The problem of accurately measuring the similarity between graphs is at the core of many applications in a variety of disciplines. Graph kernels have recently emerged as a promising approach to this problem. There are now many kernels, each focusing on different structural aspects of graphs. Here, we present GraKeL, a library that unifies several graph kernels into a common framework. The library is written in Python and adheres to the scikit-learn interface. It is simple to use and can be naturally combined with scikit-learn's modules to build a complete machine learning pipeline for tasks such as graph classification and clustering. The code is BSD licensed and is available at: https://github.com/ysig/GraKeL.

Graph clustering or community detection constitutes an important task forinvestigating the internal structure of graphs, with a plethora of applications in several domains. Traditional tools for graph clustering, such asspectral methods, typically suffer from high time and space complexity. In thisarticle, we present CoreCluster, an efficient graph clusteringframework based on the concept of graph degeneracy, that can be used along withany known graph clustering algorithm. Our approach capitalizes on processing thegraph in a hierarchical manner provided by its core expansion sequence, anordered partition of the graph into different levels according to the k-coredecomposition. Such a partition provides a way to process the graph inan incremental manner that preserves its clustering structure, whilemaking the execution of the chosen clustering algorithm much faster due to thesmaller size of the graph's partitions onto which the algorithm operates.

Lately, there has been an increased interest in signed networks with applications in trust, security, or social computing. This paper focuses on the issue of defining models and metrics for reciprocity in signed graphs. In unsigned directed networks, reciprocity quantifies the predisposition of network members in creating mutual connections. On the other hand, this concept has not yet been investigated in the case of signed graphs. We capitalize on the graph degeneracy concept to identify subgraphs of the signed network in which reciprocity is more likely to occur. This enables us to assess reciprocity at a global level, rather than at an exclusively local one as in existing approaches. The large scale experiments we perform on real world data sets of trust networks lead to both interesting and intuitive results. We believe these reciprocity measures can be used in various social applications such as trust management, community detection and evaluation of individual nodes. The global reciprocity we define in this paper is closely correlated to the clustering structure of the graph, more than the local reciprocity as it is indicated by the experimental evaluation we conducted.

Graphs constitute a dominant data structure and appear essentially in all forms of information. Examples are the Web graph, numerous social networks, protein interaction networks, terms dependency graphs and network topologies. The main features of these graphs are their huge volume and rate of change. Presumably, there is important hidden knowledge in the macroscopic topology and features of these graphs. A cornerstone issue here is the detection and evaluation of communities -- bearing multiple and diverse semantics. The tutorial reports the basic models of graph structures for undirected, directed and signed graphs and their properties. Next we offer a thorough review of fundamental methods for graph clustering and community detection, on both undirected and directed graphs. Then we survey community evaluation measures, including both the individual node based ones as well as those that take into account aggregate properties of communities. A special mention is made on approaches that capitalize on the concept of degeneracy (k-cores and extensions), as a novel means of community detection and evaluation. We justify the above foundational framework with applications on citation graphs, trust networks and protein graphs.

We demonstrate a system that supports the visual exploration of collaboration networks. The system leverages the notion of fractional cores introduced in earlier work to rank vertices in a collaboration network and filter vertices' neighborhoods. Fractional cores build on the idea of graph degeneracy as captured by the notion of k-cores in graph theory and extend it to undirected edge-weighted graphs. In a co-authorship network, for instance, the fractional core index of an author intuitively reflects the degree of collaboration with equally or higher-ranked authors. Our system has been deployed on a real-world co-authorship network derived from DBLP, demonstrating that the idea of fractional cores can be applied even to large-scale networks. The system provides an easy-to-use interface to query for the fractional core index of an author, to see who the closest equally or higher-ranked co-authors are, and explore the entire co-authorship network in an incremental manner.

Community sub graphs are characterized by dense connections or interactions among its nodes. Community detection and evaluation is an important task in graph mining. A variety of measures have been proposed to evaluate the quality of such communities. In this paper, we evaluate communities based on the k-core concept, as means of evaluating their collaborative nature - a property not captured by the single node metrics or by the established community evaluation metrics. Based on the k-core, which essentially measures the robustness of a community under degeneracy, we extend it to weighted graphs, devising a novel concept of k-cores on weighted graphs. We applied the k-core approach on large real world graphs - such as DBLP and report interesting results.

Community detection and evaluation is an important task in graph mining. In many cases, a community is defined as a sub graph characterized by dense connections or interactions among its nodes. A large variety of measures have been proposed to evaluate the quality of such communities - in most cases ignoring the directed nature of edges. In this paper, we introduce novel metrics for evaluating the collaborative nature of directed graphs - a property not captured by the single node metrics or by other established community evaluation metrics. In order to accomplish this objective, we capitalize on the concept of graph degeneracy and define a novel D-core framework, extending the classic graph-theoretic notion of k-cores for undirected graphs to directed ones. Based on the D-core, which essentially can be seen as a measure of the robustness of a community under degeneracy, we devise a wealth of novel metrics used to evaluate graph collaboration features of directed graphs. We applied the D-core approach on large real-world graphs such as Wikipedia and DBLP and report interesting results at the graph as well at node level.