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

"In such a universe as this what significance could there be in our fortuitous, our frail, our evanescent community?" (Olaf Stapledon, "Star Maker", 1937)

"A man is morally responsible to his community. That’s a good idea. But his community is also morally responsible to him." (Philip K Dick, "The Man Who Japed", 1956)

"Science fiction is the branch of literature that deals with the effects of change on people in the real world as it can be projected into the past, the future, or to distant places. It often concerns itself with scientific or technological change, and it usually involves matters whose importance is greater than the individual or the community; often civilization or the race itself is in danger." (James Gunn, "Road to Science Fiction" Vol. 2, 1979)

"I think that most of us, anyway, read these stories that we know are not 'true' because we're hungry for another kind of truth: the mythic truth about human nature in general, the particular truth about those life-communities that define our own identity, and the most specific truth of all: our own self-story. Fiction, because it is not about someone who lived in the real world, always has the possibility of being about oneself." (Orson Scott Card, "Ender’s Game", [introduction] 1985)

"Universities are meant to pass the torch of civilization, not just download data into student skulls, and the values of the academic community are strongly at odds with those of all would-be information empires. Teachers at all levels, from kindergarten up, have proven to be shameless and persistent software and data pirates. Universities do not merely ‘leak information' but vigorously broadcast free thought." (Bruce Sterling, "The Hacker Crackdown", 1992)

"Here is the truth. What human life is, what it’s for, what we do, is create communities." (Orson Scott Card, "Pastwatch: The Redemption of Christopher Columbus", 1996)

"We need to bridge our sense of loneliness and disconnection with a sense of community and continuity even if we must manufacture it from our time on the Web and our use of calling cards to connect long distance. We must log on somewhere, and if it is only in cyberspace, that is still far better than nowhere at all." (Julia Cameron, "God is No Laughing Matter", 2001)

"Human evolution is driven by community needs […]. How is that possible, since genetic information is passed only by and to individuals?" (Orson Scott Card, "First Meetings in Ender's Universe", 2002)

"Most human communities demand anti-survival behavior from large numbers of their members." (Orson Scott Card, "First Meetings in Ender's Universe", 2002)

"That would be true […] and indeed is true in the main, except that there are only a few types of human communities that actually survive long enough to improve the chances of individual survival." (Orson Scott Card, "First Meetings in Ender's Universe", 2002)

"The real training ground for leadership is in the game." (Orson Scott Card, "First Meetings in Ender's Universe", 2002)

"There are community traits that promote survival of the individual, and individual traits that promote the survival of the community." (Orson Scott Card, "First Meetings in Ender's Universe", 2002)

On Scientific Communities

"A theoretical science unaware that those of its constructs considered relevant and momentous are destined eventually to be framed in concepts and words that have a grip on the educated community and become part and parcel of the general world picture - a theoretical science, I say, where this is forgotten, and where the initiated continue musing to each other in terms that are, at best, understood by a small group of close fellow travelers, will necessarily be cut off from the rest of cultural mankind; in the long run it is bound to atrophy and ossify however virulently esoteric chat may continue within its joyfully isolated groups of experts." (Erwin Schrödinger, "Are There Quantum Jumps?", The British Journal for the Philosophy of Science Vol. 3, 1952)

"A theoretical science unaware that those of its constructs considered relevant and momentous are destined eventually to be framed in concepts and words that have a grip on the educated community and become part and parcel of the general world picture - a theoretical science [...]" (Erwin Schrödinger, "Are There Quantum Jumps?", The British Journal for the Philosophy of Science Vol. 3, 1952)

"The subject matter of the scientist is a crowd of natural events at all times; he presupposes that this crowd is not real but apparent, and seeks to discover the true place of events in the system of nature. The subject matter of the poet is a crowd of historical occasions of feeling recollected from the past; he presupposes that this crowd is real but should not be, and seeks to transform it into a community. Both science and art are primarily spiritual activities, whatever practical applications may be derived from their results. Disorder, lack of meaning, are spiritual not physical discomforts, order and sense spiritual not physical satisfactions." (Wystan H Auden, "The Dyer’s Hand and Other Essays", 1965)

"Scientific knowledge is not created solely by the piecemeal mining of discrete facts by uniformly accurate and reliable individual scientific investigations. The process of criticism and evaluation, of analysis and synthesis, are essential to the whole system. It is impossible for each one of us to be continually aware of all that is going on around us, so that we can immediately decide the significance of every new paper that is published. The job of making such judgments must therefore be delegated to the best and wisest among us, who speak, not with their own personal voices, but on behalf of the whole community of Science. […] It is impossible for the consensus - public knowledge - to be voiced at all, unless it is channeled through the minds of selected persons, and restated in their words for all to hear." (John M Ziman, "Public Knowledge: An Essay Concerning the Social Dimension of Science", 1968)

"A scientist should not cheat or falsify data or quote out of context or do any other thing that is intellectually dishonest. Of course, as always, some individuals fail; but science as a whole disapproves of such action. Indeed, when transgressors are detected, they are usually expelled from the community." (Michael Ruse, "Response to the Commentary: Pro Judice", Science, Technology and Human Values Vol. 7" (41), 1982)

"A theory is only ‘true’ for as long as the majority of the scientific community maintain the view that the theory is the one best able to explain the observations." (Jim Baggott, "The Meaning of Quantum Theory", 1992)

"It is in the nature of theoretical science that there can be no such thing as certainty. A theory is only ‘true’ for as long as the majority of the scientific community maintain the view that the theory is the one best able to explain the observations." (Jim Baggott,"The Meaning of Quantum Theory", 1992)

"There are many possible definitions of chaos. In fact, there is no general agreement within the scientific community as to what constitutes a chaotic dynamical system." (Robert L Devaney, "A First Course in Chaotic Dynamical Systems: Theory and Experiment", 1992)

"In the new systems thinking, the metaphor of knowledge as a building is being replaced by that of the network. As we perceive reality as a network of relationships, our descriptions, too, form an interconnected network of concepts and models in which there are no foundations. For most scientists such a view of knowledge as a network with no firm foundations is extremely unsettling, and today it is by no means generally accepted. But as the network approach expands throughout the scientific community, the idea of knowledge as a network will undoubtedly find increasing acceptance." (Fritjof Capra," The Web of Life: a new scientific understanding of living systems", 1996)

🪸On Systems Thinking: On Communities

"Unlike its predecessor, the new cybernetics concerns itself with the interaction of autonomous political actors and subgroups, and the practical and reflexive consciousness of the subjects who produce and reproduce the structure of a political community. A dominant consideration is that of recursiveness, or self-reference of political action both with regards to the expression of political consciousness and with the ways in which systems build upon themselves." (Peter Harries-Jones, "The Self-Organizing Policy: An Epistemological Analysis of Political Life by Laurent Dobuzinskis, Canadian Journal of Political Science 21 (2), 1988)

"In the new systems thinking, the metaphor of knowledge as a building is being replaced by that of the network. As we perceive reality as a network of relationships, our descriptions, too, form an interconnected network of concepts and models in which there are no foundations. For most scientists such a view of knowledge as a network with no firm foundations is extremely unsettling, and today it is by no means generally accepted. But as the network approach expands throughout the scientific community, the idea of knowledge as a network will undoubtedly find increasing acceptance." (Fritjof Capra, "The Web of Life: a new scientific understanding of living systems", 1996)

"Understanding ecological interdependence means understanding relationships. It requires the shifts of perception that are characteristic of systems thinking - from the parts to the whole, from objects to relationships, from contents to patterns. [...]  Nourishing the community means nourishing those relationships." (Fritjof Capra, "The Web of Life: A New Scientific Understanding of Living Systems", 1996)

"What is sustained in a sustainable community is not economic growth, development, market share, or competitive advantage, but the entire web of life on which our long-term survival depends. In other words, a sustainable community is designed in such a way that its ways of life, businesses, economy, physical structures, and technologies do not interfere with nature’s inherent ability to sustain life." (Fritjof Capra, "Ecoliteracy: The Challenge for Education in the Next Century", 1999)

"The living world can be viewed as a biological hierarchy that starts with subcellular particles, and continues up through cells, tissues and organs. Ecology deals with the next three levels: the individual organism, the population (consisting of individuals of the same species) and the community (consisting of a greater or lesser number of species populations). At the level of the organism, ecology deals with how individuals are affected by" (and how they affect) their environment. At the level of the population, ecology is concerned with the presence or absence of particular species, their abundance or rarity, and with the trends and fluctuations in their numbers. Community ecology then deals with the composition and organization of ecological communities." (Michael Begon et al, "Ecology: From individuals to ecosystems", 2006)

"The butterfly effect demonstrates that complex dynamical systems are highly responsive and interconnected webs of feedback loops. It reminds us that we live in a highly interconnected world. Thus our actions within an organization can lead to a range of unpredicted responses and unexpected outcomes. This seriously calls into doubt the wisdom of believing that a major organizational change intervention will necessarily achieve its pre-planned and highly desired outcomes. Small changes in the social, technological, political, ecological or economic conditions can have major implications over time for organizations, communities, societies and even nations." (Elizabeth McMillan, "Complexity, Management and the Dynamics of Change: Challenges for practice", 2008)

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

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

🥀On Style in Science

"I think that science may be styled the knowledge of universals, or abstract wisdom; and art is science reduced to practice - or science is reason, and art the mechanism of it - and may be called practical science. Science, in fine, is the theorem, and art the problem." (Richard Griffith, 1779) [attributed to Laurence Sterne] 

"Both science and art form in the course of the centuries a human language by which we can speak about the more remote parts of reality, and the coherent sets of concepts as well as the different styles of art are different words or groups of words in this language." (Werner K Heisenberg, "Physics and Philosophy", 1958)

"When terms [...] evolve and change definition with time; and when the social reality which terms are intended to organize and render intelligible is also seen to be in flux, capturing the truth in a net of words becomes a matter of intuition and style more than of any scientific method that can be replicated by others and made to achieve the same result every time someone asks the same question, or undertakes the same operations." (William H McNeill, "Discrepancies among the Social Sciences", 1981)

"The body of mathematics to which the calculus gives rise embodies a certain swashbuckling style of thinking, at once bold and dramatic, given over to large intellectual gestures and indifferent, in large measure, to any very detailed description of the world. It is a style that has shaped the physical but not the biological sciences, and its success in Newtonian mechanics, general relativity and quantum mechanics is among the miracles of mankind. But the era in thought that the calculus made possible is coming to an end. Everyone feels this is so and everyone is right." (David Berlinski, "A Tour of the Calculus", 1995)

"Modeling involves a style of scientific thinking in which the argument is structured by the model, but in which the application is achieved via a narrative prompted by an external fact, an imagined event or question to be answered." (Uskali Mäki, "Fact and Fiction in Economics: Models, Realism and Social Construction", 2002)

"[...] the scientific models of concrete things are symbolic rather than iconic: they are systems of propositions, not pictures. Besides, such models are seldom if ever completely accurate, if only because they involve more or less brutal simplifications, such as pretending that a metallic surface is smooth, a crystal has no impurities, a biopopulation has a single predator, or a market is in equilibrium. These are all fictions. However, they are stylizations rather than wild fantasies. Hence, introducing and using them to account for real existents does not commit us to fictionism, just as defending the role of experience need not make us empiricists, nor is admitting the role of intuition enough to qualify as intuitionist." (Mario Bunge, "Chasing Reality: Strife over Realism", 2006)

"Scientists reach their conclusions for the damnedest of reasons: intuition, guesses, redirections after wild-goose chases, all combing with a dollop of rigorous observation and logical reasoning to be sure […] This messy and personal side of science should not be disparaged, or covered up, by scientists for two major reasons. First, scientists should proudly show this human face to display their kinship with all other modes of creative human thought […] Second, while biases and references often impede understanding, these mental idiosyncrasies may also serve as powerful, if quirky and personal, guides to solutions." (Stephen J Gould,"Dinosaur in a Haystack: Reflections in natural history", 1995)

🥀On Style in Mathematics

"The figures which excite in us the ideas of beauty seem to be those in which there is uniformity amidst variety. […] What we call beautiful in objects, to speak in the mathematical style, seems to be in compound ratio of uniformity and variety: so that where the uniformity of bodies is equal, the beauty is as the variety; and where the variety is equal, the beauty is as the uniformity." (Francis Hutcheson,"An Inquiry Concerning Beauty, Order, Harmony, Design", 1973)

"Numeracy has two facets-reading and writing, or extracting numerical information and presenting it. The skills of data presentation may at first seem ad hoc and judgemental, a matter of style rather than of technology, but certain aspects can be formalized into explicit rules, the equivalent of elementary syntax." (Andrew Ehrenberg,"Rudiments of Numeracy", Journal of Royal Statistical Society, 1977)

"Moreover, ‘fact’ doesn’t mean ‘absolute certainty’; there ain’t no such animal in an exciting and complex world. The final proofs of logic and mathematics flow deductively from stated premises and achieve certainty only because they are NOT about the empirical world. Evolutionists make no claim for perpetual truth, though creationists often do" (and then attack us falsely for a style of argument that they themselves favor)." (Stephen J Gould, "Evolution as Fact and Theory", Discover, 1981)

"[...] probability is a style of thinking." (Richard W Hamming, "The Art of Probability for Scientists and Engineers", 1991)

"Mathematics is not just a collection of results, often called theorems; it is a style of thinking. Computing is also basically a style of thinking. Similarly, probability is a style of thinking." (Richard W Hamming, "The Art of Probability for Scientists and Engineers", 1991)

"The sequence for the understanding of mathematics may be: intuition, trial, error, speculation, conjecture, proof. The mixture and the sequence of these events differ widely in different domains, but there is general agreement that the end product is rigorous proof – which we know and can recognize, without the formal advice of the logicians. […] Intuition is glorious, but the heaven of mathematics requires much more. Physics has provided mathematics with many fine suggestions and new initiatives, but mathematics does not need to copy the style of experimental physics. Mathematics rests on proof - and proof is eternal." (Saunders Mac Lane,"Reponses to …", Bulletin of the American Mathematical Society Vol. 30" (2), 1994)

"Allowing more than two players into the game and/or postulating payoff structures in which one player's gain does not necessarily equal the other player's loss brings us much closer to the type of games played in real life. Unfortunately, it's generally the case that the closer you get to the messiness of the real world, the farther you move from the stylized and structured world of mathematics. Game theory is no exception." (John L Casti, "Five Golden Rules", 1995)

"The body of mathematics to which the calculus gives rise embodies a certain swashbuckling style of thinking, at once bold and dramatic, given over to large intellectual gestures and indifferent, in large measure, to any very detailed description of the world. It is a style that has shaped the physical but not the biological sciences, and its success in Newtonian mechanics, general relativity and quantum mechanics is among the miracles of mankind. But the era in thought that the calculus made possible is coming to an end. Everyone feels this is so and everyone is right." (David Berlinski, "A Tour of the Calculus", 1995)

"I see some parallels between the shifts of fashion in mathematics and in music. In music, the popular new styles of jazz and rock became fashionable a little earlier than the new mathematical styles of chaos and complexity theory. Jazz and rock were long despised by classical musicians, but have emerged as art-forms more accessible than classical music to a wide section of the public. Jazz and rock are no longer to be despised as passing fads. Neither are chaos and complexity theory. But still, classical music and classical mathematics are not dead. Mozart lives, and so does Euler. When the wheel of fashion turns once more, quantum mechanics and hard analysis will once again be in style." (Freeman J Dyson, "Book Review of ‘Nature’s Numbers’", The American Mathematical Monthly, Vol. 103 (7), 1996)

"Physicists have been drawn to elegant mathematical relationships that bind the subject together with economy and style, melding disparate qualities in subtle and harmonious ways. But this is to import a new factor into the argument - questions of aesthetics and taste. We are then on shaky ground indeed. It may be that M theory looks beautiful to its creators, but ugly to N theorists, who think that their theory is the most elegant. But then the O theorists disagree with both groups [...]" (Paul C W Davies, "Cosmic Jackpot: Why Our Universe Is Just Right for Life", 2007) 

💠On Problem Solving: On Elegance

"It is true that not every geometric construction is elegant, for each particular problem has its own refinements. It is also true that [that construction] is preferred to any other that makes clear not the structure of a work from an equation but the equation from the structure; thus the structure demonstrates itself. So a skillful geometer, although thoroughly versed in analysis, conceals the fact and, while thinking about the accomplishment of his work, sheds light on and explains his problem Then, as an aid to the arithmeticians, he sets out and demonstrates his theorem with the equation or proportion he sees in it." (François Viète, "On the Meaning and Components of Analysis and on Matters Useful to Zetetics", 1591)

"The problem of distinguishing prime numbers from composite numbers, and of resolving the latter into their prime factors, is known to be one of the most important and useful in arithmetic. It has engaged the industry and wisdom of ancient and modern geometers to such an extent that it would be superfluous to discuss the problem at length. Nevertheless we must confess that all methods that have been proposed thus far are either restricted to very special cases or are so laborious and difficult that even for numbers that do not exceed the limits of tables constructed by estimable men, they try the patience of even the practiced calculator. And these methods do not apply at all to larger numbers. […] Further, the dignity of the science itself seems to require that every possible means be explored for the solution of a problem so elegant and so celebrated." (Carl F Gauss, "Disquisitiones Arithmeticae” [“Arithmetical Researches”], 1801)

"To appreciate the living spirit rather than the dry bones of mathematics, it is necessary to inspect the work of a master at first hand.Textbooks and treatises are an unavoidable evil [...] The very crudities of the first attack on a significant problem by a master are more illuminating than all the pretty elegance of the standard texts which has been won at the cost of perhaps centuries of finicky polishing." (Eric T Bell, "Mathematics: Queen and Servant of Science", 1951)

"Mathematical problems, or puzzles, are important to real mathematics (like solving real-life problems), just as fables, stories, and anecdotes are important to the young in understanding real life. Mathematical problems are ‘sanitized’ mathematics, where an elegant solution has already been found (by someone else, of course), the question is stripped of all superfluousness and posed in an interesting and (hopefully) thought-provoking way. If mathematics is likened to prospecting for gold, solving a good mathematical problem is akin to a ‘hide-and-seek’ course in gold-prospecting: you are given a nugget to find, and you know what it looks like, that it is out there somewhere, that it is not too hard to reach, that it is unearthing within your capabilities, and you have conveniently been given the right equipment (i.e. data) to get it. It may be hidden in a cunning place, but it will require ingenuity rather than digging to reach it." (Terence Tao, "Solving Mathematical Problems: A Personal Perspective", 2006)

"This is what it means to do mathematics. To make a discovery (by whatever means, including playing around with physical models like paper, string, and rubber bands), and then to explain it in the simplest and most elegant way possible. This is the art of it, and this is why it is so challenging and fun." (Paul Lockhart, "Measurement", 2012)

"What makes a great mathematical problem great? Intellectual depth, combined with simplicity and elegance. Plus: it has to be hard. Anyone can climb a hillock; Everest is another matter entirely. A great problem is usually simple to state, although the terms required may be elementary or highly technical." (Ian Stewart, "Symmetry: A Very Short Introduction", 2013)

🦋On Chaos Theory: On Paths

"The term chaos is used in a specific sense where it is an inherently random pattern of behaviour generated by fixed inputs into deterministic" (that is fixed) rules" (relationships). The rules take the form of non-linear feedback loops. Although the specific path followed by the behaviour so generated is random and hence unpredictable in the long-term, it always has an underlying pattern to it, a 'hidden' pattern, a global pattern or rhythm. That pattern is self-similarity, that is a constant degree of variation, consistent variability, regular irregularity, or more precisely, a constant fractal dimension. Chaos is therefore order" (a pattern) within disorder" (random behaviour)." (Ralph D Stacey, "The Chaos Frontier: Creative Strategic Control for Business", 1991)

"One reason nature pleases us is its endless use of a few simple principles: the cube-square law; fractals; spirals; the way that waves, wheels, trig functions, and harmonic oscillators are alike; the importance of ratios between small primes; bilateral symmetry; Fibonacci series, golden sections, quantization, strange attractors, path-dependency, all the things that show up in places where you don’t expect them [...] these rules work with and against each other ceaselessly at all levels, so that out of their intrinsic simplicity comes the rich complexity of the world around us. That tension - between the simple rules that describe the world and the complex world we see - is itself both simple in execution and immensely complex in effect. Thus exactly the levels, mixtures, and relations of complexity that seem to be hardwired into the pleasure centers of the human brain - or are they, perhaps, intrinsic to intelligence and perception, pleasant to anything that can see, think, create? - are the ones found in the world around us." (John Barnes, "Mother of Storms", 1994)

"Complexity is the characteristic property of complicated systems we don’t understand immediately. It is the amount of difficulties we face while trying to understand it. In this sense, complexity resides largely in the eye of the beholder - someone who is familiar with s.th. often sees less complexity than someone who is less familiar with it. [...] A complex system is created by evolutionary processes. There are multiple pathways by which a system can evolve. Many complex systems are similar, but each instance of a system is unique." (Jochen Fromm, The Emergence of Complexity, 2004)

"Chaos can leave statistical footprints that look like noise. This can arise from simple systems that are deterministic and not random. [...] The surprising mathematical fact is that most systems are chaotic. Change the starting value ever so slightly and soon the system wanders off on a new chaotic path no matter how close the starting point of the new path was to the starting point of the old path. Mathematicians call this sensitivity to initial conditions but many scientists just call it the butterfly effect. And what holds in math seems to hold in the real world - more and more systems appear to be chaotic." (Bart Kosko, "Noise", 2006)

"'Chaos' refers to systems that are very sensitive to small changes in their inputs. A minuscule change in a chaotic communication system can flip a 0 to a 1 or vice versa. This is the so-called butterfly effect: Small changes in the input of a chaotic system can produce large changes in the output. Suppose a butterfly flaps its wings in a slightly different way. can change its flight path. The change in flight path can in time change how a swarm of butterflies migrates." (Bart Kosko, "Noise", 2006)

⚛️On Physics: On Paths

"The state of a system at a given moment depends on two things - its initial state, and the law according to which that state varies. If we know both this law and this initial state, we have a simple mathematical problem to solve, and we fall back upon our first degree of ignorance. Then it often happens that we know the law and do not know the initial state. It may be asked, for instance, what is the present distribution of the minor planets? We know that from all time they have obeyed the laws of Kepler, but we do not know what was their initial distribution. In the kinetic theory of gases we assume that the gaseous molecules follow rectilinear paths and obey the laws of impact and elastic bodies; yet as we know nothing of their initial velocities, we know nothing of their present velocities. The calculus of probabilities alone enables us to predict the mean phenomena which will result from a combination of these velocities. This is the second degree of ignorance. Finally it is possible, that not only the initial conditions but the laws themselves are unknown. We then reach the third degree of ignorance, and in general we can no longer affirm anything at all as to the probability of a phenomenon. It often happens that instead of trying to discover an event by means of a more or less imperfect knowledge of the law, the events may be known, and we want to find the law; or that, instead of deducing effects from causes, we wish to deduce the causes." (Henri Poincaré, "Science and Hypothesis", 1902)

"We frequently find that nature acts in such a way as to minimize certain magnitudes. The soap film will take the shape of a surface of smallest area. Light always follows the shortest path, that is, the straight line, and, even when reflected or broken, follows a path which takes a minimum of time. In mechanical systems we find that the movements actually take place in a form which requires less effort in a certain sense than any other possible movement would use. There was a period, about 150 years ago, when physicists believed that the whole of physics might be deduced from certain minimizing principles, subject to calculus of variations, and these principles were interpreted as tendencies - so to say, economical tendencies of nature. Nature seems to follow the tendency of economizing certain magnitudes, of obtaining maximum effects with given means, or to spend minimal means for given effects." (Karl Menger, "What Is Calculus of Variations and What Are Its Applications?" [James R Newman, "The World of Mathematics" Vol. II], 1956)

"Experimental physicists [...] walk a narrow path with pitfalls on either side. If we spend all our time developing equipment, we risk the appellation of 'plumber', and if we merely use the tools developed by others, we risk the censure of our peers for being parasitic." (Luis W Alvarez, "Recent Developments in Particle Physics", [Nobel] 1968)

"Although we expect to find eddies in turbulent flow, we do not know when any specific eddy will come into being or die away . We cannot yet predict how eddies interact. Similarly, we know as a general rule that any particle within a turbulent flow gets knocked about in an aimless fashion by the swirls, so that it describes an erratic meandering path, but at any given moment we cannot predict the precise location or velocity of the particle." (Peter B Stevens, "Patterns in Nature", 1974)

"The Feynman method has the virtue that it provides us with a vivid picture of nature’s quantum trickery at work. The idea is that the path of a particle through space is not generally well defined in quantum mechanics. […] So when an electron arrives at a point in space - say a target screen - many different histories must be integrated together to create this one event. Feynman’s so-called path-integral, or sum-over-histories approach to quantum mechanics, set this remarkable concept out as a mathematical procedure. It remained more or less a curiosity for many years, but as physicists pushed quantum mechanics to its limits - applying it to gravitation and even cosmology - so the Feynman approach turned out to offer the best calculational tool for describing a quantum universe. History may well judge that, among his many outstanding contributions to physics, the path-integral formulation of quantum mechanics is the most significant." (Paul Davies , [Introduction to Richard P Feynman's "Six Easy Pieces"] 1994)

"What was clearly useful was the use of diagrams to prove certain results either in algebraic topology, homological algebra or algebraic geometry. It is clear that doing category theory, or simply applying category theory, implies manipulating diagrams: constructing the relevant diagrams, chasing arrows by going via various paths in diagrams and showing they are equal, etc. This practice suggests that diagram manipulation, or more generally diagrams, constitutes the natural syntax of category theory and the category-theoretic way of thinking. Thus, if one could develop a formal language based on diagrams and diagrams manipulation, one would have a natural syntactical framework for category theory. However, moving from the informal language of categories which includes diagrams and diagrammatic manipulations to a formal language based on diagrams and diagrammatic manipulations is not entirely obvious." (Jean-Pierre Marquis, "From a Geometrical Point of View: A Study of the History and Philosophy of Category Theory", 2009)

🕸️On Graph Theory: On Paths

"Euler's proof that in Königsberg there is no path crossing all seven bridges only once was based on a simple observation. Nodes with an odd number of links must be either the starting or the end point of the journey. A continuous path that goes through all the bridges can have only one starting and one end point. Thus, such a path cannot exist on a graph that has more than two nodes with an odd number of links. As the Königsberg graph had four such nodes, one could not find the desired path." (Albert-László Barabási, "Linked: How Everything Is Connected to Everything Else and What It Means for Business, Science, and Everyday Life", 2002)

"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)

"The concept of path-connectedness, in which it is required that it be possible to reach any point in the space from any other point along a continuous path is necessary for the notion of fundamental group. This approach is especially useful in studying connectivity properties from an algebraic point of view, e.g., via homotopy theory." (D Chatterjee, "Topology: General & Algebraic", 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 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 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)

"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)

"A widely adopted technique for solving the data sparsity issue and the cold-start problem is based on graph representation, navigation, and processing. Graph navigation methods (like the pathfinding example [...]) and graph algorithms (such as PageRank) are applied to fill some gaps and create a denser representation of [a] dataset." (Alessandro Negro, "Graph-Powered Machine Learning", 2021)

Mark Needham - Collected Quotes

 "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)

"Adding graph features and context improves predictions, especially in situations where connections matter. [...] Unfortunately, many machine learning approaches today miss a lot of rich contextual information. This stems from ML’s reliance on input data built from tuples, leaving out a lot of predictive relationships and network data. Furthermore, contextual information is not always readily available or is too difficult to access and process. Even finding connections that are four or more hops away can be a challenge at scale for traditional methods. Using graphs, we can more easily reach and incorporate connected data." (Mark Needham & Amy E Hodler, "Graph Algorithms: Practical Examples in Apache Spark and Neo4j", 2019)

"At the most abstract level, graph analytics is applied to forecast behavior and prescribe action for dynamic groups. Doing this requires understanding the relationships and structure within the group. Graph algorithms accomplish this by examining the overall nature of networks through their connections. With this approach, you can understand the topology of connected systems and model their processes." (Mark Needham & Amy E Hodler, "Graph Algorithms: Practical Examples in Apache Spark and Neo4j", 2019)

"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)

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

"Feature extraction is a way to distill large volumes of data and attributes down to a set of representative descriptive attributes. The process derives numerical values (features) for distinctive characteristics or patterns in input data so that we can differentiate categories in other data. It’s used when data is difficult for a model to analyze directly - perhaps because of size, format, or the need for incidental comparisons. Feature selection is the process of determining the subset of extracted features that are most important or influential to a target goal. It’s used to surface predictive importance as well as for efficiency." (Mark Needham & Amy E Hodler, "Graph Algorithms: Practical Examples in Apache Spark and Neo4j", 2019)

"Graph algorithms provide one of the most potent approaches to analyzing connected data because their mathematical calculations are specifically built to operate on relationships. They describe steps to be taken to process a graph to discover its general qualities or specific quantities. Based on the mathematics of graph theory, graph algorithms use the relationships between nodes to infer the organization and dynamics of complex systems. Network scientists use these algorithms to uncover hidden infomation, test hypotheses, and make predictions about behavior." (Mark Needham & Amy E Hodler, "Graph Algorithms: Practical Examples in Apache Spark and Neo4j", 2019)

"Graph embedding is the representation of the nodes and relationships in a graph asfeature vectors. [...] Graph embedding uses graph data slightly differently than in connected feature extraction. It enables us to represent entire graphs, or subsets of graph data, in a numerical format ready for machine learning tasks. This is especially useful for unsupervised learning, where the data is not categorized because it pulls in more contextual information through relationships. Graph embedding is also useful for data exploration, computing similarity between entities, and reducing dimensionality to aid in statistical analysis." (Mark Needham & Amy E Hodler, "Graph Algorithms: Practical Examples in Apache Spark and Neo4j", 2019)

"In classic graph theory, an acyclic graph that is undirected is called a tree. In computer science, trees can also be directed. A more inclusive definition would be a graph where any two nodes are connected by only one path. Trees are significant for understanding graph structures and many algorithms. They play a key role in designing networks, data structures, and search optimizations to improve categorization or organizational hierarchies." (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)

"Machine learning is not artificial intelligence (AI), but a method for achieving AI. ML uses algorithms to train software through specific examples and progressive improvements based on expected outcome - without explicit programming of how to accomplish these better results. Training involves providing a lot of data to a model and enabling it to learn how to process and incorporate that information." (Mark Needham & Amy E Hodler, "Graph Algorithms: Practical Examples in Apache Spark and Neo4j", 2019)

"Make it a habit to run Connected Components to test whether a graph is connected as a preparatory step for general graph analysis. Performing this quick test can avoid accidentally running algorithms on only one disconnected component of a graph and getting incorrect results." (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)

"Putting together the right mix of features can increase accuracy because it fundamentally influences how our models learn. Because even modest improvements can make a significant difference [...]. Connected features are features extracted from the structure of the data. These features can be derived from graph-local queries based on parts of the graph surrounding a node, or graph-global queries that use graph algorithms to identify predictive elements within data based on relationships for connected feature extraction." (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)

"Using graphs for connected feature extraction can significantly improve our predictions. The ideal graph features and algorithms vary depending on the attributes of the data, including the network domain and graph shape. We suggest first considering the predictive elements within your data and testing hypotheses with different types of connected features before fine-tuning." (Mark Needham & Amy E Hodler, "Graph Algorithms: Practical Examples in Apache Spark and Neo4j", 2019)

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

"Without peripheral and related information, solutions that attempt to predict behavior or make recommendations for varying circumstances require more exhaustive training and prescriptive rules. This is partly why AI is good at specific, well-defined tasks, but struggles with ambiguity. Graph-enhanced ML can help fill in that missing contextual information that is so important for better decisions." (Mark Needham & Amy E Hodler, "Graph Algorithms: Practical Examples in Apache Spark and Neo4j", 2019)