Graphs are ubiquitous in nature and can therefore serve as models for many practical but also theoretical problems. Based on this, the young research field of Graph Neural Networks (GNNs) has emerged. Despite the youth of the field and the speed in which new models are developed, many good surveys have been published in the last years. Nevertheless, an overview on which graph types can be modeled by GNNs is missing. In this survey, we give a detailed overview of already existing GNNs and, unlike previous surveys, categorize them according to their ability to handle different graph types. We consider GNNs operating on static as well as on dynamic graphs of different structural constitutions, with or without node or edge attributes. Moreover in the dynamic case, we separate the models in discrete-time and continuous-time dynamic graphs based on their architecture. According to our findings, there are still graph types, that are not covered by existing GNN models. Specifically, models concerning heterogeneity in attributes are missing and the deletion of nodes and edges is only covered rarely.Keywords Graph Neural Networks ⋅ Graph Types ⋅ Graph Properties ⋅ Survey * Corresponding author. 2 [1] provides a historical overview up to the end of the 1980s.

Dynamic Graph Neural Networks recently became more and more important as graphs from many scientific fields, ranging from mathematics, biology, social sciences, and physics to computer science, are dynamic by nature. While temporal changes (dynamics) play an essential role in many real-world applications, most of the models in the literature on Graph Neural Networks (GNN) process static graphs. The few GNN models on dynamic graphs only consider exceptional cases of dynamics, e.g., node attribute-dynamic graphs or structure-dynamic graphs limited to additions or changes to the graph's edges, etc. Therefore, we present a novel Fully Dynamic Graph Neural Network (FDGNN) that can handle fully-dynamic graphs in continuous time. The proposed method provides a node and an edge embedding that includes their activity to address added and deleted nodes or edges, and possible attributes. Furthermore, the embeddings specify Temporal Point Processes for each event to encode the distributions of the structure-and attribute-related incoming graph events. In addition, our model can be updated efficiently by considering single events for local retraining.

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Graph representations have gained importance in almost every scientific field, ranging from mathematics, biology, social sciences and physics to computer science. In contrast to other data formats, graphs propose the possibility to model relations between entities. Together with the continuously rising amount of available data, graphs therefore open up a wide range of modeling capabilities for theoretical and real-world problems. However, the modeling possibilities of graphs have not been fully exploited. One reason for this is that there is neither an easily comprehensible overview of graph types nor an analysis of their modeling capacities available. As a result, neither the potential of modeling with certain graph types is exhausted nor higher modeling freedom and more efficient computing of graphs after transformation to another graph type is in scope of view of many users. In order to clarify the modeling possibilities of graphs, we order the different graph types, collate their memory complexity and provide an expressivity measure on them. Furthermore, we introduce transformation algorithms between the graph types from which equal expressivity of all graph types can be inferred, i.e., they are able to represent the same information or properties respectively. Finally, we provide a guideline for the question when a graph type transformation is efficient by defining a cost function dependend on the memory complexity and the transformation runtime as a decision-making tool.