Graphs in machine learning: an introduction

Graphs are commonly used to characterise interactions between objects of interest. Because they are based on a straightforward formalism, they are used in many scientific fields from computer science to historical sciences. In this paper, we give an introduction to some methods relying on graphs for learning. This includes both unsupervised and supervised methods. Unsupervised learning algorithms usually aim at visualising graphs in latent spaces and/or clustering the nodes. Both focus on extracting knowledge from graph topologies. While most existing techniques are only applicable to static graphs, where edges do not evolve through time, recent developments have shown that they could be extended to deal with evolving networks. In a supervised context, one generally aims at inferring labels or numerical values attached to nodes using both the graph and, when they are available, node characteristics. Balancing the two sources of information can be challenging, especially as they can disagree locally or globally. In both contexts, supervised and un-supervised, data can be relational (augmented with one or several global graphs) as described above, or graph valued. In this latter case, each object of interest is given as a full graph (possibly completed by other characteristics). In this context, natural tasks include graph clustering (as in producing clusters of graphs rather than clusters of nodes in a single graph), graph classification, etc. 1 Real networks One of the first practical studies on graphs can be dated back to the original work of Moreno [51] in the 30s. Since then, there has been a growing interest in graph analysis associated with strong developments in the modelling and the processing of these data. Graphs are now used in many scientific fields. In Biology [54, 2, 7], for instance, metabolic networks can describe pathways of biochemical reactions [41], while in social sciences networks are used to represent relation ties between actors [66, 56, 36, 34]. Other examples include powergrids [71] and the web [75]. Recently, networks have also been considered in other areas such as geography [22] and history [59, 39]. In machine learning, networks are seen as powerful tools to model problems in order to extract information from data and for prediction purposes. This is the object of this paper. For more complete surveys, we refer to [28, 62, 49, 45]. In this section, we introduce notations and highlight properties shared by most real networks. In Section 2, we then consider methods aiming at extracting information from a unique network. We will particularly focus on clustering methods where the goal is to find clusters of vertices. Finally, in Section 3, techniques that take a series of networks into account, where each network i

Batch and median neural gas

Neural Gas (NG) constitutes a very robust clustering algorithm given euclidian data which does not suffer from the problem of local minima like simple vector quantization, or topological restrictions like the self-organizing map. Based on the cost function of NG, we introduce a batch variant of NG which shows much faster convergence and which can be interpreted as an optimization of the cost function by the Newton method. This formulation has the additional benefit that, based on the notion of the generalized median in analogy to Median SOM, a variant for non-vectorial proximity data can be introduced. We prove convergence of batch and median versions of NG, SOM, and k-means in a unified formulation, and we investigate the behavior of the algorithms in several experiments.Comment: In Special Issue after WSOM 05 Conference, 5-8 september, 2005, Pari

String Measure Applied to String Self-Organizing Maps and Networks of Evolutionary Processors

* Supported by projects CCG08-UAM TIC-4425-2009 and TEC2007-68065-C03-02This paper shows some ideas about how to incorporate a string learning stage in self-organizing algorithms. T. Kohonen and P. Somervuo have shown that self-organizing maps (SOM) are not restricted to numerical data. This paper proposes a symbolic measure that is used to implement a string self-organizing map based on SOM algorithm. Such measure between two strings is a new string. Computation over strings is performed using a priority relationship among symbols; in this case, symbolic measure is able to generate new symbols. A complementary operation is defined in order to apply such measure to DNA strands. Finally, an algorithm is proposed in order to be able to implement a string self-organizing map

On-line relational and multiple relational SOM

International audienceIn some applications and in order to address real-world situations better, data may be more complex than simple numerical vectors. In some examples, data can be known only through their pairwise dissimilarities or through multiple dissimilarities, each of them describing a particular feature of the data set. Several variants of the Self Organizing Map (SOM) algorithm were introduced to generalize the original algorithm to the framework of dissimilarity data. Whereas median SOM is based on a rough representation of the prototypes, relational SOM allows representing these prototypes by a virtual linear combination of all elements in the data set, referring to a pseudo-euclidean framework. In the present article, an on-line version of relational SOM is introduced and studied. Similarly to the situation in the Euclidean framework, this on-line algorithm provides a better organization and is much less sensible to prototype initialization than standard (batch) relational SOM. In a more general case, this stochastic version allows us to integrate an additional stochastic gradient descent step in the algorithm which can tune the respective weights of several dissimilarities in an optimal way: the resulting \emph{multiple relational SOM} thus has the ability to integrate several sources of data of different types, or to make a consensus between several dissimilarities describing the same data. The algorithms introduced in this manuscript are tested on several data sets, including categorical data and graphs. On-line relational SOM is currently available in the R package SOMbrero that can be downloaded at http://sombrero.r-forge.r-project.org or directly tested on its Web User Interface at http://shiny.nathalievilla.org/sombrero

XML Matchers: approaches and challenges

Schema Matching, i.e. the process of discovering semantic correspondences between concepts adopted in different data source schemas, has been a key topic in Database and Artificial Intelligence research areas for many years. In the past, it was largely investigated especially for classical database models (e.g., E/R schemas, relational databases, etc.). However, in the latest years, the widespread adoption of XML in the most disparate application fields pushed a growing number of researchers to design XML-specific Schema Matching approaches, called XML Matchers, aiming at finding semantic matchings between concepts defined in DTDs and XSDs. XML Matchers do not just take well-known techniques originally designed for other data models and apply them on DTDs/XSDs, but they exploit specific XML features (e.g., the hierarchical structure of a DTD/XSD) to improve the performance of the Schema Matching process. The design of XML Matchers is currently a well-established research area. The main goal of this paper is to provide a detailed description and classification of XML Matchers. We first describe to what extent the specificities of DTDs/XSDs impact on the Schema Matching task. Then we introduce a template, called XML Matcher Template, that describes the main components of an XML Matcher, their role and behavior. We illustrate how each of these components has been implemented in some popular XML Matchers. We consider our XML Matcher Template as the baseline for objectively comparing approaches that, at first glance, might appear as unrelated. The introduction of this template can be useful in the design of future XML Matchers. Finally, we analyze commercial tools implementing XML Matchers and introduce two challenging issues strictly related to this topic, namely XML source clustering and uncertainty management in XML Matchers.Comment: 34 pages, 8 tables, 7 figure

Current State of Ontology Matching. A Survey of Ontology and Schema Matching

Ontology matching is an important task when data from multiple data sources is integrated. Problems of ontology matching have been studied widely in the researchliterature and many different solutions and approaches have been proposed alsoin commercial software tools. In this survey, well-known approaches of ontologymatching, and its subtype schema matching, are reviewed and compared. The aimof this report is to summarize the knowledge about the state-of-the-art solutionsfrom the research literature, discuss how the methods work on different application domains, and analyze pros and cons of different open source and academic tools inthe commercial world.Siirretty Doriast

Relaxation of Subgraph Queries Delivering Empty Results

Graph databases with the property graph model are used in multiple domains including social networks, biology, and data integration. They provide schema-flexible storage for data of a different degree of a structure and support complex, expressive queries such as subgraph isomorphism queries. The exibility and expressiveness of graph databases make it difficult for the users to express queries correctly and can lead to unexpected query results, e.g. empty results. Therefore, we propose a relaxation approach for subgraph isomorphism queries that is able to automatically rewrite a graph query, such that the rewritten query is similar to the original query and returns a non-empty result set. In detail, we present relaxation operations applicable to a query, cardinality estimation heuristics, and strategies for prioritizing graph query elements to be relaxed. To determine the similarity between the original query and its relaxed variants, we propose a novel cardinality-based graph edit distance. The feasibility of our approach is shown by using real-world queries from the DBpedia query log

Discriminative prototype selection methods for graph embedding

Graphs possess a strong representational power for many types of patterns. However, a main limitation in their use for pattern analysis derives from their difficult mathematical treatment. One way of circumventing this problem is that of transforming the graphs into a vector space by means of graph embedding. Such an embedding can be conveniently obtained by using a set of prototype graphs and a dissimilarity measure. However, when we apply this approach to a set of class-labelled graphs, it is challenging to select prototypes capturing both the salient structure within each class and inter-class separation. In this paper, we introduce a novel framework for selecting a set of prototypes from a labelled graph set taking their discriminative power into account. Experimental results showed that such a discriminative prototype selection framework can achieve superior results in classification compared to other well-established prototype selection approaches. © 2012 Elsevier Ltd