Erasure-Resilient Sublinear-Time Graph Algorithms

We investigate sublinear-time algorithms that take partially erased graphs represented by adjacency lists as input. Our algorithms make degree and neighbor queries to the input graph and work with a specified fraction of adversarial erasures in adjacency entries. We focus on two computational tasks: testing if a graph is connected or ε-far from connected and estimating the average degree. For testing connectedness, we discover a threshold phenomenon: when the fraction of erasures is less than ε, this property can be tested efficiently (in time independent of the size of the graph); when the fraction of erasures is at least ε, then a number of queries linear in the size of the graph representation is required. Our erasure-resilient algorithm (for the special case with no erasures) is an improvement over the previously known algorithm for connectedness in the standard property testing model and has optimal dependence on the proximity parameter ε. For estimating the average degree, our results provide an “interpolation” between the query complexity for this computational task in the model with no erasures in two different settings: with only degree queries, investigated by Feige (SIAM J. Comput. ‘06), and with degree queries and neighbor queries, investigated by Goldreich and Ron (Random Struct. Algorithms ‘08) and Eden et al. (ICALP ‘17). We conclude with a discussion of our model and open questions raised by our work.

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Molecular Graph Contrastive Learning with Parameterized Explainable Augmentations

10.1101/2021.12.03.471150 ◽

2021 ◽

Author(s):

Yingheng Wang ◽

Yaosen Min ◽

Erzhuo Shao ◽

Ji Wu

Keyword(s):

Neural Networks ◽

Deep Neural Networks ◽

Structural Information ◽

Molecular Graph ◽

Representation Learning ◽

Graph Representation ◽

Input Graph ◽

Recent Success ◽

Real World Datasets ◽

Comparative Results

ABSTRACTLearning generalizable, transferable, and robust representations for molecule data has always been a challenge. The recent success of contrastive learning (CL) for self-supervised graph representation learning provides a novel perspective to learn molecule representations. The most prevailing graph CL framework is to maximize the agreement of representations in different augmented graph views. However, existing graph CL frameworks usually adopt stochastic augmentations or schemes according to pre-defined rules on the input graph to obtain different graph views in various scales (e.g. node, edge, and subgraph), which may destroy topological semantemes and domain prior in molecule data, leading to suboptimal performance. Therefore, designing parameterized, learnable, and explainable augmentation is quite necessary for molecular graph contrastive learning. A well-designed parameterized augmentation scheme can preserve chemically meaningful structural information and intrinsically essential attributes for molecule graphs, which helps to learn representations that are insensitive to perturbation on unimportant atoms and bonds. In this paper, we propose a novel Molecular Graph Contrastive Learning with Parameterized Explainable Augmentations, MolCLE for brevity, that self-adaptively incorporates chemically significative information from both topological and semantic aspects of molecular graphs. Specifically, we apply deep neural networks to parameterize the augmentation process for both the molecular graph topology and atom attributes, to highlight contributive molecular substructures and recognize underlying chemical semantemes. Comprehensive experiments on a variety of real-world datasets demonstrate that our proposed method consistently outperforms compared baselines, which verifies the effectiveness of the proposed framework. Detailedly, our self-supervised MolCLE model surpasses many supervised counterparts, and meanwhile only uses hundreds of thousands of parameters to achieve comparative results against the state-of-the-art baseline, which has tens of millions of parameters. We also provide detailed case studies to validate the explainability of augmented graph views.CCS CONCEPTS• Mathematics of computing → Graph algorithms; • Applied computing → Bioinformatics; • Computing methodologies → Neural networks; Unsupervised learning.

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Explicit relation between all lower bound techniques for quantum query complexity

International Journal of Quantum Information ◽

10.1142/s0219749913500597 ◽

2015 ◽

Vol 13 (04) ◽

pp. 1350059

Author(s):

Loïck Magnin ◽

Jérémie Roland

Keyword(s):

Lower Bound ◽

Boolean Functions ◽

Query Complexity ◽

Polynomial Method ◽

Quantum Query Complexity ◽

State Generation ◽

Adversary Method ◽

Explicit Relation ◽

Special Case ◽

Quantum Query

The polynomial method and the adversary method are the two main techniques to prove lower bounds on quantum query complexity, and they have so far been considered as unrelated approaches. Here, we show an explicit reduction from the polynomial method to the multiplicative adversary method. The proof goes by extending the polynomial method from Boolean functions to quantum state generation problems. In the process, the bound is even strengthened. We then show that this extended polynomial method is a special case of the multiplicative adversary method with an adversary matrix that is independent of the function. This new result therefore provides insight on the reason why in some cases the adversary method is stronger than the polynomial method. It also reveals a clear picture of the relation between the different lower bound techniques, as it implies that all known techniques reduce to the multiplicative adversary method.

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The polymerization of styrene by sulphuric acid - I. Theory of fast-initiated non-stationary polymerization

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1961.0144 ◽

1961 ◽

Vol 263 (1312) ◽

pp. 58-62 ◽

Cited By ~ 13

Keyword(s):

Sulphuric Acid ◽

Time Course ◽

Monomer Concentration ◽

Degree Of Polymerization ◽

Average Degree ◽

Chain Reaction ◽

First Order ◽

Chain Lengths ◽

Special Case ◽

Termination Process

Theoretical relationships are derived for the time course, yield, and number-average degree of polymerization in a non-stationary chain reaction having a very rapid initiation process and a first-order termination process. The distribution of chain lengths is derived for the special case of constant monomer concentration.

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Idempotent lifting and ring extensions

Journal of Algebra and Its Applications ◽

10.1142/s0219498816501127 ◽

2016 ◽

Vol 15 (06) ◽

pp. 1650112 ◽

Cited By ~ 2

Author(s):

Alexander J. Diesl ◽

Samuel J. Dittmer ◽

Pace P. Nielsen

Keyword(s):

Jacobson Radical ◽

General Theorem ◽

Positive Side ◽

Open Questions ◽

Morita Invariant ◽

Ring Extensions ◽

Special Case

We answer multiple open questions concerning lifting of idempotents that appear in the literature. Most of the results are obtained by constructing explicit counter-examples. For instance, we provide a ring [Formula: see text] for which idempotents lift modulo the Jacobson radical [Formula: see text], but idempotents do not lift modulo [Formula: see text]. Thus, the property “idempotents lift modulo the Jacobson radical” is not a Morita invariant. We also prove that if [Formula: see text] and [Formula: see text] are ideals of [Formula: see text] for which idempotents lift (even strongly), then it can be the case that idempotents do not lift over [Formula: see text]. On the positive side, if [Formula: see text] and [Formula: see text] are enabling ideals in [Formula: see text], then [Formula: see text] is also an enabling ideal. We show that if [Formula: see text] is (weakly) enabling in [Formula: see text], then [Formula: see text] is not necessarily (weakly) enabling in [Formula: see text] while [Formula: see text] is (weakly) enabling in [Formula: see text]. The latter result is a special case of a more general theorem about completions. Finally, we give examples showing that conjugate idempotents are not necessarily related by a string of perspectivities.

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A theory of spectral partitions of metric graphs

Calculus of Variations and Partial Differential Equations ◽

10.1007/s00526-021-01966-y ◽

2021 ◽

Vol 60 (2) ◽

Author(s):

James B. Kennedy ◽

Pavel Kurasov ◽

Corentin Léna ◽

Delio Mugnolo

Keyword(s):

Spectral Gaps ◽

Qualitative Properties ◽

Metric Graphs ◽

One Dimensional ◽

Graph Partitions ◽

Planar Domains ◽

Open Questions ◽

The One ◽

Abstract Framework ◽

Special Case

AbstractWe introduce an abstract framework for the study of clustering in metric graphs: after suitably metrising the space of graph partitions, we restrict Laplacians to the clusters thus arising and use their spectral gaps to define several notions of partition energies; this is the graph counterpart of the well-known theory of spectral minimal partitions on planar domains and includes the setting in Band et al. (Commun Math Phys 311:815–838, 2012) as a special case. We focus on the existence of optimisers for a large class of functionals defined on such partitions, but also study their qualitative properties, including stability, regularity, and parameter dependence. We also discuss in detail their interplay with the theory of nodal partitions. Unlike in the case of domains, the one-dimensional setting of metric graphs allows for explicit computation and analytic—rather than numerical—results. Not only do we recover the main assertions in the theory of spectral minimal partitions on domains, as studied in Conti et al. (Calc Var 22:45–72, 2005), Helffer et al. (Ann Inst Henri Poincaré Anal Non Linéaire 26:101–138, 2009), but we can also generalise some of them and answer (the graph counterparts of) a few open questions.

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Multi-Scale Contrastive Siamese Networks for Self-Supervised Graph Representation Learning

Proceedings of the Thirtieth International Joint Conference on Artificial Intelligence ◽

10.24963/ijcai.2021/204 ◽

2021 ◽

Author(s):

Ming Jin ◽

Yizhen Zheng ◽

Yuan-Fang Li ◽

Chen Gong ◽

Chuan Zhou ◽

...

Keyword(s):

State Of The Art ◽

Representation Learning ◽

Vital Role ◽

Graph Representation ◽

Input Graph ◽

Global Perspectives ◽

Multi Scale ◽

Recent Success ◽

Real World Datasets ◽

Siamese Networks

Graph representation learning plays a vital role in processing graph-structured data. However, prior arts on graph representation learning heavily rely on labeling information. To overcome this problem, inspired by the recent success of graph contrastive learning and Siamese networks in visual representation learning, we propose a novel self-supervised approach in this paper to learn node representations by enhancing Siamese self-distillation with multi-scale contrastive learning. Specifically, we first generate two augmented views from the input graph based on local and global perspectives. Then, we employ two objectives called cross-view and cross-network contrastiveness to maximize the agreement between node representations across different views and networks. To demonstrate the effectiveness of our approach, we perform empirical experiments on five real-world datasets. Our method not only achieves new state-of-the-art results but also surpasses some semi-supervised counterparts by large margins. Code is made available at https://github.com/GRAND-Lab/MERIT

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Popularity, Mixed Matchings, and Self-Duality

Mathematics of Operations Research ◽

10.1287/moor.2020.1063 ◽

2021 ◽

Author(s):

Chien-Chung Huang ◽

Telikepalli Kavitha

Keyword(s):

Perfect Matching ◽

The Self ◽

Extended Formulation ◽

Input Graph ◽

List Ranking ◽

Self Duality ◽

Special Case ◽

Popular Matching ◽

Input Instance ◽

Matching Polytope

Our input instance is a bipartite graph G where each vertex has a preference list ranking its neighbors in a strict order of preference. A matching M is popular if there is no matching N such that the number of vertices that prefer N to M outnumber those that prefer M to N. Each edge is associated with a utility and we consider the problem of matching vertices in a popular and utility-optimal manner. It is known that it is NP-hard to compute a max-utility popular matching. So we consider mixed matchings: a mixed matching is a probability distribution or a lottery over matchings. Our main result is that the popular fractional matching polytope PG is half-integral and in the special case where a stable matching in G is a perfect matching, this polytope is integral. This implies that there is always a max-utility popular mixed matching which is the average of two integral matchings. So in order to implement a max-utility popular mixed matching in G, we need just a single random bit. We analyze the popular fractional matching polytope whose description may have exponentially many constraints via an extended formulation with a linear number of constraints. The linear program that gives rise to this formulation has an unusual property: self-duality. The self-duality of this LP plays a crucial role in our proof. Our result implies that a max-utility popular half-integral matching in G and also in the roommates problem (where the input graph need not be bipartite) can be computed in polynomial time.

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Who is the infector? General multi-type epidemics and real-time susceptibility processes

Advances in Applied Probability ◽

10.1017/apr.2019.25 ◽

2019 ◽

Vol 51 (2) ◽

pp. 606-631

Author(s):

Tom Britton ◽

Ka Yin Leung ◽

Pieter Trapman

Keyword(s):

Real Time ◽

Random Graph ◽

Branching Processes ◽

Large Population ◽

Graph Representation ◽

Type Model ◽

Random Weights ◽

Stochastic Epidemic ◽

Special Case ◽

Large Population Limit

AbstractWe couple a multi-type stochastic epidemic process with a directed random graph, where edges have random weights (traversal times). This random graph representation is used to characterise the fractions of individuals infected by the different types of vertices among all infected individuals in the large population limit. For this characterisation, we rely on the theory of multi-type real-time branching processes. We identify a special case of the two-type model in which the fraction of individuals of a certain type infected by individuals of the same type is maximised among all two-type epidemics approximated by branching processes with the same mean offspring matrix.

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A Sharp Threshold Phenomenon in String Graphs

Discrete & Computational Geometry ◽

10.1007/s00454-021-00279-3 ◽

2021 ◽

Author(s):

István Tomon

Keyword(s):

The Geometry of Lecture Hall Partitions and Quadratic Permutation Statistics

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.2875 ◽

2010 ◽

Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽

Author(s):

Katie L. Bright ◽

Carla D. Savage

Keyword(s):

Weyl Group ◽

Generating Function ◽

Lecture Hall ◽

Permutation Statistics ◽

Linear Statistics ◽

Joint Distributions ◽

Open Questions ◽

International Audience ◽

Lecture Hall Partitions ◽

Special Case

International audience We take a geometric view of lecture hall partitions and anti-lecture hall compositions in order to settle some open questions about their enumeration. In the process, we discover an intrinsic connection between these families of partitions and certain quadratic permutation statistics. We define some unusual quadratic permutation statistics and derive results about their joint distributions with linear statistics. We show that certain specializations are equivalent to the lecture hall and anti-lecture hall theorems and another leads back to a special case of a Weyl group generating function that "ought to be better known.'' Nous regardons géométriquement les partitions amphithéâtre et les compositions planétarium afin de résoudre quelques questions énumératives ouvertes. Nous découvrons un lien intrinsèque entre ces familles des partitions et certaines statistiques quadratiques de permutation. Nous définissons quelques statistiques quadratiques peu communes des permutations et dérivons des résultats sur leurs distributions jointes avec des statistiques linéaires. Nous démontrons que certaines spécialisations sont équivalentes aux théorèmes amphithéâtre et planétarium. Une autre spécialisation mène à un cas spécial de la série génératrice d'un groupe de Weyl qui "devrait être mieux connue''.

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