Kernel Spectral Clustering and applications

In this chapter we review the main literature related to kernel spectral clustering (KSC), an approach to clustering cast within a kernel-based optimization setting. KSC represents a least-squares support vector machine based formulation of spectral clustering described by a weighted kernel PCA objective. Just as in the classifier case, the binary clustering model is expressed by a hyperplane in a high dimensional space induced by a kernel. In addition, the multi-way clustering can be obtained by combining a set of binary decision functions via an Error Correcting Output Codes (ECOC) encoding scheme. Because of its model-based nature, the KSC method encompasses three main steps: training, validation, testing. In the validation stage model selection is performed to obtain tuning parameters, like the number of clusters present in the data. This is a major advantage compared to classical spectral clustering where the determination of the clustering parameters is unclear and relies on heuristics. Once a KSC model is trained on a small subset of the entire data, it is able to generalize well to unseen test points. Beyond the basic formulation, sparse KSC algorithms based on the Incomplete Cholesky Decomposition (ICD) and $L_0$, $L_1, L_0 + L_1$, Group Lasso regularization are reviewed. In that respect, we show how it is possible to handle large scale data. Also, two possible ways to perform hierarchical clustering and a soft clustering method are presented. Finally, real-world applications such as image segmentation, power load time-series clustering, document clustering and big data learning are considered.Comment: chapter contribution to the book "Unsupervised Learning Algorithms

Complex Networks

An outline of recent work on complex networks is given from the point of view of a physicist. Motivation, achievements and goals are discussed with some of the typical applications from a wide range of academic fields. An introduction to the relevant literature and useful resources is also given.Comment: Review for Contemporary Physics, 31 page

Intrinsically Dynamic Network Communities

Community finding algorithms for networks have recently been extended to dynamic data. Most of these recent methods aim at exhibiting community partitions from successive graph snapshots and thereafter connecting or smoothing these partitions using clever time-dependent features and sampling techniques. These approaches are nonetheless achieving longitudinal rather than dynamic community detection. We assume that communities are fundamentally defined by the repetition of interactions among a set of nodes over time. According to this definition, analyzing the data by considering successive snapshots induces a significant loss of information: we suggest that it blurs essentially dynamic phenomena - such as communities based on repeated inter-temporal interactions, nodes switching from a community to another across time, or the possibility that a community survives while its members are being integrally replaced over a longer time period. We propose a formalism which aims at tackling this issue in the context of time-directed datasets (such as citation networks), and present several illustrations on both empirical and synthetic dynamic networks. We eventually introduce intrinsically dynamic metrics to qualify temporal community structure and emphasize their possible role as an estimator of the quality of the community detection - taking into account the fact that various empirical contexts may call for distinct `community' definitions and detection criteria.Comment: 27 pages, 11 figure

Discriminative Link Prediction using Local Links, Node Features and Community Structure

A link prediction (LP) algorithm is given a graph, and has to rank, for each node, other nodes that are candidates for new linkage. LP is strongly motivated by social search and recommendation applications. LP techniques often focus on global properties (graph conductance, hitting or commute times, Katz score) or local properties (Adamic-Adar and many variations, or node feature vectors), but rarely combine these signals. Furthermore, neither of these extremes exploit link densities at the intermediate level of communities. In this paper we describe a discriminative LP algorithm that exploits two new signals. First, a co-clustering algorithm provides community level link density estimates, which are used to qualify observed links with a surprise value. Second, links in the immediate neighborhood of the link to be predicted are not interpreted at face value, but through a local model of node feature similarities. These signals are combined into a discriminative link predictor. We evaluate the new predictor using five diverse data sets that are standard in the literature. We report on significant accuracy boosts compared to standard LP methods (including Adamic-Adar and random walk). Apart from the new predictor, another contribution is a rigorous protocol for benchmarking and reporting LP algorithms, which reveals the regions of strengths and weaknesses of all the predictors studied here, and establishes the new proposal as the most robust.Comment: 10 pages, 5 figure