Dear poets and lovers of literature,

I have been asked by William to give a short introduction to the world of graphs.

Perhaps it will be easier for me to express my own reason for becoming interested in graph theory. If one looks at the most common mathematical structures, e.g. groups, rings, fields, vector spaces, Hilbert Spaces, etc. it is difficult to escape the impression that they are all objects of a very specialized kind.

Sometimes five, ten or even fifteen axioms are needed to define these objects. Of course one may easily feel that the choice of these axioms is somehow arbitrary. Therefore it seems that pursuing the study of these structures would result in putting a strong bias to one’s mathematical activity in the long term. I was attracted by graphs because they express a concept that may be described by a single axiom: friendship. We are either friends or not, there is no alternative, no third excluded. This is in some sense what the structure of a graph conveys: when you see two points connected by a line that is the geometrical expression of this relation, whereby the two points can be considered “friends”, or as we say: “neighbours”. Of course what one immediately learns is that, if A is a friend of B and B is a friend of C, that does not mean that A should be a friend of C: that is, the relation of friendship is not transitive (as we know very well).

Now the word friendship may be changed, if you wish, with any symmetric relation that you like, or even asymmetric, provided we are ready to switch to the world of digraphs, or directed graphs.

The relation may not be limited to two people only, it may involve any number of people, provided we pass from the world of graphs to the world of hypergraphs.

Of course the more general is the concept, the more difficult is to establish theorems about these objects. If any of you has a genuine interest in graph theory, he or she is welcome to join the discussion group that I run on Yahoo, at the web site

http://www.yahoogroups.com/group/Graphs

Happy continuation of the activities in this blog.

Cheers,

David

# On Graph Theory. David Cariolaro

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Welcome to David, lecturer at the Academia Sinica in Taiwan. For more information on his acedemic career click the following link:

http://www.math.sinica.edu.tw/post-doctor/cariolaro/

The whole thing reminds me of the Tractatus.. But later in life Wittgenstein abandoned the idea of a ‘mathematisation’ of the world. Fascinating though.

VERY interesting David! Thanks Luca for hosting such ‘alternative’ posts.

I like the idea of the graph and how you associate frienship to mathematical concepts; it almost gives the illusion to read reality (hence feelings and humans) with the simple aid of numbers. Perhaps I only claim this since mathematics is alien to me!

The idea of the graph is really intriguing; I picture it as an externalization of a concept, as a way to place in space, visually, a combination of numbers and letters. It is like producing a text, a way to express an idea, emotion or theory through writing, being it prose or poetry.

However there can be a bad text, umblalanced, poorly written piece of writing – but graph represent perfection in actualization. What each text seeks to attain.

Perhps here I am only blethering….though I like to think of writing as a perennial attempt to achieve the immaculate perfection of a graph

thanks for the insightful reflections mariangela.. I also found striking similarities between the graph theory and language as it is seen after post-modernism..

Mathematics , after all, is a language as well.

It is a language indeed Luca, you are right.

What i think is remarkable is the idea that whilst numbers/letters are a language, graphs constitute the form – what we would call the narrative sturctures. Each graphs has its own curves, trajectory, direction, each one is different, and yet there is always a perfect unmistakeble balance between numbers and letters and the line/picture.

Forms and content are perfectly entwined in a way unique to that very sequence of numbers.

Can we say the same for words? isn’t it the case that narrative forms are not always or necessarly crafted in relation to the content? Can mathematics be a model for writing? As you claimed earlier, not rarely philosophers or thinkers were/are laso mathematicians – yet Wittgenstein did give up….

Throughout history of Philosophy those interested in the idea of a philosophia universalis flirted with a vis geometrica which would combine shapes & numbers (or reality & words): Pythagoras, Plato, Nicholas of Cusa, Marsilio Ficino.. the list is endless.

But Maths is a closed and self-referential system. When it is applied to the natural world it becomes a series of approximations, working hypotheses, and best guesses. That these approximations and guesses actually work in most cases is a tribute to the system and its thinkers. But absolute perfection is no more than a powerful illusion. Language too can be seen as a closed, self-referential system capable of perfection. But language is required to interact with the natural world all the time. One cannot really study language as distinct from its speakers, its context, its culture.

There is, however, a greater sense in which we, as human beings, are the reasonably happy prisoners of all languages. We cannot now think ourselves into a pre-linguistic or extra-linguistic state, and therefore

ourworld, is in essence a linguistic construct and Mathematics is one of our languages. But, when I say ‘reasonably happy’ I refer to those times when language is inadequate (O quanto è corto il dire,’ as our master Dante said). These are often the most intense experiences, the most personal. And they point to a general ‘bad-fit’ between language and experience that goes unnoticed in humdrum matters. I think that this ‘bad fit’ is indicative of a pre-linguistic experience in which self-consciousness has not yet been achieved, the babbling stage of infancy perhaps, in which man was at ease with his world and before a consciousness of her/his self intruded between the self and the environment. We are half-remembering that there is another way to be, one that we no longer have access to because we are prisoners of language.Amalagamation fascinating concept of the graph theory (see Crispin St.John Alvah Nash-William) is part, Luca, of the project on creole poems.

will

Will explain these terms as you use them: ‘amalgamation’ and ‘creole poem’.

Bill see my interview to Vocativo on Faraeditore:

Luigi Metropoli: Più che nello stile, la creolità, il tema ricorrente della tua raccolta di poesia fin dal titolo (con rimando anche alla materia musicale), sembra emergere dallo sfondo: luoghi, rimandi, alcuni paesaggi e frutti tropicali che abbondano tra i versi. Tuttavia lo spagnolo, il portoghese-brasiliano e l’inglese si innestano sull’italiano. “La verità sta nel contrappunto”, scrivi nella prima poesia. Cosa è per te la creolità e come un poeta dell’Occidente “Sviluppato” può renderla nei suoi versi e/o farsene portavoce? Quale il suo valore politico?

William Stabile: Per quanto mi riguarda, la creolità è un moto interno; il veicolo di espressione del mio essere poeta nel mondo, di trasformarmi ogni volta in uomo nuovo.

Non molto tempo fa, con un amico ben grasso, che oramai ha scelto di trasformarsi in un sottile ectoplasma metropolitano, mettemmo su carta questi assiomi che tengo sul mio tavolo di lavoro:

“Sense can transmit between languages. There is no need for communication to be held within one set of symbols. Order is not significant… Inspiration of Amalgamation is shared and boundless.”

Ciò che mi interessa è esplorare il concetto di creolità o quello che io chiamo (mutuandolo dalla Teoria dei Grafi che ho conosciuto grazie al Prof. David Cariolaro e al grande scienziato Prof. Crispin St. John Alvah Nash-William) Amalgamation.

“Amalgamation means that communications through symbols can have multiple authors.”

L’obiettivo finale è quello di poter scrivere, un giorno, un poema, in varie lingue, un opera che tutti possano leggere ed intendere, che a tutti possa comunicare…

Una opera dove possano fondersi tutte le discipline, le Arti, senza distinzioni né gerarchie. Un opera “creola”.

Un poeta dell’Occidente “Sviluppato” come dici tu, ed io aggiungo la S di “Sfruttatore” a maggior ragione, oggi, non può ingannare o ingannarsi. Non possiamo confondere il concetto di globalizzazione con creolità. Oggi, si crede, o tendiamo a credere, che il mondo è globale e quindi avvicina le differenze e le sa miscelare. In verità, le distrugge. La creolità esalta le differenze e si scontra con l’uniformità e la standardizzazione imperante.

Personalmente, quando parlo di creolità penso ad una regione specifica della terra: i Caraibi. Chi abbia esperito i Caraibi ha gli strumenti adatti per leggere la creolità. Sa che lì forme di espressione artistiche hanno sempre un valore politico.

Non c’è altra via per il creolo di essere nel mondo che la via politica. Il poeta creolo è un uomo politico. Anche se non è quello che i partiti le idee e gli ideologi di sinistra vogliono imbrigliare. Poi, un valore artistico-politico della creolità è stato già espresso nella Conferenza pronunciata domenica 22 maggio 1988 al Festival Caraibico della Seine-Saint-Denis contenuta nel libro Elogio della Creolità. Questo libro è stato uno dei miei testi di formazione, un testo che invito a leggere.

Eccone un passo:

ELOGIO DELLA CREOLITÀ

di Bernabé, Chamoiseau, Confiant

Noi non siamo europei né africani né asiatici, ci dichiariamo creoli. Il nostro sarà un atteggiamento interiore, una vigilanza, meglio ancora, una specie di involucro mentale al cui interno costruiremo il nostro mondo nella piena consapevolezza del mondo. Queste parole non si fondano su una teoria o su principi scientifici. Sono una testimonianza. Derivano da un’esperienza infruttuosa che abbiamo fatto prima di impegnarci a riattivare il nostro potenziale creativo e a esprimere quello che in realtá siamo. Non ci rivolgiamo solo agli scrittori, ma a tutti gli artisti delle nostre terre […] a qualunque disciplina appartengano, che siano alla ricerca dolorosa di un pensiero più fecondo, un’espressione più adeguata, un’estetica più vera. Con l’augurio che questa prospettiva possa essere utile ad altri, come è utile a noi, e possa far emergere in qualcuno dei nostri paesi grandi personalità che abbiano radici nell’identità creola e sappiano rappresentarla, aprendoci contemporaneamente le vie del mondo e della libertà. […]

La Creolitá ci libera dal mondo antico. Ma, in questo nuovo movimento, cercheremo il massimo di comunicabilitá compatibile con l’espressione estrema di una particolarità […]

La nostra immersione nella creolità non sarà incomunicabile ma non sarà nemmeno totalmente comunicabile. Lo sarà con le sue opacità, l’opacità che restituiamo ai processi della comunicazione tra gli uomini. Rinchiudersi nella Creolità sarebbe contraddire il proprio principio costruttivo – negarla. Sarebbe trasformare l’emozione iniziale in una meccanica vuota, che gira a vuoto, che si impoverisce progressivamente, come quelle civiltà dominatrici oggi scomparse. […] La nostra creolità dovrà diventare patrimonio nostro, dovrà strutturarsi, preservarsi, pur modificandosi e arricchendosi. Sopravvivere nella diversità. L’integrazione di questo doppio movimento favorirà la nostra vitalità creativa in piena autenticità. Ci eviterà anche un ritorno all’ordine totalitario del vecchio mondo, irrigidito dalla tentazione dell’Uno e del definitivo. […] La cultura viva, e ancor più la Creolità, è una eccitazione permanente di desiderio conviviale. E se raccomandiamo ai nostri artisti questa esplorazione delle nostre caratteristiche peculiari è perché essa riconduce alla naturalità del mondo, lontano dall’Identico e dall’Uno e perché oppone all’Universalità, tutte le opportunità del mondo diffratto ma ricomposto, l’armonizzazione cosciente delle diversità preservate: la DIVERSALITÀ.

Il mondo si muove verso uno stato diffuso di creolità. Le vecchie tensioni nazionali cedono di fronte all’avanzata di federazioni le quali forse non vivranno a lungo. Sotto la scorza universale totalitaria, il Diverso è sopravvissuto in piccoli popoli, in piccole lingue, in piccole culture. Il mondo standardizzato brulica contradditoriamente nel Diverso. Tutto in relazione con tutto, gli spazi si allargano, determinando il paradosso di una tendenza all’uniformità generale e di una contemporanea esaltazione delle differenze. E abbiamo il presentimento che Babele è irrespirabile solo per gli spazi ristretti. Che non sarà una preoccupazione per la grande voce dell’Europa che si parli bretone in Bretagna, corso in corsica, che non sará un problema per il Maghreb unificato che si parli berbero in Cabilia, o che le popolazioni tuareg seguano le proprie abitudini nel proprio paese. La capacità di integrare il diverso è sempre stata appannaggio delle grandi potenze. Le culture si fondono, si diffondono in subculture, generano esse stesse nuovi aggregati culturali. Pensare oggi il mondo, l’identità di un uomo, il principio di un popolo o di una cultura, con i criteri di valutazione del diciottesimo o del diciannovesimo secolo sarebbe riduttivo. Con sempre maggior forza emergerà una nuova umanità creola: tutta la complessità della Creolità.

Il figlio nato e residente a Pechino, di un tedesco che ha sposato una haitiana, sarà diviso e combattuto tra più lingue, più storie, preso nell’ambiguitù torrenziale di un’identità mosaico. Dovrà, pena la morte creativa, pensarla in tutta la sua complessità. Sarà nella condizione creola

Ah, Will. Ho pensato che tu debba rispondere in Inglese! Allora, devo trovare un dizionario! Sei crudele!

I wish to suggest an interesting reading on this topic: Albert-László Barabási, Linked: The New Science of Networks (Perseus, Cambridge, MA, 2002). It is an introductory yet precise description of what from the graphs theory can be applied to various kinds of networks (which are graphs), from Internet to power lines to epidemics to relationships among people (you perhaps know the so called six-degrees of separation principle).

Mathemes, if that’s what graphs might be interpreted to be (in essence), are, indeed, analogous to the search for that perfect language of forms that gets rid of expression (subjective residue), after Wittgenstein, yes!, but also after all quests for a pure language of logic (e.g., Hegel’s ‘Science’ as sign of the Absolute) … Life is far more messy — as is Art … But there was that marvellous Modern attempt nonetheless, in literature as well, to reach quiddity, or things themselves (which is the parallel, I think, to the beauty of graphs and graphic knowledge) … It is just that this quest returns always to fail — as in Badiou’s use of set theory to explain (explain away) the ultimately impure nature of metaphysical discourse (which is always part existential) … GK … It is odd, then, that the Pre-Socratics (Parmenides and such) are often cited as the precursors to such a stripped-down, material (natural) language/knowledge …

Will, ‘La Creolitá ci libera dal mondo antico.’ I think I have believed in this concept for years without knowing it. Creolità – I must find an English equivalent for the term. Creolity doesn’t work. (or does it?) We have discussed your usage of ‘Contrappunto’ (in

Contrappunti E Tre Poesie Creole) in relation to the concept as used by Edward Said. It think it fits well. ‘We areofthe connecctions, not outside them,’ as Said wrote.Many thanks to doctor Vincenzo della Mea who works at Dept. of Mathematics and Computer Science AND the Faculty of Medicine in Udine.

Welcome back to Gavin..

Life is funadmentally chaotic. We share approximations, we negotiate meaning. Language reflects this quality of messiness. We live within and as part of our language and so tend not to notice, except in egregious circumstances, the extent to which we misunderstand each other. Imagination and a residual human solidarity holds us together. No pure language of logic can arise out of such an existence. All our languages are mixed. All our thoughts are impure. Art celebrates the impure. This is what I think Will’s creolità (creolism?) means. I see quests for pure language as fundamentally misguided and repressive. I think, if I understand it rightly, that various developments in mathematics have been towards creolism – chaos theory, Godel’s Incompleteness theorem, etc. (Although, I hasten to add I know nothing about mathematics)., in the sense that they have pointed to the limits of perfectability in mathematical discourse.

I agree with your reading William. Even in the ‘exact scence’ of mathematics the fruitful seed of Doubt has been planted…

Dear all, sorry to get into this at this late stage. Of course, the first thing I thought about when reading about graphs was, more than language, music (as a language). Musical intervals are, after all, nothing but the relationships between different frequencies of vibration, that end up in different sounds. Counterpoint (Said’s counterpoint) may be visualized as a ‘perfect’ set of graphs, in which every interval (the relationship between two notes) is governed by a mathematical constant. Moreover, the ‘friendships’ between notes, accounting for ‘good/consonant’ and ‘bad/dissonant’ intervals (also called ‘wolf intervals’), are culturally constructed by the Western canon – something Said himself had to deal with in his Musical Elaborations, where he speaks about music (his beloved music) as a “police regime of the signifier”. Still, the silence between two contemporary played notes is both countable as a mathematical figure and uncountable as something that does not ‘sound’, yet exists – in fact it is the very source of existence of the chord itself. So music is not (only) pure organized sound as much as it is not only the place of counterpoint and creolité (or metissage…Balibar has his say here). After all where does the zero (the quantity that does not exist, the note missing in the chord) on which maths and computer science are based comes from?

I hope I have tackled on many of the issues this discussion has fueled. If I have wandered…blame the flu! xs

“Amalgamation means that communications through symbols can have multiple authors.”

wonderful project…

welcome back Serena! Very insightful post ( as usual..)

Bill,

what hold us together is also dreams and nights allucinations…in the whale’s mouth! but you got it right with creolism!

will stabile

sorry correction: are also dreams

Walking home last night, in a lovely rain, I was thinking about how graphs (as graphic things) have as their base ‘Zero’ … Was also thinking about how a lecture I once witnessed by Ernesto Laclau (years ago) used a graph (i.e., two intersecting axes, X, Y, if you wish) to diagram hegemony … The vertical axis is, in such a ‘post-Marxist’ critique, the — um — paradigmatic; the horizontal is the syntagmatic … One is nominally ‘male’, and one is nominally ‘female’ … You guess which … Anyway … Was also struck by Serena’s comments, this morning, on music and silence as the analogue for intersubjectivity and Zero … Was it Barenboim who said all music relies on silence? Referring most likely to Mahler? … GK … Oh, yes … And then there’s Badiou (again) whittling away at things, approaching Zero (the null set); a weird ‘reduction’ (approach to ‘ground’), to say the least …

Take this:

On thursady night I was so tired/messed up that my right eye’s nerve was pumping blood out of my skull and I had some allucinations and dreams very good for poems! I was walking on the railtrack close to a station and a voice was telling me: wait, wait, see the train is coming. wait and learn, look at the timetable, wait …with a yellow pistillo’s flower squashed in your hand.

please translate pistillo for me !

will

The central female reproductive organ around which the other flower parts are arranged is called pistil Will..

Timetables & flowers, maths &nature, music&silence. Interesting dicotomies..

Yes, Gavin, it was Barenboim, but also Trinh Minh-ha says more or less the same thing…both with Said in mind and the idea that hegemony and subalternity are intertwined like sound and silence in a piece of music, both carefully chosen and interwoven…was it Althusser who compared the Ideological State Apparatus to a concert? Yes, but still (as he writes) “so silent”…Foucault I believe would beg to differ, but using music as an analytic/critical tool (as Said did, as I often do myself) sometimes does away with dichotomies altogether, to the point that even silence is not (just) silent. Or maybe silent only as the echo that one can perceive at the bottom of the uterus (for Irigaray, Plato’s grotto), thinking also about the fact that music, most rational and mathematical of languages, has for centuries now been associated with the ‘feminine’, irrational voice of feelings…the singer, with her/his orifice open for all the world to see (anf hear), is always ‘feminine’, irrespective of sex. love, guys…xs

Fascinating post. This is really taking us into (un)dreamt of areas. As regards, dreams, I include them in that residual human solidarity I mentioned. Plato’s grotto as uterus, Will’s pistillic hallucinations, the resounding silence, and Serena’s ‘flu. All human life is here.

Serena did a post-doctoral thesis on voices & music…

Indeed. My point there (that takes me back to other posts and comments, and William’s and my idea of an un-gendered party) was that music defined a body gendered ‘differently’, through organs reproductive of sound, like the throat and mouth, with their ‘feminine’, say uterine shape and echo, but also the voice itself as phallic, ‘penetrating’ sound. As a consequence, un-gendered here would not mean ‘neutral’ or ‘wasexual’, but an expression of dynamics of desire through a language and imaginary different from the heterosexual matrix. That is what brought me to consider music itself as a system and counter-system, language of hegemony (my interest was mainly in classical music) but also site of resistence, much as graphs appear to be…xs

I am surprised by the wealth of comments that my message has generated! Thank you! Indeed it was meant to be an isolated message to you by me as modest representative of the mathematical world just to let you know that we, too, are working for and in accordance to the same aesthetic principles of rigour and beauty that you follow. But now it seems appropriate that I write at least this second message in order to clarify and make once for all unambiguous what is meant by William and me by the word GRAPH.

Informally speaking a graph is a set of points (conventionally called vertices) together with a set of lines (conventionally called edges) such that each line joins precisely two of the points. It does not really matter how this line (in a diagrammatic representation) joins the two points: all what matters is that the two points are “connected” by a line. Obviously this line can represent a multitude of different things. For example the points could be people, and the lines are representing those pairs of people who know each other, or the vertices could be airports, with lines joining pairs of airports if there are flights scheduled between one airport and the other. Despite this model being of wide applicability, we are not necessarily concerned with the interpretation of the edges and vertices of graphs. These are just the basic elements of our theory of graphs: all what we need is an unequivocal, self-contained, mathematically rigorous definition of GRAPH. Once the object is created, we can spend all our life investigating its beutiful properties, but I will not go on now to talk about these. Despite this being a literary circle, I want to present a mathematically consistent definition of graph, that has taken about ten years of research by one of the pioneers of the field, the late Prof. Nash-Williams (1932-2001), to be formulated (its beauty is unmatched in my opinion). A graph G is an ordered triple (V,E,psi), where V is a set (called the set of vertices), E is another set, disjoint from V (called the set of edges) and psi is a function which associates to each edge lambda (i.e. to each element lambda of E) an unordered pair of distinct vertices u,v which lambda is said to join. This definition is all what we need to start a theory of graphs, and I would like to take you along with me in this wonderful journey, but reasons of space limitation and the non-mathematical orientation of this group make it necessary for me to stop here.

Thank you for your attention.

David

Thanks to serena and David.

David. i’m afraid in tha last part of your comment you lost me 😉

You took me back to the unespected questions from the Maths teacher at the Liceo…

By the way, Graph Theory was accidentally invented (or, better, initiated) by the well known mathematician Leonhard Euler, while trying to solve a silly problem. In the city of Königsberg, now Kaliningrad, there were seven bridges crossing a river and connecting an island. Citizens were trying to understand whether a walk around the city passing through all bridges, but just one time for each, was possible. He abstracted the situation by substituting places (i.e., banks and island) with vertices and bridges with edges (this was the novelty of the approach). Then demonstrated that such a walk was not possible in his paper entitled Solutio problematis ad geometriam situs pertinentis, 1736.

You can find here: http://mathforum.org/isaac/problems/bridges1.html a not-so-much mathematical explanation of that demonstration.

thanks Enzo. Fascinating resonance with Kant, obviously..

The number of the ice crystal is 6

the hard-edged, hard-nosed hexagon.

But the language of ice is domestic –

needles & flakes & blankets.

All the prismatic clarity of numerals

culminates in the snow-verb

to flocculate & then we have snow.

William Wall

flocculate is a new verb to me…

It’s a poem from my first collection (Mathematics & Other Poems). It just seemed to contain the elements of what we’re talking about – the ordinary and the extraordinary, numbers and more homely things like friendship. ‘Flocculate’ means to come together to form fluffy clumps (believe it or not). It contrasts with the ordinary word ‘snow’.

Very evocative poem Bill. Thanks for the explanation on flocculate…

William: I’m interested in that collection (which I found following your links), as I’m interested in contaminations between science and poetry. Here a poem from my book entitled Algoritmi (Algorithms), where another kind of graph is present (in Italian, sorry – if someone wants to translate…):

Scrivere

Questa ordinata danza delle dita

è una vista parziale sullo stato

dei nodi nello strato nascosto

della mia rete neurale: si è persa

la traccia degli ingressi, e l’effetto

pure chiaro non è più correlabile,

rimanendo come consolazione

una matrice sparsa di sinapsi

con la sua attivazione imprevedibile.

Grazie, VDM, per questia poesia. Provo tradurrla.

Bill

Ecco, Vincenzo, la prima bozza:

This orderly finger-dance

is a partial view of the state

of the nodes in the hidden layers

of my neural net; the trace

of inputs is lost and the clear

effect cannot be correlated;

there remains like a consolation,

a matrix of scattered synapses

and their unpredictable discharge.

Ho visitato il suo blog. Un bel sito, molto interessante.

wow, great ‘draft’Bill..

David Cariolaro wrote: “but reasons of space limitation and the non-mathematical orientation of this group make it necessary for me to stop here”.

The point here is that we should try to lead the discussion/ or the blog itself towards an amalgamated stage/zone where areas of knowledge can creolise with no hierarchy.

We should broaden it, amplify again & again…

Guys these blogs Come on! become so boring after a while. post after post…

Luca where is the real rizomatic structure?

And here I make a connection with the broad question/argument we had:

is internet useful to improve our knowledge, to go deep into problems?

can we really communicate/understand or is just a contemporary tool given to us by the technology.

It is clear for me as (Baricco says in “I Barbari”) that we surf, only merely surf (as the verb says) on wawes of information with no historic prospective before and after and NO MEMORY.

But probably this is what the new human beeing want, Oh mio Luca?: no memory. a chip in the ear will do…

I am very calm not walking on the fire. Asd you know I am just naturally provocative.

Will Stabile

Caro Will,

I confess to being ignorant of mathematics. As a result, David’s post interested me in a general way, but I can’t see how to apply it, or how it might be a useful way to look on society or literature or even philosophy. It is, I feel, one of those areas of mathematics that is fruitful for mathematicians but not of obvious applicability. In this respect, I am happy to be led into new areas, to follow the leader. I am open to expansion, but incapable of doing any expanding myself.

So, let the graph theory specialists amplify. Where does this theory go? How does it relate to my life, my work, my family, my friends, my problems, my politics.

Bill

Dear Wills (and all),

this is really a tough point. David had indeed the delicacy we don’t always have – to limit the outburst of the passion he clearly feels for what he does. the other side of the coin Stabile speaks about is exactly the unrestrained, narcissistic flow of ‘information’ – poetry (good or bad), ‘inner’ thoughts, pseudo-autobiography…and so on and so forth. Internet thus becomes only a huge resonance chamber for our thwarted, frustrated egos – which also bears upon, just to open and immediately close another parenthesis, the way relationships develop on the internet, between people who may find difficulties to perceive each other as ‘people’, human beings with a sort of identity (whatever that means, that’s still how most people are educated to perceive ouselves), and yes, Stabile, a (one, diverse, several) memory.

I know that launching a pseudo-philosophical discussion about what is memory, how we recreate it in the present, and how the internet is in a sense a concrete (if virtual) embodiment of its fleeting nature may eventually prove unpopular. Yet, let’s give it a try…xs

Yes. The internet and memory. Google as global memory. Internet communities instead of real communities of people. Surfing rather than travel, and travelling as if surfing travel blogs.

Thank you William. The only change I would do, just to remain in the neural networks language, is ‘hidden’ instead of ‘secret’ (although the sense is mostly the same).

Regarding the real-life meanings of graph theory, many are described in the book I cited in some post above. Graphs are just a way to model relationships of many kinds – but after modeled that way, there are mathematical properties that can be applied. Sometimes such properties are already known as common sense, of course. One of the chapters of the Barabasi book is entitled ‘Rich get richer’: it is common sense, but can be formally justified in terms of graphs (starting from web sites – the more links you have the more you will have). We usually live in so-called ‘small-world networks’: clusters of intimate friends, each one knowing all the others, with some loose connection with external people. One consequence, experimented by Stanley Milgram, is that you may need six different ‘contacts’ (edges) among persons (arcs) to reach any other person you do not know (the six degrees of separation principle). This can be in turn and grossly related to epidemics, depending on how strict are clusters. Small world networks describe also other networks (on the web, we have about 19 degrees of separation).

So Lunatagliente let’s speak about Memory !

Would you anyone remember me when i will not be a blogger anymore? Would you, lunatagliente?

Probabily I will write me/myself my name down on the list of the desaparecidos i am filing now in London.

Yes I am writing my own “list of desaparecidos” here in UK. You dont need to go to Argentina… Here are the human beeings that you meet once a year altough they live 5 min from your house because they are lost. Lost in web !??

and as one of my friend said: Guglielmo I dont know! I dont know anything about [this world].

Please note: I am not someone suffering of solitude!

Stabile

VDM, ho cambiato ‘secret’ a ‘hidden’.

Grazie!

Dear poets and writers,

I am very glad to see that the broad horizons you are opening up with the discussions in this group are not circumscribed to one or a few particular topics or line of thoughts, but are of a very general and often philosophical type, which suits so well me and many other readers with different backgrounds. You know, there are only two types of graphs, the recurrent ones, where a particle which starts at a point and performs a random walk will pass through any vertex of the graph an infinite number of times with probability one, and the transient ones, where the same phenomenon will occur with probability zero (that is essentially never). This was proven by Nash-Williams in his first paper [C.St.J.A.Nash-Williams, Random Walks and Electric currents in networks, Proc. Cambridge Philosophical Society,1959] written at the age of 26, which has become immediately a classical. Well, I think this literary circle IS a graph of the first kind (recurrent) and I am one of the particles who, in his random walk, am coming back to it again and again. I must thank VdM, or Vincenzo della Mea, who has very kindly supplied many interesting informations about the origin of graph theory. Indeed Vincenzo (who I am getting to know through his home page and I am very glad has so many diverse skills both in literature and in science) may well be equally if not better qualified than me in giving delucidations about graph theory, since graph theory is a discipline which is at the interface of different areas of knowledge, including mathematics and computer science (but also economics, social science, psychology, evolutionary biology just to name a few). Well, William has in different occasions named this marvellous and evocative word “amalgamation”, which he claims he has taken from graph theory (at least from the few aspects of graph theory that I have been talking to him about).

I believe this concept has captured his imagination well before he tried to have an understanding of it- a task which is challenging even for a professional mathematician-. I am probably not even entitled to speak with a sufficient degree of accuracy about this concept which still appears to be too deep in my eyes (at the present moment) to allow for an appreciation or even vague estimate of its power as a mathematical tool, and I shall only try in this message to offer a rough, necessarily informal and therefore inaccurate geometrical description of the idea which is behind the definition of amalgamation. This is not in order to throw in somebody’s face some uninteresting and cumbersome mathematical jargon, only to try to at least partially demistify what has been said and has been left to one’s imagination about this misterious expression, which William so bravely has captured and assimilated during his mathematical detours and now generously donated to the world of poetry.

It is probably better if I start to explain what is meant by a DETACHMENT of a graph. As usual, it is useful to think of a graph as a geometrical object constituted by a set of points some pairs of which are joined by lines (as we said last time).

So, if one looks at a single point (i.e. a vertex of the graph) one sees a set of lines which are incident to this point, each line connecting this point to some other point. Now imagine that we wish to SPLIT this set of lines into various groups, maybe for different reasons, such as distinguishing those connections which are really important from those that are not. When we create the detachment we are indeed replicating each vertex a certain number of times, where each of the replicas gets only one of the groups of edges which were originally incident to the original vertex. Therefore we may have, e.g. two replicas of each vertex, one being incident with only the “important” connections, the other only with the “unimportant”. Repeating this process for any vertex will give rise to what we call a detachment of the original graph. Now we come to the concept of amalgamation. Once it is understood what is a detachment, the concept of amalgamation can be defined as follows. If a graph H is a detachment of a graph G, then G is called an amalgamation of H. In other words, G (the amalgamation) is just the result of the inverse process than a detachment, namely it can be obtained from H (the detachment) by (so-to-speak) “gluing together” some vertices and making sure that the connections with other vertices of the graph are not lost in the process.

I hope this can provide you at least with a cursory understanding of this term as a mathematical entity- of course it is not by any means an attempt to condition what might or could be the interpretation and etymology of the term in literature-that is largely left to William’s and your imagination, or whoever will decide to use this beautiful world in prose and poetry.

Thanks for your long ‘comment’ David. This gives me the opportunity to talk about the structure of Rizoma. As you can see with the recent comments option one can choose to join a thread which was posted days before.. This makes, I believe, things interesting. We can shift, like we do in normal life, from a topic to another or choose to focus our attention on one. We can go back and decide to read a poem, a short story a review. We can also exert our choice to comment on the post or just read it.

I find the ‘comments’ part of blogs the most problematic and intriguing one. Indeed comments are often a great tool to explain without burdening the body of the main text with explanations & footnotes. It is also true that more often than not comments become prattle, useless junk. But that happens very often when we communicate with a person in front of us. In both cases we have the choice of stepping out of the conversation. We should be wise in our considering the function of comments. Sometimes they are incredibly insightful, sometimes incredibly banal. I would say that the most important part must remain the post, otherwise instead of a rhizomatic structure we have a monstrous one( growth per se is not always a good thing) This is just to state, to myself first, that we shouldn’t take comments too seriously or too lightheartedly but use them to build a ‘virtual’ community of thinkers..

As regards memory and forgetting, Will, on the internet we are like fireflies. They make a small light against the darkness, buit when they are gone the darkness is still there. I’m not speaking now of any value in what we write, merely of our ‘presence’ on the web, which is brief and transient. But we should bear in mind that there are more important things than this so-called ‘world’, this so-called community. Friendship, writing, thinking – these need no validation.

But if, at the same time, we can also open another avenue of communication in which we can raise ideas, consider each others writing, even provide a limited form of support for like-minded people, then this participation on the internet, although no substitute for life, can be a valuable thing.

Ditto Bill. I think that a lot of frustration for the internet is given by our unrealistic expectations. We should used it as tool to facilitate communication. If and when it get in our way we might as well get rid of it..

Caro Luca, ho postato il mio comment primo di leggere il tuo. Sono d’accordo.

What is your policy regarding the frequency of posts? I seem to remember you asking that time be given to each one to develop.

I think we should limit ourselves to no more than 2 posts a week. the number of comments or clicks should not be our main concern – although I understand it could be rewarding at narcissistic level-

We need time for the post to ‘sediment’ and create a resonance.

AMALGAMATION IS SHARED AND BOUNDLESS.

WE SHARE THE FUZZY WORLD.

WHEN YOU HAVE ALL THE ANSWERS THE QUESTIONS ARE CHANGED !

You share interesting things here. I think that your page can go viral easily,

but you must give it initial boost and i know how to do it, just type in google

for – mundillo traffic increase go viral