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Unstructured meshes are used in many large-scale scientific and engineering problems, including finite-volume methods for computational fluid dynamics and finite-element methods for structural analysis. If unstructured problems such as these are to be solved on distributed-memory parallel computers, their data structures must be partitioned and distributed across processors; if they are to be solved efficiently, the partitioning must maximize load balance and minimize interprocessor communication. Recently the recursive spectral bisection method (RSB) has been shown to be very effective for such partitioning problems compared to alternative methods. Unfortunately, RSB in its simplest form is rather expensive. In this report we shall describe a multilevel implementation of RSB that can attain about an order-of-magnitude improvement in run time on typical examples. Keywords: graph partitioning, domain decomposition, MIMD machines, multilevel algorithm, spectral bisection, sp...

In this paper, we discuss consensus problems for a network of dynamic agents with fixed and switching topologies. We analyze three cases: i) networks with switching topology and no time-delays, ii) networks with fixed topology and communication time-delays, and iii) max-consensus problems (or leader determination) for groups of discrete-time agents. In each case, we introduce a linear/nonlinear consensus protocol and provide convergence analysis for the proposed distributed algorithm. Moreover, we establish a connection between the Fiedler eigenvalue of the information flow in a network (i.e. algebraic connectivity of the network) and the negotiation speed (or performance) of the corresponding agreement protocol. It turns out that balanced digraphs play an important role in addressing average-consensus problems. We introduce disagreement functions that play the role of Lyapunov functions in convergence analysis of consensus protocols. A distinctive feature of this work is to address consensus problems for networks with directed information flow. We provide analytical tools that rely on algebraic graph theory, matrix theory, and control theory. Simulations are provided that demonstrate the effectiveness of our theoretical results.

Accrue Software, Inc. Clustering is a division of data into groups of similar objects. Representing the data by fewer clusters necessarily loses certain fine details, but achieves simplification. It models data by its clusters. Data modeling puts clustering in a historical perspective rooted in mathematics, statistics, and numerical analysis. From a machine learning perspective clusters correspond to hidden patterns, the search for clusters is unsupervised learning, and the resulting system represents a data concept. From a practical perspective clustering plays an outstanding role in data mining applications such as scientific data exploration, information retrieval and text mining, spatial database applications, Web analysis, CRM, marketing, medical diagnostics, computational biology, and many others. Clustering is the subject of active research in several fields such as statistics, pattern recognition, and machine learning. This survey focuses on clustering in data mining. Data mining adds to clustering the complications of very large datasets with very many attributes of different types. This imposes unique

Spectral partitioning methods use the Fiedler vector---the eigenvector of the secondsmallest eigenvalue of the Laplacian matrix---to find a small separator of a graph. These methods are important components of many scientific numerical algorithms and have been demonstrated by experiment to work extremely well. In this paper, we show that spectral partitioning methods work well on bounded-degree planar graphs and finite element meshes--- the classes of graphs to which they are usually applied. While naive spectral bisection does not necessarily work, we prove that spectral partitioning techniques can be used to produce separators whose ratio of vertices removed to edges cut is O( p n) for bounded-degree planar graphs and two-dimensional meshes and O i n 1=d j for well-shaped d-dimensional meshes. The heart of our analysis is an upper bound on the second-smallest eigenvalues of the Laplacian matrices of these graphs. 1. Introduction Spectral partitioning has become one of the mos...

An important application of graph partitioning is data clustering using a graph model | the pairwise similarities between all data objects form a weighted graph adjacency matrix that contains all necessary information for clustering. Here we propose a new algorithm for graph partition with an objective function that follows the min-max clustering principle. The relaxed version of the optimization of the min-max cut objective function leads to the Fiedler vector in spectral graph partition. Theoretical analyses of min-max cut indicate that it leads to balanced partitions, and lower bonds are derived. The min-max cut algorithm is tested on newsgroup datasets and is found to outperform other current popular partitioning/clustering methods. The linkagebased re nements in the algorithm further improve the quality of clustering substantially. We also demonstrate that the linearized search order based on linkage di erential is better than that based on the Fiedler vector, providing another e ective partition method.

We propose a motion segmentation algorithm that aims to break a scene into its most prominent moving groups. A weighted graph is constructed on the ira. age sequence by connecting pixels that arc in the spatio-temporal neighborhood of each other. At each pizel, we define motion profile vectors which capture the probability distribution of the image veloczty. The distance between motion profiles is used to assign a weight on the graph edges. 5rsmg normalized cuts we find the most salient partitions of the spatiotemporaI graph formed by the image sequence. For swmenting long image sequences,' we have developed a recursire update procedure that incorporates knowledge of segmentation in previous frames for efficiently finding the group correspondence in the new frame.

Abstract. The paper is essentially a survey of known results about the spectrum of the Laplacian matrix of graphs with special emphasis on the second smallest Lapla-cian eigenvalue λ2 and its relation to numerous graph invariants, including connectivity, expanding properties, isoperimetric number, maximum cut, independence number, genus, diameter, mean distance, and bandwidth-type parameters of a graph. Some new results and generalizations are added. † This article appeared in “Graph Theory, Combinatorics, and Applications”, Vol. 2,

Parametrization of 3D mesh data is important for many graphics applications, in particular for texture mapping, remeshing and morphing. Closed manifold genus-0 meshes are topologically equivalent to a sphere, hence this is the natural parameter domain for them. Parametrizing a triangle mesh onto the sphere means assigning a 3D position on the unit sphere to each of the mesh vertices, such that the spherical triangles induced by the mesh connectivity do not overlap. Satisfying the non-overlapping requirement is the most difficult and critical component of this process. We present a generalization of the method of barycentric coordinates for planar parametrization which solves the spherical parametrization problem, prove its correctness by establishing a connection to spectral graph theory and describe efficient numerical methods for computing these parametrizations.

Fitness landscapes are an important concept in molecular evolution. Many important examples of landscapes in physics and combinatorial optimation, which are widely used as model landscapes in simulations of molecular evolution and adaptation, are "elementary", i.e., they are (up to an additive constant) eigenfuctions of a graph Laplacian. It is shown that elementary landscapes are characterized by their correlation functions. The correlation functions are in turn uniquely determined by the geometry of the underlying configuration space and the nearest neighbor correlation of the elementary landscape. Two types of correlation functions are investigated here: the correlation of a time series sampled along a random walk on the landscape and the correlation function with respect to a partition of the set of all vertex pairs.