23 research outputs found
Combinatorial approach to Modularity
Communities are clusters of nodes with a higher than average density of
internal connections. Their detection is of great relevance to better
understand the structure and hierarchies present in a network. Modularity has
become a standard tool in the area of community detection, providing at the
same time a way to evaluate partitions and, by maximizing it, a method to find
communities. In this work, we study the modularity from a combinatorial point
of view. Our analysis (as the modularity definition) relies on the use of the
configurational model, a technique that given a graph produces a series of
randomized copies keeping the degree sequence invariant. We develop an approach
that enumerates the null model partitions and can be used to calculate the
probability distribution function of the modularity. Our theory allows for a
deep inquiry of several interesting features characterizing modularity such as
its resolution limit and the statistics of the partitions that maximize it.
Additionally, the study of the probability of extremes of the modularity in the
random graph partitions opens the way for a definition of the statistical
significance of network partitions.Comment: 8 pages, 4 figure
Size reduction of complex networks preserving modularity
The ubiquity of modular structure in real-world complex networks is being the
focus of attention in many trials to understand the interplay between network
topology and functionality. The best approaches to the identification of
modular structure are based on the optimization of a quality function known as
modularity. However this optimization is a hard task provided that the
computational complexity of the problem is in the NP-hard class. Here we
propose an exact method for reducing the size of weighted (directed and
undirected) complex networks while maintaining invariant its modularity. This
size reduction allows the heuristic algorithms that optimize modularity for a
better exploration of the modularity landscape. We compare the modularity
obtained in several real complex-networks by using the Extremal Optimization
algorithm, before and after the size reduction, showing the improvement
obtained. We speculate that the proposed analytical size reduction could be
extended to an exact coarse graining of the network in the scope of real-space
renormalization.Comment: 14 pages, 2 figure
Enhance the Efficiency of Heuristic Algorithm for Maximizing Modularity Q
Modularity Q is an important function for identifying community structure in
complex networks. In this paper, we prove that the modularity maximization
problem is equivalent to a nonconvex quadratic programming problem. This result
provide us a simple way to improve the efficiency of heuristic algorithms for
maximizing modularity Q. Many numerical results demonstrate that it is very
effective.Comment: 9 pages, 3 figure
Modularity clustering is force-directed layout
Two natural and widely used representations for the community structure of
networks are clusterings, which partition the vertex set into disjoint subsets,
and layouts, which assign the vertices to positions in a metric space. This
paper unifies prominent characterizations of layout quality and clustering
quality, by showing that energy models of pairwise attraction and repulsion
subsume Newman and Girvan's modularity measure. Layouts with optimal energy are
relaxations of, and are thus consistent with, clusterings with optimal
modularity, which is of practical relevance because both representations are
complementary and often used together.Comment: 9 pages, 7 figures, see http://code.google.com/p/linloglayout/ for
downloading the graph clustering and layout softwar
Unveiling Relations in the Industry 4.0 Standards Landscape based on Knowledge Graph Embeddings
Industry~4.0 (I4.0) standards and standardization frameworks have been
proposed with the goal of \emph{empowering interoperability} in smart
factories. These standards enable the description and interaction of the main
components, systems, and processes inside of a smart factory. Due to the
growing number of frameworks and standards, there is an increasing need for
approaches that automatically analyze the landscape of I4.0 standards.
Standardization frameworks classify standards according to their functions into
layers and dimensions. However, similar standards can be classified differently
across the frameworks, producing, thus, interoperability conflicts among them.
Semantic-based approaches that rely on ontologies and knowledge graphs, have
been proposed to represent standards, known relations among them, as well as
their classification according to existing frameworks. Albeit informative, the
structured modeling of the I4.0 landscape only provides the foundations for
detecting interoperability issues. Thus, graph-based analytical methods able to
exploit knowledge encoded by these approaches, are required to uncover
alignments among standards. We study the relatedness among standards and
frameworks based on community analysis to discover knowledge that helps to cope
with interoperability conflicts between standards. We use knowledge graph
embeddings to automatically create these communities exploiting the meaning of
the existing relationships. In particular, we focus on the identification of
similar standards, i.e., communities of standards, and analyze their properties
to detect unknown relations. We empirically evaluate our approach on a
knowledge graph of I4.0 standards using the Trans family of embedding
models for knowledge graph entities. Our results are promising and suggest that
relations among standards can be detected accurately.Comment: 15 pages, 7 figures, DEXA2020 Conferenc
Fast unfolding of communities in large networks
We propose a simple method to extract the community structure of large
networks. Our method is a heuristic method that is based on modularity
optimization. It is shown to outperform all other known community detection
method in terms of computation time. Moreover, the quality of the communities
detected is very good, as measured by the so-called modularity. This is shown
first by identifying language communities in a Belgian mobile phone network of
2.6 million customers and by analyzing a web graph of 118 million nodes and
more than one billion links. The accuracy of our algorithm is also verified on
ad-hoc modular networks. .Comment: 6 pages, 5 figures, 1 table; new version with new figures in order to
clarify our method, where we look more carefully at the role played by the
ordering of the nodes and where we compare our method with that of Wakita and
Tsurum
Community Structure in Large Complex Networks
In this paper, we establish the definition of community fundamentally different from what was commonly accepted in previous studies, where communities were typically assumed to be densely connected internally but sparsely connected to the rest of the network. A community should be considered as a densely connected subset in which the probability of an edge between two randomly-picked vertices is higher than average. Moreover, a community should also be well connected to the remaining network, that is, the number of edges connecting a community to the rest of the graph should be significant. In order to identify a well-defined community, we provide rigorous definitions of two relevant terms: "whiskers" and the "core". Whiskers correspond to subsets of vertices that are barely connected to the rest of the network, while the core exclusively contains the type of community we are interested in. We have proven that detecting whiskers, or equivalently, extracting the core, is an NP-complete problem for weighted graphs. Then, three heuristic algorithms are proposed for finding an approximate core and are evaluated for their performance on large networks, which reveals the common existence of the core structure in both random and real-world graphs. Further, well-defined communities can be extracted from the core using a number of techniques, and the experimental results not only justify our intuitive notion of community, but also demonstrate the existence of large-scale communities in various complex networks.This research was partially supported by the U.S. Air Force Office of Scientific Research under Grant FA9550-09-1-0675