Axioms for graph clustering quality functions

We investigate properties that intuitively ought to be satisfied by graph clustering quality functions, that is, functions that assign a score to a clustering of a graph. Graph clustering, also known as network community detection, is often performed by optimizing such a function. Two axioms tailored for graph clustering quality functions are introduced, and the four axioms introduced in previous work on distance based clustering are reformulated and generalized for the graph setting. We show that modularity, a standard quality function for graph clustering, does not satisfy all of these six properties. This motivates the derivation of a new family of quality functions, adaptive scale modularity, which does satisfy the proposed axioms. Adaptive scale modularity has two parameters, which give greater flexibility in the kinds of clusterings that can be found. Standard graph clustering quality functions, such as normalized cut and unnormalized cut, are obtained as special cases of adaptive scale modularity. In general, the results of our investigation indicate that the considered axiomatic framework covers existing `good' quality functions for graph clustering, and can be used to derive an interesting new family of quality functions.Comment: 23 pages. Full text and sources available on: http://www.cs.ru.nl/~T.vanLaarhoven/graph-clustering-axioms-2014

Why multi-tracer surveys beat cosmic variance

Galaxy surveys that map multiple species of tracers of large-scale structure can improve the constraints on some cosmological parameters far beyond the limits imposed by a simplistic interpretation of cosmic variance. This enhancement derives from comparing the relative clustering between different tracers of large-scale structure. We present a simple but fully generic expression for the Fisher information matrix of surveys with any (discrete) number of tracers, and show that the enhancement of the constraints on bias-sensitive parameters are a straightforward consequence of this multi-tracer Fisher matrix. In fact, the relative clustering amplitudes between tracers are eigenvectors of this multi-tracer Fisher matrix. The diagonalized multi-tracer Fisher matrix clearly shows that while the effective volume is bounded by the physical volume of the survey, the relational information between species is unbounded. As an application, we study the expected enhancements in the constraints of realistic surveys that aim at mapping several different types of tracers of large-scale structure. The gain obtained by combining multiple tracers is highest at low redshifts, and in one particular scenario we analyzed, the enhancement can be as large as a factor of ~3 for the accuracy in the determination of the redshift distortion parameter, and a factor ~5 for the local non-Gaussianity parameter. Radial and angular distance determinations from the baryonic features in the power spectrum may also benefit from the multi-tracer approach.Comment: New references included; 9 pages, 9 figure

Partitioning Complex Networks via Size-constrained Clustering

The most commonly used method to tackle the graph partitioning problem in practice is the multilevel approach. During a coarsening phase, a multilevel graph partitioning algorithm reduces the graph size by iteratively contracting nodes and edges until the graph is small enough to be partitioned by some other algorithm. A partition of the input graph is then constructed by successively transferring the solution to the next finer graph and applying a local search algorithm to improve the current solution. In this paper, we describe a novel approach to partition graphs effectively especially if the networks have a highly irregular structure. More precisely, our algorithm provides graph coarsening by iteratively contracting size-constrained clusterings that are computed using a label propagation algorithm. The same algorithm that provides the size-constrained clusterings can also be used during uncoarsening as a fast and simple local search algorithm. Depending on the algorithm's configuration, we are able to compute partitions of very high quality outperforming all competitors, or partitions that are comparable to the best competitor in terms of quality, hMetis, while being nearly an order of magnitude faster on average. The fastest configuration partitions the largest graph available to us with 3.3 billion edges using a single machine in about ten minutes while cutting less than half of the edges than the fastest competitor, kMetis

Distributed Graph Clustering using Modularity and Map Equation

We study large-scale, distributed graph clustering. Given an undirected graph, our objective is to partition the nodes into disjoint sets called clusters. A cluster should contain many internal edges while being sparsely connected to other clusters. In the context of a social network, a cluster could be a group of friends. Modularity and map equation are established formalizations of this internally-dense-externally-sparse principle. We present two versions of a simple distributed algorithm to optimize both measures. They are based on Thrill, a distributed big data processing framework that implements an extended MapReduce model. The algorithms for the two measures, DSLM-Mod and DSLM-Map, differ only slightly. Adapting them for similar quality measures is straight-forward. We conduct an extensive experimental study on real-world graphs and on synthetic benchmark graphs with up to 68 billion edges. Our algorithms are fast while detecting clusterings similar to those detected by other sequential, parallel and distributed clustering algorithms. Compared to the distributed GossipMap algorithm, DSLM-Map needs less memory, is up to an order of magnitude faster and achieves better quality.Comment: 14 pages, 3 figures; v3: Camera ready for Euro-Par 2018, more details, more results; v2: extended experiments to include comparison with competing algorithms, shortened for submission to Euro-Par 201

Comparing clusterings and numbers of clusters by aggregation of calibrated clustering validity indexes

A key issue in cluster analysis is the choice of an appropriate clustering method and the determination of the best number of clusters. Different clusterings are optimal on the same data set according to different criteria, and the choice of such criteria depends on the context and aim of clustering. Therefore, researchers need to consider what data analytic characteristics the clusters they are aiming at are supposed to have, among others within-cluster homogeneity, between-clusters separation, and stability. Here, a set of internal clustering validity indexes measuring different aspects of clustering quality is proposed, including some indexes from the literature. Users can choose the indexes that are relevant in the application at hand. In order to measure the overall quality of a clustering (for comparing clusterings from different methods and/or different numbers of clusters), the index values are calibrated for aggregation. Calibration is relative to a set of random clusterings on the same data. Two specific aggregated indexes are proposed and compared with existing indexes on simulated and real data.Comment: 42 pages, 11 figure