Random walk kernels have been introduced in seminal work on graph learning and were later largely superseded by kernels based on the Weisfeiler-Leman test for graph isomorphism. We give a unified view on both classes of graph kernels. We study walk-based node refinement methods and formally relate them to several widely-used techniques, including Morgan's algorithm for molecule canonization and the Weisfeiler-Leman test. We define corresponding walk-based kernels on nodes that allow fine-grained parameterized neighborhood comparison, reach Weisfeiler-Leman expressiveness, and are computed using the kernel trick. From this we show that classical random walk kernels with only minor modifications regarding definition and computation are as expressive as the widely-used Weisfeiler-Leman subtree kernel but support non-strict neighborhood comparison. We verify experimentally that walk-based kernels reach or even surpass the accuracy of Weisfeiler-Leman kernels in real-world classification tasks.

Neural Information Processing Systems http://nips.cc/

Random walk kernels have been introduced in seminal work on graph learning and were later largely superseded by kernels based on the Weisfeiler-Leman test for graph isomorphism. We give a unified view on both classes of graph kernels. We study walk-based node refinement methods and formally relate them to several widely-used techniques, including Morgan's algorithm for molecule canonization and the Weisfeiler-Leman test. We define corresponding walk-based kernels on nodes that allow fine-grained parameterized neighborhood comparison, reach Weisfeiler-Leman expressiveness, and are computed using the kernel trick. From this we show that classical random walk kernels with only minor modifications regarding definition and computation are as expressive as the widely-used Weisfeiler-Leman subtree kernel but support non-strict neighborhood comparison.

Random walks play an important role in probing the structure of complex networks. On traditional networks, they can be used to extract community structure, understand node centrality, perform link prediction, or capture the similarity between nodes. On signed networks, where the edge weights can be either positive or negative, it is non-trivial to design a random walk which can be used to extract information about the signed structure of the network, in particular the ability to partition the graph into communities with positive edges inside and negative edges in between. Prior works on signed network random walks focus on the case where there are only two such communities (strong balance), which is rarely the case in empirical networks. In this paper, we propose a signed network random walk which can capture the structure of a network with more than two such communities (weak balance). The walk results in a similarity matrix which can be used to cluster the nodes into antagonistic communities. We compare the characteristics of the so-called strong and weak random walks, in terms of walk length and stationarity. We show through a series of experiments on synthetic and empirical networks that the similarity matrix based on weak walks can be used for both unsupervised and semi-supervised clustering, outperforming the same similarity matrix based on strong walks when the graph has more than two communities, or exhibits asymmetry in the density of links. These results suggest that other random-walk based algorithms for signed networks could be improved simply by running them with weak walks instead of strong walks.

Random walk kernels measure graph similarity by counting matching walks in two graphs.

Random walk kernels have been introduced in seminal work on graph learning and were later largely superseded by kernels based on the Weisfeiler-Leman test for graph isomorphism. We give a unified view on both classes of graph kernels. We study walk-based node refinement methods and formally relate them to several widely-used techniques, including Morgan's algorithm for molecule canonization and the Weisfeiler-Leman test. We define corresponding walk-based kernels on nodes that allow fine-grained parameterized neighborhood comparison, reach Weisfeiler-Leman expressiveness, and are computed using the kernel trick. From this we show that classical random walk kernels with only minor modifications regarding definition and computation are as expressive as the widely-used Weisfeiler-Leman subtree kernel but support non-strict neighborhood comparison. We verify experimentally that walk-based kernels reach or even surpass the accuracy of Weisfeiler-Leman kernels in real-world classification tasks.

We introduce a family of multilayer graph kernels and establish new links between graph convolutional neural networks and kernel methods. Our approach generalizes convolutional kernel networks to graph-structured data, by representing graphs as a sequence of kernel feature maps, where each node carries information about local graph substructures. On the one hand, the kernel point of view offers an unsupervised, expressive, and easy-to-regularize data representation, which is useful when limited samples are available. On the other hand, our model can also be trained end-to-end on large-scale data, leading to new types of graph convolutional neural networks. We show that our method achieves competitive performance on several graph classification benchmarks, while offering simple model interpretation. Our code is freely available at https://github.com/claying/GCKN.

Non-linear kernel methods can be approximated by fast linear ones using suitable explicit feature maps allowing their application to large scale problems. To this end, explicit feature maps of kernels for vectorial data have been extensively studied. As many real-world data is structured, various kernels for complex data like graphs have been proposed. Indeed, many of them directly compute feature maps. However, the kernel trick is employed when the number of features is very large or the individual vertices of graphs are annotated by real-valued attributes. Can we still compute explicit feature maps efficiently under these circumstances? Triggered by this question, we investigate how general convolution kernels are composed from base kernels and construct corresponding feature maps. We apply our results to widely used graph kernels and analyze for which kernels and graph properties computation by explicit feature maps is feasible and actually more efficient. In particular, we derive feature maps for random walk and subgraph matching kernels and apply them to real-world graphs with discrete labels. Thereby, our theoretical results are confirmed experimentally by observing a phase transition when comparing running time with respect to label diversity, walk lengths and subgraph size, respectively. Moreover, we derive approximative, explicit feature maps for state-of-the-art kernels supporting real-valued attributes including the GraphHopper and Graph Invariant kernels. In extensive experiments we show that our approaches often achieve a classification accuracy close to the exact methods based on the kernel trick, but require only a fraction of their running time.

Random walk kernels measure graph similarity by counting matching walks in two graphs. In their most popular form of geometric random walk kernels, longer walks of length $k$ are downweighted by a factor of $\lambda^k$ ($\lambda < 1$) to ensure convergence of the corresponding geometric series. We know from the field of link prediction that this downweighting often leads to a phenomenon referred to as halting: Longer walks are downweighted so much that the similarity score is completely dominated by the comparison of walks of length 1. This is a naive kernel between edges and vertices. We theoretically show that halting may occur in geometric random walk kernels. We also empirically quantify its impact in simulated datasets and popular graph classification benchmark datasets. Our findings promise to be instrumental in future graph kernel development and applications of random walk kernels.