A simple graph on $n$ vertices may contain a lot of maximum cliques. But how many can it potentially contain? We will show that the maximum number of maximum cliques is taken over so-called cliqueful graphs, more specifically, later we will show that it is taken over saturated composite cliqueful graphs, if $n \ge 15$. Using this we will show that the graph that contains $3^{\lfloor n/3 \rfloor}c$ maxcliques has the most number of maxcliques on $n$ vertices, where $c\in\{1,\frac{4}{3},2\}$, depending on $n \text{ mod } 3$.
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In this article, we study nonparametric inference for a covariate-adjusted regression function. This parameter captures the average association between a continuous exposure and an outcome after adjusting for other covariates. In particular, under certain causal conditions, this parameter corresponds to the average outcome had all units been assigned to a specific exposure level, known as the causal dose-response curve. We propose a debiased local linear estimator of the covariate-adjusted regression function, and demonstrate that our estimator converges pointwise to a mean-zero normal limit distribution. We use this result to construct asymptotically valid confidence intervals for function values and differences thereof. In addition, we use approximation results for the distribution of the supremum of an empirical process to construct asymptotically valid uniform confidence bands. Our methods do not require undersmoothing, permit the use of data-adaptive estimators of nuisance functions, and our estimator attains the optimal rate of convergence for a twice differentiable function. We illustrate the practical performance of our estimator using numerical studies and an analysis of the effect of air pollution exposure on cardiovascular mortality.
Trajectory prediction in traffic scenes involves accurately forecasting the behaviour of surrounding vehicles. To achieve this objective it is crucial to consider contextual information, including the driving path of vehicles, road topology, lane dividers, and traffic rules. Although studies demonstrated the potential of leveraging heterogeneous context for improving trajectory prediction, state-of-the-art deep learning approaches still rely on a limited subset of this information. This is mainly due to the limited availability of comprehensive representations. This paper presents an approach that utilizes knowledge graphs to model the diverse entities and their semantic connections within traffic scenes. Further, we present nuScenes Knowledge Graph (nSKG), a knowledge graph for the nuScenes dataset, that models explicitly all scene participants and road elements, as well as their semantic and spatial relationships. To facilitate the usage of the nSKG via graph neural networks for trajectory prediction, we provide the data in a format, ready-to-use by the PyG library. All artefacts can be found here: //github.com/boschresearch/nuScenes_Knowledge_Graph
We solve several first questions in the table of small parameters of completely regular (CR) codes in Hamming graphs $H(n,q)$. The most uplifting result is the existence of a $\{13,6,1;1,6,9\}$-CR code in $H(n,2)$, $n\ge 13$. We also establish the non-existence of a $\{11,4;3,6\}$-code and a $\{10,3;4,7\}$-code in $H(12,2)$ and $H(13,2)$. A partition of the complement of the quaternary Hamming code of length~$5$ into $4$-cliques is found, which can be used to construct completely regular codes with covering radius $1$ by known constructions. Additionally we discuss the parameters $\{24,21,10;1,4,12\}$ of a putative completely regular code in $H(24,2)$ and show the nonexistence of such a code in $H(8,4)$. Keywords: Hamming graph, equitable partition, completely regular code
Smart meters are of the basic elements in the so-called Smart Grid. These devices, connected to the Internet, keep bidirectional communication with other devices in the Smart Grid structure to allow remote readings and maintenance. As any other device connected to a network, smart meters become vulnerable to attacks with different purposes, like stealing data or altering readings. Nowadays, it is becoming more and more popular to buy and plug-and-play smart meters, additionally to those installed by the energy providers, to directly monitor the energy consumption at home. This option inherently entails security risks that are under the responsibility of householders. In this paper, we focus on an open solution based on Smartpi 2.0 devices with two purposes. On the one hand, we propose a network configuration and different data flows to exchange data (energy readings) in the home. These flows are designed to support collaborative among the devices in order to prevent external attacks and attempts of corrupting the data. On the other hand, we check the vulnerability by performing two kind of attacks (denial of service and stealing and changing data by using a malware). We conclude that, as expected, these devices are vulnerable to these attacks, but we provide mechanisms to detect both of them and to solve, by applying cooperation techniques
Active reconfigurable intelligent surface (RIS) is a new RIS architecture that can reflect and amplify communication signals. It can provide enhanced performance gain compared to the conventional passive RIS systems that can only reflect the signals. On the other hand, the design problem of active RIS-aided systems is more challenging than the passive RIS-aided systems and its efficient algorithms are less studied. In this paper, we consider the sum rate maximization problem in the multiuser massive multiple-input single-output (MISO) downlink with the aid of a large-scale active RIS. Existing approaches usually resort to general optimization solvers and can be computationally prohibitive in the considered settings. We propose an efficient block successive upper bound minimization (BSUM) method, of which each step has a (semi) closed-form update. Thus, the proposed algorithm has an attractive low per-iteration complexity. By simulation, our proposed algorithm consumes much less computation than the existing approaches. In particular, when the MIMO and/or RIS sizes are large, our proposed algorithm can be orders-of-magnitude faster than existing approaches.
Given any finite set equipped with a probability measure, one may compute its Shannon entropy or information content. The entropy becomes the logarithm of the cardinality of the set when the uniform probability is used. Leinster introduced a notion of Euler characteristic for certain finite categories, also known as magnitude, that can be seen as a categorical generalization of cardinality. This paper aims to connect the two ideas by considering the extension of Shannon entropy to finite categories endowed with probability, in such a way that the magnitude is recovered when a certain choice of "uniform" probability is made.
Forecast combination involves using multiple forecasts to create a single, more accurate prediction. Recently, feature-based forecasting has been employed to either select the most appropriate forecasting models or to optimize the weights of their combination. In this paper, we present a multi-task optimization paradigm that focuses on solving both problems simultaneously and enriches current operational research approaches to forecasting. In essence, it incorporates an additional learning and optimization task into the standard feature-based forecasting approach, focusing on the identification of an optimal set of forecasting methods. During the training phase, an optimization model with linear constraints and quadratic objective function is employed to identify accurate and diverse methods for each time series. Moreover, within the training phase, a neural network is used to learn the behavior of that optimization model. Once training is completed the candidate set of methods is identified using the network. The proposed approach elicits the essential role of diversity in feature-based forecasting and highlights the interplay between model combination and model selection when optimizing forecasting ensembles. Experimental results on a large set of series from the M4 competition dataset show that our proposal enhances point forecast accuracy compared to state-of-the-art methods.
The dichromatic number of a digraph is the minimum size of a partition of its vertices into acyclic induced subgraphs. Given a class of digraphs $\mathcal C$, a digraph $H$ is a hero in $\mc C$ if $H$-free digraphs of $\mathcal C$ have bounded dichromatic number. In a seminal paper, Berger at al. give a simple characterization of all heroes in tournaments. In this paper, we give a simple proof that heroes in quasi-transitive oriented graphs are the same as heroes in tournaments. We also prove that it is not the case in the class of oriented multipartite graphs, disproving a conjecture of Aboulker, Charbit and Naserasr. We also give a full characterisation of heroes in oriented complete multipartite graphs up to the status of a single tournament on $6$ vertices.
Graph Neural Networks (GNNs) are state-of-the-art models for performing prediction tasks on graphs. While existing GNNs have shown great performance on various tasks related to graphs, little attention has been paid to the scenario where out-of-distribution (OOD) nodes exist in the graph during training and inference. Borrowing the concept from CV and NLP, we define OOD nodes as nodes with labels unseen from the training set. Since a lot of networks are automatically constructed by programs, real-world graphs are often noisy and may contain nodes from unknown distributions. In this work, we define the problem of graph learning with out-of-distribution nodes. Specifically, we aim to accomplish two tasks: 1) detect nodes which do not belong to the known distribution and 2) classify the remaining nodes to be one of the known classes. We demonstrate that the connection patterns in graphs are informative for outlier detection, and propose Out-of-Distribution Graph Attention Network (OODGAT), a novel GNN model which explicitly models the interaction between different kinds of nodes and separate inliers from outliers during feature propagation. Extensive experiments show that OODGAT outperforms existing outlier detection methods by a large margin, while being better or comparable in terms of in-distribution classification.
Graph representation learning for hypergraphs can be used to extract patterns among higher-order interactions that are critically important in many real world problems. Current approaches designed for hypergraphs, however, are unable to handle different types of hypergraphs and are typically not generic for various learning tasks. Indeed, models that can predict variable-sized heterogeneous hyperedges have not been available. Here we develop a new self-attention based graph neural network called Hyper-SAGNN applicable to homogeneous and heterogeneous hypergraphs with variable hyperedge sizes. We perform extensive evaluations on multiple datasets, including four benchmark network datasets and two single-cell Hi-C datasets in genomics. We demonstrate that Hyper-SAGNN significantly outperforms the state-of-the-art methods on traditional tasks while also achieving great performance on a new task called outsider identification. Hyper-SAGNN will be useful for graph representation learning to uncover complex higher-order interactions in different applications.
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