Spectral Partitioning Algorithm Research Articles

Hypergraph partitioning using tensor eigenvalue decomposition.

Hypergraphs have gained increasing attention in the machine learning community lately due to their superiority over graphs in capturing super-dyadic interactions among entities. In this work, we propose a novel approach for the partitioning of k-uniform hypergraphs. Most of the existing methods work by reducing the hypergraph to a graph followed by applying standard graph partitioning algorithms. The reduction step restricts the algorithms to capturing only some weighted pairwise interactions and hence loses essential information about the original hypergraph. We overcome this issue by utilizing tensor-based representation of hypergraphs, which enables us to capture actual super-dyadic interactions. We extend the notion of minimum ratio-cut and normalized-cut from graphs to hypergraphs and show that the relaxed optimization problem can be solved using eigenvalue decomposition of the Laplacian tensor. This novel formulation also enables us to remove a hyperedge completely by using the "hyperedge score" metric proposed by us, unlike the existing reduction approaches. We propose a hypergraph partitioning algorithm inspired from spectral graph theory and also derive a tighter upper bound on the minimum positive eigenvalue of even-order hypergraph Laplacian tensor in terms of its conductance, which is utilized in the partitioning algorithm to approximate the normalized cut. The efficacy of the proposed method is demonstrated numerically on synthetic hypergraphs generated by stochastic block model. We also show improvement for the min-cut solution on 2-uniform hypergraphs (graphs) over the standard spectral partitioning algorithm.

User-centric base station clustering and resource allocation for cell-edge users in 6G ultra-dense networks

Ultra-Dense Networks (UDNs) have been proposed to meet the ultra-high system capacity and ultra-high user experience rate requirements of sixth generation (6G) mobile networks. However, ultra-dense base stations (BSs) deployment poses challenges for cell-edge users such as BS selection, severe interference, resource allocation, and inter-cell handover. To tackle these obstacles, this paper formulates a joint BS clustering and resource allocation problem for cell-edge users in a 6G heterogeneous UDN system. For efficient resolution, this problem is decoupled into two sub-problems to be solved independently. We first propose a user-centric BS clustering algorithm based on many-to-many matching for the BS clustering problem in the heterogeneous UDN system; This algorithm considers the constraint of the system capacity. A many-to-many stable matching between users and small cell BSs is constructed under the optimization objective of maximizing the achievable user rate. Furthermore, we analyze the effectiveness, stability, convergence, and complexity of the BS clustering algorithm. Given established small cell BS clusters, we also propose a user-centric resource allocation algorithm based on network partitioning. This algorithm accounts for the interference of all users in the heterogeneous UDN system and maximizes intra-sub-network interference while minimizing inter-sub-network interference via a spectral clustering-based sub-network partitioning algorithm. Next, orthogonal allocation of resource blocks (RBs) is implemented within each sub-network, and RBs are spatially multiplexed between sub-networks. Numerical results confirm the benefits of the proposed methods: our algorithms outperform benchmark solutions in terms of the sum achievable system rate, average achievable user rate, and user spectral efficiency.

SMA: A constructive partitioning based mapping approach for Networks-on-Chip

This paper proposes a constructive partitioning approach as an effective mapping approach for 2D Networks-on-Chip. Cluster-based mapping approach offers a promising improvement on the fastest initial mapping against the standard Networks-on-Chip (NoC) architecture. In this paper, the spectral partitioning algorithm is adopted for the fastest mapping with NoC, named SMA. The proposed spectral partitioning-based mapping approach (SMA) explores the efficacious NoC mapping concerning communication bandwidth, average hop counts, and total power consumption NoC architecture. The experimental analysis indicates that the proposed SMA outperforms existing partitioning approaches.

An efficient supply management in water flow network using graph spectral techniques.

An efficient way of distributing water in urban cities is one of the major global challenges of our time. A Water Distribution Network (WDN) is a vital infrastructure, intended to provide fresh water to households within a city or designated boundary. Given the WDN's complexity, effective numerical techniques are required to aid in the development of an ideal monitoring system. Flow meters and gate valves should be placed in low-connected areas of the WDN with water flow reaching several regions of the network. This study proposes a general strategy for assisting water utilities in making decisions for effective water supply. The aim of the research is to use weighted spectral clustering algorithms to outline water districts while addressing hydraulic restrictions via weighted adjacency and Laplacian matrices of the weighted network. This work aims to identify influenced nodes in the network based on Eigen centrality and effectively distribute water across those nodes. This project also looks at how to measure the network's connection strength to avoid water leaks. The best clusters are found using Eigenvalues and Eigenvectors of weighted adjacency matrices and Laplacian matrices of the water network in the proposed graph spectral framework. In order to establish the optimum water network division approach, topological and graph metrics were used to compare multiple spectral clustering techniques. The proposed graph spectral approach is tested using a genuine water distribution network serving an urban area of Coimbatore city in India and offering a method for partitioning complex networks that employ the spectral graph partitioning algorithm.

Efficient Task Scheduling of Virtual Machines using Novel Spectral Partitioning and Differential Evaluation Algorithm

Abstract Task-scheduling is a major challenge in cloud computing environment that degrades the performance of the system. To enhance the performance of the system, an effective task-scheduling algorithm is needed. Hence an effective task-partitioning and taskscheduling algorithm is introduced for enhancing the system performance. To create resources (datacentre, broker, Virtual Machine - VM and cloudlet) in a dynamic way through the use of CloudSim. In addition, this study intended to perform taskpartitioning and task-scheduling in an effective manner by utilizing the novel spectral partitioning - (SP) and differential evaluation algorithm - (DEA). At first, the task and datacentre is initialized. Subsequently, task-partitioning is performed using the novel SP. It includes a series of steps in which a Laplacian matrix is computed initially. Then based on the Eigen-values and Eigen-vectors of the Laplacian matrix the tasks are partitioned. Followed by this, taskscheduling is performed with the employment of proposed novel DEA. The process comprise the following series of steps such as threshold calculation, mutation, crossover, selection and knee solution for achieving efficient task-partitioning and scheduling. The performance of the proposed system is evaluated by comparing it with other traditional methods. And validated in terms of service cost, load balancing, makespan and energy consumption. The results proved the efficacy of the introduced system. The overall results obtained from comparative analysis also reveal that proposed method outperformed other traditional techniques thereby accomplishing effective task scheduling of VMs in cloud computing environment. Keywords: Cloud computing environment, Virtual Machines, Task Scheduling, Novel Spectral Partitioning and Differential Evaluation Algorithm.

Swell and sea characteristics along the south-west coast of India during the monsoon

Swell and wind–seas generally co-exist in the open ocean, and are often difficult to segregate. Swell and wind–sea characteristics of observed wave spectra at off-Kochi, located along India’s south-west coast in the Arabian Sea during the south-west monsoon are presented. This paper attempts to split the observed wave spectra into individual wave partitions, thereby enabling the computation of integral wave parameters of each partition. The spectra with five partitions is found occur more frequently in the considered season. To extract wind–sea and swell information from the observed directional spectrum, a consistent overshooting phenomenon (OP), Modified Overshooting phenomenon (MOP) and Wave age (WA) method are employed in each partition. The results show a prevalence of mixed sea states. An analysis from WA method shows the absence of young wind seas. The derived wind–wave parameters of the spectral partitioning algorithm agree well with Toba’s 3/2 law with a good correlation. The effectiveness of interaction between the derived swell and sea components is discussed through nonlinear interactions that are occurring between the set of four waves.

Spectral partitioning identifies individual heterogeneity in the functional network topography of ventral and anterior medial prefrontal cortex

Regions of human medial prefrontal cortex (mPFC) and posterior cingulate cortex (PCC) are part of the default network (DN), and additionally are implicated in diverse cognitive functions ranging from autobiographical memory to subjective valuation. Our ability to interpret the apparent co-localization of task-related effects with DN-regions is constrained by a limited understanding of the individual-level heterogeneity in mPFC/PCC functional organization. Here we used cortical surface-based meta-analysis to identify a parcel in human PCC that was more strongly associated with the DN than with valuation effects. We then used resting-state fMRI data and a data-driven network analysis algorithm, spectral partitioning, to partition mPFC and PCC into “DN” and “non-DN” subdivisions in individual participants (n = 100 from the Human Connectome Project). The spectral partitioning algorithm identified individual-level cortical subdivisions that varied markedly across individuals, especially in mPFC, and were reliable across test/retest datasets. Our results point toward new strategies for assessing whether distinct cognitive functions engage common or distinct mPFC subregions at the individual level.

A spectral partitioning algorithm for maximum directed cut problem

We investigate the maximum directed cut (MaxDC) problem by designing a spectral partitioning algorithm. Given a directed graph with nonnegative arc weights, we wish to obtain a bipartition of the vertices such that the total weight of the directed cut arcs is maximized. Relaxing the MaxDC problem as a quadratic program allows us to explore combinatorial properties of the optimal solution, leading to a 0.272-approximation algorithm via the technique of spectral partitioning rounding.

Network Partitioning Algorithms for Solving the Traffic Assignment Problem using a Decomposition Approach

Recent methods in the literature to parallelize the traffic assignment problem consider partitioning a network into subnetworks to reduce the computation time. In this article, a partitioning method is sought that generates subnetworks minimizing the computation time of a decomposition approach for solving the traffic assignment (DSTAP). The aim is to minimize the number of boundary nodes, the interflow between subnetworks, and the computation time when the traffic assignment problem is solved in parallel on the subnetworks. Two different methods for partitioning are tested. The first is an agglomerative clustering heuristic that reduces the subnetwork boundary nodes. The second is a flow weighted spectral partitioning algorithm that uses the normalized graph Laplacian to partition the network. The performance of both algorithms is assessed on different test networks. The results indicate that the agglomerative heuristic generates subnetworks with a lower number of boundary nodes, which reduces the per iteration computation time of DSTAP. However, the partitions generated may be heavily imbalanced leading to a higher computation time when the subnetworks are solved in parallel separately at a particular DSTAP iteration. For the Austin network partitioned into four subnetworks, the agglomerative heuristic requires 3.5 times more computation time to solve the subnetworks in parallel. The results also show that the spectral partitioning method is superior for minimizing the interflow between subnetworks. This leads to a faster convergence rate of the DSTAP algorithm.

Deep sparse representation-based mid-level visual elements discovery in fine-grained classification

In this paper, we propose a new mid-level visual elements discovery method and apply it to the fine-grained classification. We present the duality between image patches and features extracted by the convolutional winner-take-all autoencoder (CONV-WTA-AE). The sparsity constraints used by CONV-WTA-AE make a group of objects sharing the same feature components. Hence, the image patches could be clustered by their sharing feature components and the feature components can be clustered by their co-occurrence in the image patches. We propose formulating the mid-level visual elements mining as a bipartite graph partitioning problem. The spectral partitioning algorithm is employed to co-cluster image patches and feature components. The CONV-WTA-AE is an unsupervised feature learning method. Hence, it avoids using expensive annotations. Our experiments demonstrate that the spectral partitioning method is very efficient but only the confident instances in a cluster are well discriminated. The similarity metric used by this algorithm is not accurate enough. Hence, we propose training a group of linear support vector machine (SVM) to refine the clustering results. These SVMs will be trained on the initial confident instances and provide a better discriminative similarity. Then we can re-assign instances to each clusters. To avoid overfitting, this process is iterated on many data subsets. We conduct a series of experiments on the MNIST dataset to verify our algorithm. The experimental results show that our method can discover meaningful image patch clusters. In the fine-grained classification task, visual elements are input into an ensemble of convolutional neural networks. The experiments on the CompCars dataset illustrate that our method can achieve the state-of-the-art performance.

A generalized inverse for graphs with absorption

We consider weighted, directed graphs with a notion of absorption on the vertices, related to absorbing random walks on graphs. We define a generalized inverse of the graph Laplacian, called the absorption inverse, that reflects both the graph structure as well as the absorption rates on the vertices. Properties of this generalized inverse are presented, including a basic relationship between the absorption inverse and the group inverse of a related graph, a forest theorem for interpreting the entries of the absorption inverse, as well as relationships between the absorption inverse and the fundamental matrix of the absorbing random walk. Applications of the absorption inverse for describing the structure of graphs with absorption are given, including a directed distance metric, spectral partitioning algorithm, and centrality measure.

Consistency of spectral hypergraph partitioning under planted partition model

Hypergraph partitioning lies at the heart of a number of problems in machine learning and network sciences. Many algorithms for hypergraph partitioning have been proposed that extend standard approaches for graph partitioning to the case of hypergraphs. However, theoretical aspects of such methods have seldom received attention in the literature as compared to the extensive studies on the guarantees of graph partitioning. For instance, consistency results of spectral graph partitioning under the stochastic block model are well known. In this paper, we present a planted partition model for sparse random non-uniform hypergraphs that generalizes the stochastic block model. We derive an error bound for a spectral hypergraph partitioning algorithm under this model using matrix concentration inequalities. To the best of our knowledge, this is the first consistency result related to partitioning non-uniform hypergraphs.

Improved Cheeger's Inequality and Analysis of Local Graph Partitioning using Vertex Expansion and Expansion Profile

We prove two generalizations of the Cheeger's inequality. The first generalization relates the second eigenvalue to the edge expansion and the vertex expansion of the graph $G$, $\lambda_2 = \Omega( \phi^V(G) \phi(G) )$, where $\phi^V(G)$ denotes the robust vertex expansion of $G$ and $\phi(G)$ denotes the edge expansion of $G$. The second generalization relates the second eigenvalue to the edge expansion and the expansion profile of $G$, for all $k \geq 2$, $ \lambda_2 = \Omega( \phi_k(G) \phi(G) / k )$, where $\phi_k(G)$ denotes the $k$-way expansion of $G$. These show that the spectral partitioning algorithm has better performance guarantees when $\phi^V(G)$ is large (e.g., planted random instances) or $\phi_k(G)$ is large (instances with few disjoint nonexpanding sets). Both bounds are tight up to a constant factor. Our approach is based on a method to analyze solutions of Laplacian systems, and this allows us to extend the results to local graph partitioning algorithms. In particular, we show that ou...

Probabilistic hypergraph based hash codes for social image search

With the rapid development of the Internet, recent years have seen the explosive growth of social media. This brings great challenges in performing efficient and accurate image retrieval on a large scale. Recent work shows that using hashing methods to embed high-dimensional image features and tag information into Hamming space provides a powerful way to index large collections of social images. By learning hash codes through a spectral graph partitioning algorithm, spectral hashing (SH) has shown promising performance among various hashing approaches. However, it is incomplete to model the relations among images only by pairwise simple graphs which ignore the relationship in a higher order. In this paper, we utilize a probabilistic hypergraph model to learn hash codes for social image retrieval. A probabilistic hypergraph model offers a higher order representation among social images by connecting more than two images in one hyperedge. Unlike a normal hypergraph model, a probabilistic hypergraph model considers not only the grouping information, but also the similarities between vertices in hyperedges. Experiments on Flickr image datasets verify the performance of our proposed approach.

Graph Partitioning Using Matrix Values for Preconditioning Symmetric Positive Definite Systems

Prior to the parallel solution of a large linear system, it is required to perform a partitioning of its equations/unknowns. Standard partitioning algorithms are designed using the considerations of the efficiency of the parallel matrix-vector multiplication, and typically disregard the information on the coefficients of the matrix. This information, however, may have a significant impact on the quality of the preconditioning procedure used within the chosen iterative scheme. In the present paper, we suggest a spectral partitioning algorithm, which takes into account the information on the matrix coefficients and constructs partitions with respect to the objective of enhancing the quality of the nonoverlapping additive Schwarz (block Jacobi) preconditioning for symmetric positive definite linear systems. For a set of test problems with large variations in magnitudes of matrix coefficients, our numerical experiments demonstrate a noticeable improvement in the convergence of the resulting solution scheme when using the new partitioning approach.

A Multigrid Wave Forecasting Model: A New Paradigm in Operational Wave Forecasting

Abstract A new operational wave forecasting system has been implemented at the National Centers for Environmental Prediction (NCEP) using the third public release of WAVEWATCH III. The new system uses a mosaic of grids with two-way nesting in a single model. This global system replaces a previous operational wave modeling suite (based on the second release of WAVEWATCH III). The new forecast system consists of nine grids at different resolutions to provide the National Weather Service (NWS) and NCEP centers with model guidance of suitable resolution for all areas where they have the responsibility of providing gridded forecast products. New features introduced in WAVEWATCH III, such as two-way nesting between grids and carving out selected areas of the computational domain, have allowed the operational model to increase spatial resolution and extend the global domain closer to the North Pole, while at the same time optimizing the computational cost. A spectral partitioning algorithm has been implemented to separate individual sea states from the overall spectrum, thus providing additional products for multiple sea states. Field output data are now packed in version 2 of the gridded binary (GRIB2) format and apart from the standard mean wave parameters, they also include parameters of partitioned wave spectra. The partitioning is currently limited to three fields: the wind-wave component, and primary and secondary swells. The modeling system has been validated against data using a multiyear hindcast database as well as archived forecasts. A new software tool developed by the U.S. Army Corps of Engineers (USACE) is used to extend the analysis from overall error estimates to separate skill scores for wind seas and swells.

SpecP: A tool for spectral partitioning of protein contact graph.

SpecP is an open-source Python module that performs Spectral Partitioning on Protein Contact Graphs. Protein Contact Graphs are graph theory based representation of the protein structure, where each amino acid forms a ‘vertex’ and spatial contact of any two amino acids is an ‘edge’ between them. Spectral partitioning is carried out in SpecP based on the second smallest spectral value (eigen value) of the Protein Contact Graph. The eigen vector corresponding to the second smallest spectral value are partitioned into two clusters based on the sign of the corresponding vector entry. Spectral Partitioning algorithm is repeatedly carried out until the desired numbers of partitions are obtained. SpecP visualizes the spectrally partitioned clusters of protein structure along with the Protein Contact Map and Protein Contact Graph which can be saved for later use. It also possesses an interactive mode whereby the user has the ability to zoom, pan, resize and save these raster images in various image formats (.eps, .jpg, .png) manually. SpecP is a stand-alone extensible tool useful for structural analysis of proteins.

Hierarchical Consensus of Multi-Agent Systems Based on Community Decomposition

The convergence speed of multi-agent system is a focused issue in the consensus problem. The traditional consensus algorithm is generally discussed on the single-layer topology. The spectral partitioning algorithm of the complex network and NF algorithm was applied to the optimal decomposition problem of the multi-agent topology and a hierarchical consensus algorithm based on the community decomposition was proposed in this paper. By converting the single-layer consensus problem of the multi-agent system to multi-layer consensus problem, the convergence speed is effectively improved on the premise of maintaining the original topology constraints. The effectiveness of two types of algorithms is demonstrated by simulations compared with the classic model. Then the comparison between the spectral partitioning method and NF algorithm applied to the optimal decomposition problem is proposed.

Eigenvector synchronization, graph rigidity and the molecule problem.

The graph realization problem has received a great deal of attention in recent years, due to its importance in applications such as wireless sensor networks and structural biology. In this paper, we extend the previous work and propose the 3D-As-Synchronized-As-Possible (3D-ASAP) algorithm, for the graph realization problem in ℝ3, given a sparse and noisy set of distance measurements. 3D-ASAP is a divide and conquer, non-incremental and non-iterative algorithm, which integrates local distance information into a global structure determination. Our approach starts with identifying, for every node, a subgraph of its 1-hop neighborhood graph, which can be accurately embedded in its own coordinate system. In the noise-free case, the computed coordinates of the sensors in each patch must agree with their global positioning up to some unknown rigid motion, that is, up to translation, rotation and possibly reflection. In other words, to every patch, there corresponds an element of the Euclidean group, Euc(3), of rigid transformations in ℝ3, and the goal was to estimate the group elements that will properly align all the patches in a globally consistent way. Furthermore, 3D-ASAP successfully incorporates information specific to the molecule problem in structural biology, in particular information on known substructures and their orientation. In addition, we also propose 3D-spectral-partitioning (SP)-ASAP, a faster version of 3D-ASAP, which uses a spectral partitioning algorithm as a pre-processing step for dividing the initial graph into smaller subgraphs. Our extensive numerical simulations show that 3D-ASAP and 3D-SP-ASAP are very robust to high levels of noise in the measured distances and to sparse connectivity in the measurement graph, and compare favorably with similar state-of-the-art localization algorithms.

Detection of community structure in networks based on community coefficients

Determining community structure in networks is fundamental to the analysis of the structural and functional properties of those networks, including social networks, computer networks, and biological networks. Modularity function Q, which was proposed by Newman and Girvan, was once the most widely used criterion for evaluating the partition of a network into communities. However, modularity Q is subject to a serious resolution limit. In this paper, we propose a new function for evaluating the partition of a network into communities. This is called community coefficient C. Using community coefficient C, we can automatically identify the ideal number of communities in the network, without any prior knowledge. We demonstrate that community coefficient C is superior to the modularity Q and does not have a resolution limit. We also compared the two widely used community structure partitioning methods, the hierarchical partitioning algorithm and the normalized cuts (Ncut) spectral partitioning algorithm. We tested these methods on computer-generated networks and real-world networks whose community structures were already known. The Ncut algorithm and community coefficient C were found to produce better results than hierarchical algorithms. Unlike several other community detection methods, the proposed method effectively partitioned the networks into different community structures and indicated the correct number of communities.