Package 'graphkernels'

Graph Kernels

Description

A fast C++ implementation for computing various graph kernels including (1) simple kernels between vertex and/or edge label histograms, (2) graphlet kernels, (3) random walk kernels (popular baselines), and (4) the Weisfeiler-Lehman graph kernel (state-of-the-art).

Details

This library provides the following graph kernels:

• the linear kernel between vertex label histograms

• the linear kernel between edge label histograms

• the linear kernel between vertex-edge label histograms

• the linear kernel combination vertex label histograms and vertex-edge label histograms

• the Gaussian RBF kernel between vertex label histograms

• the Gaussian RBF kernel between edge label histograms

• the Gaussian RBF kernel between vertex-edge label histograms

• the graphlet kernel

• the $k$-step random walk kernel

• the geometric random walk kernel

• the exponential random walk kernel

• the shortest-path kernel

• the Weisfeiler-Lehman subtree kernel

Given a list of igraph graphs, each function calculates the corresponding kernel (Gram) matrix.

Author(s)

Mahito Sugiyama

Maintainer: Mahito Sugiyama <[email protected]>

References

Borgwardt, K. M., Kriegel, H.-P.: Shortest-Path Kernels on Graphs, Proceedings of the 5th IEEE International Conference on Data Mining (ICDM'05), 74-81 (2005) https://ieeexplore.ieee.org/document/1565664/.

Debnath, A. K., Lopez de Compadre, R. L., Debnath, G., Shusterman, A. J., Hansch, C.: Structure-activity relationship of mutagenic aromatic and heteroaromatic nitro compounds. correlation with molecular orbital energies and hydrophobicity, Journal of Medicinal Chemistry, 34(2), 786-797 (1991) https://pubs.acs.org/doi/abs/10.1021/jm00106a046.

Gartner, T., Flach, P., Wrobel, S.: On graph kernels: Hardness results and efficient alternatives, Learning Theory and Kernel Machines (LNCS 2777), 129-143 (2003) https://link.springer.com/chapter/10.1007/978-3-540-45167-9_11.

Shervashidze, N., Schweitzer, P., van Leeuwen, E. J., Mehlhorn, K., Borgwardt, K. M.: Weisfeiler-Lehman Graph Kernels, Journal of Machine Learning Research, 12, 2359-2561 (2011) https://www.jmlr.org/papers/volume12/shervashidze11a/shervashidze11a.pdf.

Shervashidze, N., Vishwanathan, S. V. N., Petri, T., Mehlhorn, K., Borgwardt, K. M.: Efficient Graphlet Kernels for Large Graph Comparison, Proceedings of the 12th International Conference on Artificial Intelligence and Statistics (AISTATS), 5, 488-495 (2009) https://proceedings.mlr.press/v5/shervashidze09a.html.

Sugiyama, M., Borgwardt, K. M.: Halting in Random Walk Kernels, Advances in Neural Information Processing Systems (NIPS 2015), 28, 1630-1638 (2015) https://papers.nips.cc/paper/5688-halting-in-random-walk-kernels.pdf.

Examples

data(mutag)
KEH <- CalculateEdgeHistKernel(mutag)
  ## compute linear kernel between edge histograms
KWL <- CalculateWLKernel(mutag, 5)
  ## compute Weisfeiler-Lehman subtree kernel

---

Connected graphlet kernel

Description

This function calculates a kernel matrix of the graphlet kernel with connected graphlets $K_{CGL}$ between unlabeled graphs.

Usage

CalculateConnectedGraphletKernel(G, par)

Arguments

G	a list of igraph graphs
par	the number $k$ of graphlet nodes ($k = 3$, 4, or 5 is supported)

Value

a kernel matrix of the connected graphlet kernel $K_{CGL}$

Author(s)

Mahito Sugiyama

References

Shervashidze, N., Vishwanathan, S. V. N., Petri, T., Mehlhorn, K., Borgwardt, K. M.: Efficient Graphlet Kernels for Large Graph Comparison, Proceedings of the 12th International Conference on Artificial Intelligence and Statistics (AISTATS), 5, 488-495 (2009) https://proceedings.mlr.press/v5/shervashidze09a.html.

Examples

data(mutag)
K <- CalculateConnectedGraphletKernel(mutag, 4)

---

Gaussian RBF kernel between edge label histograms

Description

This function calculates a kernel matrix of the Gaussian RBF kernel $K_{EH,G}$ between edge label histograms.

Usage

CalculateEdgeHistGaussKernel(G, par)

Arguments

G	a list of igraph graphs
par	$\sigma$ in the Gaussian RBF kernel

Value

a kernel matrix of the Gaussian RBF kernel $K_{EH,G}$ between edge label histograms

Author(s)

Mahito Sugiyama

References

Sugiyama, M., Borgwardt, K. M.: Halting in Random Walk Kernels, Advances in Neural Information Processing Systems (NIPS 2015), 28, 1630-1638 (2015) https://papers.nips.cc/paper/5688-halting-in-random-walk-kernels.pdf.

Examples

data(mutag)
K <- CalculateEdgeHistGaussKernel(mutag, .1)

---

Linear kernel between edge label histograms

Description

This function calculates a kernel matrix of the linear kernel $K_{EH}$ between edge label histograms.

Usage

CalculateEdgeHistKernel(G)

Arguments

G	a list of igraph graphs

Value

a kernel matrix of the linear kernel $K_{EH}$ between edge label histograms

Author(s)

Mahito Sugiyama

References

Sugiyama, M., Borgwardt, K. M.: Halting in Random Walk Kernels, Advances in Neural Information Processing Systems (NIPS 2015), 28, 1630-1638 (2015) https://papers.nips.cc/paper/5688-halting-in-random-walk-kernels.pdf.

Examples

data(mutag)
K <- CalculateEdgeHistKernel(mutag)

---

Exponential random walk kernel

Description

This function calculates a kernel matrix of the exponential random walk kernel $K_{ER}$.

Usage

CalculateExponentialRandomWalkKernel(G, par)

Arguments

G	a list of igraph graphs
par	a coefficient $\beta$, with which the weight $\lambda_k$ for each step $k$ is given as $\lambda_k = \beta^k / k!$

Value

a kernel matrix of the exponential random walk kernel $K_{ER}$

Author(s)

Mahito Sugiyama

References

Gartner, T., Flach, P., Wrobel, S.: On graph kernels: Hardness results and efficient alternatives, Learning Theory and Kernel Machines (LNCS 2777), 129-143 (2003) https://link.springer.com/chapter/10.1007/978-3-540-45167-9_11.

Examples

data(mutag)
K <- CalculateExponentialRandomWalkKernel(mutag[1:5], .1)

---

Geometric random walk kernel

Description

This function calculates a kernel matrix of the geometric random walk kernel $K_{GR}$.

Usage

CalculateGeometricRandomWalkKernel(G, par)

Arguments

G	a list of igraph graphs
par	a coefficient $\lambda$, with which the weight $\lambda_k$ for each step $k$ is given as $\lambda_k = \lambda^k$

Value

a kernel matrix of the geometric random walk kernel $K_{GR}$

Author(s)

Mahito Sugiyama

References

Gartner, T., Flach, P., Wrobel, S.: On graph kernels: Hardness results and efficient alternatives, Learning Theory and Kernel Machines (LNCS 2777), 129-143 (2003) https://link.springer.com/chapter/10.1007/978-3-540-45167-9_11.

Sugiyama, M., Borgwardt, K. M.: Halting in Random Walk Kernels, Advances in Neural Information Processing Systems (NIPS 2015), 28, 1630-1638 (2015) https://papers.nips.cc/paper/5688-halting-in-random-walk-kernels.pdf.

Examples

data(mutag)
K <- CalculateGeometricRandomWalkKernel(mutag, .1)

---

Graphlet kernel

Description

This function calculates a kernel matrix of the graphlet kernel $K_{GL}$ between unlabeled graphs.

Usage

CalculateGraphletKernel(G, par)

Arguments

G	a list of igraph graphs
par	the number $k$ of graphlet nodes ($k = 3$ or 4 is supported)

Value

a kernel matrix of the graphlet kernel $K_{GL}$

Author(s)

Mahito Sugiyama

References

Shervashidze, N., Vishwanathan, S. V. N., Petri, T., Mehlhorn, K., Borgwardt, K. M.: Efficient Graphlet Kernels for Large Graph Comparison, Proceedings of the 12th International Conference on Artificial Intelligence and Statistics (AISTATS), 5, 488-495 (2009) https://proceedings.mlr.press/v5/shervashidze09a.html.

Examples

data(mutag)
K <- CalculateGraphletKernel(mutag, 4)

---

An C++ implementation of graphlet kernels

Description

This function calculates a graphlet kernel matrix.

Usage

CalculateGraphletKernelCpp(graph_adj_all, graph_adjlist_all, k, connected)

Arguments

graph_adj_all	a list of adjacency matrices
graph_adjlist_all	a list of adjacency lists
k	the number $k$ of graphlet nodes
connected	whether or not graphlets are conneceted

Value

a kernel matrix of the respective graphlet kernel

Author(s)

Mahito Sugiyama

References

Shervashidze, N., Vishwanathan, S. V. N., Petri, T., Mehlhorn, K., Borgwardt, K. M.: Efficient Graphlet Kernels for Large Graph Comparison, Proceedings of the 12th International Conference on Artificial Intelligence and Statistics (AISTATS), 5, 488-495 (2009) https://proceedings.mlr.press/v5/shervashidze09a.html.

Examples

data(mutag)
al.list <- as.list(rep(NA, length(mutag)))
for (i in 1:length(mutag)) { al.list[[i]] <- as_adj_list(mutag[[i]]) }
K <- CalculateGraphletKernelCpp(list(), al.list, 4, 0)

---

An C++ implementation of graph kernels

Description

This function calculates a kernel matrix.

Usage

CalculateKernelCpp(graph_info_list, par_r, kernel_type)

Arguments

graph_info_list	a list of igraph graphs
par_r	parameters of kernels
kernel_type	type of kernel

Value

a kernel matrix of the respective kernel

Author(s)

Mahito Sugiyama

References

Sugiyama, M., Borgwardt, K. M.: Halting in Random Walk Kernels, Advances in Neural Information Processing Systems (NIPS 2015), 28, 1630-1638 (2015) https://papers.nips.cc/paper/5688-halting-in-random-walk-kernels.pdf.

Examples

data(mutag)
graph.info.list <- vector("list", length(mutag))
for (i in 1:length(mutag)) { graph.info.list[[i]] <- GetGraphInfo(mutag[[i]]) }
K <- CalculateKernelCpp(graph.info.list, 5, 11)

---

k-step random walk kernel

Description

This function calculates a kernel matrix of the $k$-step random walk kernel $K_{\times}^{k}$.

Usage

CalculateKStepRandomWalkKernel(G, par)

Arguments

G	a list of igraph graphs
par	a vector of coefficients $\lambda_0, \lambda_1, \dots, \lambda_k$

Value

a kernel matrix of the k-step random walk kernel $K_{\times}^{k}$

Author(s)

Mahito Sugiyama

References

Gartner, T., Flach, P., Wrobel, S.: On graph kernels: Hardness results and efficient alternatives, Learning Theory and Kernel Machines (LNCS 2777), 129-143 (2003) https://link.springer.com/chapter/10.1007/978-3-540-45167-9_11.

Sugiyama, M., Borgwardt, K. M.: Halting in Random Walk Kernels, Advances in Neural Information Processing Systems (NIPS 2015), 28, 1630-1638 (2015) https://papers.nips.cc/paper/5688-halting-in-random-walk-kernels.pdf.

Examples

data(mutag)
K <- CalculateKStepRandomWalkKernel(mutag, rep(1, 2))

---

Shortest-path kernel

Description

This function calculates a kernel matrix of the shortest-path kernel $K_{SP}$.

Usage

CalculateShortestPathKernel(G)

Arguments

G	a list of igraph graphs

Value

a kernel matrix of the shortest-path kernel $K_{SP}$

Author(s)

Mahito Sugiyama

References

Borgwardt, K. M., Kriegel, H.-P.: Shortest-Path Kernels on Graphs, Proceedings of the 5th IEEE International Conference on Data Mining (ICDM'05), 74-81 (2005) https://ieeexplore.ieee.org/document/1565664/.

Examples

data(mutag)
K <- CalculateShortestPathKernel(mutag)

---

Gaussian RBF kernel between vertex-edge label histograms

Description

This function calculates a kernel matrix of the Gaussian RBF kernel $K_{VEH,G}$ between vertex-edge label histograms.

Usage

CalculateVertexEdgeHistGaussKernel(G, par)

Arguments

G	a list of igraph graphs
par	$\sigma$ in the Gaussian RBF kernel

Value

a kernel matrix of the Gaussian RBF kernel $K_{VEH,G}$ between vertex-edge label histograms

Author(s)

Mahito Sugiyama

References

Sugiyama, M., Borgwardt, K. M.: Halting in Random Walk Kernels, Advances in Neural Information Processing Systems (NIPS 2015), 28, 1630-1638 (2015) https://papers.nips.cc/paper/5688-halting-in-random-walk-kernels.pdf.

Examples

data(mutag)
K <- CalculateVertexEdgeHistGaussKernel(mutag, .1)

---

Linear kernel between vertex-edge label histograms

Description

This function calculates a kernel matrix of the linear kernel $K_{VEH}$ between vertex-edge label histograms.

Usage

CalculateVertexEdgeHistKernel(G)

Arguments

G	a list of igraph graphs

Value

a kernel matrix of the linear kernel $K_{VEH}$ between vertex-edge label histograms

Author(s)

Mahito Sugiyama

References

Sugiyama, M., Borgwardt, K. M.: Halting in Random Walk Kernels, Advances in Neural Information Processing Systems (NIPS 2015), 28, 1630-1638 (2015) https://papers.nips.cc/paper/5688-halting-in-random-walk-kernels.pdf.

Examples

data(mutag)
K <- CalculateVertexEdgeHistKernel(mutag)

---

Gaussian RBF kernel between vertex label histograms

Description

This function calculates a kernel matrix of the Gaussian RBF kernel $K_{VH,G}$ between vertex label histograms.

Usage

CalculateVertexHistGaussKernel(G, par)

Arguments

G	a list of igraph graphs
par	$\sigma$ in the Gaussian RBF kernel

Value

a kernel matrix of the Gaussian RBF kernel $K_{VH,G}$ between vertex label histograms

Author(s)

Mahito Sugiyama

References

Sugiyama, M., Borgwardt, K. M.: Halting in Random Walk Kernels, Advances in Neural Information Processing Systems (NIPS 2015), 28, 1630-1638 (2015) https://papers.nips.cc/paper/5688-halting-in-random-walk-kernels.pdf.

Examples

data(mutag)
K <- CalculateVertexHistGaussKernel(mutag, .1)

---

Linear kernel between vertex label histograms

Description

This function calculates a kernel matrix of the linear kernel $K_{VH}$ between vertex label histograms.

Usage

CalculateVertexHistKernel(G)

Arguments

G	a list of igraph graphs

Value

a kernel matrix of the linear kernel $K_{VH}$ between vertex label histograms

Author(s)

Mahito Sugiyama

References

Sugiyama, M., Borgwardt, K. M.: Halting in Random Walk Kernels, Advances in Neural Information Processing Systems (NIPS 2015), 28, 1630-1638 (2015) https://papers.nips.cc/paper/5688-halting-in-random-walk-kernels.pdf.

Examples

data(mutag)
K <- CalculateVertexHistKernel(mutag)

---

Linear kernel combination of vertex label histograms and vertex-edge label histograms

Description

This function calculates a kernel matrix of the linear kernel combination $K_{H}$ of vertex label histograms $K_{VH}$ and vertex-edge label histograms $K_{VEH}$.

Usage

CalculateVertexVertexEdgeHistKernel(G, par)

Arguments

G	a list of igraph graphs
par	a coefficient $\lambda$, with which the resulting kernel is given as $K_{VH} + \lambda K_{VEH}$

Value

a kernel matrix that is equivalent to $K_{VH} + \lambda K_{VEH}$

Author(s)

Mahito Sugiyama

References

Sugiyama, M., Borgwardt, K. M.: Halting in Random Walk Kernels, Advances in Neural Information Processing Systems (NIPS 2015), 28, 1630-1638 (2015) https://papers.nips.cc/paper/5688-halting-in-random-walk-kernels.pdf.

Examples

data(mutag)
K <- CalculateVertexVertexEdgeHistKernel(mutag, .1)

---

Weisfeiler-Lehman subtree kernel

Description

This function calculates a kernel matrix of the Weisfeiler-Lehman subtree kernel $K_{WL}$.

Usage

CalculateWLKernel(G, par)

Arguments

G	a list of igraph graphs
par	the number $h$ of iterations

Value

a kernel matrix of the Weisfeiler-Lehman subtree kernel $K_{WL}$

Author(s)

Mahito Sugiyama

References

Shervashidze, N., Schweitzer, P., van Leeuwen, E. J., Mehlhorn, K., Borgwardt, K. M.: Weisfeiler-Lehman Graph Kernels, Journal of Machine Learning Research, 12, 2359-2561 (2011) https://www.jmlr.org/papers/volume12/shervashidze11a/shervashidze11a.pdf.

Examples

data(mutag)
K <- CalculateWLKernel(mutag, 5)

---

Necessary information of graphs for kernel computation

Description

This function extracts necessary information of graphs for kernel computation.

Usage

GetGraphInfo(g)

Arguments

g	an igraph graph

Value

a list of graph information with the following elements:

edge	a matrix of edges with their labels
vlabel	a vector of vertex labels
vsize	the number of vertices
esize	the number of edges
maxdegree	the maximum degree

Author(s)

Mahito Sugiyama

Examples

data(mutag)
ginfo <- GetGraphInfo(mutag[[1]])

---

Symbol registration

Description

This is a supplement for symbol registration.

Author(s)

Mahito Sugiyama

---

Symbol registration

Description

This is a supplement for symbol registration.

Author(s)

Mahito Sugiyama

---

The mutag dataset

Description

This is the mutag dataset, a well known benchmark dataset for graph processing algorithms.

Usage

data(mutag)

Author(s)

Mahito Sugiyama

References

Debnath, A. K., Lopez de Compadre, R. L., Debnath, G., Shusterman, A. J., Hansch, C.: Structure-activity relationship of mutagenic aromatic and heteroaromatic nitro compounds. correlation with molecular orbital energies and hydrophobicity, Journal of Medicinal Chemistry, 34(2), 786-797 (1991) https://pubs.acs.org/doi/abs/10.1021/jm00106a046.

Examples

data(mutag)
K <- CalculateWLKernel(mutag, 5)

---