"A graph is a simple and quite old mathematical concept: a data structure consisting of a set of vertices (or nodes/points) and edges (or relationships/lines) that can be used to model relationships among a collection of objects." (Alessandro Negro, "Graph-Powered Machine Learning", 2021)
"[...] graphs are extremely useful for encoding information, and data in graph format is increasingly plentiful. In many areas of machine learning - including natural language processing, computer vision, and recommendations - graphs are used to model local relationships between isolated data items (users, items, events, and so on) and to construct global structures from local information. Representing data as graphs is often a necessary step (and at other times only a desirable one) in dealing with problems arising from applications in machine learning or data mining." (Alessandro Negro, "Graph-Powered Machine Learning", 2021)
"Graphs, with multiple node types and different types of relationships, are far from being a Euclidean space. This task is where graph neural networks (GNNs) comes in. GNNs are deep learning-based methods that operate on a graph domain to perform complex tasks such as node classification (the bot example), link prediction (the disease example), and so on. Due to its convincing performance, GNN has become a widely applied graph analysis method." (Alessandro Negro, "Graph-Powered Machine Learning", 2021)
"In many areas of machine learning, graphs are used to model local relationships between data elements and to build global structures from local information. Building graphs is sometimes necessary for dealing with problems arising from applications in machine learning or data mining, and at other times, it's helpful for managing data. It's important to note that the transformation from the original data to a graph data representation can always be performed in a lossless manner. The opposite is not always true." (Alessandro Negro, "Graph-Powered Machine Learning", 2021)
"Several areas of graph theory are concerned with the likelihood or certainty of the presence in a graph of various subgraphs or, more generally, of graph properties that emerge as the number of vertices and/or the number of edges increases. Collectively they are grouped as analytic graph theory." (Jonathan L Gross et al, "Topics in Graph Theory", 2023)
"Some connected graphs are 'more connected' than others. That is, a connected graph’s vulnerability to disconnection by edge- or vertex-deletion varies. Two numerical parameters, vertex-connectivity and edge-connectivity, are useful in measuring a graph’s connectedness. Intuitively, a network’s vulnerability should be closely related also to the number of alternative paths between each pair of nodes. There is a rich body of mathematical results concerning this relationship, many of which are variations of a classical result of Menger, and some of these extend well beyond graph theory." (Jonathan L Gross et al, "Topics in Graph Theory", 2023)
"Spanning trees capture the connectedness of a graph in the most efficient way, and they provide a foundation for a systematic analysis of the cycle structure of a graph. Mathematicians regard the algebraic structures underlying the collection of cycles and edge-cuts of agraph as beautiful in their own right. Establishing connections between linear algebra and graph theory provides some powerful analytical tools for understanding a graph’s structure." (Jonathan L Gross et al, "Topics in Graph Theory", 2023)
"[...] a graph is a mathematical model that is used for describing relationships between entities. However, each complex network presents intrinsic properties. Such properties can be measured by particular metrics, and each measure may characterize one or several local and global aspects of the graph." (Aldo Marzullo et al, "Graph Machine Learning" 2nd Ed., 2025)
"A point of caution when dealing with projections: be aware of the dimension of the projected graph. In certain cases, such as the one we are considering here, projection may create extremely large numbers of edges, which makes the graph hard to be analyzed." (Aldo Marzullo et al, "Graph Machine Learning" 2nd Ed., 2025)
"Most of the complexity indeed arises from the presence of entities that appear only once or very few times, but still generate cliques within the graph. Such entities are not very informative to capture patterns and provide insights. Besides, they are possibly strongly affected by statistical variability. On the other hand, we should focus on strong correlations that are supported by larger occurrences and provide more reliable statistical results." (Aldo Marzullo et al, "Graph Machine Learning" 2nd Ed., 2025)
"The concept of temporal graphs is useful in all the real-world problems that can be represented as a graph, where the nodes and edges of the graph may change over time. For example, temporal graphs are extensively applied in modeling social networks. By capturing the evolving relationships between individuals, temporal graphs enable a more accurate representation of social dynamics. This is particularly useful for predicting changes in friendships, community structures, and the information diffusion over time." (Aldo Marzullo et al, "Graph Machine Learning" 2nd Ed., 2025)
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