  Convolutional neural networks (CNNs) can be applied to graph similarity
matching, in which case they are called graph CNNs. Graph CNNs are attracting
increasing attention due to their effectiveness and efficiency. However, the
existing convolution approaches focus only on regular data forms and require
the transfer of the graph or key node neighborhoods of the graph into the same
fixed form. During this transfer process, structural information of the graph
can be lost, and some redundant information can be incorporated. To overcome
this problem, we propose the disordered graph convolutional neural network
(DGCNN) based on the mixed Gaussian model, which extends the CNN by adding a
preprocessing layer called the disordered graph convolutional layer (DGCL). The
DGCL uses a mixed Gaussian function to realize the mapping between the
convolution kernel and the nodes in the neighborhood of the graph. The output
of the DGCL is the input of the CNN. We further implement a
backward-propagation optimization process of the convolutional layer by which
we incorporate the feature-learning model of the irregular node neighborhood
structure into the network. Thereafter, the optimization of the convolution
kernel becomes part of the neural network learning process. The DGCNN can
accept arbitrary scaled and disordered neighborhood graph structures as the
receptive fields of CNNs, which reduces information loss during graph
transformation. Finally, we perform experiments on multiple standard graph
datasets. The results show that the proposed method outperforms the
state-of-the-art methods in graph classification and retrieval.
