In this unit, we are going to introduce
the idea and concept of networks.
That is going to be central to future
discussions on the complexity of music.
Right now, it is the first time in this
course, that we actually start dealing
with concepts and ideas that come
from complexity theory.
Many of you are probably familiar with the
idea of a complex system, of complexity
but I just want to remind you, very simply
what complex means in terms of
the definition of the word and what
complexity entails, given the meaning
of the word.
So, complex in English, is an adjective;
it means something that is composed
of many interconnected parts. This is very
essential to the idea of music complexity
because for us, music is a complex system.
It is made of interconnected parts
that have a complex way of interacting
in some sort of generalized way.
So out of this definition of a
complex system, the science of complexity
is a scientific theory that starts from
this idea of a system made of inter-
connected parts that are usually simple
and they behave in a very simple way
when looked at individually, but when they
are combined, they form a system that has
properties that are different from the one
of the individual entities and that have
properties that emerge from this kind
of interaction.
So one of the central ideas of my approach
of music as a complex system is that music
itself is an emergent property of a
complex system, being that either the
systems of pitches that we have introduced
or, at a higher level, as we will see in
future units, the interaction between
individual performers, the performers in
the audience, the audience in themselves,
society and, at a larger scale,
human culture.
Networks are important in this framework,
because behind a complex system, there is
a network that describes the elements and
the interactions that form that system
itself. One of the central things that
I will be introducing in the next couple
of units is how we can represent musical
spaces as networks. Okay, but what is a
network? A network is a representation of
interactions between agents of points,
or data, and this is a typical example
here, in this slide, of a network as we
understand it in a very intuitive way.
This is a transportation network, it shows
airports on the map of the US, and
connections between airports. So what this
representation tells us is, for instance,
that there are airports that are more
important than others, because they have
more connections with other airports, and
you see that there are at least five or
six important points that are more red
than the rest. And then the network of
interaction, that is how one airport is
connected through flights to another
airport. This gives us not only a way
of visualizing the data, but most
importantly, a way of making sense of this
data, given the characteristics of this
network itself.
Networks are everywhere, many of you are
probably familiar with many of them.
We have technological networks, like
the Internet for instance, or
communication systems, or electric grids,
or water mains, and so on. We have
biological and technological networks,
neural systems, disease epidemics - we
all went through the COVID years so we
are very familiar at least with some of
these concepts. We were fed this through
our experience with the COVID pandemic.
We have economic networks, social networks
all of you and I are all part of something
Facebook, Instagram, Tik Tok, you name it
These are, again, networks where complex
interactions can be represented using
graphs.
And then, finally, we have cultural
networks, like language families, semantic
networks, like networks of words and so on
and again, this is something that we are
getting very familiar with, through the
advancement of generative AI and all these
large language models, that in a way or
another, exploit the structure of these
networks.
Now, a network as a graph looks like this:
You have nodes, these colored points,
that are individual data or agents,
and then you have connections, the grey
lines, these are called the links. This is
for instance an example of a network of
a company, where there are departments,
there are consultants, there are experts,
and how all these agents interact
with each other.
Now, in this network representation, we
are interested in mainly two specific
objects, or measures. One is the degree
and the other one is the community
structure. This will be central in the
discussion of networks in musical spaces.
So, a degree is a number of neighbors that
a single node has. For example, here at
the top, we have a few nodes connected by
links, and if we look at the different
number of connections, the node A has
only one connection, therefore has a
degree one, while the node B has four
connections, therefore has a degree four.
Networks, and this is another important
characteristic of a network or link, can
be either directed or undirected, so the
top one here is undirected, it means
that the links are equivalent left-right,
or right-left, while in a directed network
links have a direction. Basically, in the
example at the bottom, we have a link
from A to B that exists, that means that
in principle, there is an interaction that
operates from A to B, but not vice versa.
There is no interaction going from B to A.
This is also important because, as you
will see in future units, music networks
are typically directed networks.
Now, the other concept that I mention in
this slide, is the community structure.
Community structure is a complex
quantity to evaluate. There are many
different algorithms to evaluate the
community structure, but in very general
terms, it means that in a network, we
typically have nodes that are more
connected in an environment, than others,
so they form some sort of communities, or
groups, that are more well defined than
the whole network itself.
To show this in a practical example,
this is a graph of my Facebook network,
from quite a few years ago, because now
Facebook doesn't allow you to download
your network anymore, unfortunately.
But at that time I did that, I downloaded
it, and you see here, a network that has
very well defined groups of nodes and if
I look at the nodes individually, I can
identify groups: for instance a group of
all my musicians friends, or a group of
all my physicist friends, or all my
Italian friends. Some of these groups
might have interactions with each other
but they are very well isolated, and very
well defined. This is important in the
next discussions of musical networks
because, as we will see, we can use this
community structure of the network itself
to identify very specific quality of the
music, that emerges very clearly from
this kind of representation.