🪸On Systems Thinking: On Function (1960-1969)

 

"The last, and most important, characteristic of systems is that they are automatic. Although the degree of automaticity may vary over a wide spectrum, there are no systems in which human beings perform all control functions; conversely, there will probably never be systems in which no human beings are involved." (Charles D Flagle et al, "Operations Research and Systems Engineering", 1960)

"To say a system is 'self-organizing' leaves open two quite different meanings. There is a first meaning that is simple and unobjectionable. This refers to the system that starts with its parts separate" (so that the behavior of each is independent of the others' states) and whose parts then act so that they change towards forming connections of some type. Such a system is 'self-organizing' in the sense that it changes from 'parts separated' to 'parts joined'. […] In general such systems can be more simply characterized as 'self-connecting', for the change from independence between the parts to conditionality can always be seen as some form of 'connection', even if it is as purely functional […] 'Organizing' […] may also mean 'changing from a bad organization to a good one' […] The system would be 'self-organizing' if a change were automatically made to the feedback, changing it from positive to negative; then the whole would have changed from a bad organization to a good." (W Ross Ashby, "Principles of the self-organizing system", 1962)

"A system has order, flowing from point to point. If something dams that flow, order collapses. The untrained might miss that collapse until it was too late. That's why the highest function of ecology is the understanding of consequences." (Frank Herbert, "Dune", 1965)

"From a pessimistic viewpoint, it can be stated that there is no good general way of structuring a system. However, from an optimistic point of view one can say that a number of good ways of structuring systems exist and that some are better than others for any particular system. In this and the following sections, there will be a presentation of a number of structuring approaches that have merit and have been employed successfully, including functional structuring, equipment structuring, and use of various coordinate systems." (Harold Chestnut, "Systems Engineering Tools", 1965)

"The process of formulating and structuring a system are important and creative, since they provide and organize the information, which each system. establishes the number of objectives and the balance between them which will be optimized". Furthermore, they help identify and define the system parts. Furthermore, they help identify and define the system parts which make up its 'diverse, specialized structures and subfunctions'." (Harold Chestnut, "Systems Engineering Tools", 1965)

"The Systems engineering method recognizes each system is an integrated whole even though composed of diverse, specialized structures and sub-functions. It further recognizes that any system has a number of objectives and that the balance between them may differ widely from system to system. The methods seek to optimize the overall system functions according to the weighted objectives and to achieve maximum compatibility of its parts." (Harold Chestnut, "Systems Engineering Tools", 1965)

"The homeostatic principle does not apply literally to the functioning of all complex living systems, in that in counteracting entropy they move toward growth and expansion." (Daniel Katz, "The Social Psychology of Organizations", 1966)

"Traditional organizational theories have tended to view the human organization as a closed system. This tendency has led to a disregard of differing organizational environments and the nature of organizational dependency on environment. It has led also to an over-concentration on principles of internal organizational functioning, with consequent failure to develop and understand the processes of feedback which are essential to survival." (Daniel Katz, "The Social Psychology of Organizations", 1966)

"Despite two centuries of study, the integrals of general dynamical systems remain covered with darkness. To save the classical thermostatics, the practical success of which is shown by the wide use to which it has been put, we must find a way out. That is, we must find some mathematical connection between time averages of the functions of physical interest and the corresponding simple canonical averages." (Clifford Truesdell, "Six Lectures on Modern Natural Philosophy", 1966)

"Traditional organizational theories have tended to view the human organization as a closed system. This tendency has led to a disregard of differing organizational environments and the nature of organizational dependency on environment. It has led also to an over-concentration on principles of internal organizational functioning, with consequent failure to develop and understand the processes of feedback which are essential to survival." (Daniel Katz, "The Social Psychology of Organizations", 1966)

"The mathematical models for many physical systems have manifolds as the basic objects of study, upon which further structure may be defined to obtain whatever system is in question. The concept generalizes and includes the special cases of the cartesian line, plane, space, and the surfaces which are studied in advanced calculus. The theory of these spaces which generalizes to manifolds includes the ideas of differentiable functions, smooth curves, tangent vectors, and vector fields. However, the notions of distance between points and straight lines (or shortest paths) are not part of the idea of a manifold but arise as consequences of additional structure, which may or may not be assumed and in any case is not unique." (Richard L Bishop & Samuel I Goldberg, "Tensor Analysis on Manifolds", 1968)

"Perhaps the most important single characteristic of modern organizational cybernetics is this: That in addition to concern with the deleterious impacts of rigidly-imposed notions of what constitutes the application of good 'principles of organization and management' the organization is viewed as a subsystem of a larger system(s), and as comprised itself of functionally interdependent subsystems." (Richard F Ericson, "Organizational cybernetics and human values", 1969)

🪸On Systems Thinking: On Function (-1959)

"A state of equilibrium in a system does not mean, further, that the system is without tension. Systems can, on the contrary, also come to equilibrium in a state of tension" (e.g., a spring under tension or a container with gas under pressure).The occurrence of this sort of system, however, presupposes a certain firmness of boundaries and actual segregation of the system from its environment" (both of these in a functional, not a spatial, sense). If the different parts of the system are insufficiently cohesive to withstand the forces working toward displacement" (i.e., if the system shows insufficient internal firmness, if it is fluid), or if the system is not segregated from its environment by sufficiently firm walls but is open to its neighboring systems, stationary tensions cannot occur. Instead, there occurs a process in the direction of the forces, which encroaches upon the neighboring regions with diffusion of energy and which goes in the direction of an equilibrium at a lower level of tension in the total region. The presupposition for the existence of a stationary state of tension is thus a certain firmness of the system in question, whether this be its own inner firmness or the firmness of its walls." (Kurt Lewin, "A Dynamic Theory of Personality", 1935)

"An organization is a subordinate system of a specific larger system, the cooperative system, whose components are physical, biological, and personal systems. The relations with other organizations… are outside this specific cooperative system. Other organizations are a part of the social environment of the organization. It is for this reason I have used the phrase 'complex of organizations' rather than 'system'. Usually the most significant relationships of a unit organization are those with the specific cooperative system of which it is a part. It is this system which primarily and on the whole in most instances determines the chief conditions of the organization's existence." (Chester Barnard, "The Functions of the Executive The Functions of the Executive", 1938)

"The first implication of the principle of immanent change may be formulated as follows: As long as it exists and functions, any sociocultural system incessantly generates consequences which are not the results of the external factors to the system, but the consequences of the existence of the system and of its activities." (Pitirim A Sorokin, "Social and Cultural Dynamics" Vol. 4, 1937-1941)

"The first attempts to consider the behavior of so-called 'random neural nets' in a systematic way have led to a series of problems concerned with relations between the 'structure' and the 'function' of such nets. The 'structure' of a random net is not a clearly defined topological manifold such as could be used to describe a circuit with explicitly given connections. In a random neural net, one does not speak of 'this' neuron synapsing on 'that' one, but rather in terms of tendencies and probabilities associated with points or regions in the net." (Anatol Rapoport, "Cycle distributions in random nets", The Bulletin of Mathematical Biophysics 10(3), 1948)

"Systems engineering embraces every scientific and technical concept known, including economics, management, operations, maintenance, etc. It is the job of integrating an entire problem or problem to arrive at one overall answer, and the breaking down of this answer into defined units which are selected to function compatibly to achieve the specified objectives. [...] Instrument and control engineering is but one aspect of systems engineering - a vitally important and highly publicized aspect, because the ability to create automatic controls within overall systems has made it possible to achieve objectives never before attainable, While automatic controls are vital to systems which are to be controlled, every aspect of a system is essential. Systems engineering is unbiased, it demands only what is logically required. Control engineers have been the leaders in pulling together a systems approach in the various technologies." (Instrumentation Technology, 1957)

"The Systems Engineering method recognizes each system is an integrated whole even though composed of devices, specialized structures and sub-functions. It is further recognized that any system has a number of objectives and that the balance between them may differ widely from system to system. The methods seek to optimize the overall system function according to the weighted objectives and to achieve maximum capability of its parts." (Jack A Morton, "Integrating of Systems Engineering with Component Development", Electrical Manufacturing, 1959)

🦋On Chaos Theory: On Edge of Chaos

"Living systems exist in the solid regime near the edge of chaos, and natural selection achieves and sustains such a poised state." (Stuart Kauffman, "The Origins of Order: Self-organization and selection in evolution", 1993) 

"Complexity theory began with an interest on how order spring from chaos. According to complexity theory, adaption is most effective in systems that are only partially connected. The argument is that too much structure creates gridlock, while too little structure creates chaos. […] Consequently, the key to effective change is to stay poised on this edge of chaos. Complexity theory focuses managerial thinking on the interrelationships among different parts of an organization and on the trade-off of less control for greater adaptation." (Shona Brown, "Competing on the Edge, 1998)  

"The self-similarity of fractal structures implies that there is some redundancy because of the repetition of details at all scales. Even though some of these structures may appear to teeter on the edge of randomness, they actually represent complex systems at the interface of order and disorder. " (Edward Beltrami, "What is Random?: Chaos and Order in Mathematics and Life", 1999)

"The central proposition in [realistic thinking] is that human actions and interactions are processes, not systems, and the coherent patterning of those processes becomes what it becomes because of their intrinsic capacity, the intrinsic capacity of interaction and relationship, to form coherence. That emergent form is radically unpredictable, but it emerges in a controlled or patterned way because of the characteristic of relationship itself, creation and destruction in conditions at the edge of chaos." (Ralph D Stacey et al, "Complexity and Management: Fad or Radical Challenge to Systems Thinking?", 2000) 

"Networks are not en route from a random to an ordered state. Neither are they at the edge of randomness and chaos. Rather, the scale-free topology is evidence of organizing principles acting at each stage of the network formation process." (Albert-László Barabási, "Linked: How Everything Is Connected to Everything Else and What It Means for Business, Science, and Everyday Life", 2002) 

"Systems are thus seen by complexity as being ‘on the edge of chaos'. Order and chaos are in a kind of balance where the components are neither fully locked into place but yet do not fully dissolve into anarchy. chaos is not complete anarchic randomness but there is a kind of 'orderly disorder' present within all such dynamic systems." (John Urry, "Global Complexity", 2003) 

"Chaos cuts with two edges. We have seen how it is impossible to retrieve past history from current observations. We will now show that it is impossible to predict future states from the current observations." (Ivar Ekeland, "The Best of All Possible Worlds", 2006)

"Although the potential for chaos resides in every system, chaos, when it emerges, frequently stays within the bounds of its attractor(s): No point or pattern of points is ever repeated, but some form of patterning emerges, rather than randomness. Life scientists in different areas have noticed that life seems able to balance order and chaos at a place of balance known as the edge of chaos. Observations from both nature and artificial life suggest that the edge of chaos favors evolutionary adaptation." (Terry Cooke-Davies et al, "Exploring the Complexity of Projects", 2009) 

"Things evolve to evolve. Evolutionary processes are the linchpin of change. These processes of discovery represent a complexity of simple systems that flux in perpetual tension as they teeter at the edge of chaos. This whirlwind of emergence is responsible for the spontaneous order and higher, organized complexity so noticeable in biological evolution - one–celled critters beefing up to become multicellular organisms." (Lawrence K Samuels, "Defense of Chaos: The Chaology of Politics, Economics and Human Action", 2013) 

"Only at the edge of chaos can complex systems flourish. This threshold line, that edge between anarchy and frozen rigidity, is not a like a fence line, it is a fractal line; it possesses nonlinearity." (Stephen H Buhner, "Plant Intelligence and the Imaginal Realm: Beyond the Doors of Perception into the Dreaming of Earth", 2014) 

"Even more important is the way complex systems seem to strike a balance between the need for order and the imperative for change. Complex systems tend to locate themselves at a place we call 'the edge of chaos'. We imagine the edge of chaos as a place where there is enough innovation to keep a living system vibrant, and enough stability to keep it from collapsing into anarchy. It is a zone of conflict and upheaval, where the old and new are constantly at war. Finding the balance point must be a delicate matter - if a living system drifts too close, it risks falling over into incoherence and dissolution; but if the system moves too far away from the edge, it becomes rigid, frozen, totalitarian. Both conditions lead to extinction. […] Only at the edge of chaos can complex systems flourish. This threshold line, that edge between anarchy and frozen rigidity, is not a like a fence line, it is a fractal line; it possesses nonlinearity."(Stephen H Buhner, "Plant Intelligence and the Imaginal Realm: Beyond the Doors of Perception into the Dreaming of Earth", 2014) 

"To remedy chaotic situations requires a chaotic approach, one that is non-linear, constantly morphing, and continually sharpening its competitive edge with recurring feedback loops that build upon past experiences and lessons learned. Improvement cannot be sustained without reflection. Chaos arises from myriad sources that stem from two origins: internal chaos rising within you, and external chaos being imposed upon you by the environment. The result of this push/pull effect is the disequilibrium [...]." (Jeff Boss, "Navigating Chaos: How to Find Certainty in Uncertain Situations", 2015) 

𒇫On Topology: On Neighborhoods (1990-1999)

"Formally, a Cantor set is defined as a set that is totally disconnected, closed, and perfect. A totally disconnected set is a set that contains no intervals and therefore has no interior points. A closed set is one that contains all its boundary elements. (A boundary element is an element that contains elements both inside and outside the set in arbitrarily small neighborhoods.) A perfect set is a nonempty set that is equal to the set of its accumulation points. All three conditions are met by our middle-third - erasing construction, the original Cantor set." (Manfred Schroeder, "Fractals, Chaos, Power Laws Minutes from an Infinite Paradise", 1990)

"Differentiability of a function can be established by examining the behavior of the function in the immediate neighborhood of a single point a in its domain. Thus, all we need is coordinates in the vicinity of the point a. From this point of view, one might say that local coordinates have more essential qualities. However, if are not looking at individual surfaces, we cannot find a more general and universal notion than smoothness." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"Continuous functions can move freely. Graphs of continuous functions can freely branch off at any place, whereas analytic functions coinciding in some neighborhood of a point P cannot branch outside of this neighborhood. Because of this property, continuous functions can mathematically represent wildly changing wind inside a typhoon or a gentle breeze." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"Differentiability of a function can be established by examining the behavior of the function in the immediate neighborhood of a single point a in its domain. Thus, all we need is coordinates in the vicinity of the point a. From this point of view, one might say that local coordinates have more essential qualities. However, if are not looking at individual surfaces, we cannot find a more general and universal notion than smoothness." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"Similarly to the graphs of continuous functions, graphs of differentiable (smooth) functions which coincide in a neighborhood of a point P can branch off outside of the neighborhood. Because of this property, differentiable functions can represent smoothly changing natural phenomena." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

𒇫On Topology: On Neighborhoods (2000-)

"Analyzing a small part of a surface in the neighborhood of a point is called a local analysis. Though, at first glance, a local analysis may seem to be less interesting than a global analysis, time has proved otherwise." (John Tabak, "Geometry: The Language of Space and Form", 2004)

"Gauss recognized that the study of curvature of a surface in the neighborhood of a point had to be understood in order to make much progress in the study of surfaces, and one of his important contributions to differential geometry was the study of curvature. Gauss found a way to measure the curvature of a surface in a way that would enable the user to state quantitatively exactly how curved a surface is. This is harder than it may first seem, because we often confuse the curvature of a surface with that of a curve." (John Tabak, "Geometry: The Language of Space and Form", 2004)

"The fundamental concept of Gauss’s surface theory is the curvature, a quantity that is positive (and constant) for a sphere, zero for the plane and cylinder, and negative for surfaces that are 'saddle-shaped' in the neighborhood of each point." (John Stillwell, The Four Pillars of Geometry, 2000)

"Roughly speaking, a function defined on an open set of Euclidean space is differentiable at a point if we can approximate it in a neighborhood of this point by a linear map, which is called its differential (or total derivative). This differential can be of course expressed by partial derivatives, but it is the differential and not the partial derivatives that plays the central role." (Jacques Lafontaine, "An Introduction to Differential Manifolds", 2010)

"The details of the shapes of the neighborhoods are not important. If the two sets of neighborhoods satisfy the equivalence criterion, then any set that is open, closed, and so on with respect to one set of neighborhoods will be open, closed, and so on with respect to the other set of neighborhoods." (John Tabak, "Beyond Geometry: A new mathematics of space and form", 2011)

"From a narrow point of view, the precise way in which neighborhoods are specified is unimportant. As long as the resulting collection of neighborhoods satisfies the axioms that define what a topological space is, the parent set together with its neighborhoods constitute a topological space, and any such space can be studied on its own terms. From a broader point of view, given a set X, two different-looking definitions for the neighborhoods of X can sometimes result in the same topological structure - the same open sets, the same limit points, and so forth - and other times different-looking definitions for the neighborhoods of X can result in very different topological structures." (John Tabak, "Beyond geometry: a new mathematics of space and form" 2011)

"[…] topologies are determined by the way the neighborhoods are defined. Neighborhoods, not individual points, are what matter. They determine the topological structure of the parent set. In fact, in topology, the word point conveys very little information at all." (John Tabak, "Beyond Geometry: A new mathematics of space and form", 2011)

"There are many such definitions of dimension. Let me try to describe one definition of dimension. Let’s say that an 'empty object' has dimension −1. That’s an odd definition, but let’s just say that. An object has dimension n + 1 provided it has arbitrarily small neighborhoods whose boundaries have dimension n. So, for example, the real line has dimension 1 because any point in it has neighborhoods that are open intervals. The boundaries of the open intervals consist of two points, and each point is an object of dimension zero. Those two points have dimension zero because their boundary is empty." (John Tabak, "Beyond geometry: a new mathematics of space and form", 2011)

"In algebra, we defined how to operate on two elements to get another; in topology we instead define in a rough sense how close points are to each other. Specifically, beyond being a set, a topological space includes a definition of open sets or neighborhoods. This is also called defining a topology for the space." (Adam Marsh,"Mathematics for Physics: An Illustrated Handbook", 2018)

"The idea of a space covering another generalizes to any topological space: a covering space C of a space X has a continuous surjective map to X whose inverse in a neighborhood of any point of X is a union of mutually disjoint open sets homeomorphic to that neighborhood. The points that map to a point p ∈ X are called the fiber over p, and the disjoint open sets over a neighborhood of p are called sheets. Under reasonable connectivity requirements, every space then has a unique simply connected universal covering space that covers all connected covers." (Adam Marsh,"Mathematics for Physics: An Illustrated Handbook", 2018)

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𒇫On Topology: On Neighborhoods (1950-1989)

"A common and very powerful constraint is that of continuity. It is a constraint because whereas the function that changes arbitrarily can undergo any change, the continuous function can change, at each step, only to a neighbouring value." (W Ross Ashby, "An Introduction to Cybernetics", 1956)

"A manifold, roughly, is a topological space in which some neighborhood of each point admits a coordinate system, consisting of real coordinate functions on the points of the neighborhood, which determine the position of points and the topology of that neighborhood; that is, the space is locally cartesian. Moreover, the passage from one coordinate system to another is smooth in the overlapping region, so that the meaning of 'differentiable' curve, function, or map is consistent when referred to either system." (Richard L Bishop & Samuel I Goldberg, "Tensor Analysis on Manifolds", 1968)

"In the definition of a coordinate system we have required that the coordinate neighborhood and the range in Rd be open sets. This is contrary to popular usage, or at least more specific than the usage of curvilinear coordinates in advanced calculus. For example, spherical coordinates are used even along points of the z axis where they are not even 1-1. The reasons for the restriction to open sets are that it forces a uniformity in the local structure which simplifies analysis on a manifold" (there are no 'edge points') and, even if local uniformity were forced in some other way, it avoids the problem of. spelling out what we mean by differentiability at boundary points of the coordinate neighborhood; that is, one-sided derivatives need not be mentioned. On the other hand, in applications, boundary value problems frequently arise, the setting for which is a manifold with boundary. These spaces are more general than manifolds and the extra generality arises from allowing a boundary manifold of one dimension less. The points of the boundary manifold have a coordinate neighborhood in the boundary manifold which is attached to a coordinate neighborhood of the interior in much the same way as a face of a cube is attached to the interior. Just as the study of boundary value problems is more difficult than the study of spatial problems, the study of manifolds with boundary is more difficult than that of mere manifolds, so we shall limit ourselves to the latter." (Richard L Bishop & Samuel I Goldberg, "Tensor Analysis on Manifolds", 1968)

"The differential geometry of curves and surfaces has two aspects. One, which may be called classical differential geometry, started with the beginnings of calculus. Roughly speaking, classical differential geometry is the study of local properties of curves and surfaces. By local properties we mean those properties which depend only on the behavior of the curve or surface in the neighborhood of a point. The methods which have shown themselves to be adequate in the study of such properties are the methods of differential calculus. Because of this, the curves and surfaces considered in differential geometry will be defined by functions which can be differentiated a certain number of times." (Manfredo P Do Carmo, "Differential Geometry of Curves and Surfaces", 1976)

"A surface is a topological space in which each point has a neighbourhood homeomorphic to the plane, ad for which any two distinct points possess disjoint neighbourhoods. […] The requirement that each point of the space should have a neighbourhood which is homeomorphic to the plane fits exactly our intuitive idea of what a surface should be. If we stand in it at some point (imagining a giant version of the surface in question) and look at the points very close to our feet we should be able to imagine that we are standing on a plane. The surface of the earth is a good example. Unless you belong to the Flat Earth Society you believe it to,be (topologically) a sphere, yet locally it looks distinctly planar. Think more carefully about this requirement: we ask that some neighbourhood of each point of our space be homeomorphic to the plane. We have then to treat this neighbourhood as a topological space in its own right. But this presents no difficulty; the neighbourhood is after all a subset of the given space and we can therefore supply it with the subspace topology." (Mark A Armstrong, "Basic Topology", 1979)

"This notion of each point in a space having a collection of 'neighbourhoods', the neighbourhoods leading in turn to a good definition of continuous function, is the crucial one. Notice that in defining neighbourhoods in a euclidean space we used very strongly the euclidean distance between points. In constructing an abstract space we would like to retain the concept of neighbourhood but rid ourselves of any dependence on a distance function. (A topological equivalence does not preserve distances.)" (Mark A Armstrong, "Basic Topology", 1979)

"[...] the image of a stable map of a surface into space looks like in the neighborhood of each point. If no neighborhood, no matter how small, of a given point looks like a mildly bent disc, then it is a singular point. A stable map can have three kinds of singular points. In a neighborhood of a double point a surface looks like two sheets of some fabric crossing along a so-called double curve. A neighborhood of a triple point looks like three surface sheets crossing transversely. Thus triple points are isolated. You can see why a quadruple point is unstable. A slight perturbation of one of the sheets would make four sheets cross each other so as to produce a little tetrahedral cell. Double curves are either closed, extend to infinity, terminate on the border or simply end at very special points, called pinch points." (George K Francis, "A Topological Picturebook", 1987)

"Elementary functions, such as trigonometric functions and rational functions, have their roots in Euclidean geometry. They share the feature that when their graphs are 'magnified' sufficiently, locally they 'look like' straight lines. That is, the tangent line approximation can be used effectively in the vicinity of most points. Moreover, the fractal dimension of the graphs of these functions is always one. These elementary 'Euclidean' functions are useful not only because of their geometrical content, but because they can be expressed by simple formulas. We can use them to pass information easily from one person to another. They provide a common language for our scientific work. Moreover, elementary functions are used extensively in scientific computation, computer-aided design, and data analysis because they can be stored in small files and computed by fast algorithms." (Michael Barnsley, "Fractals Everwhere", 1988

🪸On Systems Thinking: On Neighborhoods

"A state of equilibrium in a system does not mean, further, that the system is without tension. Systems can, on the contrary, also come to equilibrium in a state of tension (e.g., a spring under tension or a container with gas under pressure).The occurrence of this sort of system, however, presupposes a certain firmness of boundaries and actual segregation of the system from its environment (both of these in a functional, not a spatial, sense). If the different parts of the system are insufficiently cohesive to withstand the forces working toward displacement (i.e., if the system shows insufficient internal firmness, if it is fluid), or if the system is not segregated from its environment by sufficiently firm walls but is open to its neighboring systems, stationary tensions cannot occur. Instead, there occurs a process in the direction of the forces, which encroaches upon the neighboring regions with diffusion of energy and which goes in the direction of an equilibrium at a lower level of tension in the total region. The presupposition for the existence of a stationary state of tension is thus a certain firmness of the system in question, whether this be its own inner firmness or the firmness of its walls." (Kurt Lewin, "A Dynamic Theory of Personality", 1935)

"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 characteristic of such chaotic dynamics is an extreme sensitivity to initial conditions (exponential separation of neighboring trajectories), which puts severe limitations on any forecast of the future fate of a particular trajectory. This sensitivity is known as the ‘butterfly effect’: the state of the system at time t can be entirely different even if the initial conditions are only slightly changed, i.e., by a butterfly flapping its wings." (Hans J Korsch et al, "Chaos: A Program Collection for the PC", 2008)

"A typical complex system consists of a vast number of identical copies of several generic processes, which are operating and interacting only locally or with a limited number of not necessary close neighbours. There is no global leader or controller associated to such systems and the resulting behaviour is usually very complex." (Jirí Kroc & Peter M A Sloot, "Complex Systems Modeling by Cellular Automata", Encyclopedia of Artificial Intelligence, 2009)

"Cellular automata are mathematical models for complex natural systems containing large numbers of simple identical components with local interactions. They consist of a lattice of sites, each with a finite set of possible values. The value of the sites evolve synchronously in discrete time steps according to identical rules. The value of a particular site is determined by the previous values of a neighbourhood of sites around it." (Stephen Wolfram, "Nonlinear Phenomena, Universality and complexity in cellular automata", Physica 10D, 1984)

"A mathematical construct and technique that models a system in discrete time and discrete space in which the state of a cell depends on transition rules and the states of neighboring cells." (Charles M Macal, "Agent Based Modeling and Artificial Life", 2009) 

"A spatiotemporal modeling technique in which a set of rules is applied to determine the state transitions of individual cells based on each cell’s current state and the states of its neighbors." (May Yuan, "Challenges and Critical Issues for Temporal GIS Research and Technologies", 2009) 

"Cellular Automata (CA) are discrete, spatially explicit extended dynamic systems composed of adjacent cells characterized by an internal state whose value belongs to a finite set. The updating of these states is made simultaneously according to a common local transition rule involving only a neighborhood of each cell." (Ramon Alonso-Sanz, "Cellular Automata with Memory", 2009) 

"A discrete model consisting of a grip of cells each of which have a finite number of defined states where the state of a cells is a function of the states of neighboring cells and the transition among states is according to some predefined updating rule. (Brian L Heath & Raymond R Hill, "Agent-Based Modeling: A Historical Perspective and a Review of Validation and Verification Efforts, 2010)

"A limit cycle is an isolated closed trajectory. Isolated means that neighboring trajectories are not closed; they spiral either toward or away from the limit cycle. If all neighboring trajectories approach the limit cycle, we say the limit cycle is stable or attracting. Otherwise the limit cycle is unstable, or in exceptional cases, half-stable. Stable limit cycles are very important scientifically - they model systems that exhibit self-sustained oscillations. In other words, these systems oscillate even in the absence of external periodic forcing." (Steven H Strogatz, "Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering", 2015)

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

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

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

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

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

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

🕸️On Graph Theory: On Edges (-1999)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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