Hierarchical clustering is a fundamental machine-learning technique for grouping data points into dendrograms. However, existing hierarchical clustering methods encounter two primary challenges: 1) Most methods specify dendrograms without a global objective. 2) Graph-based methods often neglect the significance of graph structure, optimizing objectives on complete or static predefined graphs. In this work, we propose Hyperbolic Continuous Structural Entropy neural networks, namely HypCSE, for structure-enhanced continuous hierarchical clustering. Our key idea is to map data points in the hyperbolic space and minimize the relaxed continuous structural entropy (SE) on structure-enhanced graphs. Specifically, we encode graph vertices in hyperbolic space using hyperbolic graph neural networks and minimize approximate SE defined on graph embeddings. To make the SE objective differentiable for optimization, we reformulate it into a function using the lowest common ancestor (LCA) on trees and then relax it into continuous SE (CSE) by the analogy of hyperbolic graph embeddings and partitioning trees. To ensure a graph structure that effectively captures the hierarchy of data points for CSE calculation, we employ a graph structure learning (GSL) strategy that updates the graph structure during training. Extensive experiments on seven datasets demonstrate the superior performance of HypCSE.

Bot detection is crucial for combating misinformation and preserving the authenticity of online interactions on social media. However, the increasing sophistication of bots in mimicking genuine accounts and evading detection has created an ongoing arms race between detection systems and modeling techniques. In this paper, we propose a novel Structural Information principles-based Adversarial framework, namely SIAMD, designed to Model bot behaviors and achieve proactive Detection. This framework begins by organizing multi-relational interactions between user accounts and social messages into a unified heterogeneous structure, incorporating structural entropy to quantify the uncertainty inherent in historical activities. The high-dimensional entropy is then minimized to uncover a layered hierarchy within account communities, which facilitates activity determination and account selection in behavioral modeling for bot accounts. For each modeled bot and its selected account, SIAMD extracts historical messages and user descriptions to construct prompts and integrates large language models to generate the associated message content. By embedding synthetic message vertices and establishing multi-relational interactions within the original heterogeneous network, SIAMD achieves network evolution in both structure and content, thereby enhancing graph-based proactive detection in an adversarial manner. Extensive comparative experiments on well-established real-world datasets demonstrate that SIAMD significantly and consistently outperforms state-of-the-art detection baselines for social bots in terms of effectiveness, generalizability, robustness, and interpretability.

Large Language Models (LLMs) excel in tasks such as retrieval and question answering but require updates to incorporate new knowledge and reduce inaccuracies and hallucinations. Traditional updating methods, like fine-tuning and incremental learning, face challenges such as overfitting and high computational costs. Knowledge Editing (KE) provides a promising alternative but often overlooks the Knowledge Element Overlap (KEO) phenomenon, where multiple triplets share common elements, leading to editing conflicts. We identify the prevalence of KEO in existing KE datasets and show its significant impact on current KE methods, causing performance degradation in handling such triplets. To address this, we propose a new formulation, Knowledge Set Editing (KSE), and introduce SetKE, a method that edits sets of triplets simultaneously. Experimental results demonstrate that SetKE outperforms existing methods in KEO scenarios on mainstream LLMs. Additionally, we introduce EditSet, a dataset containing KEO triplets, providing a comprehensive benchmark.

Classical information theory, a cornerstone of artificial intelligence, is fundamentally limited by its local perspective, often analyzing pairwise interactions while ignoring the larger, hierarchical architecture of complex systems. Structural entropy (SE) presents a paradigm shift, extending Shannon entropy to quantify information on a global scale and measure the uncertainty embedded in a system's organizational hierarchy. Although its applications have broadened significantly from its origins in community detection across diverse AI domains, a systematic synthesis of its theory, computational methods, and applications is currently lacking. This survey provides a comprehensive overview of SE to fill this critical void in the literature. We offer a detailed examination of its theoretical foundations, computational frameworks, and key learning paradigms, with a focus on its integration with graph learning and reinforcement learning. Through an exploration of its diverse applications, we highlight the power of SE to advance graph-based analysis and modeling. Finally, we discuss key challenges and future research opportunities for incorporating SE principles into the development of more interpretable and theoretically grounded AI systems.

Although Graph Neural Networks (GNNs) have shown promising potential in fake news detection, they remain highly vulnerable to adversarial manipulations within social networks. Existing methods primarily establish connections between malicious accounts and individual target news to investigate the vulnerability of graph-based detectors, while they neglect the structural relationships surrounding targets, limiting their effectiveness in robustness evaluation. In this work, we propose a novel Structural Information principles-guided Adversarial Attack Framework, namely SI2AF, which effectively challenges graph-based detectors and further probes their detection robustness. Specifically, structural entropy is introduced to quantify the dynamic uncertainty in social engagements and identify hierarchical communities that encompass all user accounts and news posts. An influence metric is presented to measure each account's probability of engaging in random interactions, facilitating the design of multiple agents that manage distinct malicious accounts. For each target news, three attack strategies are developed through multi-agent collaboration within the associated subgraph to optimize evasion against black-box detectors. By incorporating the adversarial manipulations generated by SI2AF, we enrich the original network structure and refine graph-based detectors to improve their robustness against adversarial attacks. Extensive evaluations demonstrate that SI2AF significantly outperforms state-of-the-art baselines in attack effectiveness with an average improvement of 16.71%, and enhances GNN-based detection robustness by 41.54% on average.

Large language models (LLMs) have achieved unprecedented performance by leveraging vast pretraining corpora, yet their performance remains suboptimal in knowledge-intensive domains such as medicine and scientific research, where high factual precision is required. While synthetic data provides a promising avenue for augmenting domain knowledge, existing methods frequently generate redundant samples that do not align with the model’s true knowledge gaps. To overcome this limitation, we propose a novel Structural Entropy-guided Knowledge Navigator (SENATOR) framework that addresses the intrinsic knowledge deficiencies of LLMs. Our approach employs the Structure Entropy (SE) metric to quantify uncertainty along knowledge graph paths and leverages Monte Carlo Tree Search (MCTS) to selectively explore regions where the model lacks domain-specific knowledge. Guided by these insights, the framework generates targeted synthetic data for supervised fine-tuning, enabling continuous self-improvement. Experimental results on LLaMA-3 and Qwen2 across multiple domain-specific benchmarks show that SENATOR effectively detects and repairs knowledge deficiencies, achieving notable performance improvements.

Diffusion-based generative methods have shown promising potential for modeling trajectories from offline reinforcement learning (RL) datasets, and hierarchical diffusion has been introduced to mitigate variance accumulation and computational challenges in long-horizon planning tasks. However, existing approaches typically assume a fixed two-layer diffusion hierarchy with a single predefined temporal scale, which limits adaptability to diverse downstream tasks and reduces flexibility in decision making. In this work, we propose SIHD, a novel Structural Information-based Hierarchical Diffusion framework for effective and stable offline policy learning in long-horizon environments with sparse rewards. Specifically, we analyze structural information embedded in offline trajectories to construct the diffusion hierarchy adaptively, enabling flexible trajectory modeling across multiple temporal scales. Rather than relying on reward predictions from localized sub-trajectories, we quantify the structural information gain of each state community and use it as a conditioning signal within the corresponding diffusion layer. To reduce overreliance on offline datasets, we introduce a structural entropy regularizer that encourages exploration of underrepresented states while avoiding extrapolation errors from distributional shifts. Extensive evaluations show that SIHD significantly outperforms state-of-the-art baselines in decision-making performance and demonstrates superior generalization across diverse scenarios.

Multi-agent cooperation is essential for tasks that require collaboration to achieve optimal performance or cannot be completed by individual agents alone. These tasks often necessitate a divide-and-conquer strategy, where subgoals are allocated to individual agents or groups. By integrating coalition formation concepts from cooperative game theory, we demonstrate the implicit learning of coalition formation and task assignments, resulting in emergent cooperative behavior. We propose a novel COaLition LABeling technique for Multi-Agent Reinforcement Learning (COLLAB-MARL) to encourage coalition formation and introduce a structural entropy measure to detect the emergence of coalitions and cooperative behavior. Compared to classical MARL methods, COLLAB-MARL is more effective, explainable, and easier to implement. Experiments on state-of-the-art cooperative MARL benchmarks show that our method’s mean return outperforms the strongest baselines by 8.4% on average. Additionally, visualization and structural entropy analysis reveal that COLLAB-MARL effectively learns meaningful cooperative behavior. The source code is available at https://github.com/SELGroup/collab.

Hierarchical Reinforcement Learning (HRL) is a promising approach for managing task complexity across multiple levels of abstraction and accelerating long-horizon agent exploration. However, the effectiveness of hierarchical policies heavily depends on prior knowledge and manual assumptions about skill definitions and task decomposition. In this paper, we propose a novel Structural Information principles-based framework, namely SIDM, for hierarchical Decision Making in both single-agent and multi-agent scenarios. Central to our work is the utilization of structural information embedded in the decision-making process to adaptively and dynamically discover and learn hierarchical policies through environmental abstractions. Specifically, we present an abstraction mechanism that processes historical state-action trajectories to construct abstract representations of states and actions. We define and optimize directed structural entropy—a metric quantifying the uncertainty in transition dynamics between abstract states—to discover skills that capture key transition patterns in RL environments. Building on these findings, we develop a skill-based learning method for single-agent scenarios and a role-based collaboration method for multi-agent scenarios, both of which can flexibly integrate various underlying algorithms for enhanced performance. Extensive evaluations on challenging benchmarks demonstrate that our framework significantly and consistently outperforms state-of-the-art baselines, improving the effectiveness, efficiency, and stability of policy learning by up to 32.70%, 64.86%, and 88.26%, respectively, as measured by average rewards, convergence timesteps, and standard deviations.

Graph kernels have been regarded as a successful tool for handling a variety of graph applications since they were proposed. However, most of the proposed graph kernels are based on the R-convolution framework, which decomposes graphs into a set of substructures at the same abstraction level and compares all substructure pairs equally; these methods inherently overlook the utility of the hierarchical structural information embedded in graphs. In this paper, we proposeHierarchicalAbstractingGraphKernels (HAGK), a novel set of graph kernels that compare graphs’ hierarchical substructures to capture and utilize the latent hierarchical structural information fully. Instead of generating non-structural substructures, we reveal each graph’s hierarchical substructures by constructing itshierarchical abstracting, specifically, the hierarchically organized nested node sets adhering to the principle of structural entropy minimization. To compare a pair of hierarchical abstractings, we propose two novel substructure matching approaches,Local Optimal Matching(LOM) andPriority Ordering Matching(POM), to find appropriate matching between the substructures by different strategies recursively. Extensive experiments demonstrate that the proposed kernels are highly competitive with the existing state-of-the-art graph kernels, and verify that the hierarchical abstracting plays a significant role in the improvement of the kernel performance.

Semi-supervised clustering leverages prior information in the form of constraints to achieve higher-quality clustering outcomes. However, most existing methods struggle with large-scale datasets owing to their high time and space complexity. Moreover, they encounter the challenge of seamlessly integrating various constraints, thereby limiting their applicability. In this paper, we presentScalableSemi-supervised clustering viaStructuralEntropy (SSSE), a novel method that tackles scalable datasets with different types of constraints from diverse sources to perform both semi-supervised partitioning and hierarchical clustering, which is fully explainable compared to deep learning-based methods. Specifically, we design objectives based on structural entropy, integrating constraints for semi-supervised partitioning and hierarchical clustering. To achieve scalability on data size, we develop efficient algorithms based on graph sampling to reduce the time and space complexity. To achieve generalization on constraint types, we formulate a uniform view for widely used pairwise and label constraints. Extensive experiments on real-world clustering datasets at different scales demonstrate the superiority of SSSE in clustering accuracy and scalability with different constraints. Additionally, Cell clustering experiments on single-cell RNA-seq datasets demonstrate the functionality of SSSE for biological data analysis.

As one of the most common tasks in graph data analysis, node classification is frequently solved by using graph structure learning (GSL) techniques to optimize graph structures and learn suitable graph neural networks. Most of the existing GSL methods focus on fusing different structural features (basic views) extracted from the graph, but very little graph semantics, like hierarchical communities, has been incorporated. Thus, they might be insufficient when dealing with the graphs containing noises from real-world complex systems. To address this issue, we propose a novel and effective GSL framework for node classification based on the structural information theory. Specifically, we first prove that an encoding tree with the minimal structural entropy could contain sufficient information for node classification and eliminate redundant noise via the graph's hierarchical abstraction. Then, we provide an efficient algorithm for constructing the encoding tree to enhance the basic views. Combining the community influence deduced from the encoding tree and the prediction confidence of each view, we further fuse the enhanced views to generate the optimal structure. Finally, we conduct extensive experiments on a variety of datasets. The results demonstrate that our method outperforms the state-of-the-art competitors on effectiveness and robustness.

The importance of effective detection is underscored by the fact that socialbots imitate human behavior to propagate misinformation, leading to an ongoing competition between socialbots and detectors. Despite the rapid advancement of reactive detectors, the exploration of adversarial socialbot modeling remains incomplete, significantly hindering the development of proactive detectors. To address this issue, we propose a mathematical Structural Information principles-based Adversarial Socialbots Modeling framework, namely SIASM, to enable more accurate and effective modeling of adversarial behaviors. First, a heterogeneous graph is presented to integrate various users and rich activities in the original social network and measure its dynamic uncertainty as structural entropy. By minimizing the high-dimensional structural entropy, a hierarchical community structure of the social network is generated and referred to as the optimal encoding tree. Secondly, a novel method is designed to quantify influence by utilizing the assigned structural entropy, which helps reduce the computational cost of SIASM by filtering out uninfluential users. Besides, a new conditional structural entropy is defined between the socialbot and other users to guide the follower selection for network influence maximization. Extensive and comparative experiments on both homogeneous and heterogeneous social networks demonstrate that, compared with state-of-the-art baselines, the proposed SIASM framework yields substantial performance improvements in terms of network influence (up to 16.32%) and sustainable stealthiness (up to 16.29%) when evaluated against a robust detector with 90% accuracy.

Traditional information theory provides a valuable foundation for Reinforcement Learning (RL), particularly through representation learning and entropy maximiza tion for agent exploration. However, existing methods primarily concentrate on modeling the uncertainty associated with RL’s random variables, neglecting the in herent structure within the state and action spaces. In this paper, we propose a novel Structural Information principles-based Effective Exploration framework, namely SI2E. Structural mutual information between two variables is defined to address the single-variable limitation in structural information, and an innovative embedding principle is presented to capture dynamics-relevant state-action representations. The SI2E analyzes value differences in the agent’s policy between state-action pairs and minimizes structural entropy to derive the hierarchical state-action struc ture, referred to as the encoding tree. Under this tree structure, value-conditional structural entropy is defined and maximized to design an intrinsic reward mechanism that avoids redundant transitions and promotes enhanced coverage in the state-action space. Theoretical connections are established between SI2E and classical information-theoretic methodologies, highlighting our framework’s rationality and advantage. Comprehensive evaluations in the MiniGrid, MetaWorld, and DeepMind Control Suite benchmarks demonstrate that SI2E significantly outperforms state-of-the-art exploration baselines regarding final performance and sample efficiency, with maximum improvements of 37.63% and 60.25%, respectively.

Semi-supervised clustering techniques have emerged as valuable tools for leveraging prior information in the form of constraints to improve the quality of clustering outcomes. Despite the proliferation of such methods, the ability to seamlessly integrate various types of constraints remains limited. While structural entropy has proven to be a powerful clustering approach with wide-ranging applications, it has lacked a variant capable of accommodating these constraints. In this work, we present Semi-supervised clustering via Structural Entropy (SSE), a novel method that can incorporate different types of constraints from diverse sources to perform both partitioning and hierarchical clustering. Specifically, we formulate a uniform view for the commonly used pairwise and label constraints for both types of clustering. Then, we design objectives that incorporate these constraints into structural entropy and develop tailored algorithms for their optimization. We evaluate SSE on nine clustering datasets and compare it with eleven semi-supervised partitioning and hierarchical clustering methods. Experimental results demonstrate the superiority of SSE on clustering accuracy with different types of constraints. Additionally, the functionality of SSE for biological data analysis is demonstrated by cell clustering experiments conducted on four single-cell RNA-seq datasets.

Structural Entropy (SE) measures the structural information contained in a graph. Minimizing or maximizing SE helps to reveal or obscure the intrinsic structural patterns underlying graphs in an interpretable manner, finding applications in various tasks driven by networked data. However, SE ignores the heterogeneity inherent in the graph relations, which is ubiquitous in modern networks. In this work, we extend SE to consider heterogeneous relations and propose the first metric for multi-relational graph structural information, namely, Multi-relational Structural Entropy (MrSE). To this end, we first cast SE through the novel lens of the stationary distribution from random surfing, which readily extends to multi-relational networks by considering the choices of both nodes and relation types simultaneously at each step. The resulting MrSE is then optimized by a new greedy algorithm to reveal the essential structures within a multi-relational network. Experimental results highlight that the proposed MrSE offers a more insightful interpretation of the structure of multi-relational graphs compared to SE. Additionally, it enhances the performance of two tasks that involve real-world multi-relational graphs, including node clustering and social event detection.

Structural entropy is a metric that measures the amount of information embedded in graph structure data under a strategy of hierarchical abstracting. To measure the structural entropy of a dynamic graph, we need to decode the optimal encoding tree corresponding to the best community partitioning for each snapshot. However, the current methods do not support dynamic encoding tree updating and incremental structural entropy computation. To address this issue, we propose Incre-2dSE , a novel incremental measurement framework that dynamically adjusts the community partitioning and efficiently computes the updated structural entropy for each updated graph. Specifically, Incre-2dSE includes incremental algorithms based on two dynamic adjustment strategies for two-dimensional encoding trees, i.e., the naive adjustment strategy and the node-shifting adjustment strategy , which support theoretical analysis of updated structural entropy and incrementally optimize community partitioning towards a lower structural entropy. We conduct extensive experiments on 3 artificial datasets generated by Hawkes Process and 3 real-world datasets. Experimental results confirm that our incremental algorithms effectively capture the dynamic evolution of the communities, reduce time consumption, and provide great interpretability.

Research on social bot detection plays a crucial role in maintaining the order and reliability of information dissemination while increasing trust in social interactions. The current mainstream social bot detection models rely on black-box neural network technology, for example, Graph Neural Network, Transformer, and so on, which lacks interpretability. In this work, we present UnDBot, a novel unsupervised, interpretable, yet effective, and practical framework for detecting social bots. This framework is built upon structural information theory. We begin by designing three social relationship metrics that capture various aspects of social bot behaviors: posting type distribution , posting influence , and follow-to-follower ratio . Three new relationships are utilized to construct a new, unified, and weighted social multi-relational graph, aiming to model the relevance of social user behaviors and discover long-distance correlations between users. Second, we introduce a novel method for optimizing heterogeneous structural entropy. This method involves the personalized aggregation of edge information from the social multi-relational graph to generate a two-dimensional encoding tree. The heterogeneous structural entropy facilitates decoding of the substantial structure of the social bots network and enables hierarchical clustering of social bots. Third, a new community labeling method is presented to distinguish social bot communities by computing the user’s stationary distribution, measuring user contributions to network structure, and counting the intensity of user aggregation within the community. Compared with 10 representative social bot detection approaches, comprehensive experiments demonstrate the advantages of effectiveness and interpretability of UnDBot on 4 real social network datasets.

Role-based learning is a promising approach to improving the performance of Multi-Agent Reinforcement Learning (MARL). Nevertheless, without manual assistance, current role-based methods cannot guarantee stably discovering a set of roles to effectively decompose a complex task, as they assume either a predefined role structure or practical experience for selecting hyperparameters. In this article, we propose a mathematical Structural Information principles-based Role Discovery method, namely SIRD, and then present a SIRD optimizing MARL framework, namely SR-MARL, for multi-agent collaboration. The SIRD transforms role discovery into a hierarchical action space clustering. Specifically, the SIRD consists of structuralization, sparsification, and optimization modules, where an optimal encoding tree is generated to perform abstracting to discover roles. The SIRD is agnostic to specific MARL algorithms and flexibly integrated with various value function factorization approaches. Empirical evaluations on the StarCraft II micromanagement benchmark demonstrate that, compared with state-of-the-art MARL algorithms, the SR-MARL framework improves the average test win rate by 0.17%, 6.08%, and 3.24%, and reduces the deviation by 16.67%, 30.80%, and 66.30%, under easy, hard, and super hard scenarios.

Skin lesion segmentation is a fundamental task in dermoscopic image analysis. The complex features of pixels in the lesion region impede the lesion segmentation accuracy, and existing deep learning-based methods often lack interpretability to this problem. In this work, we propose a novel unsupervised Skin Lesion sEgmentation framework based on structural entropy and isolation forest outlier Detection, namely SLED. Specifically, skin lesions are segmented by minimizing the structural entropy of a superpixel graph constructed from the dermoscopic image. Then, we characterize the consistency of healthy skin features and devise a novel multi-scale segmentation mechanism by outlier detection, which enhances the segmentation accuracy by leveraging the superpixel features from multiple scales. We conduct experiments on four skin lesion benchmarks and compare SLED with nine representative unsupervised segmentation methods. Experimental results demonstrate the superiority of the proposed framework. Additionally, some case studies are analyzed to demonstrate the effectiveness of SLED.

State abstraction optimizes decision-making by ignoring irrelevant environmental information in reinforcement learning with rich observations. Nevertheless, recent approaches focus on adequate representational capacities resulting in essential information loss, affecting their performances on challenging tasks. In this article, we propose a novel mathematical Structural Information principles-based State Abstraction framework, namely SISA, from the information-theoretic perspective. Specifically, an unsupervised, adaptive hierarchical state clustering method without requiring manual assistance is presented, and meanwhile, an optimal encoding tree is generated. On each non-root tree node, a new aggregation function and condition structural entropy are designed to achieve hierarchical state abstraction and compensate for sampling-induced essential information loss in state abstraction. Empirical evaluations on a visual gridworld domain and six continuous control benchmarks demonstrate that, compared with five SOTA state abstraction approaches, SISA significantly improves mean episode reward and sample efficiency up to 18.98 and 44.44%, respectively. Besides, we experimentally show that SISA is a general framework that can be flexibly integrated with different representation-learning objectives to improve their performances further.

Graph neural networks (GNNs) are now the mainstream method for mining graph-structured data and learning low-dimensional node- and graph-level embeddings to serve downstream tasks. However, limited by the bottleneck of interpretability that deep neural networks present, existing GNNs have ignored the issue of estimating the appropriate number of dimensions for the embeddings. Hence, we propose a novel framework called Minimum Graph Entropy principle-guided Dimension Estimation, i.e. MGEDE, that learns the appropriate embedding dimensions for both node and graph representations. In terms of node-level estimation, a minimum entropy function that counts both structure and attribute entropy, appraises the appropriate number of dimensions. In terms of graph-level estimation, each graph is assigned a customized embedding dimension from a candidate set based on the number of dimensions estimated for the node-level embeddings. Comprehensive experiments with node and graph classification tasks and nine benchmark datasets verify the effectiveness and generalizability of MGEDE.

In the deep neural network based few-shot learning, the limited training data may make the neural network extract ineffective features, which leads to inaccurate results. By Bayesian graph neural network (BGNN), the probability distributions on hidden layers imply useful features, and the few-shot learning could improved by establishing the correlation among features. Thus, in this paper, we incorporate mutual information (MI) into BGNN to describe the correlation, and propose an innovative framework by adopting the Bayesian network with continuous variables (BNCV) for effective calculation of MI. First, we build the BNCV simultaneously when calculating the probability distributions of features from the Dropout in hidden layers of BGNN. Then, we approximate the MI values efficiently by probabilistic inferences over BNCV. Finally, we give the correlation based loss function and training algorithm of our BGNN model. Experimental results show that our MI based BGNN framework is effective for few-shot learning and outperforms some state-of-the-art competitors by large margins on accuracy.

This paper focuses on the privacy risks of disclosing the community structure in an online social network. By exploiting the community affiliations of user accounts, an attacker may infer sensitive user attributes. This raises the problem of community structure deception (CSD), which asks for ways to minimally modify the network so that a given community structure maximally hides itself from community detection algorithms. We investigate CSD through an information-theoretic lens. To this end, we propose a community-based structural entropy to express the amount of information revealed by a community structure. This notion allows us to devise residual entropy minimization (REM) as an efficient procedure to solve CSD. Experimental results over 9 real-world networks and 6 community detection algorithms show that REM is very effective in obfuscating the community structure as compared to other benchmark methods.

In 1953, Shannon proposed the question of quantification of structural information to analyze communication systems. The question has become one of the longest great challenges in information science and computer science. Here, we propose the first metric for structural information. Given a graph G , we define the K-dimensional structural information of G (or structure entropy of G), denoted by HK(G) , to be the minimum overall number of bits required to determine the K-dimensional code of the node that is accessible from random walk in G. The K-dimensional structural information provides the principle for completely detecting the natural or true structure, which consists of the rules, regulations, and orders of the graphs, for fully distinguishing the order from disorder in structured noisy data, and for analyzing communication systems, solving the Shannon's problem and opening up new directions. The K-dimensional structural information is also the first metric of dynamical complexity of networks, measuring the complexity of interactions, communications, operations, and even evolution of networks. The metric satisfies a number of fundamental properties, including additivity, locality, robustness, local and incremental computability, and so on. We establish the fundamental theorems of the one- and two-dimensional structural information of networks, including both lower and upper bounds of the metrics of classic data structures, general graphs, the networks of models, and the networks of natural evolution. We propose algorithms to approximate the K-dimensional structural information of graphs by finding the K-dimensional structure of the graphs that minimizes the K-dimensional structure entropy. We find that the K-dimensional structure entropy minimization is the principle for detecting the natural or true structures in real-world networks. Consequently, our structural information provides the foundation for knowledge discovering from noisy data. We establish a black hole principle by using the two-dimensional structure information of graphs. We propose the natural rank of locally listing algorithms by the structure entropy minimization principle, providing the basis for a next-generation search engine.

We consider the problem of testing small set expansion for general graphs. A graph G is a (k,\phi)-expander if every subset of volume at most k has conductance at least \phi. Small set expansion has recently received significant attention due to its close connection to the unique games conjecture, the local graph partitioning algorithms and locally testable codes. We give testers with two-sided error and one-sided error in the \adjacency list model that allows degree and neighbor queries to the oracle of the input graph. The testers take as input an n-vertex graph G, a volume bound k, an expansion bound \phi and a distance parameter \varepsilon>0. For the two-sided error tester, with probability at least 2/3, it accepts the graph if it is a (k,\phi)-expander and rejects the graph if it is \varepsilon-far from any (k^*,\phi^*)-expander, where k^*=\Theta(k\varepsilon) and \phi^*=\Theta(\frac{\phi^4}{\min\{\log(4m/k),\log n\}\cdot(\ln k)}). The query complexity and running time of the tester are \widetilde{O}(\sqrt{m}\phi^{-4}\varepsilon^{-2}), where m is the number of edges of the graph. For the one-sided error tester, it accepts every (k,\phi)-expander, and with probability at least 2/3, rejects every graph that is \varepsilon-far from (k^*,\phi^*)-expander, where k^*=O(k^{1-\xi}) and \phi^*=O(\xi\phi^2) for any 0<\xi<1. The query complexity and running time of this tester are \widetilde{O}(\sqrt{\frac{n}{\varepsilon^3}}+\frac{k}{\varepsilon \phi^4}). We also give a two-sided error tester in the \textit{rotation map} model that allows \textit{(neighbor, index)} queries and degree queries. This tester has asymptotically almost the same query complexity and running time as the two-sided error tester in the adjacency list model, but has a better performance: it can distinguish any $(k,\phi)$-expander from graphs that are $\varepsilon$-far from $(k^*,\phi^*)$-expanders, where $k^*=\Theta(k\varepsilon)$ and $\phi^*=\Theta(\frac{\phi^2}{\min\{\log(4m/k),\log n\}\cdot(\ln k)})$. In our analysis, we introduce a new graph product called \textit{non-uniform replacement product} that transforms a general graph into a bounded degree graph, and approximately preserves the expansion profile as well as the corresponding spectral property.

We investigate several geometric models of networks that simultaneously have some nice global properties, including the small-diameter property, the small-community phenomenon, which is defined to capture the common experience that (almost) everyone in society also belongs to some meaningful small communities, and the power law degree distribution, for which our result significantly strengthens those given in van den Esker (2008) and Jordan (2010). These results, together with our previous work in Li and Peng (2011), build a mathematical foundation for the study of both communities and the small-community phenomenon in various networks. In the proof of the power law degree distribution, we develop the method of alternating concentration analysis to build a concentration inequality by alternately and iteratively applying both the sub- and super-martingale inequalities, which seems to be a powerful technique with further potential applications.

The Theory and Applications of Models of Computation (TAMC) conference series is both international and interdisciplinary in character, bringing together researchers working in computer science, mathematics (especially logic) and the physical sciences. It is this, together with its predominantly computational and computability theoretic focus, that gives the series its special character.

Theory and Applications of Models of Computation (TAMC) is an international conference series with an interdisciplinary character bringing together researchers working in computer science, mathematics (especially logic) and the physical sciences. This interdisciplinary approach, with an emphasis on the theory of computation in a broad sense, gives the series its special appeal within China and internationally. At a time when the pressures are increasingly towards narrowly ad hoc research, and scientific fragmentation, meetings that reassert the importance of theory, fundamental concepts and a wider perspective have an important role to play.

We show that there exist c.e. bounded Turing degrees a, b such that 0 < a < 0′, and that for any c.e. bounded Turing degree x, we have b ∨ x = 0′ if and only if x ≥ a. The result gives an unexpected definability theorem in the structure of bounded Turing reducibility.

Atmospheric water vapor is a major limitation for high precision interferometric synthetic aperture radar (InSAR) applications due to its significant impact on microwave signals. Temporal variability of the atmosphere delay can be estimated directly from GPS data, but spatial variations of water-vapor-induced distortions cannot easily be separated by the sparse network of GPS observations in the imaged area. The primary objective of this work is to build a hybrid model combining the GPS and Medium Resolution Imaging Spectrometer (MERIS) water vapor measurements for correcting the atmosphere effects on InSAR measurements. Moreover, a comparison between MERIS and GPS water vapor products was performed using data covering Beijing from January 2004 to February 2007. MERIS water vapor values appeared to be slightly greater than GPS values. However, MERIS water vapor agreed well with GPS water vapor retrievals under Beijing's conditions with a RMS error of 2.38 mm.

Land subsidence in Changping, Beijing of China, has been an ongoing problem for the past four decades (since the later 1970s). We use permanent scatterers interferometric synthetic aperture radar (PS-InSAR) technique to detect and measure ground movement in this area. The detail information of deformation shows that spatial extent of subsidence is controlled by geologic structures (Huangzhuang-Gaoliying and Nankou-Sunhe faults) and thickness of Quaternary sediment. Comparing the subsidence line to the aggregate clay thickness of Changping area, both of them are mostly consistent. The locations of high-subsidence areas coincided with areas of heavy groundwater use and the clay mud layer, which is thicker than 50 m.

The computational function of a matchgate is represented by its character matrix. In this article, we show that all nonsingular character matrices are closed under matrix inverse operation, so that for every $k$, the nonsingular character matrices of $k$-bit matchgates form a group, extending the recent work of Cai and Choudhary (2006) of the same result for the case of $k=2$, and that the single and the two-bit matchgates are universal for matchcircuits, answering a question of Valiant (2002).

We show that for any 2- computably enumerable Turing degree ${\bf l}$ , any computably enumerable degree ${\bf a}$ and any Turing degree ${\bf s}$ , if ${\bf l'=\boldsymbol{0}'}$ , ${\bf l<a}$ , ${\bf s\geq \boldsymbol{0}'}$ , and ${\bf s}$ is c.e. in ${\bf a}$ , then there is a 2-computably enumerable degree ${\bf x}$ with the following properties: ${\bf l<x

Abstract We show that every nonzero Δ20, e-degree bounds a minimal pair. On the other hand, there exist Σ20, e-degrees which bound no minimal pair.

Abstract We say that a computably enumerable (c. e.) degree a is plus-cupping, if for every c.e. degree x with 0 < x ≤ a, there is a c. e. degree y ≠ 0′ such that x ∨ y = 0′. We say that a is n-plus-cupping, if for every c. e. degree x, if 0 < x ≤ a, then there is a lown c. e. degree I such that x ∨ I = 0′. Let PC and PCn be the set of all plus-cupping, and n-plus-cupping c. e. degrees respectively. Then PC1 ⊆ PC2 ⊆ PC3 = PC. In this paper we show that PC1 ⊂ PC2, so giving a nontrivial hierarchy for the plus cupping degrees. The theorem also extends the result of Li, Wu and Zhang [14] showing that LC1 ⊂ LC2, as well as extending the Harrington plus-cupping theorem [8].

Abstract It is shown that there exists a low2 Harrington non-splitting base — that is, a low2 computably enumerable (c.e.) degree a such that for any c.e. degrees x, y, if 0′ = x ∨ y, then either 0′ = x ∨ a or 0′ = y ∨ a. Contrary to prior expectations, the standard Harrington non-splitting construction is incompatible with the low2-ness requirements to be satisfied, and the proof given involves new techniques with potentially wider application.

Abstract In this paper, we prove that there exist computably enumerable degrees a and b such that a > b and for any degree x, if x ≤ a and x is a minimal degree, then x < b.