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.