As we saw in the previous videos,

social networks play a very important role in helping us to

understand how information and behavior can spread through society.

In fact, as we saw,

even small changes to a network structure could have

major implications for how people within that population might behave.

Also, as we saw,

networks can be represented as very simple mathematical objects known as graphs,

which have two basic components, a set of nodes,

which often represent people and a set of edges,

which tend to represent the relationships between those people.

In this video, we're going to talk about

a very simple and special type of

network known as a lattice network or a spatial network.

These networks are known as spatial networks because

they represent the type of network you get when you place

nodes into a line or grid and simply connect

each node to the one next to it in that line or grid.

Lattice networks are often referred to as a one dimensional or a two dimensional lattice,

which refers to the type of space that the nodes are placed into.

In a one dimensional lattice,

nodes are placed into a line which is

a one dimensional mathematical object and that looks like this.

In a two dimensional lattice,

nodes are placed into two dimensional space such as a grid and it looks like this.

You could also have a three dimensional lattice that might look something like this.

And in fact, lattice with any number of dimensions can be represented mathematically.

However, in social science,

we tend to focus on

one dimensional and two dimensional lattices or line networks and grid networks.

These lattice networks are very similar to other type

of spatial networks that are important in social science.

For example, in a one dimensional network,

you might be interested in a situation where people are not only

connected to the nodes immediately adjacent to them on the line,

but might be connected to the next two nodes in the line in this example.

In a two dimensional network, similarly,

you might be interested in a situation where people are connected

to more than just the nodes above and below them in a grid.

For example, you might also be interested in

a situation where nodes are connected not only to the nodes up,

down, left and right,

but also on all four corners.

This would create a two dimensional network where a node,

each node has eight neighbors.

This node is so common and so important in

social science that has a special name and it's called a Moore neighborhood.

Both two dimensional and one dimensional spatial networks

have a very special and important property known as clustering.

Clustering refers to the number of neighbors that are shared between any two given nodes.

For a given node, local clustering

measures the proportion of neighbors who know each other.

It tends to create what we call overlapping

neighborhoods because in a network with high local clustering,

two neighboring nodes are likely to share many neighbors in common.

In social life, this captures the phenomenon where your friends know each other.

So as you can see from these examples here,

in a one dimensional network where each node

has four neighbors to the left and two to the right,

they share several neighbors with the nodes on either side of them.

And in a two dimensional network, for example,

this Moore neighborhood here,

you can also see that each node shares neighbors with the nodes on either side of it.

Before we finish up,

it's worth considering how these lattice or

spatial networks are represented and important in social science.

Imagine for example, a rumor spreading in a city.

You might think about the network formed by neighbors who live on the same block.

In one example, you might not lay out the nodes in the line,

if you imagine that each neighbor is going to

talk to the next door neighbor on either side of it,

or maybe even, the next door neighbor and the neighbor two doors down.

And that would represent a line network or a one dimensional lattice.

But maybe, people in this situation don't only talk to their next door neighbors,

but also talk to the people across the street from them.

And if they share a backyard with someone,

they might talk to the neighbors behind them.

And if they're really friendly,

they'll also talk to the neighbors across the street from their next door neighbors.

Now, we're back to a Moore network where

each person in the neighborhood has eight friends.

As a result, spatial networks are often used to

represent social situations in which we're interested in behavior

and information spreading through

a geographic network because these are the types of networks that are formed when we

think about friendships and relationships that people

have with other people who are geographically nearby.

You might also for example,

think about the networks of a student in a classroom.

If each student chats with the students in front of them,

behind them and on either side,

that would be another type of spatial or geographical network.

Naturally, the types of networks we've talked about in this video,

are very simple from a mathematical sense.

And the real world is often far more complicated.

However, by using these simplifications,

we can help to identify some of the basic properties that are important to

understanding how information and behavior can spread through a society.