergence exists. Thus, we apply NeuroMFA the diffusion model. The experimental results are shown in Appendix C.1 Fig. 13 The degree of emergence for the diffusion model (CV task) is lower than the LLM (NLP task) at the same parameter size, which indicates a lower level of emergent abilities.
We extend our analysis to CNNs, testing both ResNet-18 18 and ResNet- 152 models. As shown in Appendix Fig. [4 ResNe-t18's spectral graph lacks regular bell-shaped structure, indicating no clear multifractal structure self-organization. While ResNet-152 displays noticeable multifractal structure, its spectral graph shows irregular horizontal shifts, suggesting that a stable self-organized structure did not form. These results indicate that CNN structures alone may not produce emer- gent abilities comparable to transformer- based models. Our findings from both CNN-based and transformer- based models, presented in Appendix C.I provide preliminary validation for applying multifractal and eeff-organization research to study model emergent abilities. The contrasting results between these architectures highlight the potential of our method in differentiating and understanding various model structures in terms of their capacity for emergent abilities.
Diffusion Mocel (160M)
Pythia (180M)
0.0
Step
0.0
Stop.1
Step
2-011
Step.
Step
Step.
-0.4
Step 112
-066
Step 1000
Step 1800
9-0.4
Step 5000
Step 1000
-00
10000
10000
25000
1.0
10000
-0.6
39000
Stop_107000
100000
0.35
1.50
ppchiillndder
Upperhor
Figure 13: NeuroMFA results on stable diffusion model and the Pythia model of equivalent size.
Rachae-18 (RMB)
RusNGi-152 (300()))))))
0.22
0.2
-0.4
00.4
-0.6
-0.6
-0.8
-0.8
-1.0
112
414
-1.2
0.75
1.00
1.25
1.50
1.75
0.5
1.0
1.5
Lipschitz-Holder exponent,
Hopschitz-Holder exponent,
Figure 14: NeuroMFA results on ResNet-18 model and the ResNet- 152 model.
The analysis results for different training epochs on a Computer Vision (CV) task using the Diffusion Model are depicted in Fig. (Left). On the right, analysis outcomes for LLM of equivalent size (in terms of the number of parameters) are presented. Furthermore, we extend the NeuroMFA analysis to two typical CV models: ResNet- 18 and ResNet- 152, as shown in Fig. 14
C.2 VARYING TRAINING DATA
We explore the potential impact of different types of training data on neuron interaction dynamics. Our study primarily focuses on Pythia-deduped models, but also conduct comparative experiments
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using both Pythia-standant and Pythia- models to explore the impact of different training datasets, whose results are shown in Appendix C.2 The Pythia project offers two types of models with identical architectures but trained different datasets: token dataset) and Pythia-deduped models (trained on a 207B token dataset with duplicates removed). Applying our NeuroMFA method both model types reveals that the critical point for emergence occurs at the same training step. However, the Pythia-standard model shows a slightly higher rate of increase in the degree of emergence after this transition. This aligns with Pythia's official statement that the effect of training a model for one epoch on the standard dataset is approximately equivalent training for 1.5 epochs on the deduped dataset. These findings suggest that our method can capture the influence of different datasets on model training, reflecting how the nature of training data affects self-organization and emergence phenomena in neural networks.
models (trained on 300B
Fig. 15 Illustrates the degree of emergence of two sizes of Pythia models trained on different data by using our NeuroMFA method.
160M
Pythio
2006 4100
54000 WOM 10WK 1900
1908
2000 (903
08/8 1300
1900 1000
11:00
Spocks
Epada
Figure 15: Emergence of Pythia 160M model and Pythia 1.4B model trained on two different data.
C.3 VARYING TRAINING DIRECTIONS
To validate our proposed metric's effectiveness in capturing model performance changes, we conduct experiments using contaminated training data with BERT. We create a controlled experimental setup by deliberately contaminating the training dataset, combining 20% of Wikipedia dumps data with 80% randomly generated data following a normal distribution.
Using pre-trained BERT-large-uncasee model from HuggingFace (24 layers, 336M parameters, 1024 hidden dimensions, 16 attention heads), we conduct fine-tuning experiments with this contaminated dataset. To evaluate model performance, we employ two widely-used metrics: SQUAD 1.1 Fl score for reading comprehension and Multi NLI Accuracy for natural language inference.
As shown in Figure 16 our proposed metric effectively captures the pattern of model ability degra- dation caused by severe data contamination, highlighting its capability to track structural changes in the model under different training scenarios. Moreover, this consistency further validates that our structure-based metric can reflect the evolution of emergent properties in LLMs throughout the training process, regardless of training quality or the direction of change.
Emergence vs SOUAD F1
Emergence vs MNU
0.6
0.6
0.5
0.5
Emergence
Emergence
70
70
SQUAD
ACCUUS
0.3
0.3
50
0.2
0.2
30
250
500
750
1000 Epochs
1250
1500
1750
2000
250
500
1000 Epochss
1250
1500
2000
Figure 16: Changes in the degree of emergence during training with contaminated data.
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D ADDITIONAL RESULTS
D.1 NEUROMFA SPECTRA WITH VARIANCE
Figure [7 displays the averaged (with variation) multifractal spectra for each model, derived from
analyses across 10 SNINs under NeuroMFA. We provide how the og and width vary with the training
steps in Fig. [8
0.0
fracaa no
-0.1
Multi Frectal spectrum R10)
-0.2
-0.2
0.3
-0.4
-0.4
0.6
-0.8
0.50 6.75
1.00 1.35
1.50
0.50
0.75
1.00
1.5
1.50
1.75
Upschitz-Holeer exponart, a
Uppchitt Holder expenent,
0.0
0.0
MubHiaatt spectuum Rd)
Multi bilbirrreccll spectrum, Rol
-0.2
-0.4
-0.6
0.6
0.8
0.88
0.5
1.0
1.5
Upschite--lllder
Lipechiza-folder exponent,
Hulb Fractal specrum flet
M
Mull-rrctal sedduum e
-0.1
-0.2
-02
-033
-0.4
0.5
-0.6
0.66
-0.0
Lipschiz-Hooddr exponent,
LisschitzHolser
M
0.0
M
Not-toccall specium, [e)
-0.1
Nw. Wwit-fon -reeceltllllll spectrum, Ao)
-0.2
0.3
-0.4
-044
-0.5
-1.0
1.0
2.0 Upschitz-Holder exponent.
1.5
1.0
1.5
2.0 2.5
Lpsccitz-Holldr exponent, d
Figure 17: Multifractal analysis spectra of 10 Sampled Neuronal Interaction Networks (SNINs), illustrating the average value of the spectra with shadows indicating the standard deviation. From upper left to lower right, the model sizes are 14M, 31M, 70M, 160M, 410M, 1B, 1.4B, and 2.8B respectively.
Here, we also provide the variance across 10 sampled NINs. In Table 3 and Table 4 below, we present the variance of and (a) from 10 samples the spectra shown Fig. Furthermore, in Table and Table 6. present the variance for both the regularity metric (ag) and the heterogeneity metric (width).
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prhacitn
11
20
prfa-11
Araha
1.5
07-
podio Sin
polli 30m
pelio
polhu-3b
pottor job
pefio
50000
140000
130300
Epochs
Figure 18: ao (left) and width (right) of each epoch in different models.
Epochs
8
64
512
1000
5000
Model
4.76e-06
1.89e-06
2.47e-06
1.10e-06
1.05e-06
4.60e-06
pythia-14m
f(a)
9.41e-06
5.68e-06
5.42e-06
3.86e-06
2.12e-06
2.24e-06
pythia-31m
a
5.19e-05
5.11e-05
3.71e-05
6.07e-05
5.95e-05
3.45e-05
f(x))
4.69e-05
3.05e-05
3.61e-05
5.32e-05
3.83e-05
2.53e-05
70thia-70m
a
3.90e-06
2.35e-05
4.96e-05
6.71e-05
9.35e-06
4.88e-05
f(a)
6.47e-06
1.60e-05
5.30e-05
6.96e-05
1.28e-05
4.31e-05
4.55e-05
9.21e-06
1.14e-05
1.14e-04
4.70e-06
1.10e-04
pythia-160m
f(xx)
1.49e-05
1.74e-05
2.18e-05
3.17e-05
9.40e-06
2.10e-05
pythia-410m
a
3.12e-06
7.38e-05
6.86e-05
6.21e-05
2.34e-06
4.38e-05
((a)
3.61e-06
1.11e-05
8.61e-06
1.62e-05
3.75e-06
6.59e-06
2.14e-05
2.83e-05
2.42e-05
3.03e-05
3.84e-05
7.84e-05
1ythia-1b
f fa:)
2.23e-05
4.90c-05
3.67e-05
3.85e-05
6.23e-05
1.12c-04
pythia-1.4b
a
3.99e-05
4.57e-05
3.98c-05
2.01e-05
2.18e-05
3.69e-05
f(a)
4.19e-05
5.21e-05
3.91e-05
2.32e-05
4.22e-05
4.18e-05
pythia-2.8b
a
5.20e-06
4.19e-05
5.79e-05
3.20e-05
3.75e-05
5,47e-05
(aa
7.56e-06
2.19e-05
2.76e-05
2.34e-05
1.94e-05
2.83e-05
Table 3: Variance of a and f(a) in the first six epochs
Epochs
10000
35000
71000
107000
143000
Model
1.03e-04
6.58e-05
1.03e-05
8.35e-06
5.03e-06
pythia-14m
f(a)
2.28c-05
2.48c-05
3.19e-05
2.48c-05
2.29e-05
5,48c-06
3.56c-06
8.55e-06
6.22c-06
1.01e-05
11thia-31m
f(a)
1.07e-05
7.16e-06
2.29e-05
1.46e-05
2.28e-05
3.75e-05
5.72e-06
2.02e-04
1.78e-04
1.92e-04
pythia-70m
f(a)
2.29c-05
3.17e-06
8,42e-05
8.36e-05
2.45e-04
6.84e-05
6. 18e-05
1.40e-04
1.47e-04
1.34e-04
pythia-160m
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Epochs
10000
35000
71000
107000
143000
Model
f(a)
1.28e-05
7.15e-06
1.96e-05
2.98e-05
2.09e-04
6.00e-06
6.27e-05
3.57e-05
1.67e-04
7.35e-04
pythia-410m
f(a)
2.15c-05
5.12e-05
1.05e-04
7.49c-04
1.08c-04
1.92c-04
2.38c-04
8.94c-04
2.52c-04
3.25e-04
pythia-Ib
f(a)
1.44e-03
3.76e-03
5.56e-03
4.39e-03
3.09e-03
1.07e-03
7.00e-04
2.62e-04
3.15e-04
1.88e-04
pythia-1.4b
f(a)
1.17e-03
2.22e-03
1.97e-03
1.61e-03
1.24e-03
pythia-2.8b
a
4.34e-03
1.85e-03
9.02e-04
9.56e-04
8.59e-04
f(a)
4.74e-03
7.56e-04
6.77e-03
8.09e-03
8.00e-03
Table 4: Variance of and f(a) in the last five epochs
Epochs
8
64
512
1000
5000
Model
pythia-70m
Oxp
1.12e-05
1.12e-05
9.66e-05
2.73c-04
2.67e-05
8.31e-05
width
4.21e-04
4.27e-04
3.40e-04
5.77e-04
4.98e-04
1.41e-03
9.72e-05
1.21e-04
3.87e-05
3.60e-04
3.07e-04
1.37e-04
pythia-160m
width
1.18c-04
1.16c-04
9.18c-05
2.40c-04
4.02c-04
7.24c-04
pythia-410m
og
9.41e-05
9.41e-05
1.04e-04
9.37c-05
3.76e-04
4.6le-04
width
1.65e-04
1.65e-04
1.95e-04
4.17e-04
2.27e-03
3.05e-04
pythia-lb 0g
2.09e-05
2.19e-05
3.67e-05
2.18c-05
7.47e-04
2.34c-04
width 2.16e-04
2.15e-04
2.28e-04
6.56e-05
2.62e-03
3.12e-04
pythia- 1.4ib
Og
3.07e-05
1.36e-05
1.30e-05
1.78e-05
1.53e-05
3.42e-04
width 3.36e-04
3.36e-04
3.32e-04
1.93e-04
1.38e-03
1.63e-04
pythia-2.8b
on
2.42e-05
5.81e-05
8.53e-05
5.70e-05
3.24e-05
6.81e-04
width
4.78c-04
4.78c-04
4.84c-04
4.19c-04
1.21c-04
1.00c-04
Table 5: Variance of on and width in the first six epochs
Model Epochs
10000
35000
71000
107000
143000
Ok 8.32e-05
5.01e-05
1.26e-05
1.26e-04
2.38e-05
pythia-70m
width
1.57e-03
1.11e-03
3.71e-03
1.95e-03
2.67e-03
Ogg 4e115-05
1.48e-04
2.29e-05
5,83e-06
7.27c-05
pythia-160m
width 2.08e-03
1.53e-03
4.43e-03
2.72e-03
1.57e-03
pythia-410m
0o
1.63e-04
4.14e-04
8.14e-04
3.92e-04
5.17e-05
width 8.36e-04
1.47e-03
1.71e-03
8.93e-04
2.58e-04
Okg 1.92e-04
2.38e-04
8.94e-04
2.52e-04
3.25e-04
pythia-lb
width
1,44e-03
3.76e-03
5.56e-03
4.39e-03
3.09e-03
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Epochs
10000
35000
71000
107000
143000
Model
0x0 1.07e-03
7.00e-04
2.62e-04
3.15e-04
1.88e-04
pythia-1.4b
width 1.17e-03
2.22e-03
1.97e-03
1.61e-03
1.24e-03
pythia-2.8b
ag
4.34c-03
1.85e-03
9.02e-04
9.56c-04
8.59e-04
width
4.74c-03
7.56e-04
6.77c-03
8.09c-03
8.00c-03
Table Variance of ag and width in the last five epochs
D.2
ASSESSMENT OF THE DEGREE OF EMERGENCE ACROSS VARIOUS DATASETS
Fig.
23
Fig.
20
Fig.
22
Fig.
and Fig. show the result for degree emergence under our
metric.
2080
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tax.
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Figure 19: Results on ARC-Easy benchmark. From upper left to lower right, the model sizes are 70M, 160M, 410M, IB, 1.4B, and 2.8B respectively.
Gede
tpxts
Exor
35
Energencs
08
831
Figure 20: Results WSC henchmark. From upper left lower right, the model sizes 70M, 160M, 410M, 1B, 1.4B, and 2.8B respectively.
Graphs with all metrics shown in Fig. 24 (a) and (b) represent the metric performance of Pythia- IB and Pythia-1 1.4B, respectively. Large language models (LLMs) can be evaluated comprehensively using metrics A-H. The seven specific metrics A-H be categorized follows: LAMBADA (A) assesses language understanding: SeiQ (B) measure scientific knowledge; LogiQA (C) evaluates logical reasoning: ARC-challenge (D), ARC-easy (H) and PIQA (G) examines physical commonsense reasoning, WSC (E) and Winogrande (F) test commonsense reasoning, Detailed performance data are provided Tables and respectively.
D.3 LOG-LOG RELATIONSHIP EVALUATION
We RD (the coefficient of determination) to evaluate the log-log relationship between (r) and between (r) and 5 at different samples of SNIN. The detailed evaluation results can be found in T
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Figure 21: Results on SciQ benchmark. From upper left to lower right, the model sizes are 70M, 160M, 410M, IB, 1.4B, and 2.8B respectively.
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Figure 22: Results on ARC-Challenge benchmark. From upper left to lower right, the model sizes are 70M, 160M, 410M, 1B, 1.4B, and 2.8B respectively.
According to the coefficient determination detailed in Appendix F.2 the log-log relationships are all above 0.7, demonstrating strong log-log relationships between (r) and r, between (r) and 5-
D.4 AVERAGE WEIGHTED DEGREE DISTRIBUTION
Previous work has reported the existence heavy-tail degree distributions in the node connectivity of neural networks, which suggests the presence of hubs, playing important role in the network (Barabási Bonabeau 2003; Broido & Clauset 2019). These phenomena are commonly considered as outcomes self-organization (Park et al.) 2005 Evans & Saramāki (2055).
In our work, we first construct NINs for two versions of the pythia model: small-scale 14M parameter model (pythia 14M) and larger 1.4B parameter model (pythia 1.4B). Dealing with NINs, we compute the weighted degree of each node, defined as the average value of distances from connections. This allows to analyze the average weighted degree distributions that emerge during training.
Figure 25 shows a clear distinction between the two models. During the training process, the average degree distribution for pythia 14M exhibits dispersed trend, with lower kurtosis values (where kurtosis is commonly used to measure distribution characteristics, with higher kurtosis indicating heavier-tailed distribution, see Appendix E.I., the distribution for pythia 1.4B with left shifting indicates the trend appearing a heavy-tail distribution over multiple epochs pertaining, which is also indicated by the increasing Kurtosis value. Please refer to Fig. 26 Fig. 27 and Fig. 28 for more information and results of Kurtosis value on pythia 14M, 160M, and 1.4B. This indicates that the scale of the model increases, the average distance from most neurons the neighbors in the next layer decreases, signifying the establishment of stronger interactions. It might imply that an organizational structure with a heavy-tailed connectivity pattern systematically self-organizes. Therefore, further in-depth research into the network's 's connectivity structure is required to observe the oraaniaganitttonn process, leading us to the multifractal analysis of the network.
Fig. 26 Fig.
and Fig. 28 provide more results on different scales of models.
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2000 oo0
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Figure 23: Results LogiQA benchmark. From upper left to lower right, the model sizes are 70M, 160M, 410M, IB, 1.4B, and 2.8B respectively.
8)
Degree Emergence
b)
Degree Emergennce
1.0
243C
103K
100C
LAMBADA colanee challnngs
ccience eamm questioe
fesscring quaetions
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logirll reason perioon
53K
Winggrod Schoma Challange
13C
Ienogrud Schema Challings
136
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Figure 24: Full radar chart comparing the degree of emergence metric with conventional benchmarks.
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Figure 25: The distributions of average weighted degree for two models of different sizes (i.e., 14M and 1.4B pythia model) during the training process. The numbers the figures represent Kurtosis, indicating the tail weight and peak sharpness of the distributions. High kurtosis implies heavier tails and sharper peak, whereas low kurtosis suggests lighter tails and a flatter peak.
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Epochs
1000
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Degree of Emergence
0.00
0.00
0.00
0.23
0.53
0.65
0.72
0.73
0.72
0.71
WSC
0,63
0.63
0.37
0.37
0.37
0.38
0.37
0.37
0.37
0.37
LogiQA
0.21
0.18
0.23
0.20
0.22
0.24
0.23
0.24
0.22
0.21
SciQ
0.22
0.27
0.47
0.71
0.82
0.86
LAMBADA
0.00
0.00
0.08
0.33
0.47
0.53
0.57
0.58
0.62
0.62
ARC-casy
0.26
0.29
0.32
0,42
0.48
0.51
0.53
0.55
0.56
0.58
ARC-challenge
0.22
0.18
0.18
0.22
0.24
0.24
0.24
PIQA
0.53
0.54
0.57
0.60
0.66
0.66
0.69
0.69
0.70
0.70
Winogrande
0.49
0.47
0.49
0.51
0.52
0.53
0.51
0.54
0.54
0.53
Table Detailed performance data of Pythia- across different epochs and metrics.
Epochs
512
1000
3000
13000
23000
53000
73000
103000
143000
Degree of Emergence
0.00
0.00
0.01
0.30
0.43
0.54
0.65
0.65
0.65
0.68
WSC
0,53
0.63
0.42
0.37
0.37
0.37
0.35
0.37
0.39
0.44
LogiQA
0.23
0.18
0.21
0.22
0.22
0.20
0.21
0.22
0.22
0.22
SciQ
0.23
0.24
0.44
0.69
0.83
0.84
0.86
0.86
0.86
0.87
LAMBADA
0.00
0.00
0.08
0.33
0.47
0.53
0.58
ARC-easy
0.26
0.28
0.32
0.43
0.52
0.54
0.57
0.60
0.61
0.62
ARC-challenge
0.23
0.19
0.18
0.19
0.22
0.22
0.24
0.26
0.27
0,27
PIQA
0.52
0.54
0.61
0.68
0.71
0.71
Winogrande
0.48
0.50
0.51
0.53
0.52
0.52
0.56
0.56
0.56
0.57
Table 8: Detailed performance data of Pythia-1.44 1..B across different epochs and metrics.
R² of (g)N(r
lgr
R² of
(g
ig Std
Model
Step
Mean
Std
Mean
1
0.8249
0.0291
0,9125
0,0018
2000
0,8308
0.0376
0,9160
0.0026
pythia- 14m
5000
0.8370
0.0330
0.8943
0.0047
10000
0,8421
0.0248
0.8613
0.0118
143000
0.8948
0.0300
0.9763
0.0012
0.9155
0.0594
0.9585
0.0004
2000
0.8874
0.0419
0.9546
0.0005
pythia-160m
5000
0.8431
0.0195
0.9248
0.0023
10000
0.8132
0.0168
0.8750
0,0093
143000
0.8556
0.0391
0.9800
0.0014
0.9005
0.0002
0.9279
0.0024
2000
0.9364
0.0424
0.9287
0.0020
pythia-
5000
0.9112
0.0307
0,9700
0.0011
10000
0.8805
0.0205
0.9336
0.0035
143000
0.8558
0.0190
0.9489
0.0018
Table 9: Log-log relationship evaluation.
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Figure 26: The distributions of average weighted degree for 14M model of different sizes different layers during the training process.
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Figure 28: The distributions of average weighted degree for 1.4B model of different sizes different layers during the training process.
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E EVALUATION METRICS
E.1
KURTOSIS (122)
Kurtosis measure of the "tailedness" of the probability distribution of real-valued random variable. It provides insights into the shape of the distribution's 's tails and peak. High kurtosis in data suggests distribution with heavy tails and a sharper peak (leptokurtic), while low kurtosis indicates distribution with lighter tails and a more flattened peak (platykurtic). Kurtosis is often compared to that of a normal distribution, which has kurtosis of
The formula for kurtosis is:
B2 E X (x #1
(46)
where the variables represent the same as the skewness formula.
These statistical measures, skewness (71) and kurtosis (52), crucial for quantifying and analyzing the non-Gaussianity in image data. They provide valuable insights into the distribution characteristics of image pixel intensities, particularly in highlighting deviations from the normal distribution.
The higher the kurtosis, the greater the degree of non- Soun-Gussiantty in the distribution, indicating a distribution with heavier tails than a normal distribution.
E.2 CORFFICIENT DETERMINATION (R3)
The coefficient of determination, represented by It2, quantifies the extent to which the variance in the dependent variable can be predicted from the independent variable(s) in regression model. It offers measure of how well observed outcomes are replicated by the model, based on the proportion of the total variation outcomes explained by the model.
The formula to calculate R² is given by:
Σ, (y: D1
R²= =
(47)
E, 7)
where yi denotes the observed values, di represents predicted values from the model, and ÿ the mean of the observed values.
R² is interpreted as follows:
R < 0.3: Weak correlation. The model accounts for small fraction of the variance in the data.
0.3 < R² 0.5: Moderate correlation. The model provides moderate level of explanation for the data's variance.
0.5 < R² < 0.7: Strong correlation. The model explains a substantial portion of the variance in the data.
R² > 0.7: Very strong correlation. The model offers high degree of explanation for the variance in the data.
While higher R² value indicates model that better explain the variance observed in the dependent variable, it does not necessarily imply a causal relationship between the dependent and independent variables. It is also critical consider other statistical metrics and tests when assessing the performance of regression model, as R² alone may not provide complete picture of the model's effectiveness.
References that delve into the concept and applications of R² in regression analysis, emphasizing its significance in evaluating model fit and understanding the variability in data, including foundational texts and studies in the field of statistics and econometrics.
Limitations. While our NeuroMFA framework demonstrates correlations between the emergence metric and downstream performance by studying the eeff-organization of LLMs, it has not yet established clear causal relationship. Future work should focus on identifying specific linguistic
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phenomena captured by LLMs and demonstrating how analysis methods reveal the learning process of these phenomena during training, thereby establishing a stronger causal connection.
48


Reviews:
[{"review_id": "G3deGBo12K", "reviewer": "Reviewer_2tKM", "summary": "This paper proposes a new metric to capture emergence within a network by analyzing the structure of neurons within the network. The score depends on the width and peak of the multifractal spectrum overall quantifying the variance in the local structure of the network. This metric is shown to correlate with metrics depending on the behavior of the model, which have traditionally used to measure emergence.", "questions": "1. What are the behaviors for Pythia 14m and 31m fig 6? Do they also agree with Lambada/PIQA?\n2. Are there examples where the performance on behavioral metrics decreases significantly and the proposed metric also decreases?", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. Writing is clear, explains the new metric clearly.\n2. Moving the discussion away from only scaling model size is important\n3. The metric is well defined, making it replicable.", "weaknesses": "1. Text in Fig 1 is very small making it hard to read and there is a lot of white space that can be used to increase text size.\n2. \"While our NeuroMFA framework demonstrates correlations between the emergence metric and\ndownstream performance by studying the self-organization of LLMs, it has not yet established a\nclear causal relationship. Future work should focus on identifying specific linguistic phenomena\ncaptured by LLMs and demonstrating how our analysis methods can reveal the learning process of\nthese phenomena during training, thereby establishing a stronger causal connection.\" This should be in the main paper.\n3. While figure 6 shows the correlation between this metric and previous ones (based on the behavior), I find little reason to use this metric over the behavior (encoded by the previous metrics) of the model itself. At best this metric is correlated with those behavioral methods, at worst they disagree where the behavior would still be more convincing.\n4. We see no reason why this property isn't just linked to a further amount of training. For example, if there is a dataset that when an LLM is trained on it, there is a low performance on the behavior based metrics would we still expect to see the proposed metric increase?\n5. In figure 6 as the accuracy goes up, the proposed metric increases (across models). This is no longer always true in Appendix D.2. For example in fig 19, the 410M model has lower performance but the proposed metric is higher than both the 1B and 1.4B models. This makes sense as the score is normalized with initial training but then you cannot compare across models. This is possible for all of the traditional behavioral metrics. \n6. Fig 8 shows that the scores are lower in the beginning of training than the end for all metrics, and that the model size is only recovered at the later end. How does this address the problem in Fig 1b? As the other metrics have a similar low score after 500 epochs do they also not only rely on the model size?\n\nOverall it is unclear to me the benefit that this metric provides, especially when the paper itself acknowledges that the metric is only validated through its correlation.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "zmC5eFJsVm", "reviewer": "Reviewer_oHwU", "summary": "In this paper the authors ask the question, is there a better metric than raw parameter number to capture why qualitatively new skills develop when scaling up large transformer models. They introduce the network structural / topological measure of ‘NeuroMFA’ and show that this measure can capture how skills develop throughout training, something which raw parameter numbers necessarily cannot capture. They propose that their measure introduces a new way of studying learning of skills across time in large scale neural network architectures.", "questions": "*Baseline calculation*\n\nThis is my only comment about a possible analysis to add. As you are proposing a new measurement, I think it would be helpful to provide measurements in untrained networks. Of course, Figure 8 shows the effect of training on the metric, but I wonder whether it actually would be zero across all models at the start of training? If it isn’t, should be somehow standardise the metric so that it is zero before training? If this was explore in more detail I would be happy to raise my ‘soundness’ rating to ‘excellent’.\n\n*Communication of methods*\n\nGiven the number of results in this paper and the resulting length of the appendix, I think the authors should really aim to communicate the core information in the main text. I think the following would help:\n\n- Given that you use transformers with attention, I imagine the author analyse the effective interaction of model parts conditioned on specific dataset. Naturally the attention weights are going to change how different parts of the models interact and so it seems one could not analyse the structure of the attention weights in general but instead would look at the effective communication when processing specific samples? Or do authors actually just use the weights? It would be great if this could be clarified, ideally at the start of the methods.\n- I think I generally got the metric development, but I did find it somewhat hard to follow. I wonder whether this could be improved if every new part of the method introduced would start with a sentence or two on the high-level intuition behind the measure / what the current analyses step is for – take section 3.2 for example. It starts with “To perform multifractal analysis of LLMs, we extend the box-covering and box-growing methods (Song et al., 2005; Evertsz & Mandelbrot, 1992; Salat et al., 2017; Xiao et al., 2021) to capture the local neuron-based fractal properties of NINs.”, followed by plenty of details on methods. If you lead into this section by first giving the reader an idea why it is interesting to decompose the graph into fractals in the first place, then maybe this would be easier to follow. Generally speaking, motivate the method before jumping into details. I know that the paper is already on the page limit but frankly the appendix is so long, moving a bit more of the methods to appendix in lieu of giving readers a better intuition in the main text is probably beneficial.\n- Given there already is a bunch of structural measures which have successfully been used for structural analyses in neuroscience and neural network models (e.g. standard ones like modularity / small worldness, or entropy-based ones or Ricci curvature) I wonder whether authors could somehow motivate why their new measure was needed? Given it works, I do not want to say that the new measure was not needed but I would be curious to know why they specifically took the route they took with the development of the metric.\n\n*Stylistic choices*\n\n- I think the font on Figure 6 is very close to unreadable small.\n- Across figures, could you either use a serif or sans serif font but not mix that up?\n- I wonder whether your strong emphasize of the word ‘emergence’ is actually needed and helpful in communicating your results? In my experience it does put off some researchers and, as you make a link to neuroscience, the development of language and reasoning skills from a newborn to a teenager would simply be considered as part of ‘development’ and ‘learning’. So, I wonder whether you want to deemphasize the concept of emergence a bit here and just call it ‘the development of skills’?", "limitations": "", "rating": 8, "confidence": 5, "soundness": 4, "presentation": 3, "contribution": 4, "strengths": "I genuinely enjoyed reading this paper. It introduces a new measure to study the training of LLMs that I have not seen in this way before. The paper generally is very thorough in its analyses / experimentation. I am sure this will be work that is of interest to the ICLR community.", "weaknesses": "I think this paper should definitely appear in the proceedings. I see some weaknesses in the way that methods are introduced, and results are described, which I list in the questions below. I hope my suggestions will be helpful to create more impactful paper.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "iDtOqsOnqP", "reviewer": "Reviewer_VqLs", "summary": "This paper presents a new approach to studying the internal property of large language models. I got really excited by the discussion but a bit disappointed by the results. I believe the efforts in this paper should be generalized across the field. \n\nThe parallel between the brain and LLMs in terms of their complexity and emergent characteristics is fascinating. This paper aims to establish a connection with network analysis in neuroscience by characterizing the evolution of organizational patterns within the network during training and linking these to different (emerging) performances.\n\nI disagree with the article’s use of “emergence,” although I understand the rationale. You state that your measure is intended to quantify network regularity. The main conclusion of the article should be that this measure appears sufficient as a proxy for the emergent properties of LLMs.\n\nThe results are interesting but quite weak and merit further development. In particular I felt like many network properties could correlate (positively or negatively) with the training stages of LLMs. Nonetheless, this is an intriguing direction towards understanding network local properties and their link to emergent properties.", "questions": "* Your description of the neural network as blocks and layers seems unnecessary. Simply stating that the neural network can be described as a graph with  w_{ij}  values between neurons is sufficient for your further definitions. The layer and block definitions are not used later, except to describe weights that ultimately serve as connections between two nodes.\n\n* A reference is missing in the appendix on page 39.\n\n* Emergence is, by definition, a property observable at a higher scale that is not explainable by its components. You are measuring network properties, which is interesting, but is it truly the appropriate scale to discuss emergence? I am not convinced the term “emergence” is wisely used in this article. Instead, the focus should be on providing a quantification of network structure that enables or correlates with emergent properties.\n\nOn the more serious problems :\n\n* You don’t explain how  |P(i,j)|  is calculated. Do you add +1 for each edge along the shortest path? This leaves it unclear whether your distance is calculated exactly.\n\n* I feel that many regularity properties would undergo the same type of increase as the one you noted, such as an irregularity, heterogeneity metric or average weighted degree distribution . What were the results (Figure 5 and 24) ? If simpler metrics correlate with emergence similarly, what additional value does your metric provide?", "limitations": "", "rating": 5, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 3, "strengths": "The paper is extremely well contextualized and the presentation of the ideas and results are solids. The quality of the writing makes it really pleasant to read. \n\nIt gives a really interesting direction of research by trying to bridge a gap with neuroscience and trying to create a new framework to link emerging properties and network topologies.", "weaknesses": "Some mathematics would have deserved better explanation in the article itself and not the appendix. \n\n You are measuring the degree of organisation in the networks and show that :\n1. it increases during training\n2. it correlates with previous definitions of emergence\n\nYour are yourself comparing it with the averaged weighted degree distribution of the network in the appendix which seem to correlates as well with emergence. \n\nI feel like your \"degree of emergence\" is itself one dimensional as well, which makes it way less interesting as if it was not. It would be a bit outrageous if we were able such a complex reorganisation with only one global metric. It would much more interesting if you were able to link certain network properties with specific emerging properties.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)