arXiv subjects

Anurag Singh

Publications and source records attributed to Anurag Singh.

At least 55 records · Page 3Linked to original sources

arXiv2021

Higher matching complexes of complete graphs and complete bipartite graphs

For $r\geq 1$, the $r$-matching complex of a graph $G$, denoted $M_r(G)$, is a simplicial complex whose faces are the subsets $H \subseteq E(G)$ of the edge set of $G$ such that the degree of any vertex in the induced subgraph $G[H]$ is at most $r$. In this article, we give a closed form formula for the homotopy type of the $(n-2)$-matching complex of complete graph on $n$ vertices. We also prove that the $(n-1)$-matching complex of complete bipartite graph $K_{n,n}$ is homotopy equivalent to a sphere of dimension $(n-1)^2-1$.

math.CO↗

arXiv2021

Optimal Reservoir Operations using Long Short-Term Memory Network

A reliable forecast of inflows to the reservoir is a key factor in the optimal operation of reservoirs. Real-time operation of the reservoir based on forecasts of inflows can lead to substantial economic gains. However, the forecast of inflow is an intricate task as it has to incorporate the impacts of climate and hydrological changes. Therefore, the major objective of the present work is to develop a novel approach based on long short-term memory (LSTM) for the forecast of inflows. Real-time inflow forecast, in other words, daily inflow at the reservoir helps in efficient operation of water resources. Also, daily variations in the release can be monitored efficiently and the reliability of operation is improved. This work proposes a naive anomaly detection algorithm baseline based on LSTM. In other words, a strong baseline to forecast flood and drought for any deep learning-based prediction model. The practicality of the approach has been demonstrated using the observed daily data of the past 20 years from Bhakra Dam in India. The results of the simulations conducted herein clearly indicate the supremacy of the LSTM approach over the traditional methods of forecasting. Although, experiments are run on data from Bhakra Dam Reservoir in India, LSTM model, and anomaly detection algorithm are general purpose and can be applied to any basin with minimal changes. A distinct practical advantage of the LSTM method presented herein is that it can adequately simulate non-stationarity and non-linearity in the historical data.

cs.LG↗

arXiv2021

A combinatorial statistic for labeled threshold graphs

Consider the collection of hyperplanes in $\mathbb{R}^n$ whose defining equations are given by $\{x_i + x_j = 0\mid 1\leq i<j\leq n\}$. This arrangement is called the threshold arrangement since its regions are in bijection with labeled threshold graphs on $n$ vertices. Zaslavsky's theorem implies that the number of regions of this arrangement is the sum of coefficients of the characteristic polynomial of the arrangement. In the present article we give a combinatorial meaning to these coefficients as the number of labeled threshold graphs with a certain property, thus answering a question posed by Stanley.

math.CO↗

arXiv2021

Topology of Clique Complexes of Line Graphs

The clique complex of a graph G is a simplicial complex whose simplices are all the cliques of G, and the line graph L(G) of G is a graph whose vertices are the edges of G and the edges of L(G) are incident edges of G. In this article, we determine the homotopy type of the clique complexes of line graphs for several classes of graphs including triangle-free graphs, chordal graphs, complete multipartite graphs, wheel-free graphs, and 4-regular circulant graphs. We also give a closed form formula for the homotopy type of these complexes in several cases.

math.CO↗

arXiv2021

Pattern avoidance and dominating compositions

Jelínek, Mansour, and Shattuck studied Wilf-equivalence among pairs of patterns of the form $\{σ,τ\}$ where $σ$ is a set partition of size $3$ with at least two blocks. They obtained an upper bound for the number of Wilf-equivalence classes for such pairs. We show that their upper bound is the exact number of equivalence classes, thus solving a problem posed by them.

math.CO↗

arXiv2021

Counting regions of the boxed threshold arrangement

In this paper we consider the hyperplane arrangement in $\mathbb{R}^n$ whose hyperplanes are $\{x_i + x_j = 1\mid 1\leq i < j\leq n\}\cup \{x_i=0,1\mid 1\leq i\leq n\}$. We call it the \emph{boxed threshold arrangement} since we show that the bounded regions of this arrangement are contained in an $n$-cube and are in one-to-one correspondence with the labeled threshold graphs on $n$ vertices. The problem of counting regions of this arrangement was studied earlier by Joungmin Song. He determined the characteristic polynomial of this arrangement by relating its coefficients to the count of certain graphs. Here, we provide bijective arguments to determine the number of regions. In particular, we construct certain signed partitions of the set $\{-n,\dots, n\}\setminus\{0\}$ and also construct colored threshold graphs on $n$ vertices and show that both these objects are in bijection with the regions of the boxed threshold arrangement. We independently count these objects and provide closed form formula for the number of regions.

math.CO↗

arXiv2021

Higher Independence Complexes of graphs and their homotopy types

For $r\geq 1$, the $r$-independence complex of a graph $G$ is a simplicial complex whose faces are subset $I \subseteq V(G)$ such that each component of the induced subgraph $G[I]$ has at most $r$ vertices. In this article, we determine the homotopy type of $r$-independence complexes of certain families of graphs including complete $s$-partite graphs, fully whiskered graphs, cycle graphs and perfect $m$-ary trees. In each case, these complexes are either homotopic to a wedge of equi-dimensional spheres or are contractible. We also give a closed form formula for their homotopy types.

math.AT↗

arXiv2021

Vertex decomposability of complexes associated to forests

In this article, we discuss the vertex decomposability of three well-studied simplicial complexes associated to forests. In particular, we show that the bounded degree complex of a forest and the complex of directed trees of a multidiforest are vertex decomposable. We then prove that the non-cover complex of a forest is either contractible or homotopy equivalent to a sphere. Finally, we provide a complete characterization of forests whose non-cover complexes are vertex decomposable.

math.CO↗

arXiv2021

Self-Supervision based Task-Specific Image Collection Summarization

Successful applications of deep learning (DL) requires large amount of annotated data. This often restricts the benefits of employing DL to businesses and individuals with large budgets for data-collection and computation. Summarization offers a possible solution by creating much smaller representative datasets that can allow real-time deep learning and analysis of big data and thus democratize use of DL. In the proposed work, our aim is to explore a novel approach to task-specific image corpus summarization using semantic information and self-supervision. Our method uses a classification-based Wasserstein generative adversarial network (CLSWGAN) as a feature generating network. The model also leverages rotational invariance as self-supervision and classification on another task. All these objectives are added on a features from resnet34 to make it discriminative and robust. The model then generates a summary at inference time by using K-means clustering in the semantic embedding space. Thus, another main advantage of this model is that it does not need to be retrained each time to obtain summaries of different lengths which is an issue with current end-to-end models. We also test our model efficacy by means of rigorous experiments both qualitatively and quantitatively.

cs.MM↗

arXiv2020

Electrostatic and hydrophobic patches on Amyloid-\b{eta} oligomers govern their fractal self-assembly: Implication to proteins

We present a generic model to describe the fractal self-assembly of proteins in terms of electrostatic and hydrophobic interactions. The predictions of the model were correlated with the simulated fractals obtained using patchy diffusion-limited aggregation, and the experimentally observed Amyloid-\b{eta} (A\b{eta}) fractal self-assembly using confocal microscopy. The molecular docking was used to determine the properties of the patches on A\b{eta} oligomers. Similar patch properties were used to design the particles for simulation. In agreement with the model predictions, the free energy of formation for the simulated fractal self-assembly was proportional to its fractal dimension; moreover, their morphologies were similar to the fractal morphologies of A\b{eta} observed experimentally at different pH.

cond-mat.soft↗

arXiv2020

Bounded degree complexes of forests

Given an arbitrary sequence of non-negative integers $\vecλ=(λ_1,\dots,λ_n)$ and a graph $G$ with vertex set $\{v_1,\dots,v_n\}$, the bounded degree complex, denoted $\text{BD}^{\vecλ}(G)$, is a simplicial complex whose faces are the subsets $H\subseteq E(G)$ such that for each $i \in \{1,\dots,n\}$, the degree of vertex $v_i$ in the induced subgraph $G[H]$ is at most $λ_i$. When $λ_i=k$ for all $i$, the bounded degree complex $\text{BD}^{\vecλ}(G)$ is called the $k$-matching complex, denoted $M_k(G)$. In this article, we determine the homotopy type of bounded degree complexes of forests. In particular, we show that, for all $k\geq 1$, the $k$-matching complexes of caterpillar graphs are either contractible or homotopy equivalent to a wedge of spheres, thereby proving a conjecture of Julianne Vega \cite[Conjecture 7.3]{Vega19}. We also give a closed form formula for the homotopy type of the bounded degree complexes of those caterpillar graphs in which every non-leaf vertex is adjacent to at least one leaf vertex.

math.CO↗

arXiv2020

A Gaussian Process Upsampling Model for Improvements in Optical Character Recognition

Optical Character Recognition and extraction is a key tool in the automatic evaluation of documents in a financial context. However, the image data provided to automated systems can have unreliable quality, and can be inherently low-resolution or downsampled and compressed by a transmitting program. In this paper, we illustrate the efficacy of a Gaussian Process upsampling model for the purposes of improving OCR and extraction through upsampling low resolution documents.

cs.CV↗

arXiv2020

Distance $r$-domination number and $r$-independence complexes of graphs

For $r\geq 1$, the $r$-independence complex of a graph $G$, denoted Ind$_r(G)$, is a simplicial complex whose faces are subsets $A \subseteq V(G)$ such that each component of the induced subgraph $G[A]$ has at most $r$ vertices. In this article, we establish a relation between the distance $r$-domination number of $G$ and (homological) connectivity of Ind$_r(G)$. We also prove that Ind$_r(G)$, for a chordal graph $G$, is either contractible or homotopy equivalent to a wedge of spheres. Given a wedge of spheres, we also provide a construction of a chordal graph whose $r$-independence complex has the homotopy type of the given wedge.

math.CO↗

arXiv2019

Homotopy Type of Independence Complexes of Certain Families of Graphs

We show that the independence complexes of generalised Mycielskian of complete graphs are homotopy equivalent to a wedge sum of spheres, and determine the number of copies and the dimensions of these spheres. We also prove that the independence complexes of categorical product of complete graphs are wedge sum of circles, upto homotopy. Further, we show that if we perturb a graph $G$ in a certain way, then the independence complex of this new graph is homotopy equivalent to the suspension of the independence complex of $G$.

math.CO↗

arXiv2019

Hom complexes of graphs of diameter $1$

Given a finite simplicial complex $X$ and a connected graph $T$ of diameter $1$, in \cite{anton} Dochtermann had conjectured that $\text{Hom}(T,G_{1,X})$ is homotopy equivalent to $X$. Here, $G_{1, X}$ is the reflexive graph obtained by taking the $1$-skeleton of the first barycentric subdivision of $X$ and adding a loop at each vertex. This was proved by Dochtermann and Schultz in \cite{ds12}. In this article, we give an alternate proof of this result by understanding the structure of the cells of Hom$(K_n,G_{1,X})$, where $K_n$ is the complete graph on $n$ vertices. We prove that the neighborhood complex of $G_{1,X}$ is homotopy equivalent to $X$ and Hom$(K_n,G_{1,X})\simeq $ Hom$(K_{n-1},G_{1,X})$, for each $n\geq 3$.

math.CO↗

arXiv2019

On the topology of bi-cyclopermutohedra

Motivated by the work of Panina and her coauthors on cyclopermutohedron we study a poset whose elements correspond to equivalence classes of partitions of the set $\{1,\cdots, n+1\}$ up to cyclic permutations and orientation reversion. This poset is the face poset of a regular CW complex which we call bi-cyclopermutohedron and denote it by $\mathrm{QP}_{n+1}$. The complex $\mathrm{QP}_{n+1}$ contains subcomplexes homeomorphic to moduli space of certain planar polygons with $n+1$ sides up to isometries. In this article we find an optimal discrete Morse function on $\mathrm{QP}_{n+1}$ and use it to compute its homology with $\mathbb{Z}$ as well as $\mathbb{Z}_2$ coefficients.

math.AT↗

arXiv2019

Efficient Edge Rewiring Strategies for Enhancement in Network Capacity

The structure of the network has great impact on its traffic dynamics. Most of the real world networks follow the heterogeneous structure and exhibit scale-free feature. In scale-free network, a new node prefers to connect with hub nodes and the network capacity is curtailed by smaller degree nodes. Therefore, we propose rewiring a fraction of links in the network, to improve the network transport efficiency. In this paper, we discuss some efficient link rewiring strategies and perform simulations on scale-free networks, confirming the effectiveness of these strategies. The rewiring strategies actually reduce the centrality of the nodes having higher betweenness centrality. After the link rewiring process, the degree distribution of the network remains the same. This work will be beneficial for the enhancement of network performance.

cs.NI↗

arXiv2018

A blockchain based Secure and Trusted framework for Information Propagation on Online Social Networks

The online social networks facilitate naturally for the users to share information. On these platforms, each user shares information based on his or her interests. The particular information being shared by a user may be legitimate or fake. Sometimes a misinformation, propagated by users and group can create chaos or in some cases, might leads to cases of riots. Nowadays the third party like ALT news and Cobrapost check the information authenticity, but it takes too much time to validate the news. Therefore, a robust and new system is required to check the information authenticity within the network, to stop the propagation of misinformation. In this paper, we propose a blockchain based framework for sharing the information securely at the peer level. In the blockchain model, a chain is created by combining blocks of information. Each node of network propagates the information based on its credibility to its peer nodes. The credibility of a node will vary according to the respective information. Trust is calculated between sender and receiver in two ways:(i) Local trust used for sharing information at the peer level and (ii) global trust is used for a credibility check of each user in the network. We evaluate our framework using real dataset derived from Facebook. Our approach achieves an accuracy of 83% which shows the effectiveness of our proposed framework.

cs.SI↗