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The number of down-steps between pairs of up-steps in $k_t$-Dyck paths, a generalization of Dyck paths consisting of steps $\{(1, k), (1, -1)\}$ such that the path stays (weakly) above the line $y=-t$, is studied. Results are proved bijectively and by means of generating functions, and lead to several interesting identities as well as links to other combinatorial structures. In particular, there is a connection between $k_t$-Dyck paths and perforation patterns for punctured convolutional codes (binary matrices) used in coding theory. Surprisingly, upon restriction to usual Dyck paths this yields a new combinatorial interpretation of Catalan numbers.

Dynamic topological logic ($\mathbf{DTL}$) is a trimodal logic designed for reasoning about dynamic topological systems. It was shown by Fern\'andez-Duque that the natural set of axioms for $\mathbf{DTL}$ is incomplete, but he provided a complete axiomatisation in an extended language. In this paper, we consider dynamic topological logic over scattered spaces, which are topological spaces where every nonempty subspace has an isolated point. Scattered spaces appear in the context of computational logic as they provide semantics for provability and enjoy definable fixed points. We exhibit the first sound and complete dynamic topological logic in the original trimodal language. In particular, we show that the version of $\mathbf{DTL}$ based on the class of scattered spaces is finitely axiomatisable over the original language, and that the natural axiomatisation is sound and complete.

Computing a maximum independent set (MaxIS) is a fundamental NP-hard problem in graph theory, which has important applications in a wide spectrum of fields. Since graphs in many applications are changing frequently over time, the problem of maintaining a MaxIS over dynamic graphs has attracted increasing attention over the past few years. Due to the intractability of maintaining an exact MaxIS, this paper aims to develop efficient algorithms that can maintain an approximate MaxIS with an accuracy guarantee theoretically. In particular, we propose a framework that maintains a $(\frac{\Delta}{2} + 1)$-approximate MaxIS over dynamic graphs and prove that it achieves a constant approximation ratio in many real-world networks. To the best of our knowledge, this is the first non-trivial approximability result for the dynamic MaxIS problem. Following the framework, we implement an efficient linear-time dynamic algorithm and a more effective dynamic algorithm with near-linear expected time complexity. Our thorough experiments over real and synthetic graphs demonstrate the effectiveness and efficiency of the proposed algorithms, especially when the graph is highly dynamic.

Recently, the enactment of privacy regulations has promoted the rise of the machine unlearning paradigm. Existing studies of machine unlearning mainly focus on sample-wise unlearning, such that a learnt model will not expose user's privacy at the sample level. Yet we argue that such ability of selective removal should also be presented at the attribute level, especially for the attributes irrelevant to the main task, e.g., whether a person recognized in a face recognition system wears glasses or the age range of that person. Through a comprehensive literature review, it is found that existing studies on attribute-related problems like fairness and de-biasing learning cannot address the above concerns properly. To bridge this gap, we propose a paradigm of selectively removing input attributes from feature representations which we name `attribute unlearning'. In this paradigm, certain attributes will be accurately captured and detached from the learned feature representations at the stage of training, according to their mutual information. The particular attributes will be progressively eliminated along with the training procedure towards convergence, while the rest of attributes related to the main task are preserved for achieving competitive model performance. Considering the computational complexity during the training process, we not only give a theoretically approximate training method, but also propose an acceleration scheme to speed up the training process. We validate our method by spanning several datasets and models and demonstrate that our design can preserve model fidelity and reach prevailing unlearning efficacy with high efficiency. The proposed unlearning paradigm builds a foundation for future machine unlearning system and will become an essential component of the latest privacy-related legislation.

Most existing works of polar codes focus on the analysis of block error probability. However, in many scenarios, bit error probability is also important for evaluating the performance of channel codes. In this paper, we establish a new framework to analyze the bit error probability of polar codes. Specifically, by revisiting the error event of bit-channel, we first introduce the conditional bit error probability as a metric to evaluate the reliability of bit-channel for both systematic and non-systematic polar codes. Guided by the concept of polar subcode, we then derive an upper bound on the conditional bit error probability of each bit-channel, and accordingly, an upper bound on the bit error probability of polar codes. Based on these, two types of construction metrics aiming at minimizing the bit error probability of polar codes are proposed, which are of linear computational complexity and explicit forms. Simulation results show that the polar codes constructed by the proposed methods can outperform those constructed by the conventional methods.

Universal coding of integers~(UCI) is a class of variable-length code, such that the ratio of the expected codeword length to $\max\{1,H(P)\}$ is within a constant factor, where $H(P)$ is the Shannon entropy of the decreasing probability distribution $P$. However, if we consider the ratio of the expected codeword length to $H(P)$, the ratio tends to infinity by using UCI, when $H(P)$ tends to zero. To solve this issue, this paper introduces a class of codes, termed generalized universal coding of integers~(GUCI), such that the ratio of the expected codeword length to $H(P)$ is within a constant factor $K$. First, the definition of GUCI is proposed and the coding structure of GUCI is introduced. Next, we propose a class of GUCI $\mathcal{C}$ to achieve the expansion factor $K_{\mathcal{C}}=2$ and show that the optimal GUCI is in the range $1\leq K_{\mathcal{C}}^{*}\leq 2$. Then, by comparing UCI and GUCI, we show that when the entropy is very large or $P(0)$ is not large, there are also cases where the average codeword length of GUCI is shorter. Finally, the asymptotically optimal GUCI is presented.

Alerts are crucial for requesting prompt human intervention upon cloud anomalies. The quality of alerts significantly affects the cloud reliability and the cloud provider's business revenue. In practice, we observe on-call engineers being hindered from quickly locating and fixing faulty cloud services because of the vast existence of misleading, non-informative, non-actionable alerts. We call the ineffectiveness of alerts "anti-patterns of alerts". To better understand the anti-patterns of alerts and provide actionable measures to mitigate anti-patterns, in this paper, we conduct the first empirical study on the practices of mitigating anti-patterns of alerts in an industrial cloud system. We study the alert strategies and the alert processing procedure at Huawei Cloud, a leading cloud provider. Our study combines the quantitative analysis of millions of alerts in two years and a survey with eighteen experienced engineers. As a result, we summarized four individual anti-patterns and two collective anti-patterns of alerts. We also summarize four current reactions to mitigate the anti-patterns of alerts, and the general preventative guidelines for the configuration of alert strategy. Lastly, we propose to explore the automatic evaluation of the Quality of Alerts (QoA), including the indicativeness, precision, and handleability of alerts, as a future research direction that assists in the automatic detection of alerts' anti-patterns. The findings of our study are valuable for optimizing cloud monitoring systems and improving the reliability of cloud services.

Catastrophic forgetting refers to the tendency that a neural network "forgets" the previous learned knowledge upon learning new tasks. Prior methods have been focused on overcoming this problem on convolutional neural networks (CNNs), where the input samples like images lie in a grid domain, but have largely overlooked graph neural networks (GNNs) that handle non-grid data. In this paper, we propose a novel scheme dedicated to overcoming catastrophic forgetting problem and hence strengthen continual learning in GNNs. At the heart of our approach is a generic module, termed as topology-aware weight preserving~(TWP), applicable to arbitrary form of GNNs in a plug-and-play fashion. Unlike the main stream of CNN-based continual learning methods that rely on solely slowing down the updates of parameters important to the downstream task, TWP explicitly explores the local structures of the input graph, and attempts to stabilize the parameters playing pivotal roles in the topological aggregation. We evaluate TWP on different GNN backbones over several datasets, and demonstrate that it yields performances superior to the state of the art. Code is publicly available at \url{//github.com/hhliu79/TWP}.

We propose a novel method for automatic reasoning on knowledge graphs based on debate dynamics. The main idea is to frame the task of triple classification as a debate game between two reinforcement learning agents which extract arguments -- paths in the knowledge graph -- with the goal to promote the fact being true (thesis) or the fact being false (antithesis), respectively. Based on these arguments, a binary classifier, called the judge, decides whether the fact is true or false. The two agents can be considered as sparse, adversarial feature generators that present interpretable evidence for either the thesis or the antithesis. In contrast to other black-box methods, the arguments allow users to get an understanding of the decision of the judge. Since the focus of this work is to create an explainable method that maintains a competitive predictive accuracy, we benchmark our method on the triple classification and link prediction task. Thereby, we find that our method outperforms several baselines on the benchmark datasets FB15k-237, WN18RR, and Hetionet. We also conduct a survey and find that the extracted arguments are informative for users.

The area of Data Analytics on graphs promises a paradigm shift as we approach information processing of classes of data, which are typically acquired on irregular but structured domains (social networks, various ad-hoc sensor networks). Yet, despite its long history, current approaches mostly focus on the optimization of graphs themselves, rather than on directly inferring learning strategies, such as detection, estimation, statistical and probabilistic inference, clustering and separation from signals and data acquired on graphs. To fill this void, we first revisit graph topologies from a Data Analytics point of view, and establish a taxonomy of graph networks through a linear algebraic formalism of graph topology (vertices, connections, directivity). This serves as a basis for spectral analysis of graphs, whereby the eigenvalues and eigenvectors of graph Laplacian and adjacency matrices are shown to convey physical meaning related to both graph topology and higher-order graph properties, such as cuts, walks, paths, and neighborhoods. Next, to illustrate estimation strategies performed on graph signals, spectral analysis of graphs is introduced through eigenanalysis of mathematical descriptors of graphs and in a generic way. Finally, a framework for vertex clustering and graph segmentation is established based on graph spectral representation (eigenanalysis) which illustrates the power of graphs in various data association tasks. The supporting examples demonstrate the promise of Graph Data Analytics in modeling structural and functional/semantic inferences. At the same time, Part I serves as a basis for Part II and Part III which deal with theory, methods and applications of processing Data on Graphs and Graph Topology Learning from data.