We present a unified framework to study graph kernels, special cases of which include the random walk (Gartner et al., 2003; Borgwardt et al., 2005) and marginalized (Kashima et al., 2003, 2004; Mahet al., 2004) graph kernels. Through reduction to a Sylvester equation we improve the time complexity of kernel computation between unlabeled graphs with n vertices from O(n6) to O(n3). We find a spectral decomposition approach even more efficient when computing entire kernel matrices. For labeled graphs we develop conjugate gradient and fixed-point methods that take O(dn3) time per iteration, where d is the size of the label set. By extending the necessary linear algebra to Reproducing Kernel Hilbert Spaces (RKHS) we obtain the same result for d-dimensional edge kernels, and O(n4) in the infinite-dimensional case; on sparse graphs these algorithms only take O(n2) time per iteration in all cases. Experiments on graphs from bioinformatics and other application domains show that these techniques can speed up computation of the kernel by an order of magnitude or more. We also show that certain rational kernels (Cortes et al., 2002, 2003, 2004) when specialized to graphs reduce to our random walk graph kernel. Finally, we relate our framework to R-convolution kernels (Haussler, 1999) and provide a kernel that is close to the optimal assignment kernel of kernel of Frohlich et al. (2006) yet provably positive semi-definite.

Graph Kernels

How can we generate realistic networks? In addition, how can we do so with a mathematically tractable model that allows for rigorous analysis of network properties? Real networks exhibit a long list of surprising properties: Heavy tails for the in- and out-degree distribution, heavy tails for the eigenvalues and eigenvectors, small diameters, and densification and shrinking diameters over time. Current network models and generators either fail to match several of the above properties, are complicated to analyze mathematically, or both. Here we propose a generative model for networks that is both mathematically tractable and can generate networks that have all the above mentioned structural properties. Our main idea here is to use a non-standard matrix operation, the Kronecker product, to generate graphs which we refer to as "Kronecker graphs".

First, we show that Kronecker graphs naturally obey common network properties. In fact, we rigorously prove that they do so. We also provide empirical evidence showing that Kronecker graphs can effectively model the structure of real networks.

We then present KRONFIT, a fast and scalable algorithm for fitting the Kronecker graph generation model to large real networks. A naive approach to fitting would take super-exponential time. In contrast, KRONFIT takes linear time, by exploiting the structure of Kronecker matrix multiplication and by using statistical simulation techniques.

Experiments on a wide range of large real and synthetic networks show that KRONFIT finds accurate parameters that very well mimic the properties of target networks. In fact, using just four parameters we can accurately model several aspects of global network structure. Once fitted, the model parameters can be used to gain insights about the network structure, and the resulting synthetic graphs can be used for null-models, anonymization, extrapolations, and graph summarization.

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Kronecker Graphs: An Approach to Modeling Networks

As most ‘real-world’ data is structured, research in kernel methods has begun investigating kernels for various kinds of structured data. One of the most widely used tools for modeling structured data are graphs. An interesting and important challenge is thus to investigate kernels on instances that are represented by graphs. So far, only very specific graphs such as trees and strings have been considered.

On Graph Kernels: Hardness Results and Efficient Alternatives

Starting around the late 1950s, several research communities began relating the geometry of graphs to stochastic processes on these graphs. This book, twenty years in the making, ties together research in the field, encompassing work on percolation, isoperimetric inequalities, eigenvalues, transition probabilities, and random walks. Written by two leading researchers, the text emphasizes intuition, while giving complete proofs and more than 850 exercises. Many recent developments, in which the authors have played a leading role, are discussed, including percolation on trees and Cayley graphs, uniform spanning forests, the mass-transport technique, and connections on random walks on graphs to embedding in Hilbert space. This state-of-the-art account of probability on networks will be indispensable for graduate students and researchers alike.

Probability on Trees and Networks

Analysis and processing of very large data sets, or big data, poses a significant challenge. Massive data sets are collected and studied in numerous domains, from engineering sciences to social networks, biomolecular research, commerce, and security. Extracting valuable information from big data requires innovative approaches that efficiently process large amounts of data as well as handle and, moreover, utilize their structure. This article discusses a paradigm for large-scale data analysis based on the discrete signal processing (DSP) on graphs (DSPG). DSPG extends signal processing concepts and methodologies from the classical signal processing theory to data indexed by general graphs. Big data analysis presents several challenges to DSPG, in particular, in filtering and frequency analysis of very large data sets. We review fundamental concepts of DSPG, including graph signals and graph filters, graph Fourier transform, graph frequency, and spectrum ordering, and compare them with their counterparts from the classical signal processing theory. We then consider product graphs as a graph model that helps extend the application of DSPG methods to large data sets through efficient implementation based on parallelization and vectorization. We relate the presented framework to existing methods for large-scale data processing and illustrate it with an application to data compression.

Big Data Analysis with Signal Processing on Graphs: Representation and processing of massive data sets with irregular structure

Basic Concepts. Hypercubes. Hamming Graphs. Cartesian Products. Strong and Direct Products. Lexicographic Products. Fast Recognition Algorithms. Invariants. Appendices. Bibliography. Indexes.

Product Graphs: Structure and Recognition

Handbook of Product Graphs, Second Edition examines the dichotomy between the structure of products and their subgraphs. It also features the design of efficient algorithms that recognize products and their subgraphs and explores the relationship between graph parameters of the product and factors. Extensively revised and expanded, the handbook presents full proofs of many important results as well as up-to-date research and conjectures. Results and Algorithms New to the Second Edition: Cancellation results A quadratic recognition algorithm for partial cubes Results on the strong isometric dimension Computing the Wiener index via canonical isometric embedding Connectivity results A fractional version of Hedetniemis conjecture Results on the independence number of Cartesian powers of vertex-transitive graphs Verification of Vizings conjecture for chordal graphs Results on minimum cycle bases Numerous selected recent results, such as complete minors and nowhere-zero flows The second edition of this classic handbook provides a thorough introduction to the subject and an extensive survey of the field. The first three parts of the book cover graph products in detail. The authors discuss algebraic properties, such as factorization and cancellation, and explore interesting and important classes of subgraphs. The fourth part presents algorithms for the recognition of products and related classes of graphs. The final two parts focus on graph invariants and infinite, directed, and product-like graphs. Sample implementations of selected algorithms and other information are available on the books website, which can be reached via the authors home pages.

Handbook of Product Graphs

From specialists in the field, learn about interesting connections and recent developments in the field of graph theory by looking in particular at Cartesian products arguably the most important of the four standard graph products. Many new results in this area appear for the first time in print in this book. Written in an accessible way, this book can be used for personal study in advanced applications of graph theory or for an advanced graph theory course.

Topics in Graph Theory: Graphs and Their Cartesian Product

The distinguishing number D(G) of a graph is the least integer d such that there is a d-labeling of the vertices of G that is not preserved by any nontrivial automorphism of G. We show that the distinguishing number of the square and higher powers of a connected graph G ≠ K2, K3 with respect to the Cartesian product is 2. This result strengthens results of Albertson [Electron J Combin, 12 (2005), nN17] on powers of prime graphs, and results of KlavŽar and Zhu [Eu J Combin, to appear]. More generally, we also prove that d(G s H) = 2 if G and H are relatively prime and |H| ≤ |G| < 2|H| - |H|. Under additional conditions similar results hold for powers of graphs with respect to the strong and the direct product. © 2006 Wiley Periodicals, Inc. J Graph Theory 53: 250260, 2006

Distinguishing Cartesian powers of graphs

The distinguishing number $D(G)$ of a graph $G$ is the least cardinal number $\aleph$ such that $G$ has a labeling with $\aleph$ labels that is only preserved by the trivial automorphism We show that the distinguishing number of the countable random graph is two, that tree-like graphs with not more than continuum many vertices have distinguishing number two, and determine the distinguishing number of many classes of infinite Cartesian products For instance, $D(Q_{n}) = 2$, where $Q_{n}$ is the infinite hypercube of dimension ${n}$

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Wilfried Imrich

Papers

Product Graphs: Structure and Recognition

Handbook of Product Graphs

Topics in Graph Theory: Graphs and Their Cartesian Product

Distinguishing Cartesian powers of graphs

Distinguishing Infinite Graphs