Translator: Priel Korenfeld Reviewer: Michele Gianella
After such an introduction, I had two choices:
either enter the room or not to.
Either way, I would have ended up in one of two possible futures.
Physics of the last century
proved an ancient intuition of Eastern mysticism:
the world, the space we live in,
it is a complex web, a dynamic network of pulsating events.
As Carlo Rovelli would put it,
"Not only can things relate to one another,
but from these very relationships emerge the concept of things".
I am a researcher at the University of Udine;
and guess what, I deal with networks.
Networks are an extremely simple, very fulfilling model,
that is also pervasive.
The Internet, the Web, social networks and even our brain.
What fascinates me about networks
is that they embed three interesting models
that I will try and illustrate you during this presentation:
the Linear, the Branching and the Circular Model.
and they are all represented in this network -
or graph, as labelled by the mathematicians.
Let's start with the Linear Model. What is it about?
The Linear Model
is a ceaseless, repetitive succession of instants.
It's akin to the concept of "Fatalism"
and it can be displayed as a path,
a succession of points.
I will explain this model
by using the phenomenon
of the "Six Degrees of Separation”.
Does anybody know about the “Small World Phenomenon”?
Well, maybe, a couple of my students do since I taught it this year,
and a few of them are present here.
The Small World Phenomenon states
that if we choose a couple random person on Earth,
like a mechanic in Tihuana, Mexico
and a greengrocer in Perth, Australia,
how many degrees of separation separate us from these people?
If I knew the mechanic directly
there would only be one degree of separation,
if I knew him indirectly, that is through another person,
maybe a friend of mine who studies in Mexico,
there'd be two degree of separation; and so on.
The Small World Phenomenon teaches us
that even though the world is big,
and the number of nodes, actors in the network, is huge,
the distances between people, between actors, are actually very small.
In mathematical terms we say the number of nodes is logarithmic:
incredibly small.
At the beginning of last century, in 1929,
an almost unknown writer named Frigyes Karinthy,
wrote a short story called “Chains”,
from which I have taken this excerpt.
He imagined picking one person
among, at the time, 1.5 billion that lived on Earth,
and bet that within no more than five intermediate people,
that is, no more than six steps,
he could reach these people through one's personal network.
He is the first person to realize this phenomenon:
but on literary level,
in the sense that his intuition is pure fantasy.
I find it surprising that in 1967, no less than 40 years later,
a famous American psychologist named Stanley Milgram
ran an experiment on the Small World
and somehow confirmed Karinthy's intuition.
What did Milgram do?
He grabs a phone book - the Internet was not around back then -
he randomly selects 100 people
and asks these people to send a letter, a package, an envelope
to a recipient 1000 km away.
However, they must send it
through their personal network.
Eventually he visits a friend, collects the envelopes -
some were lost in transition, of course.
Each envelope comes with a log
of the people it was passed on to,
Then he calculates the average distances
and the result is six - 5.9, actually.
Hence, the "Six Degrees of Separation" myth:
this hypothesis, this idea of the Small World,
one that despite being so big,
is connected between individuals through small distances.
The experiment was then repeated, and recently confirmed,
in an experiment -
done in cooperation with the Politechnico di Milano -
using Facebook's network,
With a much larger sample than Milgram's one,
a sample of 1.5 billion people,
the very same number of people
Karinthy envisioned on earth 80 years earlier.
Turns our, today's degrees of separation have been reduced to 4. 5.
The second model I'd like to tell you about is the Branching Model.
It implements the concept of "Free Will",
and it can be represented as a tree of choices,
of paths, of possible futures.
A garden of forking paths, one may say.
Reflecions on time - so we are talking about time now -
has very ancient roots, and dates back to the time of the Greek philosophers.
Aristotle is the first to ponder over temporal matters,
and proposes the problem of "Future Contingents".
What are "Future Contingents"?
These are propositions that refer to future events.
Aristotle said,
Let's use this proposition: “Will there be a naval battle tomorrow? ”,
or, reformulated in modern times,
"Will there be an earthquake tomorrow?" or “Will the sun rise tomorrow?"
Aristotle asked himself:
can we define a truth-value, within a binary logic of true or false,
for these statements?
Can we tell today if these statements are true or false?
Well, in her homely wisdom my mother would say,
"God only knows".
Many centuries after Aristotle,
William of Ockham, with all due respect, gave a rather similar answer;
he stated that the answer
was not within the mind of men but only in the mind of God.
Actually, Ockham acknowledges
a men's critical ability, the ability of Free Will,
the ability to select among many -
to make choices and thus, to create
different possibilities, paths, possible futures,
And in a sense this explains, gives a solution
to, Aristotle's Problem of Future Contingents.
Or, in other words, these statements aren't inconsistent
if we interpret them on a tree structure
with many possible futures.
As I somehow joked at the beginning,
there is a future when I showed up and gave this speech;
and another one where I sat back and enjoyed my relaxing tea,
therefore it is perfectly legitimate
that in some of these futures an earthquake also happens
or a naval battle also takes place,
and in other futures they won't;
Or that in all of these futures
the sun will rise tomorrow or the Earth will end.
It is interesting to see how -
in the 1950s, Arthur Prior, a New Zealand philosopher and logician,
studied these theories, Ockhams in particular,
and defines the concept of "temporal logic",
here you see the four main operators
of so called Branching Time Logic,
and their interpretation of tree structures,
I won't delve into that,
but it is interesting for me to convey,
that from these philosophical discussions,
that might otherwise look frivolous -
a logic was born
with a lot of important applications in artificial intelligence,
for example to represent temporal knowledge;
and in computer science,
to analyse the behaviour of nondeterministic, reactive systems.
The third model I would like to tell to you about
perhaps the most important and even most surprising,
is the self-referential or circular one.
It is a path that rethreads its own steps, a cycle as we call it in graph theory,
an Oroborus, a snake which eats its own tail.
Self-referentiality, or recursion in mathematics,
is surprisingly present in nature,
and takes up the shape of the fractal.
Here you can see roman broccoli
that besides its deliciousness, is also a perfect example of a fractal.
So what is a fractal?
A fractal is a geometric figure, which repeats on differing scales.
On your right,
you have the image of the broccoli as a whole.
On your left, a detail of it:
you see the particular looks like the whole.
So there's this recurring shape,
this kind of self-referentiality, a sort of circularity.
This is self-referentiality in nature.
And this in mathematics.
It's risky to jump on the TED stage with mathematical equations, I know.
Actually, however, I care a lot about this equation:
firstly because it is so beautiful an equation
and I dare anyone to say that this isn't pure poetry.
And secondly, it is a very useful equation.
You use this equation everyday,
because this equation is at the heart of an algorithm called PageRank.
And the PageRank is the algorithm used by Google Search
to sort webpages as per your requests.
Paraphrasing the meaning of the equation,
giving an interpretation to these mathematical symbols
we could read the equation as follow:
the equation says,
an actor within the network is important if other important actors surround it.
Have you noticed the circularity of this thing?
I am defining something in terms of itself.
A self-referentiality - a tautology, we would say in logic,
even though there exists a well defined solution.
Syntactically speaking, where does the recursion lie?
Well, lambda, "λ", is a number.
It has a matrix, an algebraic structure
used to codify, to represent a network;
and " x "is a vector,
so you can't simplify the "x", as someone said yesterday.
These structures are very different from one another.
The recursive nature is within the fact
that the "x" appears to both the left and right of the equal sign;
hence I am trying to define something in terms of itself.
Pure magic, from my point of view.
One of the papers that gave me the most satisfaction
was the one of studying the history of this equation.
Who ever thought of studying the history of an equation anyway?
Turned out, the first to have used this were sociologists in the 50s.
affirming that a person is prestigious
if they receive approval from other prestigious people.
In 1970 it was discovered in bibliometrics too,
a journal is influent
if other influent journals cite it.
In the 1990s it was popular in sports:
a team is strong if it beats strong teams.
And lastly, in the 2000s,
two PhD students of Stanford,
Sergey Brin and Larry Page,
discover once again this recursive mantra
and turn it into the pulsating heart of the Google Search Engine,
which will undoubtedly become very popular.
This is not the only equation, there is also another.
At University of Udine,
me and my colleague Enrico Bozzo
we studied a variation of this equation.
It connotes the concept of power: not centrality, not importance.
And it could be defined as:
“A powerful actor it is surrounded by powerless actors”.
This too is a recursive equation,
as you can see “x” both to the left and the right of the equal sign.
We applied it in an economic context of bargaining
if I have to negotiate with someone,
my negotiation power is high if my contractors are not that powerful.
If instead my counterparts are very powerful themselves,
for example, if my contractors are called Google, Apple, Microsoft,
then my bargaining power is clearly severely limited.
In our case we applied it
to the distribution network of natural gas across Europe,
which you can see here outlined as an indirect graph.
The node's dimension is proportional to the power of that node;
hence, as you can see, large nodes are close to small nodes
and viceversa.
This is the concept of power.
Getting back - and now my T-shirt also finds an explanation -
if we rewind the path of our uroboro,
we could get back to the beginning.
And we started by saying that space - we also heard it this morning -
is a fabric:
the fabric of reality, in Francesca Vidotto's words.
The space is a network, a network of events.
So why not suggest, to wrap up,
an original idea?
Time is a network, the network of time.
After all, the network model
follow linearity, branching and even circularity.
Which certainly have to do, as we have seen, with time.
Unfortunately, the time slot
between me and the end of this talk,
is too small to investigate this brilliant intuition any further;
perhaps we could ask a few theoretical physicists
if it makes any sense.
All that is left for me is to leave you
with one of the most beautiful quotes about time
that has ever been suggested to me,
which is to wish you all the beauty that this world has to offer.
Thank you.
(Applause)