Embedding of an input graph by a (convolutional) deep neural network for graphs (NN4G), with the context composition for a vertex v through the layers, and the output mapping (from A. Micheli, “Neural Network for Graphs: A Contextual Constructive Approach", IEEE TNN 2009, see  below).
To investigate on the possibility of extending the computational capabilities of neural networks and related machine learning approaches for the treatment of structured domains (sequences, trees and graphs), making particular attention to adaptive, constructive and contextual models and to the study of their theoretical properties.
This is a traditional research aim of the CIML group in Pisa, with origins since the 90s. CIML members and collaborators were among the pioneers in the processing of structures by Recursive Neural Networks and are active since then for the progress of the field by continuously developing theoretical analysis, new models, and many applications, including pioneering approaches for the chemical domain [4,12,14,15,16]. Together with the historical origins, the current developments contribute to lead the group to an international scientific leadership in topics for learning in structured domains. The basic of Recursive Neural Networks for adaptive processing of trees, with some historical notes, can be found in  and in RNN-Wikipedia.
The CIML research results include (as regards mainly the historical perspective):
⦁ Supervised neural networks for structures: the proposers have investigated both theoretical issues and real-world applications. They have proposed a first approach to deal with contextual information in structured domains by Recursive Neural Networks (RNN). The proposed model, i.e. Contextual Recursive Cascade Correlation (CRCC) , a generalization of the Recursive Cascade Correlation (RCC) model, is able to partially remove the causality assumption by exploiting contextual information. They formally characterize the properties of CRCC showing that it is able to compute contextual transductions and also some causal supersource transductions that RCC cannot compute [5, 6, 12]. Also, comparisons with kernel based methods have been considered .
⦁ Unsupervised recursive networks for structures: they define also a general recursive framework for unsupervised processing of structured data [10,11,13]. This general framework offers a uniform notation for training mechanisms of different models and insights into theoretical issues from the SOM literature to the structure processing case.
⦁ The introduction of machine learning models for general structures by Neural Networks. In particular, the NN4G (Neural Network for Graph) model  extends the input domain of Neural Networks to general (directed/undirected, cyclic/acyclic) classes of labeled graphs by exploiting contextual information with a constructive and adaptive approach. This approach, presented in the first version at the WIRN conference in June 2005 (and then as journal IEEE TNN version in 2009 ), pioneered the field of the spatial approach to graph processing by deep Neural Networks (also known nowadays in terms of convolutional networks for graphs or, more in general, deep graph networks). See the figure above (top page) that show the model and the graph processing through the layers.
⦁ The introduction of efficient probabilistic modeling of trees [1, 2] and efficient neural network (Reservoir Computing) modeling of trees [3, 9].
⦁ The introduction of deep neural network, Reservoir Computing (RC) and generative based approaches for graphs is an ongoing research topic of CIML. For the references to the current developments, including generative kernels for trees (TNNLS 2018), deep probabilistic models for graphs (ICML 2018, JMLR 2020), deep RC for trees and graphs (Information Science 2019, AAAI 2020), comparative studies (ICLR 2020), and a survey on deep learning for graphs (Neural Networks 2020), please see into the selected paper session of the CIML site at https://ciml.di.unipi.it/publications/
See collaborators in CIML People Page
Publications on this Topic (selection restricted to the original developments on model studies before 2010, click on the reference to open the paper site)