🕸️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)

🕸️On Graph Theory: On Centrality

"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)

"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)

"Betweenness Centrality makes the assumption that all communication between nodes happens along the shortest path and with the same frequency, which isn’t always the case in real life. Therefore, it doesn’t give us a perfect view of the most influential nodes in a graph, but rather a good representation." (Mark Needham & Amy E Hodler, "Graph Algorithms: Practical Examples in Apache Spark and Neo4j", 2019)

"Centrality algorithms are used to understand the roles of particular nodes in a graph and their impact on that network. They’re useful because they identify the most important nodes and help us understand group dynamics such as credibility, accessibility, the speed at which things spread, and bridges between groups." (Mark Needham & Amy E Hodler, "Graph Algorithms: Practical Examples in Apache Spark and Neo4j", 2019)

"Keep in mind that centrality measures represent the importance of a node in comparison to other nodes. Centrality is a ranking of the potential impact of nodes, not a measure of actual impact. For example, you might identify the two people with the highest centrality in a network, but perhaps policies or cultural norms are in play that actually shift influence to others. Quantifying actual impact is an active research area to develop additional influence metrics." (Mark Needham & Amy E Hodler, "Graph Algorithms: Practical Examples in Apache Spark and Neo4j", 2019)

"Sometimes the most important cog in the system is not the one with the most overt power or the highest status. Sometimes it’s the middlemen that connect groups or the brokers who the most control over resources or the flow of information. Betweenness Centrality is a way of detecting the amount of influence a node has over the flow of information or resources in a graph. It is typically used to find nodes that serve as a bridge from one part of a graph to another." (Mark Needham & Amy E Hodler, "Graph Algorithms: Practical Examples in Apache Spark and Neo4j", 2019)

"Use Degree Centrality if you’re attempting to analyze influence by looking at the number of incoming and outgoing relationships, or find the 'popularity' of individual nodes. It works well when you’re concerned with immediate connectedness or near-term probabilities. However, Degree Centrality is also applied to global analysis when you want to evaluate the minimum degree, maximum degree, mean degree, and standard deviation across the entire graph." (Mark Needham & Amy E Hodler, "Graph Algorithms: Practical Examples in Apache Spark and Neo4j", 2019)

"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)

🕸️On Graph Theory: On Distribution

"Neural networks conserve the complexity of the systems they model because they have complex structures themselves. Neural networks encode information about their environment in a distributed form. […] Neural networks have the capacity to self-organise their internal structure." (Paul Cilliers, "Complexity and Postmodernism: Understanding Complex Systems", 1998)

"In networks belonging to the second category, the winner takes all, meaning that the fittest node grabs all links, leaving very little for the rest of the nodes. Such networks develop a star topology, in which all nodes are connected to a central hub. In such a hub-and-spokes network there is a huge gap between the lonely hub and everybody else in the system. Thus a winner-takes-all network is very different from the scale-free networks we encountered earlier, where there is a hierarchy of hubs whose size distribution follows a power law. A winner-takes-all network is not scale-free. Instead there is a single hub and many tiny nodes. This is a very important distinction." (Albert-László Barabási, "Linked: How Everything Is Connected to Everything Else and What It Means for Business, Science, and Everyday Life", 2002)

"The first category includes all networks in which, despite the fierce competition for links, the scale-free topology survives. These networks display a fit-get-rich behavior, meaning that the fittest node will inevitably grow to become the biggest hub. The winner's lead is never significant, however. The largest hub is closely followed by a smaller one, which acquires almost as many links as the fittest node. At any moment we have a hierarchy of nodes whose degree distribution follows a power law. In most complex networks, the power law and the fight for links thus are not antagonistic but can coexist peacefully."(Albert-László Barabási, "Linked: How Everything Is Connected to Everything Else and What It Means for Business, Science, and Everyday Life", 2002)

"At an anatomical level - the level of pure, abstract connectivity - we seem to have stumbled upon a universal pattern of complexity. Disparate networks show the same three tendencies: short chains, high clustering, and scale-free link distributions. The coincidences are eerie, and baffling to interpret." (Steven Strogatz, "Sync: The Emerging Science of Spontaneous Order", 2003)

"The power law distributions transcend networks as such. The quantities that are so distributed include the number of genes in a family, the number of pseudogenes per gene, the number of people per city, the number of published papers per scientist, the number of citations per paper, and much, much more. In fact, the first distributions where power laws have been noticed are the distribution of people in a society by wealth (the Pareto law^^) and the distributions of words in a text by frequency (Zipf law)." (Eugene V Koonin et al, "The Genomic Revolution, Systems Biology, Power Laws, and Scale-Free Networks", 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)

🕸️On Graph Theory: On Small-Worlds

“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)

"[...] 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)

"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)

"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)

"To put it simply, a small world is characterized by the fact that every two people can reach each other through a chain of just a handful of messages. This phenomenon is also known as the 'six degrees of separation' […]" (Maarten van Steen, "Graph Theory and Complex Networks: An Introduction", 2010)

"A small-world network is characterized by a high clustering coefficient and a short average path length, meaning that most nodes can be reached from any other node through a small number of intermediate connections. This structure often mirrors real-world social networks, where individuals are typically connected through a few mutual acquaintances, allowing for rapid information dissemination." (Aldo Marzullo et al, "Graph Machine Learning" 2nd Ed., 2025)

On Metaphors (2010-2019)

"All sorts of metaphorical interpretations are culturally ingrained. An astute designer will think about these possible interpretations and work with them, rather than against them." (Noah Iliinsky & Julie Steel, "Designing Data Visualizations", 2011)

"Metaphor lives a secret life all around us. We utter about six metaphors a minute. Metaphorical thinking is essential to how we understand ourselves and others, how we communicate and learn, discover and invent." (James Geary, "I Is an Other: The Secret Life of Metaphor and How it Shapes the Way We See the World", 2011)

"Visual metaphors are about integrating a certain visual quality in your work that somehow conveys that extra bit of connection between the data, the design, and the topic. It goes beyond just the choice of visual variable, though this will have a strong influence. Deploying the best visual metaphor is something that really requires a strong design instinct and a certain amount of experience." (Andy Kirk, "Data Visualization: A successful design process", 2012)

"When metaphors are suggestive about the nature of economic objects and economic life, they provide the raw material from which to make substantive analogies, and analogies may be formed into models. By adopting a metaphor, economists can portray the workings of the economy in terms of some other already-formed and known world with which they are familiar, maybe a machine such as a mechanical balance or a physiological system such as the human body. In doing so, economists can be said to have ‘chosen’ the world of the model. And from imagining some aspect of the economy in terms of something else, economists are able to thinkanew about the economy from their analogical model." (Mary S Morgan, "The World in the Model: How Economists Work and Think", 2012)

"Unlike symbols in poetry, mathematical symbols begin as deliberate designs created by mathematicians. That does not stop symbols from performing the same function that a poem would: to make connections between experience and the unknown and to transfer metaphorical thoughts capable of conveying meaning. As in poetry, there are archetypes in mathematics. If there are such things as self-evident truths, then there probably are things we know about the world that come with the human package at birth." (Joseph Mazur, "Enlightening Symbols: A Short History of Mathematical Notation and Its Hidden Powers", 2014)

"The sensitivity of chaotic systems to initial conditions is particularly well known under the moniker of the 'butterfly effect', which is a metaphorical illustration of the chaotic nature of the weather system in which 'a flap of a butterfly’s wings in Brazil could set off a tornado in Texas'. The meaning of this expression is that, in a chaotic system, a small perturbation could eventually cause very large-scale difference in the long run." (Hiroki Sayama, "Introduction to the Modeling and Analysis of Complex Systems", 2015)

"A model is a metaphor, a description of a system that helps us to reason more clearly. Like all metaphors, models are approximations, and will never account for every last detail. A useful mantra here is: all models are wrong, but some models are useful." (James G Scott, "Statistical Modeling: A Gentle Introduction", 2017

"The relationship of math to the real world has been a conundrum for philosophers for centuries, but it is also an inspiration for poets. The patterns of mathematics inhabit a liminal space - they were initially derived from the natural world and yet seem to exist in a separate, self-contained system standing apart from that world. This makes them a source of potential metaphor: mapping back and forth between the world of personal experience and the world of mathematical patterns opens the door to novel connections." (Alice Major, "Mapping from e to Metaphor", 2018) 

Mary S Morgan - Collected Quotes

"A new kind of representation – mathematical models – leads economic scientists to see in a particular way; and each different form of visualization, such as arithmetic, algebra or geometry, diagrams or even machines, leads to a focus on different aspects of the economic elements represented in their models. Portrayal and cognition are intimately linked and both these, in turn, to conception and perception." (Mary S Morgan, "The World in the Model: How Economists Work and Think", 2012)

"But there are deeper issues that stem from the synchronous development of matematics and modelling. New forms of expression within a discipline involve not just a change of method and the ways that scientists do their work, but also changes in the things that they express. As scientists imagine their world, and make images of that world in new forms, they also form new concepts to work and argue with. Modelling as a new way of visualizing the economy, and mathematics as a new language of expression, both prompt conceptual change. New ways of expressing economic ideas – models and mathematics – lead to new things being expressed. So the cognitive struggles these new processes of visualization entail are rewarded by the new conceptual developments that come from them." (Mary S Morgan, "The World in the Model: How Economists Work and Think", 2012)

"Despite the ubiquity of modelling in modern economics, it is not easy to say how this way of doing science works. Scientific models are not self-evident things, and it is not obvious how such research objects are made, nor how a scientist reasons with them, nor to what purpose. These difficulties of definition and understanding are exhibited in a most concrete fashion in an example that may well be the first such economic model in the history of the field." (Mary S Morgan, "The World in the Model: How Economists Work and Think", 2012)

"Economists do not start out by perceiving the world clearly and describing it in their model but by visualizing how the economic world might be and portraying that intuition in their model – imagining and imaging. In the process of such modelling, economists develop new concepts and so they – and we – come to perceive new things in the economic world. A new wayof looking at the familiar problems of exchange led to a new sense of what the phenomena entailed." (Mary S Morgan, "The World in the Model: How Economists Work and Think", 2012)

"Mathematics provides the languages of most modern economic model-making, and we know that economics became mathematized at the same time as it became a modelling science [...]" (Mary S Morgan, "The World in the Model: How Economists Work and Think", 2012)

"Model-making – as we have already seen – is an activity of creating small worlds expressed in another medium. The economist represents his/her ideas about certain elements of the economy: the system as a whole, or people’s economic behaviour, that they want to investigate or understand into other forms: into bits of mathematics, diagrams, machines, and even – sometimes – strictly defined verbal portraits. The models have certain qualities – they are smaller-scale, and it is supposed, simpler, than the real world, made of quite different materials, and their sense of representation, imitation, or similarity might be quite opaque." (Mary S Morgan, "The World in the Model: How Economists Work and Think", 2012)

"[...] modelling as a style of reasoning in economics works as a method of enquiry comprising probing questions, manipulations to provide demonstrations that are both deductive and experimental, and informal inference arguments involving elements of narrative that offer explanatory or interpretative services." (Mary S Morgan, "The World in the Model: How Economists Work and Think", 2012)

"Models are not easy objects either to define or, in general terms, describe, but those reproduced here, some of the most important models from the history of economics, exemplify the sort of things that count as models in economics: either real objects, or pen-and-paper objects that are diagrammatic, algebraic, or arithmetic in form. Despite their variations in form, these objects share recognisable characteristics: each depicts, renders, denotes, or in some way provides, some kind of representation of ideas about some aspects of the economy. Yet, and this is a very important point to stress, these representations are not just pictures." (Mary S Morgan, "The World in the Model: How Economists Work and Think", 2012)

"Once accepted by a group of scientists, a style of reasoning comes to seem natural to them, so natural that they do not question it. They neither question its historical origins, nor the objectivity of the knowledge gained from using the method, nor do they appeal to any outside or higher level for its justification." (Mary S Morgan, "The World in the Model: How Economists Work and Think", 2012)

"Rules for reasoning with a model come from two distinct aspects of the model. First, when an economist reasons with any model, he or she must obey certain reasoning rules according to the kind of the stuff that the model is made from, or language it is written in, or the format it has. So, these rules could be those of geometry or algebra, of mechanics or hydraulics, etc. depending on the model. [....] Second, and in contrast, allowable manipulations of the model are also determined and constrained by the economics subject matter represented in the model." (Mary S Morgan, "The World in the Model: How Economists Work and Think", 2012)

"When metaphors are suggestive about the nature of economic objects and economic life, they provide the raw material from which to make substantive analogies, and analogies may be formed into models. By adopting a metaphor, economists can portray the workings of the economy in terms of some other already-formed and known world with which they are familiar, maybe a machine such as a mechanical balance or a physiological system such as the human body. In doing so, economists can be said to have ‘chosen’ the world of the model. And from imagining some aspect of the economy in terms of something else, economists are able to thinkanew about the economy from their analogical model." (Mary S Morgan, "The World in the Model: How Economists Work and Think", 2012)

"When storytelling with a model succeeds in offering accounts that are both meaningful in their theoretical terms and plausible, and perhaps striking, in terms of the explanations these narratives offer for real-world events, economists are rather pleased with their efforts. Such a model may be understood by economists as one that has already offered suggestions about the way that the economic world might work, and might be used to generate additional insights. But what counts as meaningful and plausible both change over time." (Mary S Morgan, "The World in the Model: How Economists Work and Think", 2012)

On Exponential Growth (2020-)

"Exponentially growing systems are prevalent in nature, spanning all scales from biochemical reaction networks in single cells to food webs of ecosystems. How exponential growth emerges in nonlinear systems is mathematically unclear. […] The emergence of exponential growth from a multivariable nonlinear network is not mathematically intuitive. This indicates that the network structure and the flux functions of the modeled system must be subjected to constraints to result in long-term exponential dynamics." (Wei-Hsiang Lin et al, "Origin of exponential growth in nonlinear reaction networks", PNAS 117 (45), 2020)

"With cloud data lakes you typically pay for what you use, so your costs always align with your data volumes. Since there is only a single storage layer, less data movement across different systems, availability settings, and decoupled storage versus compute, you have isolated and minimized costs for just data storage. For greater cost allocation, most cloud data lakes offer buckets, or containers (filesystems, not to be confused with application containers), to store different layers of the data (e.g., raw versus transformed data). These containers allow you to have finer-grained cost allocation for different areas of your organization. Since data sources and volumes are growing exponentially, it is extremely important to allocate and optimize costs without limiting the volume or variety of data that can be stored." (Bennie Haelen & Dan Davis, "Delta Lake: Up and Running - Modern Data Lakehouse Architectures with Delta Lake", 2023)

"In effect, AI is transforming the entire globe into a gigantic meta-brain – a vast, interconnected web of intelligence, continuously fed by humans, AI models, robots, self-driving cars, and autonomous systems of every kind. As this process accelerates, critical data density reaches unprecedented levels, driving AI into a self-reinforcing spiral of perpetual improvement, where each new insight compounds into an unstoppable force of exponential progress." (Lars Tvede et al, "Hyperintelligence How the Universe Engineers Its Own Mind", 2025

"When we explore the mathematics behind the natural world, we encounter a fascinating phenomenon: the principle of combinatorial complexity. This principle explains the exponential increase in possibilities that arise when we combine even a small number of elements." (Lars Tvede et al, "Hyperintelligence How the Universe Engineers Its Own Mind", 2025)

"While the universe as a whole tends towards greater disorder, there are local, pinched pockets where complexity spontaneously increases. And such processes can, as we have seen, continue even over billions of years and come a very long way. We live in such a pocket, and the complexity in it is growing exponentially – and on some fronts hyperexponentially." (Lars Tvede et al, "Hyperintelligence How the Universe Engineers Its Own Mind", 2025)

On Exponential Growth (-1989)

"Anyone who believes that exponential growth can go on forever in a finite world is either a madman or an economist." (Kenneth E Boulding, "General Systens Theory - The skeleton of science", Management Science Vol.2 (3), 1956)

"The greatest shortcoming of the human race is our inability to understand the exponential function." (Albert A Bartlett, "Arithmetic, Population and Energy", 1969)

"However, and conversely, our models fall far short of representing the world fully. That is why we make mistakes and why we are regularly surprised. In our heads, we can keep track of only a few variables at one time. We often draw illogical conclusions from accurate assumptions, or logical conclusions from inaccurate assumptions. Most of us, for instance, are surprised by the amount of growth an exponential process can generate. Few of us can intuit how to damp oscillations in a complex system." (Donella H Meadows, "Limits to Growth", 1972) 

"Taking no action to solve these problems is equivalent of taking strong action. Every day of continued exponential growth brings the world system closer to the ultimate limits of that growth. A decision to do nothing is a decision to increase the risk of collapse." (Donella Meadows et al, "The Limits to Growth", 1972) 

"Every day of continued exponential growth brings the world system closer to the ultimate limits of that growth." (Mihajlo D Mesarovic, "Mankind at the Turning Point", 1974)

"The world's present industrial civilization is handicapped by the coexistence of two universal, overlapping, and incompatible intellectual systems: the accumulated knowledge of the last four centuries of the properties and interrelationships of matter and energy; and the associated monetary culture which has evolved from folkways of prehistoric origin. […] Despite their inherent incompatibilities, these two systems during the last two centuries have had one fundamental characteristic in common, namely exponential growth, which has made a reasonably stable coexistence possible. But, for various reasons, it is impossible for the matter-energy system to sustain exponential growth for more than a few tens of doublings, and this phase is by now almost over. The monetary system has no such constraints, and according to one of its most fundamental rules, it must continue to grow by compound interest." (Marion K Hubbert, "Two Intellectual Systems: Matter-energy and the Monetary Culture", [seminar] 1981)

"In fact, in all those cases in which the initial state is given with limited precision (if we assume that the space-time is continuous this is always the case because a generic point turns out to be completely specified only by an infinite amount of information, for example by an infinite string of numbers), we can observe a situation in which, when time becomes large, two trajectories emerge from the 'same' initial point. So, even though there is a deterministic situation from a mathematical point of view (the uniqueness theorem for ordinary differential equations is not in question), nevertheless the exponential growth of errors makes the time evolution self-independent from its past history and then nondeterministic in any practical sense." (David Ruelle, "Chaotic Evolution and Strange Attractors: The statistical analysis of time series for deterministic nonlinear systems", 1989)

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

"And now the trumpets terribly, from far,

With rattling clangor, rouse the sleepy war.

The soldiers' shouts succeed the brazen sounds;

And heaven, from pole to pole, the noise rebounds." (Virgil, "Aeneid", 29–19 BC)

"There was no wind; there was no passing shadow on the deep shade of the night; there was no noise. The city lay behind him, lighted here and there, and starry worlds were hidden by the masonry of spire and roof that hardly made out any shapes against the sky. Dark and lonely distance lay around him everywhere, and the clocks were faintly striking two."(Charles Dickens, "Dombey and Son", cca. 1848)

"Noise is the most impertinent of all forms of interruption. It is not only an interruption, but also a disruption of thought." (Arthur Schopenhauer, "Parerga and Paralipomena", ["On Noise"] 1851)

"Everybody has their taste in noises as well as in other matters; and sounds are quite innoxious, or most distressing, by their sort rather than their quantity." (Jane Austen, "Northanger abbey, and Persuasion", 1853)

"Never had the sky been more studded with stars and more charming, the trees more trembling, the odor of the grass more penetrating; never had the birds fallen asleep among the leaves with a sweeter noise; never had all the harmonies of universal serenity responded more thoroughly to the inward music of love; never had Marius been more captivated, more happy, more ecstatic." (Victor Hugo, "Saint Denis", 1887)

"Noise proves nothing. Often a hen who has merely laid an egg cackles as if she had laid an asteroid." (Mark Twain, "Pudd'nhead Wilson's New Calendar", 1897)

"Real miracles make little noise! Essential events are so simple!" (Antoine de Saint-Exupéry, "Letter to a Hostage", 1943)

"It is loneliness that makes the loudest noise." (Eric Hoffer, The New York Times, 1971)

"There are millions of chords. There are millions of numbers. And everyone forgets the one that is a zero. But without the zero, numbers are just arithmetic. Without the empty chord, music is just noise." (Terry Pratchett, "Soul Music", 1994)

"She had come to Aka to learn how to sing this world's tune, to dance its dance; and at last, she thought, away from the city's endless noise, she was beginning to hear the music and to learn how to move to it." (Ursula K Le Guin, "Hainish Cycle: The Telling", 2000)

"Well, all information looks like noise until you break the code." (Neal Stephenson, "Snow Crash", 2003)

See more quotes on noise in spiritual writings, Data Science, Systems Engineering, Graphical Representation, Business Intelligence

Cecil Powell - Collected Quotes

"Coming out of space and incident on the high atmosphere, there is a thin rain of charged particles known as the primary cosmic radiation." (Cecil Powell, "The Cosmic Radiation", [Nobel lecture] 1950)

"For the investigation of the cosmic radiation, it is necessary to solve two principal technical problems: First, to detect the radiation, to determine the masses, energy and transformation properties of the particles of which it is composed, and to study the nuclear transmutations which they produce. Second, to develop methods of making such observations throughout the atmosphere and at depths underground." (Cecil Powell, "The Cosmic Radiation", [Nobel lecture] 1950)

"In the first class are found the trigger mechanisms such as the Geiger counter and the scintillation counter. Such devices record the instants of pas sage of individual particles through the apparatus. Their most important advantages are (a) that they allow observations to be made of great statistical weight; and (b) that the relationship in time of the instants of passage of associated particles can be established. With modern instruments of this type, the time interval between the arrival of two charged particles can be  measured even although this is as small as one or two hundredths of a micro second. These devices have made possible contributions of the greatest importance to our knowledge of the subject, and they have proved especially valuable when the nature of the physical processes being studied has been well understood." (Cecil Powell, "The Cosmic Radiation", [Nobel lecture] 1950)

"In the second class of detectors are the devices for making manifest the tracks of particles; namely, the Wilson expansion chamber and the photo graphic plate. These instruments have the particular advantage, amongst others, that they allow a direct and detailed insight into the physical pro cesses which accompany the passage of charged particles through matter. On the other hand, it is arduous to employ them to obtain observations of great statistical weight. The two classes of instruments thus provide complemen tary information, and each has made a decisive contribution." (Cecil Powell, "The Cosmic Radiation", [Nobel lecture] 1950)

"The detailed study of the 'mass spectrum' of the incoming nuclei has an important bearing on the problem of the origin of the primary particles; but it is complicated by the fact that, because of their large charge, the particles rapidly lose energy in the atmosphere by making atomic and nuclear collisions." (Cecil Powell, "The Cosmic Radiation", [Nobel lecture] 1950)

"Today the study of the cosmic radiation is, in essence, the study of nuclear physics of the extreme high-energy region. Although the number of in coming particles is very small in comparison with those which are produced by the great machines, most of them are much more energetic than any which we can yet generate artificially; and in nuclear collisions they produce effects which cannot be simulated in the laboratory. The study of the resulting transmutations is therefore complementary to that which can be made at lower energies with the aid of the cyclotrons and synchrotrons." (Cecil Powell, "The Cosmic Radiation", [Nobel lecture] 1950)

"We are only at the beginning of our penetration into what appears to be a rich field of discovery. Already, however, it seems certain that our present theoretical approach has been limited by lack of essential information; and that the world of the mesons is far more complex than has hitherto been visualized in the most brilliant theoretical speculations. The fast protons and α-particles generated by the cyclotrons are not sufficiently energetic to pro duce these more massive mesons, but this may become possible when the proton synchrotrons now under construction come into operation." (Cecil Powell, "The Cosmic Radiation", [Nobel lecture] 1950)