🕸️On Graph Theory: On Clusters (2020-2029)

"[...] scatterplots had advantages over earlier graphic forms: the ability to see clusters, patterns, trends, and relations in a cloud of points. Perhaps most importantly, it allowed the addition of visual annotations (point symbols, lines, curves, enclosing contours, etc.) to make those relationships more coherent and tell more nuanced stories." (Michael Friendly & Howard Wainer, "A History of Data Visualization and Graphic Communication", 2021)

"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)

"A tree-like structure or a tree diagram, one showing taxonomic (classification-related) relationships, is created. It is known as a dendrogram. It has branches pointing toward categories or classes. It is a pattern formed by a series of splits or segments of a given quantity of data over a set of compartments and the flow components. From this pattern, a description can be made of the profile of data allocation over the set of functional or structural compartments. The merger of clusters is terminated after every data point lies in one single cluster at the top of the tree." (Vinod K Khanna, "Introduction to Machine Learning Algorithms:  Basic Principles and Mathematics", 2026)

"Hierarchical clustering is summarised by a dendrogram, which sequentially shows points being joined to form a cluster, with the corresponding distances. Breaking the data into clusters is done by cutting the dendrogram at the long edges. [...] Plotting the dendrogram in the data space can help you understand how the hierarchical clustering has collected the points together into clusters. You can learn if the algorithm has been confused by nuisance patterns in the data, and how different choices of linkage method affect the result." (Dianne Cook & Ursula Laa, "Interactively Exploring High-Dimensional Data and Models in R", 2026)

"Viewing the dendrograms in high dimensions provides insight into how the algorithm has joined points to clusters. For example, single linkage often has edges leading to a single focal point, which might not yield a useful clustering but might help to identify outliers. If the edges point to multiple focal points, with long edges bridging gaps in the data, the result is more likely yielding a useful clustering." (Dianne Cook & Ursula Laa,  "Interactively Exploring High-Dimensional Data and Models in R", 2026)

"ML is a useful - and under-utilized - framework for studying LLMs. For one thing, LLMs are literally composed of simple ML algorithms (linear weighted averages and nonlinear transformations). Furthermore, using ML techniques like regression, classification, and clustering, can help reveal how concepts like grammar rules are represented inside LLMs. And finally, many people find LLMs to be intimidatingly complicated while finding ML to be much more approachable. Thus, using ML to study LLMs involves using simple tools to understand complicated tools." (Mike X Cohen,"50 ML Projects To Understand LLMs", 2026)

🕸️On Graph Theory: On Clusters (2010-2019)

"Where many known, measurable factors are involved, statisticians can use mathematical techniques to account for all the variables and try to find which are the truly important predictors. The terms for this include multiple regression, multivariate analysis, and discriminant analysis, and factor, cluster, path, and two-stage least-squares analyses." (Victor Cohn & Lewis Cope, "News & Numbers: A writer’s guide to statistics" 3rd Ed, 2012)

"Hierarchical clustering creates a hierarchy of clusters which can be represented in a treelike diagram, called a dendrogram. In the dendrogram, units in the same cluster are joined by a horizontal line, with the scale on the y-axis of the dendrogram reflecting a measure of the distances of the units within the cluster. The leaves at the bottom of the dendrogram represent the individual units; leaves are combined to form small branches, small branches are combined into larger branches, until one reaches the trunk or root of the tree that represents a single cluster containing all units. Dendrograms are quite useful as they give us a visual representation of the clusters." (Johannes Ledolter, "Data mining and business analytics with R", 2013)

"Different joining/linkage rules change how the final hierarchical clustering is presented. [...] Since the barrier for merging observations and clusters is lowest with the single linkage approach, the clustering dendrogram may contain chains of clusters as well as clusters that are spread out. The barrier to joining clusters is highest with complete linkage; however, it is possible that an observation is closer to observations in other clusters than the cluster to which it has been assigned. The average linkage approach moderates the tendencies of the single or complete linkage approaches." (Glenn J Myatt & Wayne P Johnson, "Making sense of data I: a practical guide to exploratory data analysis and data mining" 2nd. Ed., 2014)

"Linking is a powerful dynamic interactive graphics technique that can help us better understand high-dimensional data. This technique works in the following way: When several plots are linked, selecting an observation's point in a plot will do more than highlight the observation in the plot we are interacting with - it will also highlight points in other plots with which it is linked, giving us a more complete idea of its value across all the variables. Selecting is done interactively with a pointing device. The point selected, and corresponding points in the other linked plots, are highlighted simultaneously. Thus, we can select a cluster of points in one plot and see if it corresponds to a cluster in any other plot, enabling us to investigate the high-dimensional shape and density of the cluster of points, and permitting us to investigate the structure of the disease space." (Forrest W Young et al, "Visual Statistics: Seeing data with dynamic interactive graphics", 2016)

"Agglomerative clustering produces what is known as a hierarchical clustering. The clustering proceeds iteratively, and every point makes a journey from being a single point cluster to belonging to some final cluster. Each intermediate step provides a clustering of the data (with a different number of clusters). It is sometimes helpful to look at all possible clusterings jointly. [...] While this visualization provides a very detailed view of the hierarchical clustering, it relies on the two-dimensional nature of the data and therefore cannot be used on datasets that have more than two features. There is, however, another tool to visualize hierarchical clustering, called a dendrogram, that can handle multidimensional datasets." (Andreas C Müller & Sarah Guido, "Introduction to Machine Learning with Python: A Guide for Data Scientists", 2017)

"Cluster analysis refers to the grouping of observations so that the objects within each cluster share similar properties, and properties of all clusters are independent of each other. Cluster algorithms usually optimize by maximizing the distance among clusters and minimizing the distance between objects in a cluster. Cluster analysis does not complete in a single iteration but goes through several iterations until the model converges. Model convergence means that the cluster memberships of all objects converge and don’t change with every new iteration." (Danish Haroon, "Python Machine Learning Case Studies", 2017)

"Hierarchical clustering is comprised of a general family of clustering algorithms that construct nested clusters by successive merging or splitting of data. The hierarchy of clusters is represented as a tree. The tree is usually called a dendrogram. The root of the dendrogram is the single cluster that contains all the samples; the leaves are the clusters containing only one sample each. This is a nice tool, since it can be straightforwardly interpreted: it 'explains' how clusters are formed and visualizes clusters at different scales. The tree that results from the technique shows the similarity between the samples. Partitioning is computed by selecting a cut on the tree at a certain level." (Laura Igual & Santi Seguí, "Introduction to Data Science: A Python Approach to Concepts, Techniques and Applications", 2017)

"Unsupervised learning or clustering is a way of discovering hidden structures in unlabeled data. Clustering algorithms aim to discover latent patterns in unlabeled data using features to organize instances into meaningfully dissimilar groups." (Benjamin Bengfort et al, "Applied Text Analysis with Python: Enabling Language-Aware Data Products with Machine Learning", 2018)

"Community formation is common in all types of networks, and identifying them is essential for evaluating group behavior and emergent phenomena. The general prin‐ ciple in finding communities is that its members will have more relationships within the group than with nodes outside their group. Identifying these related sets reveals clusters of nodes, isolated groups, and network structure. This information helps infer similar behavior or preferences of peer groups, estimate resiliency, find nested relationships, and prepare data for other analyses. Community detection algorithms are also commonly used to produce network visualization for general inspection." (Mark Needham & Amy E Hodler, "Graph Algorithms: Practical Examples in Apache Spark and Neo4j", 2019)

"When using community detection algorithms, be conscious of the density of the relationships. If the graph is very dense, you may end up with all nodes congregating in one or just a few clusters. You can counteract this by filtering by degree, relationship weights, or similarity metrics. On the other hand, if the graph is too sparse with few connected nodes, you may end up with each node in its own cluster. In this case, try to incorporate additional relationship types that carry more relevant information." (Mark Needham & Amy E Hodler, "Graph Algorithms: Practical Examples in Apache Spark and Neo4j", 2019)

🕸️On Graph Theory: On Clusters (-1999)

"To the untrained eye, randomness appears as regularity or tendency to cluster." (William Feller, "An Introduction to Probability Theory and its Applications", 1950) 

"An average is sometimes called a 'measure of central tendency' because individual values of the variable usually cluster around it. Averages are useful, however, for certain  types of data in which there is little or no central tendency." (William A Spirr & Charles P Bonini, "Statistical Analysis for Business Decisions" 3rd Ed., 1967)

"In comparison with Predicate Calculus encoding s of factual knowledge, semantic nets seem more natural and understandable. This is due to the one-to-one correspondence between nodes and the concepts they denote, to the clustering about a particular node of propositions about a particular thing, and to the visual immediacy of 'interrelationships' between concepts, i.e., their connections via sequences of propositional links." (Lenhart K Schubert, "Extending the Expressive Power of Semantic Networks", Artificial Intelligence 7, 1976)

"Unlike the theory, models are restricted by technical considerations to a few components at a time, even in systems which are complex. Thus a satisfactory theory is usually a cluster of models. These models are related to each other in several ways : as coordinate alternative models for the same set of phenomena, they jointly produce robust theorems; as complementary models they can cope with different aspects of the same problem and give complementary as well as overlapping results; as hierarchically arranged 'nested' models, each provides an interpretation of the sufficient parameters of the next higher level where they are taken as given." (Richard Levins, "The Strategy of Model Building in Population Biology", American Scientist 54(4), 1966)

"Randomness is a difficult notion for people to accept. When events come in clusters and streaks, people look for explanations and patterns. They refuse to believe that such patterns - which frequently occur in random data - could equally well be derived from tossing a coin. So it is in the stock market as well." (Burton G Malkiel, "A Random Walk Down Wall Street", 1989)

"In physics, there are numerous phenomena that are said to be 'true on all scales', such as the Heisenberg uncertainty relation, to which no exception has been found over vast ranges of the variables involved (such as energy versus time, or momentum versus position). But even when the size ranges are limited, as in galaxy clusters (by the size of the universe) or the magnetic domains in a piece of iron near the transition point to ferromagnetism (by the size of the magnet), the concept true on all scales is an important postulate in analyzing otherwise often obscure observations." (Manfred Schroeder, "Fractals, Chaos, Power Laws Minutes from an Infinite Paradise", 1990)

"[…] semantic nets fail to be distinctive in the way they (1) represent propositions, (2) cluster information for access, (3) handle property inheritance, and (4) handle general inference; in other words, they lack distinctive representational properties (i.e., 1) and distinctive computational properties (i.e., 2-4). Certain propagation mechanisms, notably 'spreading activation', 'intersection search', or 'inference propagation' have sometimes been regarded as earmarks of semantic nets, but since most extant semantic nets lack such mechanisms, they cannot be considered criterial in current usage." (Lenhart K Schubert, "Semantic Nets are in the Eye of the Beholder", 1990)

"Black-noise phenomena govern natural and unnatural catastrophes like floods, droughts, bear markets, and various outrageous outages, such as those of electrical power. Because of their black spectra, such disasters often come in clusters." (Manfred R Schroeder, "Fractals, Chaos, Power Laws", 1991)

"Many of the basic functions performed by neural networks are mirrored by human abilities. These include making distinctions between items (classification), dividing similar things into groups (clustering), associating two or more things (associative memory), learning to predict outcomes based on examples (modeling), being able to predict into the future (time-series forecasting), and finally juggling multiple goals and coming up with a good- enough solution (constraint satisfaction)." (Joseph P Bigus,"Data Mining with Neural Networks: Solving business problems from application development to decision support", 1996)

"Regression to the mean occurs when the process produces results that are statistically independent or negatively correlated. With strong negative serial correlation, extremes are likely to be reversed each time (which would reinforce the instructors' error). In contrast, with strong positive dependence, extreme results are quite likely to be clustered together." (Dan Trietsch, "Statistical Quality Control : A loss minimization approach", 1998)