Speeding up graph clustering via modular decomposition based compression

Abstract

Nowadays, massive data sets of graph-like data arise in various application domains ranging from bioinformatics to social networks and communication networks analysis. The abundance of such kind of data calls for innovative techniques for storing, managing and processing graph-like data. In order to fulfill these requirements, in this paper we propose: (i) a model for representing compressed weighted graphs, and (ii) an efficient and effective compression algorithm which, leveraging on modular decomposition theory, is capable of exploiting structural properties of graphs in order to obtain highly compact and accurate compressed representations. Such compressed graphs can be used in place of the original graphs in order to enhance the performance of graph clustering algorithms in all contexts where a little inaccuracy in the results is acceptable in order to gain computational efficiency. The paper is completed by an experimental study which shows the effectiveness of the proposed approach in the context of graph clustering.

An information-theoretic framework for the lossy compression of link streams

“…The notion of structural equivalence is more generally interested in groups of vertices with similar relational patterns, that is in any block of equal-density within the adjacency matrix (not necessarily dense and not necessarily on the diagonal, as in other work focusing on block compression [6,2,3]). Hence, the GCP is more strongly related to the family of edge compression techniques [8] such as modular decomposition 1 [3], matching neighbours, and power graph analysis [9]. In the latter, one is searching for groups of vertices that have similar relation patterns of any sort.…”

“…Even if not explicitly formalised, research perspectives in that direction are sometimes provided [9]. Yet, other approaches natively deals with multigraph compression by directly taking into account, within the compression scheme, the presence of multiple edges [2,3]. Statistical methods for variable co-clustering also offers compression frameworks that are designed for numerical (non-binary) matrices [23,15,14].…”

“…However, because it is often much easier to interpret the result of compression when it is based on "hard partitioning", especially in the case of vertex partitioning, we focus in this article on this simpler setting. 3 By applying the law of total probability:…”

“…More generally, this perspective is related to the density profile of edges within the graph. Hence, density-based measures [5] seems more relevant than other traditional connectivity measures: E.g., the Euclidean distance or the mean squared error between the two density matrices [2,3], the average variance within matrix blocks [13], and many other measures inherited from traditional block modelling methods [20]. Note that, when it comes to the latter, the stochastic model underlying our compression framework is similar, but not equivalent to the one of block modelling.…”

“…Among abstraction techniques, graph compression [2,3], also known as graph simplification or graph summarisation [4,5,6], consists in replacing parts of the graph by more general structural patterns in order to reduce its description length. For example, one "can replace a dense cluster by a single node, so the overall structure of the network becomes clearer" [1], or more generally replace any frequent subgraph pattern (e.g., cliques, stars, loops) by a label of that pattern.…”