arXiv subjects

Amit Roy

Publications and source records attributed to Amit Roy.

27 records · Page 2Linked to original sources

arXiv2021

Twisted conjugacy classes in twisted Chevalley groups

Let G be a group and ϕ be an automorphism of G. Two elements x, y of G are said to be ϕ-twisted if y = gxϕ(g)^{-1} for some g in G. We say that a group G has the R_{\infty}-property if the number of ϕ-twisted conjugacy classes is infinite for every automorphism ϕ of G. In this paper, we prove that twisted Chevalley groups over the field k of characteristic zero have the R_{\infty}-property as well as S_{\infty}-property if k has finite transcendence degree over \mathbb{Q} or Aut(k) is periodic.

math.GR↗

arXiv2021

Unified Spatio-Temporal Modeling for Traffic Forecasting using Graph Neural Network

Research in deep learning models to forecast traffic intensities has gained great attention in recent years due to their capability to capture the complex spatio-temporal relationships within the traffic data. However, most state-of-the-art approaches have designed spatial-only (e.g. Graph Neural Networks) and temporal-only (e.g. Recurrent Neural Networks) modules to separately extract spatial and temporal features. However, we argue that it is less effective to extract the complex spatio-temporal relationship with such factorized modules. Besides, most existing works predict the traffic intensity of a particular time interval only based on the traffic data of the previous one hour of that day. And thereby ignores the repetitive daily/weekly pattern that may exist in the last hour of data. Therefore, we propose a Unified Spatio-Temporal Graph Convolution Network (USTGCN) for traffic forecasting that performs both spatial and temporal aggregation through direct information propagation across different timestamp nodes with the help of spectral graph convolution on a spatio-temporal graph. Furthermore, it captures historical daily patterns in previous days and current-day patterns in current-day traffic data. Finally, we validate our work's effectiveness through experimental analysis, which shows that our model USTGCN can outperform state-of-the-art performances in three popular benchmark datasets from the Performance Measurement System (PeMS). Moreover, the training time is reduced significantly with our proposed USTGCN model.

cs.LG↗

arXiv2021

Structure-Aware Hierarchical Graph Pooling using Information Bottleneck

Graph pooling is an essential ingredient of Graph Neural Networks (GNNs) in graph classification and regression tasks. For these tasks, different pooling strategies have been proposed to generate a graph-level representation by downsampling and summarizing nodes' features in a graph. However, most existing pooling methods are unable to capture distinguishable structural information effectively. Besides, they are prone to adversarial attacks. In this work, we propose a novel pooling method named as {HIBPool} where we leverage the Information Bottleneck (IB) principle that optimally balances the expressiveness and robustness of a model to learn representations of input data. Furthermore, we introduce a novel structure-aware Discriminative Pooling Readout ({DiP-Readout}) function to capture the informative local subgraph structures in the graph. Finally, our experimental results show that our model significantly outperforms other state-of-art methods on several graph classification benchmarks and more resilient to feature-perturbation attack than existing pooling methods.

cs.LG↗

arXiv2021

Node Embedding using Mutual Information and Self-Supervision based Bi-level Aggregation

Graph Neural Networks (GNNs) learn low dimensional representations of nodes by aggregating information from their neighborhood in graphs. However, traditional GNNs suffer from two fundamental shortcomings due to their local ($l$-hop neighborhood) aggregation scheme. First, not all nodes in the neighborhood carry relevant information for the target node. Since GNNs do not exclude noisy nodes in their neighborhood, irrelevant information gets aggregated, which reduces the quality of the representation. Second, traditional GNNs also fail to capture long-range non-local dependencies between nodes. To address these limitations, we exploit mutual information (MI) to define two types of neighborhood, 1) \textit{Local Neighborhood} where nodes are densely connected within a community and each node would share higher MI with its neighbors, and 2) \textit{Non-Local Neighborhood} where MI-based node clustering is introduced to assemble informative but graphically distant nodes in the same cluster. To generate node presentations, we combine the embeddings generated by bi-level aggregation - local aggregation to aggregate features from local neighborhoods to avoid noisy information and non-local aggregation to aggregate features from non-local neighborhoods. Furthermore, we leverage self-supervision learning to estimate MI with few labeled data. Finally, we show that our model significantly outperforms the state-of-the-art methods in a wide range of assortative and disassortative graphs.

cs.LG↗

arXiv2021

SST-GNN: Simplified Spatio-temporal Traffic forecasting model using Graph Neural Network

To capture spatial relationships and temporal dynamics in traffic data, spatio-temporal models for traffic forecasting have drawn significant attention in recent years. Most of the recent works employed graph neural networks(GNN) with multiple layers to capture the spatial dependency. However, road junctions with different hop-distance can carry distinct traffic information which should be exploited separately but existing multi-layer GNNs are incompetent to discriminate between their impact. Again, to capture the temporal interrelationship, recurrent neural networks are common in state-of-the-art approaches that often fail to capture long-range dependencies. Furthermore, traffic data shows repeated patterns in a daily or weekly period which should be addressed explicitly. To address these limitations, we have designed a Simplified Spatio-temporal Traffic forecasting GNN(SST-GNN) that effectively encodes the spatial dependency by separately aggregating different neighborhood representations rather than with multiple layers and capture the temporal dependency with a simple yet effective weighted spatio-temporal aggregation mechanism. We capture the periodic traffic patterns by using a novel position encoding scheme with historical and current data in two different models. With extensive experimental analysis, we have shown that our model has significantly outperformed the state-of-the-art models on three real-world traffic datasets from the Performance Measurement System (PeMS).

cs.LG↗

arXiv2020

Graded Betti numbers of some circulant graphs

Let $G$ be the circulant graph $C_n(S)$ with $S \subseteq \{1, 2, \dots, \lfloor \frac{n}{2} \rfloor\}$, and let $I(G)$ denote the edge ideal in the polynomial ring $R=\mathbb{K}[x_0, x_1, \dots, x_{n-1}]$ over a field $\mathbb{K}$. In this paper, we compute the $\mathbb{N}$-graded Betti numbers of the edge ideals of three families of circulant graphs $C_n(1,2,\dots,\widehat{j},\dots,\lfloor \frac{n}{2} \rfloor)$, $C_{lm}(1,2,\dots,\widehat{2l},\dots, \widehat{3l},\dots,\lfloor \frac{lm}{2} \rfloor)$ and $C_{lm}(1,2,\dots,\widehat{l},\dots,\widehat{2l},\dots, \widehat{3l},\dots,\lfloor \frac{lm}{2} \rfloor)$. Other algebraic and combinatorial properties like regularity, projective dimension, induced matching number and when such graphs are well-covered, Cohen-Macaulay, Sequentially Cohen-Macaulay, Buchsbaum and $S_2$ are also discussed.

math.CO↗

arXiv2020

Standard Monomials of 1-Skeleton Ideals of Graphs and Their Signless Laplace Matrices

Let $G$ be a (multi) graph on the vertex set $V=\{0,1,\ldots ,n\}$ with root $0$. The $G$-parking function ideal $\mathcal{M}_G$ is a monomial ideal in the polynomial ring $R=\mathbb{K}[x_1,\ldots ,x_n]$ over a field $\mathbb{K}$ such that $\dim_{\mathbb K}\left(\frac{R}{\mathcal{M}_G}\right)=\det\left(\widetilde{L}_G\right)$, where $\widetilde{L}_G$ is the truncated Laplace matrix of $G$ and $\det\left(\widetilde L_G\right)$ is the determinant of $\widetilde L_G$. In other words, standard monomials of the Artinian quotient $\frac{R}{\mathcal{M}_G}$ correspond bijectively with the spanning trees of $G$. For $0\leq k\leq n-1$, the $k$-skeleton ideal $\mathcal{M}_G^{(k)}$ of $G$ is the monomial subideal $\mathcal{M}_G^{(k)}=\left\langle m_A:\emptyset\neq A\subseteq[n]\text{ and }|A|\leq k+1\right\rangle$ of the $G$-parking function ideal $\mathcal{M}_G=\left\langle m_A : \emptyset \neq A\subseteq[n]\right\rangle\subseteq R$. For a simple graph $G$, Dochtermann conjectured that $\dim_{\mathbb K}\left(\frac{R}{\mathcal{M}_G^{(1)}}\right)\geq\det\left(\widetilde{Q}_G\right)$, where $\widetilde Q_G$ is the truncated signless Laplace matrix of $G$. We show that Dochtermann conjecture holds for any (simple or multi) graph $G$ on $V$.

math.CO↗

arXiv2020

Integer Sequences and Monomial Ideals

Let $\mathfrak{S}_n$ be the set of all permutations of $[n]=\{1,\ldots,n\}$ and let $W$ be the subset consisting of permutations $σ\in \mathfrak{S}_n$ avoiding 132 and 312-patterns. The monomial ideal $I_W = \left\langle \mathbf{x}^σ = \prod_{i=1}^n x_i^{σ(i)} : σ\in W \right\rangle $ in the polynomial ring $R = k[x_1,\ldots,x_n]$ over a field $k$ is called a hypercubic ideal in the article (Certain variants of multipermutohedron ideals, Proc. Indian Acad. Sci.(Math Sci. Vol. 126, No.4, (2016), 479-500). The Alexander dual $I_W^{[\mathbf{n}]}$ of $I_W$ with respect to $\mathbf{n}=(n,\ldots,n)$ has the minimal cellular resolution supported on the first barycentric subdivision $\mathbf{Bd}(Δ_{n-1})$ of an $n-1$-simplex $Δ_{n-1}$. We show that the number of standard monomials of the Artinian quotient $\frac{R}{I_W^{[\mathbf{n}]}}$ equals the number of rooted-labelled unimodal forests on the vertex set $[n]$. In other words, \[ \dim_k\left(\frac{R}{I_W^{[\mathbf{n}]}}\right) = \sum_{r=1}^n r!~s(n,r) = {\rm Per}\left([m_{ij}]_{n \times n} \right),\] where $s(n,r)$ is the (signless) Stirling number of the first kind and ${\rm Per}([m_{ij}]_{n \times n})$ is the permanent of the matrix $[m_{ij}]$ with $m_{ii}=i$ and $m_{ij}=1$ for $i \ne j$. For various subsets $S$ of $\mathfrak{S}_n$ consisting of permutations avoiding patterns, the corresponding integer sequences $\left\lbrace \dim_k\left(\frac{R}{I_S^{[\mathbf{n}]}}\right) \right\rbrace_{n=1}^{\infty}$ are identified.

math.CO↗

arXiv2019

Smart Load Node for Nonsmart Load Under Smart Grid Paradigm: A New Home Energy Management System

This article presents a novel approach for efficient operation of non-smart household appliances under smart grid environment using the proposed smart load node (SLN). In real world scenario, there are so many non-smart loads currently in use and embedding appliance specific intelligence into them to make them as smart loads will be more expensive compared to the proposed SLN, which is a common solution for all types of non-smart loads. This makes the proposed low-cost SLN, which neither requires any infrastructural change in the electrical wiring of a house nor any constructional change in home appliances at the manufacturing stage and at the consumer end, as a feasible solution for intelligent operation of non-smart home appliances under smart grid environment. The SLNs, which are placed in a home like distributed wireless sensor nodes, form a home area network (HAN). The HAN includes a load management unit (LMU) which acts as master for all distributed SLNs. Wi-Fi is chosen as a medium of communication in the HAN. The LMU incorporates load management algorithm which is written in Python script.

eess.SP↗