Table 1 Important notations

|.|

Cardinality of a set

\(\Vert . \Vert\)

Norm of a vector

\({\mathcal{D}}=\{G_1,\ldots ,G_n\}\)

Given graph dataset, \(G_i\) denotes the i-th graph in the dataset

\({\mathbf{y}}=[y_1,\ldots ,y_n]^\top\)

Class label vector for graphs in \({\mathcal{D}}\), \(y_i\in \{-1,+1\}\)

\({\mathcal{S}}=\{g_1,\ldots ,g_m\}\)

Set of all subgraph patterns in the graph dataset \({\mathcal{D}}\)

\({\mathbf{f}}_i=[f_{i1},\ldots ,f_{in}]^\top\)

Binary vector for subgraph pattern \(g_i\), \(f_{ij}=1\) iff \(g_i\subseteq G_j\), otherwise \(f_{ij}=0\)

\({\mathbf{x}}_j=[x_{1j},\ldots ,x_{mj}]^\top\)

Binary vector for \(G_j\) using subgraph patterns in \({\mathcal{S}}\), \(x_{ij}=1\) iff \(g_i\subseteq G_j\), otherwise \(x_{ij}=0\)

\(X=[x_{ij}]^{m\times n}\)

Matrix of all binary vectors in the dataset, \(X=[{\mathbf{x}}_1,\ldots ,{\mathbf{x}}_n]=[{\mathbf{f}}_1,\ldots ,{\mathbf{f}}_m]^\top \in \{0,1\}^{m\times n}\)

\({\mathcal{T}}\)

Set of selected subgraph patterns, \({\mathcal{T}}\subseteq {\mathcal{S}}\)

\({\mathcal{I}}_{\mathcal{T}}\in \{0,1\}^{m\times m}\)

Diagonal matrix indicating which subgraph patterns are selected from \({\mathcal{S}}\) into \({\mathcal{T}}\)

min_sup

Minimum frequency threshold; frequent subgraphs are contained by at least min_sup \(\times |{\mathcal{D}}|\) graphs

k

Number of subgraph patterns to be selected

\(\lambda ^{(p)}\)

Weight of the p-th side view (default: 1)

\(\kappa ^{(p)}\)

Kernel function on the p-th side view (default: RBF kernel)