🕸️On Graph Theory: On Edges (2000-2009)

"When the graph is embedded within a surface it is called a map. The nature of the surface needed to embed the graph without crossing edges is a topological feature, and the topological structure of this payoff space is not only useful, but also beautiful." (David Robinson & David Goforth, "The Topology of the 2×2 Games: A New Periodic Table", 2005)

"The Simplicial Approximation Theorem is a concise statement of the general result for functions between any two triangulated spaces. It says that on a suitable subdivision of the domain, any continuous function can be homotopically deformed by an arbitrarily small amount so that the modified function sends vertices to vertices and is linear on each edge, face, tetrahedron, and higher-dimensional cell of the triangulation." (Robert Messer & Philip Straffin, "Topology Now!", 2006)

"In modern terminology, a collection of points in space (called vertices) and lines (called edges) joining selected pairs of those points is called a graph, and the study of graphs is called graph theory. A graph that can be drawn on the plane so that the joining lines intersect only at vertices is called a planar graph." (David Gay, "Explorations in Topology", 2007) 

"The ingredients of a map - on sphere, a rectangle, or an island - are countries, borders, and vertices. A country is the interior of a polygon or distorted polygon. The countries of a map do not overlap. A border of the map is an edge of one or more of the country polygons. Two countries may meet along one of these borders. A vertex of the map is where two or more borders meet. The map is the union of all these elements." (David Gay, "Explorations in Topology", 2007)

"A graph in mathematics is a set of nodes and a set of edges between pairs of those nodes; the edges are ordered or nonordered pairs, or a relation, that defines the pairs of nodes for which the relation being examined is valid. […] The edges can either be undirected or directed; directed edges depict a relation that requires the nodes to be ordered while an undirected edge defines a relation in which no ordering of the edges is implied." (Dennis M Buede, "The Engineering Design of Systems: Models and methods", 2009)

🕸️On Graph Theory: On Edges (-1999)

 "A fuzzy cognitive map or FCM draws a causal picture. It ties facts and things and processes to values and policies and objectives. And it lets you predict how complex events interact and play out. [...] Neural nets give a shortcut to tuning an FCM. The trick is to let the fuzzy causal edges change as if they were synapses in a neural net. They cannot change with the same math laws because FCM edges stand for causal effect not signal flow. We bombard the FCM nodes with real data. The data state which nodes are on or off and to which degree at each moment in time. Then the edges grow among the nodes."  (Bart Kosko, "Fuzzy Thinking: The new science of fuzzy logic", 1993)

"The euler characteristic is strikingly easy to compute, and it really seems odd that it should be a topological invariant. After all, it is computed from a cell decomposition, and it is very easy to come up with a lot of different complexes on a single surface, all with varying numbers of faces, edges, and vertices. But somehow by taking the alternating sum, one arrives at a quantity which depends only on the underlying shape and not on the particular complex." (L Christine Kinsey. "Topology of Surfaces", 1993)

"The four-color map theorem is an assertion about graph theory, which is the study of discrete points and the lines that connect them; each point is called a vertex and each line is called an edge." (Alexander Humez et al, "Zero to Lazy Eight: The romance of numbers", 1993)

"Depth First Search is especially appropriate for threading mazes, because it is possible to use it without having a map of the maze. It involves only local rules at nodes, plus a record of nodes and edges already used, so you can explore the graph and traverse it as you go. The name indicates the basic idea: give top priority to pushing deeper into the maze. The number of steps required is at most twice the number of passages in the maze." (Ian Stewart, "The Magical Maze: Seeing the world through mathematical eyes", 1997) 

"One very effective approach is to represent all the possible actions as a maze, and try to find a route through it. It is a logical maze rather than a real one, touched with that magic genius of mathematical transformations in which a problem that seems unassailable in one form becomes trivial in another, logically equivalent one. The idea is to represent the problem in a visual manner, using a diagram called a graph. A graph consists of a number of nodes (dots) linked by edges (lines), possibly with arrows on them. Each 'state' of the puzzle - position of the items of produce relative to the river - is represented by a node. Each 'legal' move between states is represented as an edge joining the corresponding nodes. If necessary, arrows can be added to the edges to show which is the starting state and which is the end state. The solution of the puzzle then reduces to tracing a path through its graph, starting from the initial state of the problem and finishing at the desired final state. The graph is a kind of conceptual map of the puzzle - a maze of possible states whose passages are the edges of the graph and whose junctions are its nodes." (Ian Stewart, "The Magical Maze: Seeing the World Through Mathematical Eyes", 1997) 

"We wrote down all the states and legal moves (here it turned out to be helpful to have a systematic notation, but that's not essential). Then we formed a graph whose nodes correspond to states and whose edges correspond to legal moves. The solution of the puzzle is then a path through the graph that joins the start to the finish. Such a path is usually obvious to the eye, provided the puzzle is sufficiently simple for the entire graph to be drawn. Puzzles of this type are really mazes, for a maze is just a graph drawn in a slightly different fashion. Metaphorically, they are logical mazes - you have to find the right sequence of moves to solve them. The graph turns the logical maze into a genuine maze, turning the metaphor into reality. The fact that solving the real maze also solves the logical maze is one of the magical features of the maze that is mathematics." (Ian Stewart, "The Magical Maze: Seeing the World Through Mathematical Eyes", 1997) 

"The key to generating the small-world phenomenon is the presence of a small fraction of [...] edges, which contact otherwise distant parts of the graph" (Duncan J Watts, "Small Worlds: The Dynamics of Networks Between Order and Randomness", 1999)

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

"A feedback loop is a cycle in a directed graph whose edges can represent either positive or negative inputs." (Maria C A Leite & Yunjiao Wang, "Multistability, oscillations and bifurcations in feedback loops", Mathematical Biosciences and Engineering Vol 7 (1), 2010)

"First, what are the 'graphs' studied in graph theory? They are not graphs of functions as studied in calculus and analytic geometry. They are (usually finite) structures consisting of vertices and edges. As in geometry, we can think of vertices as points (but they are denoted by thick dots in diagrams) and of edges as arcs connecting pairs of distinct vertices. The positions of the vertices and the shapes of the edges are irrelevant: the graph is completely specified by saying which vertices are connected by edges. A common convention is that at most one edge connects a given pair of vertices, so a graph is essentially just a pair of sets: a set of objects." (John Stillwell, "Mathematics and Its History", 2010)

"[...] the contexts in which a social network is embedded will generally have significant effects on its structure. Each individual in a social network has a distinctive set of personal characteristics, and similarities and compatibilities between two people’s characteristics can strongly influence whether a link forms between them. Each individual also engages in a set of behaviors and activities that can shape the formation of links within the network. These considerations suggest what we mean by a network’s surrounding contexts: factors that exist outside the nodes and edges of a networks, but which nonetheless affect how the network’s structure evolves." (David Easley & Jon Kleinberg, "Networks, Crowds, and Markets: Reasoning about a Highly Connected World", 2010) 

"A network (or graph) consists of a set of nodes (or vertices, actors) and a set of edges (or links, ties) that connect those nodes. [...] The size of a network is characterized by the numbers of nodes and edges in it." (Hiroki Sayama, "Introduction to the Modeling and Analysis of Complex Systems", 2015)

"Discrete Mathematics is a branch of mathematics dealing with finite or countable processes and elements. Graph Theory is an area in Discrete Mathematics which studies configurations involving a set of vertices interconnected by edges (called graphs). From humble beginnings and almost recreational type problems, Graph Theory has found its calling in the modern world of complex systems and especially of the computer. Graph Theory and its applications can be found not only in other branches of mathematics, but also in scientific disciplines such as engineering, computer science, operational research, management sciences and the life sciences." (Khee Meng Koh et al, " Graph theory: Undergraduate mathematics", 2015)

"A network (or graph) consists of a set of nodes (or vertices, actors) and a set of edges (or links, ties) that connect those nodes. [...] The size of a network is characterized by the numbers of nodes and edges in it." (Hiroki Sayama, "Introduction to the Modeling and Analysis of Complex Systems", 2015)

"Our assumptions are unquestionably interconnected. They are nodes with connections (edges) to other nodes. The more foundational the assumption, the more strongly connected it is. What I’m suggesting is that our assumptions and the highly sensitive network of responses, perceptions, behaviors, thoughts, and ideas they create and interact with are a complex system. One of the most basic features of such a network is that when you move or disrupt one thing that is strongly connected, you don’t just affect that one thing, you affect all the other things that are connected to it. Hence small causes can have massive effects (but they don’t have to, and usually don’t actually). In a system of high tension, simple questions targeting basic assumptions have the potential to transform perception in radical  and unpredictable ways." (Beau Lotto, "Deviate: The Science of Seeing Differently", 2017) 

"Network theory confirms the view that information can take on 'a life of its own'. In the yeast network my colleagues found that 40 per cent of node pairs that are correlated via information transfer are not in fact physically connected; there is no direct chemical interaction. Conversely, about 35 per cent of node pairs transfer no information between them even though they are causally connected via a 'chemical wire' (edge). Patterns of information traversing the system may appear to be flowing down the 'wires' (along the edges of the graph) even when they are not. For some reason, 'correlation without causation' seems to be amplified in the biological case relative to random networks." (Paul Davies, "The Demon in the Machine: How Hidden Webs of Information Are Solving the Mystery of Life", 2019) 

"Node-link, matrix, and implicit representations are suited for different graph data. Node-link diagrams are good for sparse networks, which have a moderate number of edges. Dense networks with many edges are best visualized using a matrix. Trees, as we just said, are nicely represented by implicit approaches." (Christian Tominski & Heidrun Schumann, "Interactive Visual Data Analysis", 2019) 

"[...] the Game of Life, in which a few simple rules executed repeatedly can generate a surprising degree of complexity. Recall that the game treats squares, or pixels, as simply on or off (filled or blank) and the update rules are given in terms of the state of the nearest neighbours. The theory of networks is closely analogous. An electrical network, for example, consists of a collection of switches with wires connecting them. Switches can be on or off, and simple rules determine whether a given switch is flipped, according to the signals coming down the wires from the neighbouring switches. The whole network, which is easy to model on a computer, can be put in a specific starting state and then updated step by step, just like a cellular automaton. The ensuing patterns of activity depend both on the wiring diagram (the topology of the network) and the starting state. The theory of networks can be developed quite generally as a mathematical exercise: the switches are called ‘nodes’ and the wires are called ‘edges’. From very simple network rules, rich and complex activity can follow." (Paul Davies, "The Demon in the Machine: How Hidden Webs of Information Are Solving the Mystery of Life", 2019) 

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

"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 useful for representing how things are either physically or logically linked in simple or complex structures. A graph in which we assign names and meanings to the edges and vertices becomes what is known as a network. In these cases, a graph is the mathematical model for describing a network, whereas a network is a set of relations between objects, which could include people, organizations, nations, items found in a Google search, brain cells, or electrical transformers." (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) 

"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) 

"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)

🅷James S Hodges - Collected Quotes

"A bad model is a combination of assertions, some factual, others conjectural, and others plainly false but convenient. [..]  By definition, a bad model does not give power to see-accurately, deeply, or at all-into the actual situa- tion, but only into the assertions embodied in the model. Thus, if the use of a bad model provides insight, it does so not by revealing truth about the world but by revealing its own assumptions and thereby causing its user to go learn something about the world." (James S Hodges, "Six (or So) Things You Can Do with a Bad Model", 1991)

"A management system driven by a bad model must not be tested by using the model as if it were true. By presumption, the model is a deficient picture of reality, and it presents the management system with the easiest possible test because it, unlike the cruel world, satisfies the system's assumptions. But a bad model can be used as a vehicle for a fortiori arguments in an evaluation of a system of which it is a part [...]" (James S Hodges, "Six (or So) Things You Can Do with a Bad Model", 1991)

"A scale model is a bad model in the sense used here: It is grossly discrepant with reality, if only because it is far too small for anyone to live in. Nonetheless, it can do a good job of selling the idea-the project-of which it is but an illustration, by conveying aspects of the idea concretely." (James S Hodges, "Six (or So) Things You Can Do with a Bad Model", 1991)

"Just because a model is bad, however, does not mean it is useless. [...] A bad model can be used to construct correct paths from premises to conclusions, but because its relations to reality are questionable, it can only do so in a few ways-at least, ways that permit useful conclusions with respect to reality." (James S Hodges, "Six (or So) Things You Can Do with a Bad Model", 1991)

"Often, though, a policy or systems analyst is stuck with a bad model, that is, one that appeals to the analyst as adequately realistic but which is either: 1) contradicted by some data or is grossly implausible in some aspect it purports to represent, or 2) conjectural, that is, neither supported nor contradicted by data, either because data do not exist or because they are equivocal. [...] A model may have component parts that are not bad, but if, taken as a whole, it meets one of these criteria, it is a bad model." (James S Hodges, "Six (or So) Things You Can Do with a Bad Model", 1991)

"One might hope for a standard of consistency instead of a lack of inconsistency, but, as a practical matter, no one can make every consistency check, so a stingent lack of inconsistency is the most one can ask for. Even this need not be simple." (James S Hodges, "Six (or So) Things You Can Do with a Bad Model", 1991)

"Some readers have argued that the criticism implied by the term "bad models" is undeserved because they can be used appropriately in some cases. [...] If the logic works, the use is appropriate; if it fails, the use in inappropriate: (Cost effectiveness is a separate issue.) As for the pejorative connotation of the term bad model, perhaps we should admit that many useful models would be embarrassments to scientists, from whom we got the idea of a model, but whose job is to improve the match between models and reality." (James S Hodges, "Six (or So) Things You Can Do with a Bad Model", 1991)

"Sometimes the proprietors of a bad model claim that parts of it are facts, not just beliefs. Evaluation then amounts to determining if facts support the claims, and disciplines like statistics have tools for this task. The difficulty of using statistical tools will vary depending on the problem." (James S Hodges, "Six (or So) Things You Can Do with a Bad Model", 1991)

"This stricture - that a bad model can only suggest-is stronger than it may appear. Bad models produce numbers, and thus present an unbearable temptation to use those numbers as if they do more than suggest. They cannot. If a model is bad as defined here, and the specific numbers it produces cannot be buttressed by some other arguments, then the numbers have no meaning except as illustration of the consequences that flow from the model's assumptions." (James S Hodges, "Six (or So) Things You Can Do with a Bad Model", 1991)

🪷On Mind: On Viruses

"Like  computer viruses, successful mind viruses will tend to be hard for their victims to detect. If you are the victim of one, the chances are that you won't know it, and may even vigorously deny it. Accepting that a virus might be difficult to detect in your own mind, what tell-tale signs might you look out for? I shall answer by imaging how a medical textbook might describe the typical symptoms of a sufferer (arbitrarily assumed to be male)." (Richard Dawkins,"Viruses of the Mind", 1993) 

"A virus is anything that takes external copying equipment and puts it to work making copies of itself." (Richard Brodie, "Virus of the Mind", 2009)

"A virus of the mind is something out in the world that infects people with memes. Those memes, in turn, influence the infected people’s behavior so that they help perpetuate and spread the virus." (Richard Brodie, "Virus of the Mind", 2009)

"If we want to combat the mind viruses responsible for the decline of culture, we need to be conscious of our own programming, consciously adopting memes that take us in the direction we want to go." (Richard Brodie, "Virus of the Mind", 2009)

"If you currently believe in any concepts or subcultures or dogmas that meet these requirements, and you didn’t consciously choose to program yourself with these memes, you are infected with a mind virus." (Richard Brodie, "Virus of the Mind", 2009)

"The universe contains many mechanisms for copying and dispersing information, and viruses are some of the things that are often copied and dispersed." (Richard Brodie, "Virus of the Mind", 2009)

"The world is full of memes spread by mind viruses, all competing for a share of your mind, your perception, your attention. They care nothing for your well-being, but instead add to your confusion and subtract from your fulfillment." (Richard Brodie, "Virus of the Mind", 2009)

"When you plant a fertile meme in my mind you literally parasitize my brain, turning it into a vehicle for the meme's propagation in just the way that a virus may parasitize the genetic mechanism of a host cell." (Richard Dawkins, "The Selfish Gene: 40th Anniversary", 2016)

🪸On Systems Thinking: On Viruses

"General systems theory is a series of related definitions, assumptions, and postulates about all levels of systems from atomic particles through atoms, molecules, crystals, viruses, cells, organs, individuals, small groups, societies, planets, solar systems, and galaxies. General behavior systems theory is a subcategory of such theory, dealing with living systems, extending roughly from viruses through societies. A significant fact about living things is that they are open systems, with important inputs and outputs. Laws which apply to them differ from those applying to relatively closed systems." (James G Miller, "General behavior systems theory and summary", Journal of Counseling Psychology 3 (2), 1956)

"It is thought that the virus is a degeneration from a more complex life form. It may at one time have been capable of independent life. Now has fallen to the borderline between living and dead matter. It can exhibit living qualities only in a host, by using the life of another-the renunciation of life itself, a falling towards inorganic, inflexible machine, towards dead matter." (William S Burroughs, "Letters to Allen Ginsberg, 1953-1957", 1959) 

"Epidemics are another example of geometric progression: when a virus spreads through a population, it doubles and doubles again, until it has (figuratively) grown from a single sheet of paper all the way to the sun in fifty steps. As human beings we have a hard time with this kind of progression, because the end result - the effect - seems far out of proportion to the cause. To appreciate the power of epidemics, we have to abandon this expectation about proportionality. We need to prepare ourselves for the possibility that sometimes big changes follow from small events, and that sometimes these changes can happen very quickly." (Malcolm T Gladwell, "The Tipping Point: How Little Things Can Make a Big Difference", 2000)

"It is not only a metaphor to transform the Internet to a superbrain with self-organizing features of learning and adapting. Information retrieval is already realized by neural networks adapting to the information preferences of a human user with synaptic plasticity. In sociobiology, we can 1 earn from populations of ants and termites how to organize traffic and information processing by swarm intelligence. From a technical point of view, we need intelligent programs distributed in the nets. There are already more or less intelligent virtual organisms {'agents'), learning, self-organizing and adapting to our individual preferences of information, to select our e-mails, to prepare economic transactions or to defend the attacks of hostile computer viruses, like the immune system of our body." (Klaus Mainzer, "Complexity Management in the Age of Globalization", 2006)

"A side effect of computing in a small world web is vulnerability to viruses or other malicious software. A computer virus may be broadly defined as a program that can replicate itself while executing instructions on infected computers, without the user's knowledge. Modelling computer viruses may lead to new methods to combat them. How can we model computer viruses loose on scale-free networks?" (Anthony Bonato, "A Course on the Web Graph", 2008)

🧊On Geometry: On Vertices

 "[…] the way in which I have proceeded does not lead to the desired goal, the goal that you declare you have reached, but instead to a doubt of the validity of [Euclidean] geometry. I have certainly achieved results which most people would look upon as proof, but which in my eyes prove almost nothing; if, for example, one can prove that there exists a right triangle whose area is greater than any given number, then I am able to establish the entire system of [Euclidean] geometry with complete rigor. Most people would certainly set forth this theorem as an axiom; I do not do so, though certainly it may be possible that, no matter how far apart one chooses the vertices of a triangle, the triangle's area still stays within a finite bound. I am in possession of several theorems of this sort, but none of them satisfy me." (Carl F Gauss, 1799) [answer to a letter from Farkas Bolyai in which Bolyai claimed to have proved Euclid's fifth postulate]

"If a continuum, say, a two-dimensional closed manifold, a surface, is to be the subject of mathematícal investigation, then we must think of it as being subdivided into finitely many 'elementary pieces' whose topological nature is that of a circular disk. These pieces are further fragmented by repeated subdivision in accordance with a fixed scheme, and thus a particular spot in the continuum is ever more precisely intercepted by an infinite sequence 

of nested fragments that arise in the course of successive subdivisions. In the one-dimensional case, the repeated 'normal subdivision' of an elementary segment is its bipartition. In the two-dirnensional case, each edge is first bipartitioned, then each piece of surface is divided into triangles by means of Unes in the surface that lead from an arbitrary center to the (old and new) vertices." (Hermann Weyl, "Levels of Infinity", 1930)

"It was found [in the 1970s], unexpectedly and without anyone really having a concept for it, that the rules of perturbation theory can be changed in a way that makes relativistic quantum gravity inevitable rather than impossible. The change is made by replacing point particles by strings. Then Feynman graphs are replaced by Riemann surfaces, which are smooth - unlike the graphs, which have singularities at interaction vertices. The Riemann surfaces can degenerate to graphs in many different ways. In field theory, the interactions occur at the vertices of a Feynman graph. By contrast, in string theory, the interaction is encoded globally, in the topology of a Riemann surface, any small piece of which is like any other. This is reminiscent of how non-linearities are encoded globally in twistor theory." (Edward Witten,"The Past and Future of String Theory", [in  W Gibbons et al, "The Future of Theoretical Physics and Cosmology: Celebrating Stephen Hawking's Contributions to Physics", 2003)

"The basic geometric construction of the Sierpinski gasket goes as follows. We begin with a triangle in the plane and then apply a repetitive scheme of operations to it (when we say triangle here, we mean a blackened, ‘filled-in’ triangle). Pick the midpoints of its three sides. Together with the old vertices of the original triangle, these midpoints define four congruent triangles of which we drop the center one. This completes the basic construction step. In other words, after the first step we have three congruent triangles whose sides have exactly half the size of the original triangle and which touch at three points which are common vertices of two contiguous triangles. Now we follow the same procedure with the three remaining triangles and repeat." (Heinz-Otto Peitgen et al, "Chaos and Fractals: New Frontiers of Science" 2nd Ed., 2004)

"The Simplicial Approximation Theorem is a concise statement of the general result for functions between any two triangulated spaces. It says that on a suitable subdivision of the domain, any continuous function can be homotopically deformed by an arbitrarily small amount so that the modified function sends vertices to vertices and is linear on each edge, face, tetrahedron, and higher-dimensional cell of the triangulation." (Robert Messer & Philip Straffin, "Topology Now!", 2006)

🕸️On Graph Theory: On Knowledge Graphs

"Knowledge graphs are a specific type of graph with an emphasis on contextual understanding. Knowledge graphs are interlinked sets of facts that describe real-world entities, events, or things and their interrelations in a human- and machine-understandable format." (Jesús Barrasa et al, "Knowledge Graphs: Data in Context for Responsive Businesses", 2021)

"[…] knowledge graphs are useful because they provide contextualized understanding of data. They achieve this by adding a layer of metadata that imposes rules for structure and interpretation." (Jesús Barrasa et al, "Knowledge Graphs: Data in Context for Responsive Businesses", 2021)

"Knowledge graphs use an organizing principle so that a user" (or a computer system) can reason about the underlying data. The organizing principle gives us an additional layer of organizing data (metadata) that adds connected context to support reasoning and knowledge discovery. […] Importantly, some processing can be done without knowledge of the domain, just by leveraging the features of the property graph model" (the organizing principle)." (Jesús Barrasa et al,Knowledge Graphs: Data in Context for Responsive Businesses", 2021)

"Data Fabric architecture utilizes active metadata, knowledge graphs, and semantic enrichment, combining intelligent information integration and transformation technologies to intelligently support data consumers, for example, business users."  (Eberhard Hechler et al, "Data Fabric and Data Mesh Approaches with AI", 2023)

"In Exploiting semantic knowledge graphs can support interpretability and explainability of nearly all AI model types (including DL models) by discovering and depicting semantic and non-obvious relationships or depicting an ML model in a simplified and more readable, explainable way., a Data Mesh solution organizes data around business domain owners and transforms relevant data assets (data sources) to data products that can be consumed by distributed business users from various business domains or functions. These data products are created, governed, and used in an autonomous, decentralized, and self-service manner. Self-service capabilities, which we have already referenced as a Data Fabric capability, enable business organizations to entertain a data marketplace with shopping-for-data characteristics." (Eberhard Hechler et al, "Data Fabric and Data Mesh Approaches with AI", 2023)

"It is essential to realize that the Data Fabric architecture enables the Data Mesh solution via its rich knowledge catalog, semantic search and discovery, smart integration capabilities, and semantic knowledge graphs. Trustworthy AI, for instance, is enabled via the Data Fabric as well." (Eberhard Hechler et al, "Data Fabric and Data Mesh Approaches with AI", 2023)

"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)

"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)

"Despite their effectiveness and advantages in supporting the development of intelligent systems, KGs haven’t been widely adopted for several reasons, including the following: (•) They are expensive to build and maintain in terms of time, effort, and money. (•) Intricate access patterns are required to navigate multiple hops. (•) Their results scatter information across multiple nodes and relationships." (Alessandro Negro, "Graph-Powered Machine Learning", 2021)

"Generative artificial intelligence (GenAI), powered by large language models (LLMs) like Google’s Gemini and OpenAI’s GPT, has transformed how we work and live, revolutionizing business after business. Despite this success, generative AI falls short in domains where specific domain knowledge, high accuracy, and explainability are essential. And it has other significant limitations, including hallucinations and a lack of context and relations. This is where knowledge graphs (KGs) come in, provid-ing contextual information - such as experiences, environmental characteristics, cultural aspects, and social normsneeded to build the 'third wave of AI' for mission-critical applications." (Alessandro Negro et al, "Knowledge Graphs and LLMs in Action", 2026)

"KGs are sophisticated graph structures that represent real-world entities (people, places, diseases, proteins), define meaningful connections between them, and provide context. KGs provide structured, explainable knowledge representation but are challenging to build and query; LLMs offer natural language processing capabilities but suffer from hallucinations, stale information, and a lack of domain-specific grounding. Together, they are a 'killer combination': LLMs can extract entities and relationships from unstructured text to build KGs more efficiently, providing more autonomous and powerful graph querying and analysis. Meanwhile, KGs provide reliable, up-to-date domain knowledge to ground LLM responses and prevent hallucinations." (Alessandro Negro et al, "Knowledge Graphs and LLMs in Action", 2026)

"The KG serves as the central reference for all structured and unstructured data related to a domain. Because a KG represents information by focusing on the meaning of data, users can overcome challenges related to data types, formats, and provenance, connecting information from multiple data sources. [...] A KG represents the core information and big picture of a domain. Humans can analyze, visualize, and query graph data to extract insights. Inference rules and machine learning algorithms are performed on top of the KG to infer new information not explicitly encoded within the KG. Analysts can use methods such as centrality and connectivity analysis to identify influential nodes, network analysis to detect the shortest path between nodes, and community analysis to recognize groups of similar nodes." (Alessandro Negro et al, "Knowledge Graphs and LLMs in Action", 2026) 

"Traditional paradigms build systems for specific purposes with structured, homogeneous databases. This approach works for tailored needs but is impractical for complex domains that need to adapt to user characteristics and integrate heterogeneous data. KGs capture connections, enabling relationship discovery through graph pattern matching and traversal. Both the Resource Description Framework (RDF) and Labeled Property Graphs (LPGs) provide machine-readable formats that humans can interpret. KGs emphasize rich, meaningful data representations usable by both humans and machines, enabling a paradigm shift where intelligent behavior is encoded in a unique source of truth." (Alessandro Negro et al, "Knowledge Graphs and LLMs in Action", 2026)

🕸️On Graph Theory: On Vertices

"The euler characteristic is strikingly easy to compute, and it really seems odd that it should be a topological invariant. After all, it is computed from a cell decomposition, and it is very easy to come up with a lot of different complexes on a single surface, all with varying numbers of faces, edges, and vertices. But somehow by taking the alternating sum, one arrives at a quantity which depends only on the underlying shape and not on the particular complex." (L Christine Kinsey. "Topology of Surfaces", 1993)

"The four-color map theorem is an assertion about graph theory, which is the study of discrete points and the lines that connect them; each point is called a vertex and each line is called an edge." (Alexander Humez et al, "Zero to Lazy Eight: The romance of numbers", 1993)

"Moonshine forms a way of explaining the mysterious connection between the monster finite group and modular functions from classical number theory. The theory has evolved to describe the relationship between finite groups, modular forms and vertex operator algebras." (Terry Gannon, "Moonshine Beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics", 2006)

"We investigate structure in the network by characterizing motifs that represent order. A simple motif is the existence of a triangle, three vertices connected one to the next. The ratio of the number of observed to expected triangles is synonymous with the standard definition of the clustering coefficient for a small world network. This statistic is sensitive to organization over short length scales. To investigate organization over longer length scales, we investigate the distribution of longer cycles. This distribution may be measured for an empirical network We introduce a simple mathematical model for a network organized to have one level of clustering and show that this model is sufficient to explain the observed cycle distribution. Thus, there is no need to invoke a continuous distribution of length scales. Moreover, the one-level model immediately yields a characteristic, testable scaling length for the network, which again stands in contrast to scale-free behavior." (J S Bader, "The Drosophila Protein Interaction Network May Be neither Power-Law nor Scale-Free", 2006)

"A major idea [...] is the equivalency of two maps, one a stretchedshrunk version of the other. This is an instance of 'rubber sheet' geometry. Print your map on a thin sheet of rubber, and then distort the rubber sheet to get other equivalent maps. The distortion preserves the countries, borders, vertices, and their relationships to each other. However, the distortion does not preserve exact distances or angles. The investigation of geometric properties preserved under this notion of geometric equivalence was first proposed by Leibnitz [...]. He called the study of such properties, geometria situs. The modern word is topology." (David Gay, "Explorations in Topology", 2007)

"In modern terminology, a collection of points in space (called vertices) and lines (called edges) joining selected pairs of those points is called a graph, and the study of graphs is called graph theory. A graph that can be drawn on the plane so that the joining lines intersect only at vertices is called a planar graph." (David Gay, "Explorations in Topology", 2007)

"The ingredients of a map - on sphere, a rectangle, or an island - are countries, borders, and vertices. A country is the interior of a polygon or distorted polygon. The countries of a map do not overlap. A border of the map is an edge of one or more of the country polygons. Two countries may meet along one of these borders. A vertex of the map is where two or more borders meet. The map is the union of all these elements." (David Gay, "Explorations in Topology", 2007)

"In the web graph, each vertex acts as an independent agent, which will base its decision on how to link to the existing network on local knowledge. As a result, the neighbourhood of a new vertex will often be an imperfect copy of the neighbourhood of an existing vertex. This aspect of web page generation indicates a weakness of preferential attachment models: we assume global knowledge of all vertex degrees, a clearly unrealistic hypothesis when faced with a massive set of vertices. Both the copying models of the web graph, and the duplication model for biological networks incorporate this notion of copying in their definitions." (Anthony Bonato, "A Course on the Web Graph", 2008)

"These networks arise naturally in many diverse disciplines (such as biology, computer science, and social science), but all share the common characteristics of being massive, sparse graphs with a power law degree distribution and small world structure. Such networks have now been coined scale-free, complex, self-organizing, or heterogeneous. Our own preferred term is self-organizing, since this is suggestive of the generative mechanisms underlying the network: over time, vertices choose their neighbour according to their own predisposition, rather than necessarily following global rules." (Anthony Bonato, "A Course on the Web Graph", 2008)

"First, what are the 'graphs' studied in graph theory? They are not graphs of functions as studied in calculus and analytic geometry. They are (usually finite) structures consisting of vertices and edges. As in geometry, we can think of vertices as points (but they are denoted by thick dots in diagrams) and of edges as arcs connecting pairs of distinct vertices. The positions of the vertices and the shapes of the edges are irrelevant: the graph is completely specified by saying which vertices are connected by edges. A common convention is that at most one edge connects a given pair of vertices, so a graph is essentially just a pair of sets: a set of objects." (John Stillwell, "Mathematics and Its History", 2010)

"A network (or graph) consists of a set of nodes (or vertices, actors) and a set of edges (or links, ties) that connect those nodes. [...] The size of a network is characterized by the numbers of nodes and edges in it." (Hiroki Sayama, "Introduction to the Modeling and Analysis of Complex Systems", 2015)

"Discrete Mathematics is a branch of mathematics dealing with finite or countable processes and elements. Graph Theory is an area in Discrete Mathematics which studies configurations involving a set of vertices interconnected by edges (called graphs). From humble beginnings and almost recreational type problems, Graph Theory has found its calling in the modern world of complex systems and especially of the computer. Graph Theory and its applications can be found not only in other branches of mathematics, but also in scientific disciplines such as engineering, computer science, operational research, management sciences and the life sciences." (Khee Meng Koh et al, " Graph theory: Undergraduate mathematics", 2015

"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 useful for representing how things are either physically or logically linked in simple or complex structures. A graph in which we assign names and meanings to the edges and vertices becomes what is known as a network. In these cases, a graph is the mathematical model for describing a network, whereas a network is a set of relations between objects, which could include people, organizations, nations, items found in a Google search, brain cells, or electrical transformers." (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)