"[...] 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)
"A small-world network is characterized by a high clustering coefficient and a short average path length, meaning that most nodes can be reached from any other node through a small number of intermediate connections. This structure often mirrors real-world social networks, where individuals are typically connected through a few mutual acquaintances, allowing for rapid information dissemination." (Aldo Marzullo et al, "Graph Machine Learning" 2nd Ed., 2025)
"Although creating simple subgraphs and merging them is a way to generate new graphs of increasing complexity, networks may also be generated by means of probabilistic models and/or generative models that let a graph grow by itself. Such graphs usually share interesting properties with real networks and have long been used to create benchmarks and synthetic graphs, especially in times when the amount of data available was not as overwhelming as today." (Aldo Marzullo et al, "Graph Machine Learning" 2nd Ed., 2025)
"As with many other deep learning-based approaches, another major challenge is in interpretability. While knowledge graphs provide a structured and transparent way to store relationships, LLMs operate as a black box, making it difficult to understand how specific outputs are generated. [...] Data alignment is also a key issue, as structured knowledge graphs and unstructured text data must be carefully preprocessed to ensure consistency. Differences in data formats, ontology mismatches, and information redundancy can create inefficiencies when integrating these two paradigms. Developing robust pipelines that seamlessly connect graph-based insights with LLM-generated text remains an open challenge." (Aldo Marzullo et al, "Graph Machine Learning" 2nd Ed., 2025)
"Assortativity is used to quantify the tendency of nodes being connected to similar nodes, which can impact the network’s ability to withstand failures or 'attacks'. High assortativity indicates that nodes of similar degrees are more likely to be connected, leading to a resilient structure where the failure of some nodes does not significantly disrupt overall connectivity. Conversely, networks with low assortativity tend to have nodes connecting with dissimilar degrees, making them more vulnerable to targeted attacks on high-degree nodes [...]" (Aldo Marzullo et al, "Graph Machine Learning" 2nd Ed., 2025)
"Besides allowing us to compress a sparse representation into a denser vector, autoencoders are also widely used to process a signal in order to filter out noise and extract only a relevant (characteristic) signal. This can be very useful in many applications, especially when identifying anomalies and outliers." (Aldo Marzullo et al, "Graph Machine Learning" 2nd Ed., 2025)
"Despite their impressive capabilities, LLMs are not without limitations. One of the most significant challenges is the problem of hallucination, where an LLM generates factually incorrect or misleading information that appears plausible. This is particularly problematic in domains requiring high factual accuracy, such as healthcare, finance, and legal applications. To mitigate hallucinations and enhance the reliability of LLM outputs, Retrieval-Augmented Generation (RAG) has emerged as a powerful technique. RAG works by dynamically retrieving relevant information from an external knowledge source (such as a knowledge graph) at inference time, rather than just relying on pre-trained knowledge. This approach ensures that the model has access to up-to-date and accurate data, grounding answers in verified information rather than generating content purely from its internal representations." (Aldo Marzullo et al, "Graph Machine Learning" 2nd Ed., 2025)
"Generally speaking, all the unsupervised embedding algorithms based on matrix factorization use the same principle. They all factorize an input graph expressed as a matrix in different components. The main difference between each method lies in the loss function used during the optimization process and the constraints/formulations posed for the V and H matrices. Indeed, different loss functions allow the creation of an embedding space that emphasizes specific properties of the input graph." (Aldo Marzullo et al, "Graph Machine Learning" 2nd Ed., 2025)
"Graph analytics is generally very effective in clustering users, merchants, and communities to provide an effective implementation of behavior analytics. On the other hand, second-party fraud can be identified with the implementation of monitoring employee behavior, as well as compliance checks. Graph analytics can indeed be useful for these use cases. Similar to the first-party models, employee behavior can also be analyzed using graph machine learning, although the dataset may need to encode a number of other sources of information besides transactional data. From a compliance standpoint, process mining techniques that still rely on a graph representation of the various procedural steps/pathways can be effective in identifying fraudulent behavior or non-compliant processes. Finally, third-party fraud, especially in the form of phishing attacks, can also be addressed using graph machine learning. In this context, understanding the network from which the phishing attack comes as well as the URLs being used (which can also benefit from a graph representation) can be critical for building an effective phishing classifier." (Aldo Marzullo et al, "Graph Machine Learning" 2nd Ed., 2025)
"Graphs are mathematical structures that are used for describing relationships between entities, and they are used almost everywhere." (Aldo Marzullo et al, "Graph Machine Learning" 2nd Ed., 2025)
"In a nutshell, training LLMs involves optimizing a large number of parameters on very large datasets. This process, known as pretraining, typically employs unsupervised learning objectives, such as predicting missing words in a sentence (masked language modeling) or forecasting subsequent words (causal language modeling). As a side effect, the pre-training phase lets the model learn and 'understand' a language, resulting in a remarkable ability to generalize across various tasks, often achieving state-of-the-art performance. LLMs have demonstrated proficiency in a diverse array of applications, reflecting their versatility and depth of language understanding. Key areas include text generation, language translation, question answering, and summarization, among many others." (Aldo Marzullo et al, "Graph Machine Learning" 2nd Ed., 2025)
"Machine learning is a subset of artificial intelligence that aims to provide systems with the ability to learn and improve from data. It has achieved impressive results in many different applications, especially where it is difficult or unfeasible to explicitly define rules to solve a specific task." (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)
"Overfitting is one of the main problems that affect machine learning practitioners. It can occur due to several reasons. Some of the reasons can be as follows: The dataset can be ill-defined or not sufficiently representative of the task. In this case, adding more data could help to mitigate the problem. The mathematical model used for addressing the problem is too powerful for the task. In this case, proper constraints can be added to the loss function in order to reduce the model’s 'power'. Such constraints are called regularization terms.(Aldo Marzullo et al, "Graph Machine Learning" 2nd Ed., 2025)
"RAG is a framework that combines the strengths of traditional information retrieval systems with the generative capabilities of LLMs. In this setup, an LLM is augmented with a retrieval component that fetches relevant information from external data sources, such as knowledge bases or databases, to produce more accurate and contextually relevant responses. This method enhances the LLM’s output by grounding it in authoritative, up-to-date information." (Aldo Marzullo et al, "Graph Machine Learning" 2nd Ed., 2025)
"Reinforcement learning is used for training machine learning agents to make a sequence of decisions. The artificial intelligence algorithm faces a game-like situation, where the agent gets penalties or rewards based on the actions performed The goal of the agent is to understand how to act in order to maximize rewards and minimize penalties". (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)
"The label spreading algorithm is another semi-supervised shallow embedding algorithm. It was built in order to overcome one big limitation of the label propagation method: the initial labeling. Indeed, according to the label propagation algorithm, the initial labels cannot be modified in the training process and, in each iteration, they are forced to be equal to their original value. This constraint could generate incorrect results when the initial labeling is affected by errors or noise. As a consequence, the error will be propagated in all nodes of the input graph." (Aldo Marzullo et al, "Graph Machine Learning" 2nd Ed., 2025)
"The main difference between unsupervised and supervised embedding methods essentially lies in the task they attempt to solve. Indeed, if unsupervised shallow embedding algorithms try to learn a good graph, node, or edge representation in order to understand the underlying structure, the supervised algorithms try to find the best solution for a prediction task such as node classification, label prediction, or graph classification." (Aldo Marzullo et al, "Graph Machine Learning" 2nd Ed., 2025)
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