Bipartite Graph Partitioning and Data Clustering* Hongyuan Zha Xiaofeng He Dept. of Comp. Sci. & Eng. Penn State Univ. State College, PA 16802 {zha,xhe}@cse.psu.edu Chris Ding Horst Simon NERSC Division Berkeley National Lab. Berkeley, CA 94720 {chqding,hdsimon} Qlbl. gov Ming Gu Dept. of Math. U.C. Berkeley Berkeley, CA 94720 mgu@math.berkeley.edu ABSTRACT M a n y data types arising from data mining applications can be modeled as bipartite graphs, examples include terms and documents in a text corpus, customers and purchasing items in market basket analysis and reviewers and movies in a movie recommender system. In this paper, we propose a new data clustering method based on partitioning the underlying bipartite graph. The partition is constructed by minimizing a normalized sum of edge weights between unmatched pairs of vertices of the bipartite graph. We show that an approxi­ mate solution to the minimization problem can be obtained by computing a partial singular value decomposition ( S V D ) of the associated edge weight matrix of the bipartite graph. We point out the connection of our clustering algorithm to correspondence analysis used in multivariate analysis. We also briefly discuss the issue of assigning data objects to multiple clusters. In the experimental results, we apply our clustering algorithm to the problem of document clustering to illustrate its effectiveness and efficiency. 1. INTRODUCTION Cluster analysis is an important tool for exploratory data mining applications arising from many diverse disciplines. Informally, cluster analysis seeks to partition a given data set into compact clusters so that data objects within a clus­ ter are more similar than those in distinct clusters. The liter­ ature on cluster analysis is enormous including contributions from many research communities, (see [6, 9] for recent sur­ veys of some classical approaches.) M a n y traditional clus­ tering algorithms are based on the assumption that the given dataset consists of covariate information (or attributes) for each individual data object, and cluster analysis can be cast as a problem of grouping a set of n-dimensional vectors each representing a data object in the dataset. A familiar ex­ ample is document clustering using the vector space model [1]. Here each document is represented by an n-dimensional vector, and each coordinate of the vector corresponds to a term in a vocabulary of size n. This formulation leads to the so-called term-document matrix A = (oy) for the rep­ resentation of the collection of documents, where o y is the so-called term frequency, i.e., the number of times term i occurs in document j. In this vector space model terms and documents are treated asymmetrically with terms consid­ ered as the covariates or attributes of documents. It is also possible to treat both terms and documents as first-class citizens in a symmetric fashion, and consider a y as the fre­ quency of co-occurrence of term i and document j as is done, for example, in probabilistic latent semantic indexing [12]. In this paper, we follow this basic principle and propose a new approach to model terms and documents as vertices in a bipartite graph with edges of the graph indicating the co-occurrence of terms and documents. In addition we can optionally use edge weights to indicate the frequency of this co-occurrence. Cluster analysis for document collections in this context is based on a very intuitive notion: documents are grouped by topics, on one hand documents in a topic tend to more heavily use the same subset of terms which form a term cluster, and on the other hand a topic usually is characterized by a subset of terms and those documents heavily using those terms tend to be about that particular topic. It is this interplay of terms and documents which gives rise to what we call bi-clustering by which terms and documents are simultaneously grouped into semantically co- Categories and Subject Descriptors 11.3.3 [ I n f o r m a t i o n S e a r c h a n d R e t r i e v a l ] : Clustering; G.1.3 [ N u m e r i c a l L i n e a r A l g e b r a ] : Singular value de­ composition; G.2.2 [ G r a p h T h e o r y ] : G r a p h algorithms General Terms Algorithms, theory Keywords document clustering, bipartite graph, graph partitioning, spectral relaxation, singular value decomposition, correspon­ dence analysis *Part of this work was done while Xiaofeng He was a grad­ uate research assistant at N E R S C , Berkeley National Lab. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. CIKM '01 November 5-10, 2001, Atlanta, Georgia. U S A Copyright 2001 A C M X - X X X X X - X X - X / X X / X X ...$5.00. O u r clustering algorithm computes an approximate global optimal solution while probabilistic latent semantic indexing relies on the E M algorithm and therefore might be prone to local m i n i m a even with the help of some annealing process. x

Bipartite graph partitioning and data clustering

In recent years, a new “fine-grained” theory of computational hardness has been developed, based on “fine-grained reductions” that focus on exact running times for problems. Mimicking NP-hardness, the approach is to (1) select a key problem X that for some function t , is conjectured to not be solvable by any O(t(n)1 ") time algorithm for " > 0, and (2) reduce X in a fine-grained way to many important problems, thus giving tight conditional time lower bounds for them. This approach has led to the discovery of many meaningful relationships between problems, and to equivalence

On some fine-grained questions in algorithms and complexity

We introduce the notion of a stable instance for a discrete optimization problem, and argue that in many practical situations only sufficiently stable instances are of interest. The question then arises whether stable instances of NP-hard problems are easier to solve, and in particular, whether there exist algorithms that solve in polynomial time all sufficiently stable instances of some NP-hard problem. The paper focuses on the Max-Cut problem, for which we show that this is indeed the case.

Are stable instances easy

We study the problem of finding frequent itemsets in a continuous stream of transactions. The current frequency of an itemset in a stream is defined as its maximal frequency over all possible windows in the stream from any point in the past until the current state that satisfy a minimal length constraint. Properties of this new measure are studied and an incremental algorithm that allows, at any time, to immediately produce the current frequencies of all frequent itemsets is proposed. Experimental and theoretical analysis show that the space requirements for the algorithm are extremely small for many realistic data distributions.

Mining frequent itemsets in a stream

We introduce a new sublinear space data structure--the count-min sketch--for summarizing data streams. Our sketch allows fundamental queries in data stream summarization such as point, range, and inner product queries to be approximately answered very quickly; in addition, it can be applied to solve several important problems in data streams such as finding quantiles, frequent items, etc. The time and space bounds we show for using the CM sketch to solve these problems significantly improve those previously known--typically from 1/e2 to 1/e in factor.

An improved data stream summary: The Count-Min Sketch and its applications

We study the problem of finding planted clusters in bipartite graphs. We present a simple two-step algorithm which provably finds even tiny clusters of size O(n^e), where n is the number of vertices in the graph and e > 0. Previous algorithms were only able to identify clusters of size Ω( sqrt(n) ). We evaluated the algorithm on synthetic and on real-world data; the experiments show that the algorithm can find extremely small clusters even in presence of high destructive noise.

/pdf/bipartite-stochastic-block-models-with-tiny-clusters-30dbf5yd9v.pdf

Bipartite Stochastic Block Models with Tiny Clusters

Modern networked systems are increasingly reconfigurable, enabling demand-aware infrastructures whose resources can be adjusted according to the workload they currently serve. Such dynamic adjustments can be exploited to improve network utilization and hence performance, by moving frequently interacting communication partners closer, e.g., collocating them in the same server or datacenter. However, dynamically changing the embedding of workloads is algorithmically challenging: communication patterns are often not known ahead of time, but must be learned. During the learning process, overheads related to unnecessary moves (i.e., re-embeddings) should be minimized. This paper studies a fundamental model which captures the tradeoff between the benefits and costs of dynamically collocating communication partners on $\ell$ servers, in an online manner. Our main contribution is a distributed online algorithm which is asymptotically almost optimal, i.e., almost matches the lower bound (also derived in this paper) on the competitive ratio of any (distributed or centralized) online algorithm. As an application, we show that our algorithm can be used to solve a distributed union find problem in which the sets are stored across multiple servers.

pdf/efficient-distributed-workload-re-embedding-2vixvb1qji.pdf

Efficient Distributed Workload (Re-)Embedding

In recent years it has become popular to study dynamic problems in a sensitivity setting: Instead of allowing for an arbitrary sequence of updates, the sensitivity model only allows to apply batch updates of small size to the original input data. The sensitivity model is particularly appealing since recent strong conditional lower bounds ruled out fast algorithms for many dynamic problems, such as shortest paths, reachability, or subgraph connectivity. 
In this paper we prove conditional lower bounds for sensitivity problems. For example, we show that under the Boolean Matrix Multiplication (BMM) conjecture combinatorial algorithms cannot compute the (4/3 - {\epsilon})-approximate diameter of an undirected unweighted dense graph with truly subcubic preprocessing time and truly subquadratic update/query time. This result is surprising since in the static setting it is not clear whether a reduction from BMM to diameter is possible. We further show under the BMM conjecture that many problems, such as reachability or approximate shortest paths, cannot be solved faster than by recomputation from scratch even after only one or two edge insertions. We give more lower bounds under the Strong Exponential Time Hypothesis and the All Pairs Shortest Paths Conjecture. Many of our lower bounds also hold for static oracle data structures where no sensitivity is required. Finally, we give the first algorithm for the (1 + {\epsilon})-approximate radius, diameter, and eccentricity problems in directed or undirected unweighted graphs in case of single edges failures. The algorithm has a truly subcubic running time for graphs with a truly subquadratic number of edges; it is tight w.r.t. the conditional lower bounds we obtain.

Conditional Hardness for Sensitivity Problems

https://dl.acm.org/doi/pdf/10.1145/3376930.3376959

We consider the following online optimization problem. We are given a graph $G$ and each vertex of the graph is assigned to one of $\ell$ servers, where servers have capacity $k$ and we assume that the graph has $\ell \cdot k$ vertices. Initially, $G$ does not contain any edges and then the edges of $G$ are revealed one-by-one. The goal is to design an online algorithm $\operatorname{ONL}$, which always places the connected components induced by the revealed edges on the same server and never exceeds the server capacities by more than $\varepsilon k$ for constant $\varepsilon>0$. Whenever $\operatorname{ONL}$ learns about a new edge, the algorithm is allowed to move vertices from one server to another. Its objective is to minimize the number of vertex moves. More specifically, $\operatorname{ONL}$ should minimize the competitive ratio: the total cost $\operatorname{ONL}$ incurs compared to an optimal offline algorithm $\operatorname{OPT}$. 
Our main contribution is a polynomial-time randomized algorithm, that is asymptotically optimal: we derive an upper bound of $O(\log \ell + \log k)$ on its competitive ratio and show that no randomized online algorithm can achieve a competitive ratio of less than $\Omega(\log \ell + \log k)$. We also settle the open problem of the achievable competitive ratio by deterministic online algorithms, by deriving a competitive ratio of $\Theta(\ell \lg k)$; to this end, we present an improved lower bound as well as a deterministic polynomial-time online algorithm. 
Our algorithms rely on a novel technique which combines efficient integer programming with a combinatorial approach for maintaining ILP solutions. We believe this technique is of independent interest and will find further applications in the future.

Tight Bounds for Online Graph Partitioning

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Bipartite Stochastic Block Models with Tiny Clusters

Efficient Distributed Workload (Re-)Embedding

Conditional Hardness for Sensitivity Problems

Efficient Distributed Workload (Re-)Embedding

Tight Bounds for Online Graph Partitioning