🪸On System 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)

📓On Literature: On Viruses (From Fiction to Science-Fiction)

"I have remarked, in the course of such air travel as I have done, that the airmen of all nations have a common resemblance to each other and that the patriotic virus in their blood is largely corrected by a wider professionalism." (Herbert G Wells, "The Outlook for Homo Sapiens", 1942)

"A man writes to throw off the poison which he has accumulated because of his false way of life. He is trying to recapture his innocence, yet all he succeeds in doing is to inoculate the world with a virus of his disillusionment. No man would set a word down on paper if he had the courage to live out what he believed in [...]" (Henry Miller, "The Rosy Crucifixion I : Sexus", 1949)

"Democracy is cancerous, and bureaus are its cancer. A bureau takes root anywhere in the state, turns malignant like the Narcotic Bureau, and grows and grows, always reproducing more of its own kind, until it chokes the host if not controlled or excised. [...] Bureaucracy is wrong as a cancer, a turning away from the human evolutionary direction of infinite potentials and differentiation and independent spontaneous action, to the complete parasitism of a virus." (William S Burroughs, "Naked Lunch", 1959)

"The broken image of Man moves in minute by minute and cell by cell  [...] Poverty, hatred, war, police-criminals, bureaucracy, insanity, all symptoms of The Human Virus. The Human Virus can now be isolated and treated." (William S Burroughs, "Naked Lunch", 1959)

"Surrealism is merely the reflection of the death process. It is one of the manifestations of a life becoming extinct, a virus which quickens the inevitable end." (Henry Miller, "The Cosmological Eye (ed. New Directions Publishing, 1961

"When the virus of restlessness begins to take possession of a wayward man, and the road away from Here seems broad and straight and sweet, the victim must first find himself a good and sufficient reason for going." (John Steinbeck, "Travels With Charley: In Search of America", 1962)

"Others have developed cries, songs, words as weapons. Words that cut like buzz saws. Words that vibrate the entrails to jelly. Cold strange words that fall like icy nets on the mind. Virus words that eat the brain to muttering shreds." (William S Burroughs, "The Wild Boys: A Book of the Dead", 1971)

"'Ideology', growled one of his new friends. 'It’s a virus. The world is dying of it.'" (Brian W Aldiss, "Three Ways", 1978)

"To fight the Empire is to be infected by its derangement […] Whoever defeats the Empire becomes the Empire; it proliferates like a virus […] thereby it becomes its enemies." (Philip K Dick, "VALIS", 1981)

"Any information system of sufficient complexity will inevitably become infected with viruses - viruses generated from within itself." (Neal Stephenson, "Snow Crash", 1992)

"Your brain has an immune system, just like your body. The more you use it - the more viruses you get exposed to - the better your immune system becomes." (Neal Stephenson, "Snow Crash", 1992)

"Fiction is the great virus waiting to do away with fact - that is one of the most ominous meanings of the film." (David Thomson, "Rosebud: the Story of Orson Welles", [on "Of Citizen Kane", 1941 film] 1996)

"You’re not actually mammals. Every mammal on this planet instinctively develops a natural equilibrium with the surrounding environment, but you humans do not. You move to an area, and you multiply, and multiply, until every natural resource is consumed. The only way you can survive is to spread to another area. There is another organism on this planet that follows the same pattern. Do you know what it is? A virus. Human beings are a disease, a cancer of this planet, you are a plague, and we are the cure." (Andy Wachowski & Larry Wachowski, "The Matrix", [film] 1999)

"Only one form of contagion travels faster than a virus. And that's fear." (Dan Brown, "Inferno: A Novel", 2013)

Anthony Bonato - Collected Quotes

"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)

"An open problem is to design a rigorous graph model which simulates properties of physical networks such as the various levels of the internet graph. For example, any model of the internet should take into account the technological and economic constraints at work (which are more important in this context than in W)." (Anthony Bonato, "A Course on the Web Graph", 2008)

"As any mathematician can tell you, there is more to probability than gam- bling. Indeed, applications of probability theory are now common in both graph theory and theoretical computer science." (Anthony Bonato, "A Course on the Web Graph", 2008)

"Graph theory contains a myriad of elegant proofs using probabilistic methods, especially when other techniques are not applicable. An advantage of randomized methods is their ability to prove the existence of some object without explicitly constructing it." (Anthony Bonato, "A Course on the Web Graph", 2008)

"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)

"Infinite graphs possess their own rich theory which is quite different in many aspects from the finite case. [...] we will discuss some of these aspects, then attempt to apply infinite graphs to web graph theory. The approach is new and different, but we think it has and will lead to interesting mathematics and to better insights into Wand its models. As is often the case in internet mathematics, results are Inore suggestive than definitive." (Anthony Bonato, "A Course on the Web Graph", 2008)

"Preferential attachment is one of the main paradigms used in the design of web graph models. Hence, it is natural to consider limits of on-line preferential attachment models." (Anthony Bonato, "A Course on the Web Graph", 2008)

"Regardless of the finiteness of the universe, the view of W as an infinite graph presents an interesting perspective for the mathematician. Infinite graphs are fascinating creatures whose properties are often bewildering and quite unlike finite ones." (Anthony Bonato, "A Course on the Web Graph", 2008)

"Speaking of the web graph is somewhat mis- leading, as the web is an evolving structure with pages and links appearing and disappearing continuously over time. We will overlook such concerns." (Anthony Bonato, "A Course on the Web Graph", 2008)

"The presence of power law degree distributions reflects a certain undemocratic aspect of W: while most pages have a small number of links, a few pages have a large number. This is in hindsight not surprising, since the choice of links from new pages to existing ones is determined by the users' own interests. For example, it seems plausible that popular pages attract more new links than unpopular ones." (Anthony Bonato, "A Course on the Web Graph", 2008)

"The web contains a seemingly infinite ocean of information. Traversing this ocean unaided would be difficult if not impossible. As most of us experience on a daily basis, search engines are an invaluable tool to surf the vast storehouse of information available online." (Anthony Bonato, "A Course on the Web Graph", 2008)

"The web contains many communities: sets of pages sharing a common interest or topic. However, there is no consensus for a precise definition of a community in the web graph."  (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)

John Scott - Collected Quotes

"A component is a subgraph in which all points are directly or indirectly connected to each other and there are no connec-tions to points outside the subgraph. Information or resources can, therefore, flow along a path through all the members of the component but cannot reach any other points in the graph. A graph may comprise one or more components of varying size, and the number and size distribution of components is a fundamental measure of network differentiation and of the existence of boundaries to the flow of information and resources." (John Scott," What is Social Network Analysis?", 2012)

"A final measure of centrality is one that measures the extent to which a point is able to act as an intermediary in a large number of network flows. This measure has been called ‘betweenness’ and refers to the extent to which a particular point is able to serve as an intermediate point of contact between any two other points." (John Scott," What is Social Network Analysis?", 2012)

"A further refinement of the simple component idea is that of the cyclic component built from intersecting cycles of connection. A cycle is a directed path that returns to its starting point. The overlapping of such cycles produces a cyclic compo-nent in which all points are connected by one or more cycles and no points have cyclic connections outside the component. A cyclic component is a structural element within a strong component and may be connected to other members of the strong component through ‘bridges’ that do not lie on the cycle itself." (John Scott," What is Social Network Analysis?", 2012)

"A further measure of global cohesion is the centralisation of a network. Where centrality relates to the position of particular points, centralisation relates to the overall structure of a network. Centralisation measures the extent to which the cohesion of a network is organised around a specific point or set of connected points. The spokes on a bicycle wheel, for example, form a highly centralised network around its hub. Measures of centralisation can be based on the degree, distance, or betweenness of points, and extensions of these concepts have involved the idea that it is possible to identify the sets of points that comprise the centre, margin, and periphery of the network as a whole." (John Scott," What is Social Network Analysis?", 2012)

"A whole collection of global measures cluster around the idea of the centrality of points within their graphs. The degree of a point - its total of incoming and outgoing lines - is the most basic measure and has been termed local centrality. Calculating the degrees of all points in a network and ranking them from highest to lowest gives a rank order of local centrality. This centrality is ‘local’ because it highlights points that are well-connected in their immediate neighbourhoods. Such points may not, however, be central in the more global sense that a circle or sphere has a unique centre that can be understood in quasi-spatial terms. Locally central points are well-connected within particular parts of the network but may not be at all well-connected in a global sense." (John Scott," What is Social Network Analysis?", 2012)

"Inclusiveness simply measures the number or proportion of the whole set of points that are actually connected into one or more parts of the graph. Some points may be isolates, having no ties to other points, while others will be connected, to a greater or lesser extent, into larger structures. The inclusiveness of a graph is simply the total number of non-isolated points, gener-ally expressed as a percentage of the total number of points. Inclusiveness is a rough and ready approximation to cohesion, but it is usually more informative to measure the actual density of the graph." (John Scott," What is Social Network Analysis?", 2012)

"Social network analysis conceptualises individuals or groups as ‘points’ and their relations to each other as ‘lines’. It is concerned with the patterns formed by the points and lines and involves exploring these patterns, mathematically or visually, in order to assess their effects on the individuals and organisations that are the members of the ‘networks’ formed by the intersecting lines that connect them. It therefore takes the metaphorical idea of interaction as forming a network of connections and gives this idea a more formal representation in order to model structures of social relations. Treating a social structure as a network is the cornerstone of social network analysis." (John Scott," What is Social Network Analysis?", 2012)

"The converse of centrality is peripherality and it can be useful to know those points that are least close to other members of their networks. Such points are not isolated but are poorly integrated into their network. They are likely to have little influ-ence and to be uninvolved in significant communication flows through the network." (John Scott," What is Social Network Analysis?", 2012)

"The density of a graph is a very useful and direct measure of its cohesion, but it has one major limitation as a comparative measure of social structure. In real situations, density varies with the size of a network and this limits the possibilities of using the measure to compare different types of network. It is highly unlikely that agents are able to sustain more than a certain number of relationships: our ability to be ‘friends’ with people, for example, has its limits." (John Scott," What is Social Network Analysis?", 2012)

"The sociometric uses of graph theory measured the ‘distance’ from one individual to another by the number of links that must be traversed to connect the two. This is a useful measure of closeness, but it does not correspond to the everyday idea of distance as something measured across a physical space. In a sociogram, the physical arrangement of points is arbitrary, limited only by the aesthetic attempt to minimise overlaps among the lines. A measure of physical distance, however, requires a non-arbitrary representation of the data." (John Scott," What is Social Network Analysis?", 2012)

🕸️On Graph Theory: On Cycles

"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 graph enables us to visualize a relation over a set, which makes the characteristics of relations such as transitivity and symmetry easier to understand. […] Notions such as paths and cycles are key to understanding the more complex and powerful concepts of graph theory. There are many degrees of connectedness that apply to a graph; understanding these types of connectedness enables the engineer to understand the basic properties that can be defined for the graph representing some aspect of his or her system. The concepts of adjacency and reachability are the first steps to understanding the ability of an allocated architecture of a system to execute properly." (Dennis M Buede, "The Engineering Design of Systems: Models and methods", 2009)

"A further refinement of the simple component idea is that of the cyclic component built from intersecting cycles of connection. A cycle is a directed path that returns to its starting point. The overlapping of such cycles produces a cyclic compo-nent in which all points are connected by one or more cycles and no points have cyclic connections outside the component. A cyclic component is a structural element within a strong component and may be connected to other members of the strong component through ‘bridges’ that do not lie on the cycle itself." (John Scott," What is Social Network Analysis?", 2012)

"The first and most obvious property of any network is its nonlinearity – it goes in all directions. Thus the relationships in a network pattern are nonlinear relationships. In particular, an influence, or message, may travel along a cyclical path, which may become a feedback loop. In living networks, the concept of feedback is intimately connected with the network pattern." (Fritjof Capra, "The Systems View of Life: A Unifying Vision", 2014)

"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)

🕸️On Graph Theory: On Density

"To understand, how noise is related to scale-freeness, we have to do some mathematics again. Noise is usually characterized by a mathematical trick. The seemingly random fluctuation of the signal is regarded as a sum of sinusoidal waves. The components of the million waves giving the final noise structure are characterized by their frequency. To describe noise, we plot the contribution (called spectral density) of the various waves we use to model the noise as a function of their frequency. This transformation is called a Fourier transformation [...]" (Péter Csermely, "Weak Links: The Universal Key to the Stabilityof Networks and Complex Systems", 2009)

"Inclusiveness simply measures the number or proportion of the whole set of points that are actually connected into one or more parts of the graph. Some points may be isolates, having no ties to other points, while others will be connected, to a greater or lesser extent, into larger structures. The inclusiveness of a graph is simply the total number of non-isolated points, gener-ally expressed as a percentage of the total number of points. Inclusiveness is a rough and ready approximation to cohesion, but it is usually more informative to measure the actual density of the graph." (John Scott," What is Social Network Analysis?", 2012)

"The density of a graph is a very useful and direct measure of its cohesion, but it has one major limitation as a comparative measure of social structure. In real situations, density varies with the size of a network and this limits the possibilities of using the measure to compare different types of network. It is highly unlikely that agents are able to sustain more than a certain number of relationships: our ability to be ‘friends’ with people, for example, has its limits." (John Scott," What is Social Network Analysis?", 2012) 

"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)

"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 Neighborhoods

"[...] the transition to a small world is essentially undetectable at a local level. If you were living through the morph, nothing about your immediate neighborhood would tell you that the world had become small." (Steven Strogatz, "Sync: The Emerging Science of Spontaneous Order", 2003)

"Average path length reflects the global structure; it depends on the way the entire network is connected, and cannot be inferred from any local measurement. Clustering reflects the local structure; it depends only on the interconnectedness of a typical neighborhood, the inbreeding among nodes tied to a common center. Roughly speaking, path length measures how big the network is. Clustering measures how incestuous it is." (Steven Strogatz, "Sync: The Emerging Science of Spontaneous Order", 2003)

"If a network is solely composed of neighborhood connections, information must traverse a large number of connections to get from place to place. In a small-world network, however, information can be transmitted between any two nodes using, typically, only a small number of connections. In fact, just a small percentage of random, long-distance connections is required to induce such connectivity. This type of network behavior allows the generation of 'six degrees of separation' type results, whereby any agent can connect to any other agent in the system via a path consisting of only a few intermediate nodes." (John H Miller & Scott E Page, "Complex Adaptive Systems", 2007)

"A whole collection of global measures cluster around the idea of the centrality of points within their graphs. The degree of a point - its total of incoming and outgoing lines—is the most basic measure and has been termed local centrality. Calculating the degrees of all points in a network and ranking them from highest to lowest gives a rank order of local centrality. This centrality is ‘local’ because it highlights points that are well-connected in their immediate neighbourhoods. Such points may not, however, be central in the more global sense that a circle or sphere has a unique centre that can be understood in quasi-spatial terms. Locally central points are well-connected within particular parts of the network but may not be at all well-connected in a global sense." (John Scott," What is Social Network Analysis?", 2012)

"Typically, most outlier detection algorithms use some quantified measure of the outlierness of a data point, such as the sparsity of the underlying region, nearest neighbor based distance, or the fit to the underlying data distribution. Every data point lies on a continuous spectrum from normal data to noise, and finally to anomalies [...] The separation of the different regions of this spectrum is often not precisely defined, and is chosen on an ad-hoc basis according to application-specific criteria. Furthermore, the separation between noise and anomalies is not pure, and many data points created by a noisy generative process may be deviant enough to be interpreted as anomalies on the basis of the outlier score. Thus, anomalies will typically have a much higher outlier score than noise, but this is not a distinguishing factor between the two as a matter of definition. Rather, it is the interest of the analyst, which regulates the distinction between noise and an anomaly." (Charu C Aggarwal, "Outlier Analysis", 2013)

"The great strength of node–link layouts is that for sufficiently small networks they are extremely intuitive for supporting many ofthe abstract tasks that pertain to network data. They particularly shine for tasks that rely on understanding the topological structure of the network, such as path tracing and searching local topological neighborhoods a small number of hops from a target node, and can also be very effective for tasks such as general overview or finding similar substructures. The effectiveness of the general idiom varies considerably depending on the specific visual encoding idiom used [...]" (Tamara Munzner, "Visualization Analysis and Design", 2014)

"Decision trees are also discriminative models. Decision trees are induced by recursively partitioning the feature space into regions belonging to the different classes, and consequently they define a decision boundary by aggregating the neighboring regions belonging to the same class. Decision tree model ensembles based on bagging and boosting are also discriminative models." (John D Kelleher et al, "Fundamentals of Machine Learning for Predictive Data Analytics: Algorithms, Worked Examples, and Case Studies", 2015)

"A random walk, in general, is sometimes described as being similar to how a drunk person traverses a city. They know what direction or end point they want to reach but may take a very circuitous route to get there. The algorithm starts at one node and somewhat randomly follows one of the relationships forward or backward to a neighbor node. It then does the same from that node and so on, until it reaches the set path length. 'We say somewhat randomly because the number of relationships a node has, and its neighbors have, influences the probability a node will be walked through.)'" (Mark Needham & Amy E Hodler, "Graph Algorithms: Practical Examples in Apache Spark and Neo4j", 2019)

"Vector databases are designed to store and index highdimensional embeddings - dense numeric vectors that capture the semantic meaning of text, images, audio, or other content. Instead of looking for exact matches, they use approximate nearest neighbor (ANN) algorithms to return the items whose vectors lie closest to a query vector in that multidimensional space. This makes them the engine behind semantic search, recommendation systems, image-or-audio similarity matching, and retrieval augmented generation (RAG) pipelines that supply LLM prompts with relevant context in milliseconds." (Abi Aryan, "LLMOps: Managing Large Language Models in Production", 2025)