Error-Bounded Graph Anomaly Loss for GNNs

AbstractA cardinal characteristic can often be described as the smallest size of a family of sequences which has a given property. Instead of this traditional concern for a smallest realization of the given property, a basically new approach, taken in [4] and [5], asks for a realization whose members are sequences of labels that correspond to 1-way infinite paths in a labelled graph. We study this approach as such, establishing tools that are applicable to all these cardinal characteristics. As an application, we demonstrate the power of the tools developed by presenting a short proof of the bounded graph conjecture [4].

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Monadic Bounded Algebras

10.26686/wgtn.16959268.v1 ◽

2021 ◽

Author(s):

◽

Galym Akishev

Keyword(s):

Upper Bound ◽

Subdirect Product ◽

Completeness Theorem ◽

Finitely Generated ◽

Algebraic Version ◽

Finite Set ◽

Monadic Logic ◽

Object Of Study ◽

Bounded Graph

<p>The object of study of the thesis is the notion of monadic bounded algebras (shortly, MBA's). These algebras are motivated by certain natural constructions in free (first-order) monadic logic and are related to free monadic logic in the same way as monadic algebras of P. Halmos to monadic logic (Chapter 1). Although MBA's come from logic, the present work is in algebra. Another important way of approaching MBA's is via bounded graphs, namely, the complex algebra of a bounded graph is an MBA and vice versa. The main results of Chapter 2 are two representation theorems: 1) every model is a basic MBA and every basic MBA is isomorphic to a model; 2) every MBA is isomorphic to a subdirect product of basic MBA's. As a consequence, every MBA is isomorphic to a subdirect product of models. This result is thought of as an algebraic version of semantical completeness theorem for free monadic logic. Chapter 3 entirely deals with MBA-varieties. It is proved by the method of filtration that every MBA-variety is generated by its finite special members. Using connections in terms of bounded morphisms among certain bounded graphs, it is shown that every MBA-variety is generated by at most three special (not necessarily finite) MBA's. After that each MBA-variety is equationally characterized. Chapter 4 considers finitely generated MBA's. We prove that every finitely generated MBA is finite (an upper bound on the number of elements is provided) and that the number of elements of a free MBA on a finite set achieves its upper bound. Lastly, a procedure for constructing a free MBA on any finite set is given.</p>

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Estimating parameters associated with monotone properties

Combinatorics Probability Computing ◽

10.1017/s0963548320000048 ◽

2020 ◽

Vol 29 (4) ◽

pp. 616-632

Author(s):

Carlos Hoppen ◽

Yoshiharu Kohayakawa ◽

Richard Lang ◽

Hanno Lefmann ◽

Henrique Stagni

Keyword(s):

Weighted Graph ◽

Sample Complexity ◽

Input Graph ◽

Vertex Set ◽

Graph Parameters ◽

Cluster Graph ◽

Number Of Classes ◽

Substantial Interest ◽

Bounded Graph ◽

Monotone Graph

AbstractThere has been substantial interest in estimating the value of a graph parameter, i.e. of a real-valued function defined on the set of finite graphs, by querying a randomly sampled substructure whose size is independent of the size of the input. Graph parameters that may be successfully estimated in this way are said to be testable or estimable, and the sample complexity qz = qz(ε) of an estimable parameter z is the size of a random sample of a graph G required to ensure that the value of z(G) may be estimated within an error of ε with probability at least 2/3. In this paper, for any fixed monotone graph property $\mathcal{P}= \text{Forb}\!(\mathcal{F}),$ we study the sample complexity of estimating a bounded graph parameter z that, for an input graph G, counts the number of spanning subgraphs of G that satisfy$\mathcal{P}$. To improve upon previous upper bounds on the sample complexity, we show that the vertex set of any graph that satisfies a monotone property $\mathcal{P}$ may be partitioned equitably into a constant number of classes in such a way that the cluster graph induced by the partition is not far from satisfying a natural weighted graph generalization of $\mathcal{P}$. Properties for which this holds are said to be recoverable, and the study of recoverable properties may be of independent interest.

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A Bounded Graph-Connect Construction for LR-regular Parsers

Lecture Notes in Computer Science - Compiler Construction ◽

10.1007/3-540-45306-7_17 ◽

2001 ◽

pp. 244-258 ◽

Cited By ~ 7

Author(s):

Jacques Farré ◽

José Fortes Gálvez

Keyword(s):

Bounded Graph

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Monadic Bounded Algebras

10.26686/wgtn.16959268 ◽

2021 ◽

Author(s):

◽

Galym Akishev

Keyword(s):

Upper Bound ◽

Subdirect Product ◽

Completeness Theorem ◽

Finitely Generated ◽

Algebraic Version ◽

Finite Set ◽

Monadic Logic ◽

Object Of Study ◽

Bounded Graph

<p>The object of study of the thesis is the notion of monadic bounded algebras (shortly, MBA's). These algebras are motivated by certain natural constructions in free (first-order) monadic logic and are related to free monadic logic in the same way as monadic algebras of P. Halmos to monadic logic (Chapter 1). Although MBA's come from logic, the present work is in algebra. Another important way of approaching MBA's is via bounded graphs, namely, the complex algebra of a bounded graph is an MBA and vice versa. The main results of Chapter 2 are two representation theorems: 1) every model is a basic MBA and every basic MBA is isomorphic to a model; 2) every MBA is isomorphic to a subdirect product of basic MBA's. As a consequence, every MBA is isomorphic to a subdirect product of models. This result is thought of as an algebraic version of semantical completeness theorem for free monadic logic. Chapter 3 entirely deals with MBA-varieties. It is proved by the method of filtration that every MBA-variety is generated by its finite special members. Using connections in terms of bounded morphisms among certain bounded graphs, it is shown that every MBA-variety is generated by at most three special (not necessarily finite) MBA's. After that each MBA-variety is equationally characterized. Chapter 4 considers finitely generated MBA's. We prove that every finitely generated MBA is finite (an upper bound on the number of elements is provided) and that the number of elements of a free MBA on a finite set achieves its upper bound. Lastly, a procedure for constructing a free MBA on any finite set is given.</p>

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Bounded Graph Pattern Matching With View

journal of Data Intelligence ◽

10.26421/jdi2.3-4 ◽

2021 ◽

Vol 2 (3) ◽

pp. 368-387

Author(s):

Xin Wang ◽

Yang Wang ◽

Ji Zhang ◽

Yan Zhu

Keyword(s):

Pattern Matching ◽

Optimization Technique ◽

Real Life ◽

Synthetic Data ◽

Optimization Strategy ◽

Matching Algorithm ◽

Graph Pattern Matching ◽

Graph Pattern ◽

Bounded Graph ◽

Bounded Evaluation

Bounded evaluation using views is to compute the answers $Q({\cal D})$ to a query $Q$ in a dataset ${\cal D}$ by accessing only cached views and a small fraction $D_Q$ of ${\cal D}$ such that the size $|D_Q|$ of $D_Q$ and the time to identify $D_Q$ are independent of $|{\cal D}|$, no matter how big ${\cal D}$ is. Though proven effective for relational data, it has yet been investigated for graph data. In light of this, we study the problem of bounded pattern matching using views. We first introduce access schema ${\cal C}$ for graphs and propose a notion of joint containment to characterize bounded pattern matching using views. We show that a pattern query $\sq$ can be boundedly evaluated using views ${\cal V}(G)$ and a fraction $G_Q$ of $G$ if and only if the query $\sq$ is jointly contained by ${\cal V}$ and ${\cal C}$. Based on the characterization, we develop an efficient algorithm as well as an optimization strategy to compute matches by using ${\cal V}(G)$ and $G_Q$. Using real-life and synthetic data, we experimentally verify the performance of these algorithms, and show that (a) our algorithm for joint containment determination is not only effective but also efficient; and (b) our matching algorithm significantly outperforms its counterpart, and the optimization technique can further improve performance by eliminating unnecessary input.

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