Graph Theory Quotes

Though blessed with the enviable properties of a mink coat—graceful, unreasonable, and impractical no matter what she was draped over—she was nevertheless one of those people whose personality proved to be the bane of modern mathematicians. She was neither a flat nor solid shape. She showed no symmetry at all. Trigonometry, Calculus and Statistics all proved useless. Her Pie Chart was a muddle of arbitrary wedges, her Line Graph, the silhouette of the Alps. And just when one listed her under Chaos Theory—Butterfly Effects, Weather Predictions, Fractals, Bifurcation diagrams and whatnot—she showed up as an equilateral triangle, sometimes even a square.

Geometric diagrams are to geometers what board and pieces are to chessmasters: visual aids, helpful but not indispensable.

To impress my Ph.D. students with just how bizarre the quantum theory is, I sometimes ask them to calculate the probability that their atoms will suddenly dissolve and reappear on the other side of a brick wall. Such a teleportation event is impossible under Newtonian physics but is actually allowed under quantum mechanics. The answer, however, is that one would have to wait longer than the lifetime of the universe for this to occur. (If you used a computer to graph the Schrödinger wave of your own body, you would find that it very much resembles all the features of your body, except that the graph would be a bit fuzzy, with some of your waves oozing out in all directions. Some of your waves would extend even as far as the distant stars. So there is a very tiny probability that one day you might wake up on a distant planet.)

I use logic all the time in mathematics, and it seems to yield “correct” results, but in mathematics “correct” by and large means “logical”, so I’m back where I started. I can’t defend logic because I can’t remove my glasses.

There is an old debate," Erdos liked to say, "about whether you create mathematics or just discover it. In other words, are the truths already there, even if we don't yet know them?" Erdos had a clear answer to this question: Mathematical truths are there among the list of absolute truths, and we just rediscover them. Random graph theory, so elegant and simple, seemed to him to belong to the eternal truths. Yet today we know that random networks played little role in assembling our universe. Instead, nature resorted to a few fundamental laws, which will be revealed in the coming chapters. Erdos himself created mathematical truths and an alternative view of our world by developing random graph theory. Not privy to nature's laws in creating the brain and society, Erdos hazarded his best guess in assuming that God enjoys playing dice. His friend Albert Einstein, at Princeton, was convinced of the opposite: "God does not play dice with the universe.

For example, I do a little bit of data analysis, like I was saying. For tech companies mostly. They’ll give me a lot of data – say user experience data, like how long users spend on each section of a website – and I’ll spend a few hours making graphs and whatnot. Say it takes me – I don’t know, four hours to make these graphs, and I’ll pretend it took me ten hours, to get extra money. He glances over at her again, and adds: You might think that’s immoral, I don’t know. But anyway, never mind that for a second. The four hours that I actually spend making the graphs, and the ten hours that I get paid for: what is that? Like, any of that: what is it? At least when I worked as a delivery driver, I knew what I was doing. Someone wanted a Big Mac, and I brought it to them, and the amount I got paid was like, what it was worth to that person not to have to collect their own burger. The amount they will pay, not to leave the house, is the amount I will accept, yes to leave the house. Minus whatever the app is taking. If you get me. I get you. You’re making perfect sense. Oh good, he says. Because in the data analysis example, my question is, what is the money that’s being paid to me? It’s the money that the company will pay, to have their own information explained back to them in a graph. And how much money should that be? Clearly no one knows, because at the end I’ll make up a number of hours and they’ll just pay me for that number. I guess the graph is supposed to make the company more profitable, in theory, but no one knows by how much, it’s all made up.

It appears then that the essence of chess is its abstract structure. Names and shapes of pieces, colors of squares, whether the “squares” are in fact square, even the physical existence of board and pieces, are all irrelevant. What is relevant is the number and geometric arrangement of the “squares”, the number of types of piece and the number of pieces of each type, the quantitative-geometric power of each piece, etc. Everything else is a visual aid or a fairy tale.

It is a curious fact, and one to which no one knows quite how much importance to attach, that something like 85% of all known worlds in the Galaxy, be they primitive or highly advanced, have invented a drink called jynnan tonnyx, or gee-N'N-T'N-ix, or jinond-o-nicks, or any one of a thousand or more variations on the same phonetic theme. The drinks themselves are not the same, and vary between the Sivolvian 'chinanto/mnigs' which is ordinary water served at slightly above room temperature, and the Gagrakackan 'tzjin-anthony-ks' which kill cows at a hundred paces; and in fact the one common factor between all of them, beyond the fact that the names sound the same, is that they were all invented and named before the worlds concerned made contact with any other worlds. What can be made of this fact? It exists in total isolation. As far as any theory of structural linguistics is concerned it is right off the graph, and yet it persists. Old structural linguists get very angry when young structural linguists go on about it. Young structural linguists get deeply excited about it and stay up late at night convinced that they are very close to something of profound importance, and end up becoming old structural linguists before their time, getting very angry with the young ones. Structural linguistics is a bitterly divided and unhappy discipline, and a large number of its practitioners spend too many nights drowning their problems in Ouisghian Zodahs.

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.

Wild animals enjoying one another and taking pleasure in their world is so immediate and so real, yet this reality is utterly absent from textbooks and academic papers about animals and ecology. There is a truth revealed here, absurd in its simplicity. This insight is not that science is wrong or bad. On the contrary: science, done well, deepens our intimacy with the world. But there is a danger in an exclusively scientific way of thinking. The forest is turned into a diagram; animals become mere mechanisms; nature's workings become clever graphs. Today's conviviality of squirrels seems a refutation of such narrowness. Nature is not a machine. These animals feel. They are alive; they are our cousins, with the shared experience kinship implies. And they appear to enjoy the sun, a phenomenon that occurs nowhere in the curriculum of modern biology. Sadly, modern science is too often unable or unwilling to visualize or feel what others experience. Certainly science's "objective" gambit can be helpful in understanding parts of nature and in freeing us from some cultural preconceptions. Our modern scientific taste for dispassion when analyzing animal behaviour formed in reaction to the Victorian naturalists and their predecessors who saw all nature as an allegory confirming their cultural values. But a gambit is just an opening move, not a coherent vision of the whole game. Science's objectivity sheds some assumptions but takes on others that, dressed up in academic rigor, can produce hubris and callousness about the world. The danger comes when we confuse the limited scope of our scientific methods with the true scope of the world. It may be useful or expedient to describe nature as a flow diagram or an animal as a machine, but such utility should not be confused with a confirmation that our limited assumptions reflect the shape of the world. Not coincidentally, the hubris of narrowly applied science serves the needs of the industrial economy. Machines are bought, sold, and discarded; joyful cousins are not. Two days ago, on Christmas Eve, the U.S. Forest Service opened to commercial logging three hundred thousand acres of old growth in the Tongass National Forest, more than a billion square-meter mandalas. Arrows moved on a flowchart, graphs of quantified timber shifted. Modern forest science integrated seamlessly with global commodity markets—language and values needed no translation. Scientific models and metaphors of machines are helpful but limited. They cannot tell us all that we need to know. What lies beyond the theories we impose on nature? This year I have tried to put down scientific tools and to listen: to come to nature without a hypothesis, without a scheme for data extraction, without a lesson plan to convey answers to students, without machines or probes. I have glimpsed how rich science is but simultaneously how limited in scope and in spirit. It is unfortunate that the practice of listening generally has no place in the formal training of scientists. In this absence science needlessly fails. We are poorer for this, and possibly more hurtful. What Christmas Eve gifts might a listening culture give its forests? What was the insight that brushed past me as the squirrels basked? It was not to turn away from science. My experience of animals is richer for knowing their stories, and science is a powerful way to deepen this understanding. Rather, I realized that all stories are partly wrapped in fiction—the fiction of simplifying assumptions, of cultural myopia and of storytellers' pride. I learned to revel in the stories but not to mistake them for the bright, ineffable nature of the world.

A theorem is no more proved by logic and computation than a sonnet is written by grammar and rhetoric, or than a sonata is composed by harmony and counterpoint, or a picture painted by balance and perspective. Logic and computation, grammar and rhetoric, harmony and counterpoint, balance and perspective, can be seen in the work after it is created, but these forms are, in the final analysis, parasitic on, they have no existence apart from, the creativity of the work itself. Thus the relation of logic to mathematics is seen to be that of an applied science to its pure ground, and all applied science is seen as drawing sustenance from a process of creation with which it can combine to give structure, but which it cannot appropriate.

This brings me to an objection to integrated information theory by the quantum physicist Scott Aaronson. His argument has given rise to an instructive online debate that accentuates the counterintuitive nature of some IIT's predictions. Aaronson estimates phi.max for networks called expander graphs, characterized by being both sparsely yet widely connected. Their integrated information will grow indefinitely as the number of elements in these reticulated lattices increases. This is true even of a regular grid of XOR logic gates. IIT predicts that such a structure will have high phi.max. This implies that two-dimensional arrays of logic gates, easy enough to build using silicon circuit technology, have intrinsic causal powers and will feel like something. This is baffling and defies commonsense intuition. Aaronson therefor concludes that any theory with such a bizarre conclusion must be wrong. Tononi counters with a three-pronged argument that doubles down and strengthens the theory's claim. Consider a blank featureless wall. From the extrinsic perspective, it is easily described as empty. Yet the intrinsic point of view of an observer perceiving the wall seethes with an immense number of relations. It has many, many locations and neighbourhood regions surrounding these. These are positioned relative to other points and regions - to the left or right, above or below. Some regions are nearby, while others are far away. There are triangular interactions, and so on. All such relations are immediately present: they do not have to be inferred. Collectively, they constitute an opulent experience, whether it is seen space, heard space, or felt space. All share s similar phenomenology. The extrinsic poverty of empty space hides vast intrinsic wealth. This abundance must be supported by a physical mechanism that determines this phenomenology through its intrinsic causal powers. Enter the grid, such a network of million integrate-or-fire or logic units arrayed on a 1,000 by 1,000 lattice, somewhat comparable to the output of an eye. Each grid elements specifies which of its neighbours were likely ON in the immediate past and which ones will be ON in the immediate future. Collectively, that's one million first-order distinctions. But this is just the beginning, as any two nearby elements sharing inputs and outputs can specify a second-order distinction if their joint cause-effect repertoire cannot be reduced to that of the individual elements. In essence, such a second-order distinction links the probability of past and future states of the element's neighbours. By contrast, no second-order distinction is specified by elements without shared inputs and outputs, since their joint cause-effect repertoire is reducible to that of the individual elements. Potentially, there are a million times a million second-order distinctions. Similarly, subsets of three elements, as long as they share input and output, will specify third-order distinctions linking more of their neighbours together. And on and on. This quickly balloons to staggering numbers of irreducibly higher-order distinctions. The maximally irreducible cause-effect structure associated with such a grid is not so much representing space (for to whom is space presented again, for that is the meaning of re-presentation?) as creating experienced space from an intrinsic perspective.

But the pure mathematician’s goal is to have a good time, not to be efficient; and machine-building is a lot more fun than problem-solving.

Topology] is a purely qualitative subject where quantity is banned. In it two figures are always equivalent if it is possible to pass from one to the other by a continuous deformation, whose mathematical law can be of any sort whatsoever as long as continuity is respected.

Getting scientists to consider the validity of Indigenous knowledge is like swimming upstream in cold, cold water. They've been so conditioned to be skeptical of even the hardest of hard data that bending their minds toward theories that are verified without the expected graphs or equations is tough. Couple that with the unblinking assumption that science has cornered the market on truth and there's not much room for discussion.

There are also books that contain collections of papers or chapters on particular aspects of knowledge discovery—for example, Relational Data Mining edited by Dzeroski and Lavrac [De01]; Mining Graph Data edited by Cook and Holder [CH07]; Data Streams: Models and Algorithms edited by Aggarwal [Agg06]; Next Generation of Data Mining edited by Kargupta, Han, Yu, et al. [KHY+08]; Multimedia Data Mining: A Systematic Introduction to Concepts and Theory edited by Z. Zhang and R. Zhang [ZZ09]; Geographic Data Mining and Knowledge Discovery edited by Miller and Han [MH09]; and Link Mining: Models, Algorithms and Applications edited by Yu, Han, and Faloutsos [YHF10]. There are many tutorial notes on data mining in major databases, data mining, machine learning, statistics, and Web technology conferences.

Feynman converts! According to Frank Wilczek, Feynman eventually lost confidence in his particles-only view of nature: Feynman told me that when he realized that his theory of photons and electrons is mathematically equivalent to the usual theory, it crushed his deepest hopes...He gave up when, as he worked out the mathematics of his version of quantum electrodynamics, he found the fields, introduced for convenience, taking on a life of their own. He told me he lost confidence in his program of emptying space...(see quote in Chap. 2, "The Gravitational Field")-F. Wilczek (W2008, p. 84. 89) However this "conversion" is not generally known. Most physicists today routinely use Feynman graphs while promulgating and perpetuating the particle picture of nature, puzzling and paradoxical as that picture may be.

Readers who are already familiar with the basic notions of graph theory should be able to skip this section: for those who want more, there are some very readable introductions such as Bollobás (1998) and Chartrand (1985).

The proponents of chaos theory suggest that what in real life appear to be purely random measurements are, in fact, generated by some deterministic set of equations, and that these equations can be deduced from the patterns that appear in a Poincaré plot. For instance, some proponents of chaos theory have taken the times between human heartbeats and put them into Poincaré plots. They claim to see patterns in these plots, and they have found deterministic generating equations that appear to produce the same type of pattern. As of this writing, there is one major weakness to chaos theory applied in this fashion. There is no measure of how good the fit is between the plot based on data and the plot generated by a specific set of equations. The proof that the proposed generator is correct is based on asking the reader to look at two similar graphs. This eyeball test has proved to be a fallible one in statistical analysis. Those things that seem to the eye to be similar or very close to the same are often drastically different when examined carefully with statistical tools developed for this purpose.

Fractals are beautiful to look at when graphed and, thus, had been used in arts and the sciences.  It first appeared in art in the 19th century, at a painting of Mt. Fuji, which shows a great wave that threatens an open boat, wherein the dimension of the wave is the approximation of a circle’s diameter.  This is an example of a natural fractal, way before Felix Hausdorff first presented the theory of the fractal dimension in 1868.[viii]  They are used to incorporate nature into artistic elements and, thus, had been used to highlight pieces of visual arts.  It became well known when genetic programming entered the world in the 20th century, which optimized parameters of what is called “Mandelbrot sets” that are useful in generating certain biomorphs. 

An easy and even a pleasant task is it to reduce human problems to numerical figures in black and white on charts and graphs, an infinitely difficult one is it to suggest concrete solutions, or to extend true charity in individual lives. Yet life can only be lived in the individual; almost invariably the individual refuses to conform to the theories and the classifications of the statistician.

The Pythagoreans were mystics, and believed that the tetrahedron, cube, octahedron, and icosahedron respectively underlay the structure of the four elements of Greek science: fire, earth, air, and water. The dodecahedron they identified with the universe as a whole.

The Pythagoreans were mystics, and believed that the tetrahedron, cube, octahedron, and icosahedron respectively underlay the structure of the four elements of Greek science: fire, earth, air, and water. The dodecahedron they identified with the universe as a whole. Plato was quite taken by all this and spent some time in his dialogue Timaeus (named after the Pythagorean who is the chief interlocutor) discussing the connection between the five regular polyhedra and the structure of the universe. For this reason the regular polyhedra came to be known as the “platonic solids”.

A topologist enters a coffee shop, orders coffee and a doughnut, and is served. Preoccupied with topological theorems, he takes a bite out of his coffee cup and has to finish his thoughts in a nearby emergency ward. His mistake is somewhat understandable as a doughnut and coffee cup are topologically equivalent

Map coloring problem. Find the smallest number m such that the faces of every planar graph can be colored with m or fewer colors in such a way that faces sharing a border have different colors.

There has been a steady escalation of conditions since our first discussions in Chapter 2. Then we talked about plain old graphs. Subsequently we restricted our attention to planar graphs, then to planar connected graphs, then to planar connected graphs with each edge bordering two faces (polygonal graphs), and now to planar connected regular graphs with each edge bordering two faces and all faces bounded by the same number of edges (platonic graphs).

A statement that mathematicians can prove is called a “theorem”, or sometimes a “lemma” or “corollary” if it bears a certain relationship to another theorem.

A statement that mathematicians believe but cannot as yet prove is called a “conjecture”.

Algebra is another branch of mathematics; it studies sets on which there have been defined things called “operations”. An operation on a set is a rule whereby two or more elements of the set can be combined to form another element of the set. High school algebra is the algebra of one specific set, the set of real numbers, and four specific operations defined on that set, addition, subtraction, multiplication, and division. High school algebra is only the tip of the algebraic iceberg.

It has been said that geometry is the art of applying good reasoning to bad diagrams.

In particular the rules of logic tell us how to create, from the opening arrangement (the list of axioms), new arrangements (called “theorems”).

Leonhard Euler (pronounced “oiler”, 1707–1783) is judged by all to have been the most productive, and by many to have been the best, mathematician of modern times. He was Swiss, but spent much of his life in Russia because he had a big family and Catherine the Great offered him a lot of money. His paper “The Seven Bridges of Königsberg” (1736), which we will discuss in Chapter 8, is the earliest known work on the theory of graphs. The theorem now known as Euler’s Formula was proved by Euler in 1752. It is one of the classic theorems of elementary mathematics and plays a central role in the next three chapters of this book.

you should not allow yourself to be convinced by repeated failures,

The village barber shaves those and only those men who live in the village and do not shave themselves. The village barber is a man and he lives in the village. Consider the question “Who shaves the barber?

Over the years, Facebook has executed an effective playbook that does exactly this, at scale. Take Instagram as an example—in the early days, the core product tapped into Facebook’s network by making it easy to share photos from one product to the other. This creates a viral loop that drives new users, but engagement, too, when likes and comments appear on both services. Being able to sign up to Instagram using your Facebook account also increases conversion rate, which creates a frictionless experience while simultaneously setting up integrations later in the experience. A direct approach to tying together the networks relies on using the very established social graph of Facebook to create more engagement. Bangaly Kaba, formerly head of growth at Instagram, describes how Instagram built off the network of its larger parent: Tapping into Facebook’s social graph became very powerful when we realized that following your real friends and having an audience of real friends was the most important factor for long-term retention. Facebook has a very rich social graph with not only address books but also years of friend interaction data. Using that info supercharged our ability to recommend the most relevant, real-life friends within the Instagram app in a way we couldn’t before, which boosted retention in a big way. The previous theory had been that getting users to follow celebrities and influencers was the most impactful action, but this was much better—the influencers rarely followed back and engaged with a new user’s content. Your friends would do that, bringing you back to the app, and we wouldn’t have been able to create this feature without Facebook’s network. Rather than using Facebook only as a source of new users, Instagram was able to use its larger parent to build stronger, denser networks. This is the foundation for stronger network effects. Instagram is a great example of bundling done well, and why a networked product that launches another networked product is at a huge advantage. The goal is to compete not just on features or product, but to always be the “big guy” in a competitive situation—to bring your bigger network as a competitive weapon, which in turn unlocks benefits for acquisition, engagement, and monetization. Going back to Microsoft, part of their competitive magic came when they could bring their entire ecosystem—developers, customers, PC makers, and others—to compete at multiple levels, not just on building more features. And the most important part of this ecosystem was the developers.

We assume familiarity with programming, a basic understanding of computational performance issues, complexity theory, introductory level calculus and some of the terminology of graph theory.

E. M. Forster’s famous advice to “Only connect!” is beginning to look superfluous. A theory in which the building blocks of the Universe are mathematical structures—known as graphs—that do nothing but connect has just passed its first experimental test.

Four Color Conjecture is one of the most famous unsolved problems in mathematics.

Definition 20. If some new vertices of degree 2 are added to some of the edges of a graph G, the resulting graph H is called an expansion of G.

Shut up and calculate!” certainly doesn’t sound appealing if you’re not mathematically inclined. But, even if you’re a physicist, what’s the virtue in shutting up and calculating? Mermin himself provided the answer in his 1989 article. “It is a fact about the quantum theory of paramount importance which ought to be emphasized in every popular and semi-popular exposition, that it permits us to calculate measurable quantities with unprecedented precision.” Quantum physics works. The calculations enabled by the theory are astonishing in their range of applicability and the accuracy of their results. Quantum physics tells us how long it will take to heat up your frying pan to cook your eggs and how large a dying white dwarf star can be without collapsing. It reveals the exact shape of the double helix at the core of life, it tells us the age of the immortal cattle on the rock walls at Lascaux, it speaks of atoms split beneath the stone heart of Africa eons before Oppenheimer and the blinding light of Trinity. It predicts with uncanny accuracy the precise darkness of the blackest night. It shows us the history of the universe in a handful of dust. If shutting up is the price of doing these calculations, then pass the ball gag and break out the graph paper.

The theory of space and time is a cultural artifact made possible by the invention of graph paper. If we had invented the digital computer before graph paper, we might have a very different theory of information today.