The network of time | Massimo Franceschet | TEDxUdine

Massimo Franceschet proposes a reticular vision of time. This vision includes: linear time, a path of instants in a streamline, which corresponds to the idea of fatalism, a non-choice, a predetermined succession of events; branched time, a tree of temporal instants endlessly forking in different possible choices, which codifies the idea of free will, of choice between possible futures; and finally, the most paradoxical and captivating of situations, the temporal circularity, that is the idea of self-referentiality. Massimo will use these three models as a cue for just as many free associations: the phenomenon of the six degrees of separation in a social network, the branched temporal logic and the PageRank algorithm that made Google's fortune.

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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)