After the phenomenal success of the PageRank algorithm, many researchers have extended the PageRank approach to ranking graphs with richer structures in addition to the simple linkage structure. Indeed, in some scenarios we have to deal with networks modeling multi-parameters data where each node has additional features and there are important relationships between such features.This paper addresses the need of a systematic approach to deal with multi-parameter data. We propose models and ranking algorithms that can be applied to a large variety of networks (bibliographic data, patent data, twitter and social data, healthcare data). We focus on several aspects not previously addressed in the literature: (1) we propose different models for ranking multi-parameters data and a class of numerical algorithms for efficiently computing the ranking score of such models, (2) we analyze stability and convergence of the proposed numerical schemes and we derive a fast and stable ranking algorithm, (3) we analyze the robustness of our models when data are incomplete. The comparison of the rank on the incomplete data with the rank on the full structure shows that our models compute consistent rankings whose correlation is up to 60 when just 10 of the links of the attributes are maintained.
As the size of the web increases, it becomes more and more important to analyze link structure while also considering context. Multilinear algebra provides a novel tool for incorporating anchor text and other information into the authority computation used by link analysis methods such as HITS. Our recently proposed TOPHITS method uses a higher-order analogue of the matrix singular value decomposition called the PARAFAC model to analyze a three-way representation of web data. We compute hubs and authorities together with the terms that are used in the anchor text of the links between them. Adding a third dimension to the data greatly extends the applicability of HITS because the TOPHITS analysis can be performed in advance and offline. Like HITS, the TOPHITS model reveals latent groupings of pages, but TOPHITS also includes latent term information. In this paper, we describe a faster mathematical algorithm for computing the TOPHITS model on sparse data, and Web data is used to compare HITS and TOPHITS. We also discuss how the TOPHITS model can be used in queries, such as computing context-sensitive authorities and hubs. We describe different query response methodologies and present experimental results. Abstract The problem of incomplete data – i.e., data with missing or unknown values – in multi-way arrays is ubiquitous in biomedical signal processing, network traffic analysis, bibliometrics, social network analysis, chemometrics, computer vision, communication networks, etc. We consider the problem of how to factorize data sets with missing values with the goal of capturing the underlying latent structure of the data and possibly reconstructing missing values (i.e., tensor completion). We focus on one of the most well-known tensor factorizations that captures multi-linear structure, CANDECOMP PARAFAC (CP). In the presence of missing data, CP can be formulated as a weighted least squares problem that models only the known entries. We develop an algorithm called CP-WOPT (CP Weighted OPTimization) that uses a first-order optimization approach to solve the weighted least squares problem. Based on extensive numerical experiments, our algorithm is shown to successfully factorize tensors with noise and up to 99 missing data. A unique aspect of our approach is that it scales to sparse large-scale data, e.g., 1000 × 1000 × 1000 with five million known entries (0.5 dense). We further demonstrate the usefulness of CP-WOPT on two real-world applications: a novel EEG (electroencephalogram) application where missing data is frequently encountered due to disconnections of electrodes and the problem of modeling computer network traffic where data may be absent due to the expense of the data collection process. Link analysis typically focuses on a single type of connection, e.g., two journal papers are linked because they are written by the same author. However, often we want to analyze data that has multiple linkages between objects, e.g., two papers may have the same keywords and one may cite the other. The goal of this paper is to show that multilinear algebra provides a tool for multilink analysis. We analyze five years of publication data from journals published by the Society for Industrial and Applied Mathematics. We explore how papers can be grouped in the context of multiple link types using a tensor to represent all the links between them. A PARAFAC decomposition on the resulting tensor yields information similar to the SVD decomposition of a standard adjacency matrix. We show how the PARAFAC decomposition can be used to understand the structure of the document space and define paper-paper similarities based on multiple linkages. Examples are presented where the decomposed tensor data is used to find papers similar to a body of work (e.g., related by topic or similar to a particular author's papers), find related authors using linkages other than explicit co-authorship or citations, distinguish between papers written bymore » different authors with the same name, and predict the journal in which a paper was published.« less