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A Unified Approach to Structural Limits and Limits of Graphs with Bounded Tree-Depth
In this paper we introduce a general framework for the study of limits of relational structures and graphs in particular, which is
based on a combination of model theory and (functional) analysis. We show how the various approaches to graph limits fit to this
framework and that they naturally appear as “tractable cases” of a general theory. As an outcome of this, we provide extensions of known
results. We believe that this puts these into a broader context. The second part of the paper is devoted to the study of sparse
structures. First, we consider limits of structures with bounded diameter connected components and we prove that in this case the
convergence can be “almost” studied component-wise. We also propose the structure of limit objects for convergent sequences of sparse
structures. Eventually, we consider the specific case of limits of colored rooted trees with bounded height and of graphs with bounded
tree-depth, motivated by their role as “elementary bricks” these graphs play in decompositions of sparse graphs, and give an explicit
construction of a limit object in this case. This limit object is a graph built on a standard probability space with the property that
every first-order definable set of tuples is measurable. This is an example of the general concept of
eBook
Large-scale graph processing using Apache Giraph
This book takes its reader on a journey through Apache Giraph, a popular distributed graph processing platform designed to bring the power of big data processing to graph data. Designed as a step-by-step self-study guide for everyone interested in large-scale graph processing, it describes the fundamental abstractions of the system, its programming models and various techniques for using the system to process graph data at scale, including the implementation of several popular and advanced graph analytics algorithms. The book is organized as follows: Chapter 1 starts by providing a general background of the big data phenomenon and a general introduction to the Apache Giraph system, its abstraction, programming model and design architecture. Next, chapter 2 focuses on Giraph as a platform and how to use it. Based on a sample job, even more advanced topics like monitoring the Giraph application lifecycle and different methods for monitoring Giraph jobs are explained. Chapter 3 then provides an introduction to Giraph programming, introduces the basic Giraph graph model and explains how to write Giraph programs. In turn, Chapter 4 discusses in detail the implementation of some popular graph algorithms including PageRank, connected components, shortest paths and triangle closing. Chapter 5 focuses on advanced Giraph programming, discussing common Giraph algorithmic optimizations, tunable Giraph configurations that determine the system?s utilization of the underlying resources, and how to write a custom graph input and output format. Lastly, chapter 6 highlights two systems that have been introduced to tackle the challenge of large scale graph processing, GraphX and GraphLab, and explains the main commonalities and differences between these systems and Apache Giraph. This book serves as an essential reference guide for students, researchers and practitioners in the domain of large scale graph processing. It offers step-by-step guidance, with several code examples and the complete source code available in the related github repository. Students will find a comprehensive introduction to and hands-on practice with tackling large scale graph processing problems using the Apache Giraph system, while researchers will discover thorough coverage of the emerging and ongoing advancements in big graph processing systems.
Book
Weakly Modular Graphs and Nonpositive Curvature
This article investigates structural, geometrical, and topological characterizations and properties of weakly modular graphs and of
cell complexes derived from them. The unifying themes of our investigation are various “nonpositive curvature\" and “local-to-global”
properties and characterizations of weakly modular graphs and their subclasses. Weakly modular graphs have been introduced as a
far-reaching common generalization of median graphs (and more generally, of modular and orientable modular graphs), Helly graphs,
bridged graphs, and dual polar graphs occurring under different disguises (
We give a local-to-global characterization of weakly modular graphs and their subclasses in terms of simple
connectedness of associated triangle-square complexes and specific local combinatorial conditions. In particular, we revisit
characterizations of dual polar graphs by Cameron and by Brouwer-Cohen. We also show that (disk-)Helly graphs are precisely the
clique-Helly graphs with simply connected clique complexes. With
eBook
Degree-Based Topological Indices
The degree of a vertex of a molecular graph is the number of first neighbors of this vertex. A large number of molecular-graph-based structure descriptors (topological indices) have been conceived, depending on vertex degrees. We summarize their main properties, and provide a critical comparative study thereof. (doi: 10.5562/cca2294) Keywords: topological index, molecular structure descriptor, vertex-degree-based topological index, molecular graph, chemical graph theory
Journal Article
Stream graphs and link streams for the modeling of interactions over time
Graph theory provides a language for studying the structure of relations, and it is often used to study interactions over time too. However, it poorly captures the intrinsically temporal and structural nature of interactions, which calls for a dedicated formalism. In this paper, we generalize graph concepts to cope with both aspects in a consistent way. We start with elementary concepts like density, clusters, or paths, and derive from them more advanced concepts like cliques, degrees, clustering coefficients, or connected components. We obtain a language to directly deal with interactions over time, similar to the language provided by graphs to deal with relations. This formalism is self-consistent: usual relations between different concepts are preserved. It is also consistent with graph theory: graph concepts are special cases of the ones we introduce. This makes it easy to generalize higher level objects such as quotient graphs, line graphs,
k
-cores, and centralities. This paper also considers discrete versus continuous time assumptions, instantaneous links, and extensions to more complex cases.
Journal Article