A Survey of Structural Entropy: Theory, Methods, and Applications

1 Introduction

Information lies at the heart of artificial intelligence. From Shannon's entropy [Shannon, 1948] guiding learning algorithms, to the Information Bottleneck [Tishby et al., 2000] shaping representations-virtually every paradigm in AI can be seen as either compressing, transferring, organizing, or extracting meaning from information. Yet, while much of classical information theory in AI is concerned with quantities of uncertainty or relevance at a local or pairwise level, Structural Entropy (SE) offers a distinct and complementary perspective: it captures the global structure and hierarchical uncertainty embedded within complex information systems. At its core, SE provides a rigorous method to quantify and decode the uncertainty embedded in the architecture of a complex system, modeled as a graph or matrix [Li and Pan, 2016]. Its significance for the AI community is twofold, offering both a new metric and a constructive principle for optimization.

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On one hand, SE serves as a fundamental metric to quantify a system's intrinsic uncertainty. A low SE signals a clear, efficient hierarchical organization, implying greater functional coherence and robustness. Conversely, a high SE indicates a system that is difficult to decompose, approaching the behavior of a random graph. This principle is directly applied in areas like privacy-preserving graph analysis, where community structures are deliberately obscured by increasing structural entropy [Liu et al., 2019]. This analytical power enables us to measure properties like the amount of knowledge captured by a model [Wang et al., 2023a], the emergence of collaboration between agents [Su et al., 2025], or the safety and stability of an AI system [Zeng et al., 2025b].

On the other hand, and perhaps more profoundly, SE provides a principled method to discover a system's optimal hierarchical structure. The theory's central insight is that minimizing structural entropy is an optimization process that decodes a system's most efficient information-theoretic abstraction, revealing a nested, hierarchical partitioning of its components. This discovered hierarchy is not arbitrary; it represents the most concise description of the system's organization, akin to finding its "true" community structure [Li and Pan, 2016] or its most effective decision hierarchy [Zeng et al., 2025c]. This constructive capability has already found potent applications across the AI development pipeline-from designing optimal pooling layers in Graph Neural Networks (GNNs) [Wu et al., 2022; Zou et al., 2023; Duan et al., 2024; Ren et al., 2024] and guiding data augmentation [Wu et al., 2023; Wang et al., 2023b], to shaping reward functions in reinforcement learning [Huang et al., 2024; Zeng et al., 2023c; Zeng et al., 2024c] and learning more meaningful representations [Yang et al., 2023; Zeng et al., 2023b; Zeng et al., 2025c].

Despite the rapid proliferation and significant impact of SEbased methods across various AI domains, a comprehensive and systematic review that consolidates these advancements is currently lacking. This paper aims to fill this critical gap. We provide an extensive overview of the theoretical underpinnings of structural entropy, detail the computational methods developed for its calculation and optimization, and survey the learning paradigms that leverage SE for enhanced graph learning and reinforcement learning. Furthermore, we explore its diverse applications in fields such as bioinformatics, transport and geoscience studies, social network analysis, and pattern recognition, as illustrated in Figure 1.

The paper is structured as follows. Section 2 outlines the basic concepts of SE. Section 3 delves into learning methods. Section 4 explores cross-domain applications. Section 5 concludes with future directions and open challenges.

2 Basic Concepts

Structural entropy offers a theoretical foundation for quantifying the complexity and hierarchical organization of structured data. At its core, this framework models a system as a finite set of interacting elements, and characterizes its structural regularities through a tree-based recursive partitioning. In this section, we introduce the basic mathematical constructs of encoding trees and the corresponding structural entropy.

2.1 Encoding Tree of a Finite Set

Let A be a finite set representing the entities in a complex system, such as nodes in a network or data points in a dataset. A natural approach to modeling the structure of such a system is through hierarchical abstraction, in which elements are recursively grouped into nested subsets. This leads to the definition of an encoding tree, which organizes the elements of A into a tree structure where each node corresponds to a subset of A, and the leaves correspond to individual elements.

Definition 2.1 (Encoding Tree of a Finite Set). Given a finite set A, an encoding tree of A is a rooted tree T satisfying the following conditions.

First, the root node is denoted by the empty string λ, and corresponds to the entire set: Tλ = A.

Second, for every node α ∈ T , there exists a nonempty subset Tα ⊆ A. If α has children β0, β1, . . . , βl, then each child is denoted as βj = α · j, where · denotes the concatenation of the index sequence α with the integer j, forming a unique code for the node in the encoding tree. The corresponding subsets Tβ0 , . . . , Tβl form a partition of Tα; that is, Tα = ⊔lj=0 Tβj . Third, each leaf node γ ∈ T corresponds to a singleton

subset of A, that is, Tγ = {a} for some a ∈ A.

Such a tree encodes a recursive decomposition of the set A, forming a layered abstraction often referred to as a spectral hierarchy or multiscale partition.

2.2 Basic Metrics on an Encoding Tree

Consider a non-negative, irreducible matrix A ∈ Rn×n ≥0 , which represents an information system over n elements. Let V = {1, 2, . . . , n} denote the index set of rows and columns in A, corresponding to the objects in the system. An encoding tree for this system is defined over the set V .

To measure the structural properties of a system, we associate to A a stochastic process that models information flow. Let π = (π1, . . . , πn) be the stationary distribution of the Markov chain derived from A by row normalization. The probability of transitioning from object x to object y is defined by

Given a nonempty subset X ⊆ V , we define its volume as

which represents the stationary probability mass of X.

The incoming flow into X is defined by pX = ∑y/∈X ∑x∈X pyx, representing the total probability of transitions from outside X into X.

Similarly, the outgoing flow from X is given by qX = ∑x∈X ∑y /∈X pxy, which quantifies the probability of escaping fromX to the rest of the system.

These quantities provide a probabilistic characterization of how subsets of the system interact with each other under the dynamics encoded by A.

2.3 Structural Entropy under an Encoding Tree

To evaluate how well a given encoding tree T reflects the structure of the information system A, we define the structural entropy of A under T , which captures the information cost of describing the dynamics of A hierarchically along the tree.

Definition 2.2 (Structural Entropy under an Encoding Tree). Let A be a non-negative, irreducible matrix, and let T be an encoding tree over V . The structural entropy of A under T is defined by

where Tα ⊆ V is the subset associated with node α, Vα = VTα is its volume, pα = pVα is the incoming flow into Tα, and α- denotes the parent node of α in the tree.

This quantity measures how well the tree structure aligns with the natural flow of information in the system. Low entropy indicates that the tree captures well-separated and coherent groupings of elements.

To assess the intrinsic complexity of an information system, we consider the optimal encoding tree that minimizes the structural entropy.

Definition 2.3 (Structural Entropy of an Information System). Given a non-negative, irreducible matrix A, the structural entropy of the information system is defined as

where the minimum is taken over all encoding trees T over V .

This value quantifies the minimal cost of encoding the system under an optimal hierarchical abstraction. The dimension of structural entropy is characterized by the depth of the encoding tree T , representing the number of hierarchical partitioning levels. A key example is the 1-dimensional structural entropy (1D SE), corresponding to an encoding tree T1 of depth one where the root is directly partitioned into leaf nodes representing individual entities. In this case, 1D SE simplifies to the Shannon entropy of the stationary distribution π, i.e., H1 (A) = - ∑ni=1 πi log2 πi.

3 Theoretical Foundations

3.1 Foundational Theoretical Works

The theoretical framework of SE was established through pioneering works that formalized its mathematical principles and later expanded them into a broader philosophy of information.

Foundation Theory The concept of SE was introduced by [Li and Pan, 2016] as a principled, information-theoretic framework for quantifying the complexity of a graph's structure. This seminal work defined SE as the minimum code length required to describe a network's hierarchy under a random walk process, effectively bridging graph theory with Shannon's information principles. It established that minimizing SE reveals a graph's intrinsic multi-scale organization. Building on this, [Li, 2024] significantly broadened the theoretical landscape by situating SE within the "mathematical principles of the information world." This work frames SE within the broader context of the Philosophy of AI, formulating fundamental information laws and the Information Science Principles of AI (ISPAI), establishing axiomatic foundations for a comprehensive structural information theory.

Theoretic Analysis Subsequent research has focused on elucidating SE's properties and forging connections with other established graph-theoretic concepts. In social network analysis, for example, [Zhang et al., 2021a] introduced a "togetherness" metric that uses SE reduction to quantify the efficacy of network integration when adding inter-community edges. From a more fundamental perspective, [Liu et al., 2022] established a significant theoretical link between SE and spectral graph theory. Their work proved that for any undirected graph, the difference between SE and the von Neumann graph entropy (VNE) is bounded within the interval [0, γ], formally connecting SE to the spectral properties of the graph Laplacian.

ExtensionWork Recognizing that the original formulation targeted simple graphs, significant research has extended SE's principles to more complex network structures and dynamic scenarios. To handle directed graphs, multiple approaches have emerged; one introduced a localized 2D SE using a flow-based algorithm for financial networks [Yao et al., 2019], while another proposed a random walk-based entropy (RWE) with a proven theoretical connection to VNE [Zhang et al., 2024]. For heterogeneous relational data, multi-relational SE was developed using a random surfing model that jointly optimizes node and relation selection to enhance interpretability [Cao et al., 2024a]. To handle evolving networks, [Yang et al., 2024b] addressed the challenge of applying SE to dynamic graphs by proposing incremental computation paradigms, which include two distinct strategies for efficiently updating encoding trees and re-optimizing SE in response to network changes. The applicability of SE has also expanded into new domains, with extensions such as an "infected tree entropy" for source localization in information cascades [Zhang et al., 2023] and an SE-guided probabilistic coding variant that regularizes embeddings in natural language understanding tasks [Huang et al., 2024].

Discussion The evolution of SE from its core principles to a broad family of specialized variants highlights its versatility.

However, this expansion also introduces key challenges. Different SE formulations for various graph structures prioritize different network properties, and their comparative efficacy remains an open area of investigation. Incremental methods for dynamic graphs are efficient but may have limitations when faced with complex structural revolutions. A significant challenge is the development of a unified SE framework that can integrate these variants for complex, heterogeneous, and dynamic networks. While the link to spectral entropy has strengthened SE's theoretical underpinnings, a deeper exploration of its relationship with other graph-theoretic objectivessuch as modularity [Newman and Girvan, 2004], the map equation [Rosvall and Bergstrom, 2008], and various centrality measures-is needed to fully situate it within network science. Future research should prioritize comprehensive comparative analyses of existing SE variants and the development of a unified framework applicable to a broader range of structures, including hypergraphs and multigraphs, to address the complexity of modern information systems.

3.2 Computational Approaches

As illustrated in Table 1 and Figure 1, the practical algorithms for SE minimization can be categorized by their optimization strategies and the types of constraints they employ.

Heuristic Methods Initial approaches to SE minimization relied on heuristic strategies to address various flat and hierarchical clustering problems. The first SE-based chromatin domain detector, deDoc, pioneered this area by employing a greedy merging strategy to identify genomic structures [Li et al., 2018]. This principle was adapted for single-cell omics in the SEAT framework, which applies SE minimization to cell-cell graphs to reveal nested subpopulations and analyze functional diversity [Chen and Li, 2022]. In parallel, HCSE emerged as a widely used, general-purpose hierarchical clustering framework [Pan et al., 2021]. By recursively isolating the sparsest levels of a hierarchy through SE-guided optimization, HCSE automatically determines the appropriate hierarchy depth without requiring hyperparameters. Its competitive performance against established methods like LOUVAIN [Newman and Girvan, 2004] on real-world networks validated heuristic SE minimization as a practical and effective approach to graph clustering.

Constrained Optimization More recent advances in SE minimization have incorporated domain-specific constraints to tackle complex biological and computational challenges. These methods can be systematically categorized by their constraint-handling mechanisms. One major category enforces spatial or index continuity constraints, which is crucial for applications like genomic domain detection where chromatin regions have an inherent linear organization. SuperTAD established this paradigm by combining interval dynamic programming with SE minimization, ensuring that detected topologically associating domains (TADs) preserve genomic contiguity [Zhang et al., 2021b]. To improve efficiency, SuperTAD2 introduced matrix discretization and approximation strategies, achieving a significant speedup (e.g., 10x) while maintaining spatial continuity [Ling et al., 2024]. The same continuity principles were extended to single-cell analysis in deDoc2, which uses dynamic programming to resolve contiguous domains and capture cell cycle-dependent chromatin dynamics [Li et al., 2023]. A second category integrates semi-supervised constraints, such as labels and pairwise relationships, to guide the minimization process. The framework by [Zeng et al., 2024a] unifies these constraints by reformulating SE to penalize violations of must-link/cannot-link pairs during encoding tree construction. To handle massive datasets, [Zeng et al., 2025a] scaled this approach by introducing graph sampling and incremental cluster insertion, reducing time complexity from quadratic to nearly linear while preserving high accuracy through theoretically guaranteed approximations.

Gradient-Based Optimization and Differentiable SE A significant trend is the integration of SE principles with gradient-based optimization, enabled by the development of differentiable SE variants compatible with deep learning. These approaches typically transform the discrete nature of SE into a continuous objective by using soft assignment matrices, which represent cluster memberships probabilistically. This innovation allows SE to guide end-to-end learning via standard backpropagation. LSEnet exemplifies this by creating a differentiable SE formulation within hyperbolic space, facilitating cluster-free hierarchical partitioning [Sun et al., 2024]. Similarly, the DeSE framework pioneers a "soft assignment structural entropy" that enables the co-design of graph topology and cluster assignments within a unified end-to-end model [Zhang et al., 2025]. By making SE differentiable, these methods bridge the gap between information-theoretic clustering and deep learning, allowing for simultaneous optimization of representations and hierarchical structures.

Discussion The evolution of SE minimization algorithms reveals a fundamental trade-off between the efficiency of heuristics, the domain specificity of constrained optimization, and the integrative power of gradient-based methods. Heuristic approaches like HCSE and deDoc offer scalability and generality but provide less flexibility for incorporating complex domain knowledge. In contrast, constrained methods like SuperTAD and semi-supervised frameworks integrate specific knowledge to enhance relevance and accuracy, though sometimes at the cost of broader applicability. The emergence of gradient-based techniques such as LSEnet and DeSE signals a promising convergence, allowing SE to be integrated directly into deep learning pipelines. While these methods can jointly optimize representations and partitions, they introduce considerations of model complexity, hyperparameter sensitivity, and the interpretability of the learned structures. Distinct from these paradigms, game-theoretic approaches like CoDeSEG offer another promising direction, leveraging game theory for near-linear-time community detection [Xian et al., 2025]. Ultimately, open challenges persist in achieving an optimal balance of scalability, performance, and interpretability, particularly for dynamic or heterogeneous networks. Future progress may lie in hybridizing these computational paradigms to harness their respective strengths.

4 LearningMethods via Structural Entropy

4.1 Graph Learning Paradigms

SE has emerged as a powerful information-theoretic tool for graph representation learning, enabling novel architectures and frameworks. It provides a principled, quantitative measure of a graph's inherent hierarchical organization, addressing a fundamental challenge in the field: how to learn from unstructured relational data in a robust and meaningful way.

Graph Pooling In graph-level tasks, pooling operations are essential for downsampling nodes to create a compact representation of the entire graph. However, traditional methods often suffer from information loss and limited expressiveness. Since the essence of a graph's function is often encoded in its multi-level community structure, methods that iteratively pool local nodes risk destroying this larger organizational map. SE Pooling [Wu et al., 2022] directly addresses this by using global SE minimization to generate all cluster assignments simultaneously, preventing the cumulative damage to local structures seen in stepwise pooling. Further advancing this, the Hi-PART framework [Ren et al., 2024] employs SE to construct Hierarchical Partition Trees. This approach explicitly models the nested, multi-scale nature of graph structures, enabling it to capture richer detail and surpass the expressive power of the standard Weisfeiler-Lehman test.

Structure Augmentation Graph structure augmentation, a cornerstone of modern self-supervised learning on graphs, involves creating altered "views" of a graph to teach a model what features are essential versus what is noise. Random augmentations risk discarding crucial information, so SE provides a principled way to guide this process by identifying and preserving the graph's intrinsic, low-entropy structure. A stable "anchor view" representing the graph's core semantic information is created. SEGA [Wu et al., 2023] constructs such a view by finding an encoding tree with minimal SE, thus filtering out noise while maintaining the essential community hierarchy for more effective contrastive learning. Similarly, USER [Wang et al., 2023b] incorporates SE directly into its objective function to learn an "innocuous graph." By minimizing SE, it seeks an intrinsic structure with well-defined communities that is robust to perturbations. Other methods use SE in a multi-stage optimization to refine graph structures. SE-GSL [Zou et al., 2023] first maximizes 1D SE to enhance information content, then minimizes high-dimensional SE to abstract the graph into robust hierarchical trees, and finally uses deduction SE to guide an adaptive graph reconstruction. Extending this, [Duan et al., 2024] also uses SE to build minimal-SE encoding trees, leveraging this core structure to guide the fusion of multiple graph views based on community influence.

Graph Kernels & Embedding Dimensions SE has also enhanced classical graph analysis tools. Graph kernels that measure similarity between graphs often struggle to capture complex structural patterns. The Hierarchical Abstract Graph Kernel (HAGK) [Yang et al., 2024a] leverages SE to generate multi-level, abstract representations of each graph's intrinsic organization. The kernel then compares these SE-derived hierarchical structures, offering a more profound similarity measure than one based on local features alone. Furthermore, SE provides a principled solution to a persistent challenge in graph learning: determining the optimal embedding dimension. MGEDE [Yang et al., 2023] calculates a novel high-level SE from multi-order adjacency matrices and minimizes it alongside attribute entropy. This process explicitly determines the most suitable number of embedding dimensions, balancing representational capacity with model complexity.

Discussion SE provides a unifying, information-theoretic foundation for graph learning, shifting the paradigm from heuristic-driven methods to a more principled approach focused on quantifying and optimizing a graph's inherent organization. By minimizing structural uncertainty, SE acts as a powerful regularizer, guiding models to learn from the essential hierarchical information within data while filtering out stochastic noise. This approach underpins diverse advancements, from constructing robust anchor views in SEGA to enabling expressive structural comparisons in kernels like HAGK. Future work will likely focus on bridging SE with other theoretical frameworks and scaling these methods to handle the dynamics of massive real-world networks.

4.2 Reinforcement Learning Integration

Integrating SE principles into reinforcement learning RL offers novel approaches to address long-standing challenges in hierarchical learning, exploration, and multi-agent coordination. This integration equips RL agents with a mechanism to perceive and decompose complex problems, moving beyond flat state-action spaces toward a more structured, human-like understanding of their environment.

SE-Guided RL Frameworks SE provides a formal tool to automatically discover hierarchical structure in the fundamental components of an RL problem. To address the "curse of dimensionality" in State & Action Abstraction, SE helps cluster vast state or action spaces into manageable, hierarchical abstractions. The SISA framework [Zeng et al., 2023c] automatically uncovers this structure by optimizing an encoding tree from a state graph, using SE to create an efficient and task-relevant state representation. In the multi-agent domain, SIRD [Zeng et al., 2023b] reframes role discovery as hierarchical action space clustering by using SE minimization on an action graph to define distinct roles. For Intrinsic Motivation for Exploration, the SI2E framework [Zeng et al., 2024c] formulates an intrinsic reward by maximizing value-conditional SE. This reward, derived from an SE-minimized state-action graph, encourages the agent to explore states where its actions lead to the greatest reduction in uncertainty about the environment's causal structure. More comprehensive frameworks like SIDM [Zeng et al., 2025c] utilize SE at multiple stages to structure the entire Decision-Making and Skill Discovery process. COLLAB-MARL [Su et al., 2025] uses SE to quantify interaction complexity, revealing emergent coalition structures by minimizing the SE of a dynamic agent interaction graph.

Discussion The integration of SE into reinforcement learning allows for a departure from "black-box" agents that learn through brute-force trial and error. It provides the mathematical tools to build a cognitive architecture for an agent, enabling it to parse its world into a hierarchy of concepts, skills, and goals. A structured understanding allows an agent to reason abstractly, plan over longer horizons, and transfer knowledge between related tasks-hallmarks of intelligent behavior. Frameworks like SISA and SIRD demonstrate how SE can automate the discovery of abstractions, a task that has historically required significant human engineering. Methods like SI2E show that exploration need not be random; it can be a deliberate process of information seeking aimed at resolving structural uncertainty. This results in agents that are not only more sample-efficient but also more interpretable. Looking forward, a key direction is scaling these methods to more complex, partially observable environments. Furthermore, incorporating SE into meta-RL and curriculum learning is a particularly promising frontier. This would enable agents to autonomously identify the hierarchical structure across a distribution of tasks, allowing them to learn how to learn by recognizing and adapting to shared structures. This could be a critical step toward developing more general and adaptive artificial intelligence.

5 Cross-Domain Applications

5.1 Bioinformatics

SE has demonstrated notable effectiveness in biological system analysis by enabling the precise detection of hierarchical structures and patterns.

The application of SE minimization in genomics began with foundational work in cancer subtyping, which correlated entropy-minimized gene expression patterns with clinical outcomes [Li et al., 2016]. This principle was soon extended to the analysis of 3D genomic architecture. A major area of impact has been the identification of TADs from Hi-C data. The first SE-based detector, deDoc, pioneered this by identifying large-scale TAD-like structures [Li et al., 2018]. Subsequent methods focused on refining this process. SuperTAD, for instance, employed dynamic programming to better resolve hierarchical chromatin domains [Zhang et al., 2021b], and its successor, SuperTAD2, significantly accelerated this process through matrix discretization [Ling et al., 2024].

The focus on hierarchical analysis has naturally translated to the higher resolution of single-cell omics. In this domain, tools like SEAT minimize global uncertainty in cell-cell graphs to improve pseudo-time inference by detecting hierarchical subpopulations across various omics datasets [Chen and Li, 2022]. SE has also been crucial for decoding chromatin dynamics at the single-cell level. deTOKI uses SE principles to analyze regulatory dynamics within individual cells [Li et al., 2021], while the more recent deDoc2 uses dynamic programming to resolve how single-cell chromatin hierarchies change with the cell cycle, a significant advance in the field [Li et al., 2023].

5.2 Transport and Geoscience Studies

SE minimization is also being applied to complex systems in transportation and geoscience, offering novel methods for spatio-temporal modeling and knowledge quantification.

In transportation, the transformer-basedMultiSPANS framework improves traffic forecasting by using SE minimization to optimize its spatial attention mechanism. This approach generates hierarchical encoding trees of the road network, enhancing the model's interpretability and ability to capture multi-range spatial dependencies [Zou et al., 2024].

In geoscience, SE provides a structural lens for analyzing the evolution of scientific knowledge. The "Scientific X-ray" method, for example, constructs "idea trees" from citation networks to visualize how scientific ideas are inherited and to identify high-potential research directions [Li et al., 2022]. Building on this structural approach, another framework redefines knowledge quantification by measuring differences in hierarchical disorder within these same networks. This method produces a robust Knowledge Quantification Index that successfully identifies overlooked but influential works and Nobel Prize-winning topics. By focusing on information structure over semantic content, it provides a more objective measure of scientific impact that is less susceptible to manipulation [Wang et al., 2023a].

5.3 Social Networks

SE minimization principles provide interpretable, hierarchyaware frameworks that have significantly advanced forensic analysis, adversarial modeling, and knowledge discovery within social networks.

In bot detection, SE is applied in diverse ways. UnDBot offers an unsupervised approach, leveraging heterogeneous SE on multi-relational graphs to decode bot networks through community labeling [Peng et al., 2024]. For greater robustness against adversarial attacks, SEBot integrates SE with multi-view contrastive learning to reduce uncertainty via hierarchical community encoding [Yang et al., 2024c]. Moving from detection to proactive attack modeling, the SIASM framework uses conditional SE minimization to optimize follower selection, thereby maximizing a socialbot's influence while simultaneously evading detection [Zeng et al., 2024b].

SE is also pivotal for analyzing information dissemination and system resilience. The SI2AF framework, for instance, evaluates the robustness of news detection systems by analyzing uncertainty and community hierarchies in user-post interactions. It introduces an SE-based influence metric that combines content relevance with community structure, enabling more realistic adversarial simulations [Zeng et al., 2025b].

For social event detection, SE facilitates the identification of emergent topics without predefined event counts. HISEvent achieves this by incrementally building message graphs with 1D SE and then hierarchically detecting events using 2D SE optimization [Cao et al., 2024b]. Addressing privacy, ADPSEMEvent integrates adaptive differential privacy into the 2D SE minimization process, enabling accurate event detection while protecting user data [Yang et al., 2024d]. To further improve performance, HyperSED models social messages in hyperbolic space, employing differentiable, SE-guided partitioning to significantly enhance both the accuracy and speed of event detection over previous methods [Yu et al., 2025].

5.4 Pattern Recognition

SE has driven significant advances across pattern recognition by providing a principled way to model and optimize hierarchical structures in visual, textual, and speech data.

In visual computing, SE enhances feature preservation and segmentation. For object detection, SLED is an unsupervised framework that detects skin lesions by minimizing the entropy of multiscale superpixel graphs, using the resulting structural information to isolate outliers [Zeng et al., 2023a]. For generalpurpose segmentation, SIT-HSS first constructs a graph and then performs iterative merging, resulting in superior hierarchical superpixel segmentations that outperform other unsupervised methods [Xie et al., 2025].

In natural language processing, SE enables the effective integration of syntactic structures into text models. HiTIN, a hierarchy-aware tree isomorphism network, leverages SE to inject the syntactic hierarchy of a sentence directly into its graph-based representation for more accurate hierarchical text classification [Zhu et al., 2023]. Similarly, MASGCN applies an SE-based loss function to guide a graph convolutional network in capturing syntactic features for the specific task of aspect-based sentiment analysis, achieving state-of-the-art performance [Huang et al., 2025].

In speech processing, SE has spurred innovations in both compression and voice conversion by identifying meaningful, cluster-based structures in audio signals. SECodec introduces a compressive speech representation by applying SE to cluster feature nodes, which allows for efficient compression with reduced distortion [Wang et al., 2025a]. In a similar spirit, the SEVC framework for voice conversion uses 2D SE to group reference speech frames into semantic clusters. By mapping a source speaker's frames to these target clusters, it ensures a high-fidelity speaker transformation [Wang et al., 2025b].

6 Conclusion

This survey has systematically charted the landscape of SE, consolidating its theoretical underpinnings, computational methodologies, and burgeoning applications across the field of artificial intelligence. By tracing its evolution from a novel information-theoretic measure for graph complexity to a versatile optimization principle integrated into advanced learning paradigms, we have aimed to fill a critical void in the literature. Our review reveals that SE offers a distinct and powerful lens for AI, shifting the focus from local, pairwise interactions to the global, hierarchical organization of complex systems. We have detailed how minimizing SE serves as a constructive principle to uncover a system's most efficient informationtheoretic abstraction, a capability that has been effectively harnessed in graph learning for tasks like pooling and structure augmentation, and in reinforcement learning for discovering hierarchical state-action spaces and guiding exploration. The diverse cross-domain applications-from decoding genomic architectures in bioinformatics to modeling influence in social networks and enhancing pattern recognition-underscore SE's broad utility and impact.

In conclusion, Structural Entropy provides more than just a new set of tools; it offers a principled way of thinking about information and structure in a hierarchical world. By providing a rigorous method to quantify and optimize the organizational uncertainty within data, SE holds the potential to guide the development of AI systems that are not only more powerful but also more interpretable, robust, and theoretically grounded. As the complexity of our models and the systems they interact with continues to grow, the principles of Structural Entropy will be indispensable in our quest to build the next generation of artificial intelligence.

Acknowledgments

This work has been supported in part by NSFC through grants 62322202, 62441612 and 62432006, Local Science and Technology Development Fund of Hebei Province Guided by the Central Government of China through grant 246Z0102G, the "Pioneer" and "Leading Goose" R&D Program of Zhejiang through grant 2025C02044, National Key Laboratory under grant 241-HF-D07-01, Hebei Natural Science Foundation through grant F2024210008, National Key R&D Program of China through grant 2021YFB3500700, and State Key Laboratory of Complex & Critical Software Environment (CCSE2024ZX-20).