989 research outputs found
Bad Communities with High Modularity
In this paper we discuss some problematic aspects of Newman's modularity
function QN. Given a graph G, the modularity of G can be written as QN = Qf
-Q0, where Qf is the intracluster edge fraction of G and Q0 is the expected
intracluster edge fraction of the null model, i.e., a randomly connected graph
with same expected degree distribution as G. It follows that the maximization
of QN must accomodate two factors pulling in opposite directions: Qf favors a
small number of clusters and Q0 favors many balanced (i.e., with approximately
equal degrees) clusters. In certain cases the Q0 term can cause overestimation
of the true cluster number; this is the opposite of the well-known under
estimation effect caused by the "resolution limit" of modularity. We illustrate
the overestimation effect by constructing families of graphs with a "natural"
community structure which, however, does not maximize modularity. In fact, we
prove that we can always find a graph G with a "natural clustering" V of G and
another, balanced clustering U of G such that (i) the pair (G; U) has higher
modularity than (G; V) and (ii) V and U are arbitrarily different.Comment: Significantly improved version of the paper, with the help of L.
Pitsouli
Multi-level algorithms for modularity clustering
Modularity is one of the most widely used quality measures for graph
clusterings. Maximizing modularity is NP-hard, and the runtime of exact
algorithms is prohibitive for large graphs. A simple and effective class of
heuristics coarsens the graph by iteratively merging clusters (starting from
singletons), and optionally refines the resulting clustering by iteratively
moving individual vertices between clusters. Several heuristics of this type
have been proposed in the literature, but little is known about their relative
performance.
This paper experimentally compares existing and new coarsening- and
refinement-based heuristics with respect to their effectiveness (achieved
modularity) and efficiency (runtime). Concerning coarsening, it turns out that
the most widely used criterion for merging clusters (modularity increase) is
outperformed by other simple criteria, and that a recent algorithm by Schuetz
and Caflisch is no improvement over simple greedy coarsening for these
criteria. Concerning refinement, a new multi-level algorithm is shown to
produce significantly better clusterings than conventional single-level
algorithms. A comparison with published benchmark results and algorithm
implementations shows that combinations of coarsening and multi-level
refinement are competitive with the best algorithms in the literature.Comment: 12 pages, 10 figures, see
http://www.informatik.tu-cottbus.de/~rrotta/ for downloading the graph
clustering softwar
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
Analysis of Network Clustering Algorithms and Cluster Quality Metrics at Scale
Notions of community quality underlie network clustering. While studies
surrounding network clustering are increasingly common, a precise understanding
of the realtionship between different cluster quality metrics is unknown. In
this paper, we examine the relationship between stand-alone cluster quality
metrics and information recovery metrics through a rigorous analysis of four
widely-used network clustering algorithms -- Louvain, Infomap, label
propagation, and smart local moving. We consider the stand-alone quality
metrics of modularity, conductance, and coverage, and we consider the
information recovery metrics of adjusted Rand score, normalized mutual
information, and a variant of normalized mutual information used in previous
work. Our study includes both synthetic graphs and empirical data sets of sizes
varying from 1,000 to 1,000,000 nodes.
We find significant differences among the results of the different cluster
quality metrics. For example, clustering algorithms can return a value of 0.4
out of 1 on modularity but score 0 out of 1 on information recovery. We find
conductance, though imperfect, to be the stand-alone quality metric that best
indicates performance on information recovery metrics. Our study shows that the
variant of normalized mutual information used in previous work cannot be
assumed to differ only slightly from traditional normalized mutual information.
Smart local moving is the best performing algorithm in our study, but
discrepancies between cluster evaluation metrics prevent us from declaring it
absolutely superior. Louvain performed better than Infomap in nearly all the
tests in our study, contradicting the results of previous work in which Infomap
was superior to Louvain. We find that although label propagation performs
poorly when clusters are less clearly defined, it scales efficiently and
accurately to large graphs with well-defined clusters
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
Unifying Sparsest Cut, Cluster Deletion, and Modularity Clustering Objectives with Correlation Clustering
Graph clustering, or community detection, is the task of identifying groups
of closely related objects in a large network. In this paper we introduce a new
community-detection framework called LambdaCC that is based on a specially
weighted version of correlation clustering. A key component in our methodology
is a clustering resolution parameter, , which implicitly controls the
size and structure of clusters formed by our framework. We show that, by
increasing this parameter, our objective effectively interpolates between two
different strategies in graph clustering: finding a sparse cut and forming
dense subgraphs. Our methodology unifies and generalizes a number of other
important clustering quality functions including modularity, sparsest cut, and
cluster deletion, and places them all within the context of an optimization
problem that has been well studied from the perspective of approximation
algorithms. Our approach is particularly relevant in the regime of finding
dense clusters, as it leads to a 2-approximation for the cluster deletion
problem. We use our approach to cluster several graphs, including large
collaboration networks and social networks
Optimizing an Organized Modularity Measure for Topographic Graph Clustering: a Deterministic Annealing Approach
This paper proposes an organized generalization of Newman and Girvan's
modularity measure for graph clustering. Optimized via a deterministic
annealing scheme, this measure produces topologically ordered graph clusterings
that lead to faithful and readable graph representations based on clustering
induced graphs. Topographic graph clustering provides an alternative to more
classical solutions in which a standard graph clustering method is applied to
build a simpler graph that is then represented with a graph layout algorithm. A
comparative study on four real world graphs ranging from 34 to 1 133 vertices
shows the interest of the proposed approach with respect to classical solutions
and to self-organizing maps for graphs
Node-Centric Detection of Overlapping Communities in Social Networks
We present NECTAR, a community detection algorithm that generalizes Louvain
method's local search heuristic for overlapping community structures. NECTAR
chooses dynamically which objective function to optimize based on the network
on which it is invoked. Our experimental evaluation on both synthetic benchmark
graphs and real-world networks, based on ground-truth communities, shows that
NECTAR provides excellent results as compared with state of the art community
detection algorithms
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