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3.4 Threshold Models in Networks
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Из курса от партнера University of Pennsylvania
Network Dynamics of Social Behavior
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Complex Contagions and the Weakness of Long Ties
This week will begin by discussing the limitations of simple disease-like models of social contagion, introducing the idea of “complex contagions” to model people’s frequent need for social reinforcement before spreading a piece of information or behavior. While simple contagions always spread faster as networks get smaller, this week will demonstrate the paradoxical nature of complex contagions, which can spread slower (or not at all!) in the smallest networks.
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Damon CentolaAssociate Professor
Annenberg School for Communication
In this video, we're going to look at
how different network structures allow for different types of contagions to spread.
We'll be considering a threshold model and ask,
for any given network,
what's the largest threshold contagion that could spread?
So we know that a simple contagion,
which is one where a person will adopt after a single contact or exposure,
will spread easily in any network structure.
And it's easy to see also that a contagion with 100 percent threshold,
meaning that nobody will adopt until everybody else has adopted, won't spread it all.
So we want to consider the points in between and
see what types of networks the different contagions will spread on.
So let's consider first a ring lattice.
And remember, a ring lattice looks like this,
and it's created by taking a ring or a line of individuals or
nodes and connecting each node to the neighbors on either side.
And that could be one neighbor,
or two neighbors, or three neighbors,
any number of neighbors as long as each node has the same number of neighbors.
So let's consider first a single node, node 1.
Now, node 1 is connected to nodes 2 and 3,
as well as nodes 29 and 30.
Since this network wraps around,
it has no boundaries.
Now, imagine that a contagion is spreading with a threshold of one half,
which means that each agent will adopt if half of their neighbors have already adopted.
And consider a starting scenario in which node 1,
we'll call that the focal node,
and all of their neighbors have adopted.
We'll call node 1 and all of their neighbors the seed neighborhoods,
since that entire neighborhood is starting off having adopted the contagion.
Now, consider node 4,
which is the first node just outside that seed neighborhood.
We can see that node 4 shares half of their neighbors with node 1.
Now, because they share half of their neighbors with node 1,
and all of node 1's neighborhood has adopted,
node 4 is satisfied because half of their neighborhoods have adopted,
and node 4 will adopt the contagion.
Now, consider node 5.
At this point, node 5 is satisfied because half of their neighbors have adopted.
We can also see that node 2 has seen their entire neighborhood adopt,
and node 2 shares half of their neighbors with node 5.
By the same process,
we can also see that after node 5 has adopted,
node six will adopt.
And we can see the pattern unfold because whenever a given neighborhood has adopted,
the first node outside that neighborhood will always adopt,
because for any given node just outside of a neighboring neighborhood,
that node shares half of their neighbors with that adjacent neighborhood.
And this will be true in the ring lattice no matter how many neighbors each node has.
We looked in this example at a ring lattice,
where each node has two neighbors on either side,
but we could have set it up with three neighbors or four neighbors.
And the node just outside a fully adopting neighborhood will
always share half of their neighbors with that seed neighborhood,
which means that any contagion with a threshold of one half or
less will always spread easily in a ring lattice network.
However, a contagion with a threshold of greater than one half cannot spread because
any node outside an adopted neighborhood will never be
satisfied because only half of their neighbors will adopt before they do.
Now, this doesn't quite constitute a formal proof,
but it provides an intuitive demonstration of
an important property of ring lattice networks,
which is that any contagion with a threshold of one half or
less will always spread easily in ring lattice networks.
And this tells us that the critical threshold for a ring lattice network is
one half because nothing greater than one half will spread.
Now, let's consider a square lattice or a two-dimensional lattice.
In the previous example,
each node was connected to the neighbors on either side of them,
and now the nodes are connected to the neighbors to the left and right,
as well as above and below them.
And, in this example, the four corners,
which means that each node has eight neighbors.
And we gave this network a special name and called it a Moore lattice.
So, now, let's suppose as before,
that we start with an entire neighborhood having already adopted the contagion.
So that's one focal node and all of their neighbors.
As before, let's consider the node just outside the seed neighborhood.
We can see that the closest node to the focal node outside
the seed neighborhood shares three-eighths of their neighbors with that focal node.
And that means that from the outset,
three-eighths of their neighbors have already adopted.
So this tells us that if there is some contagion
spreading with a threshold of three-eighths or less,
that node will be satisfied.
We can also see that a contagion with a threshold higher than three-eighths will never
spread beyond the seed neighborhood because none of the nodes will be satisfied.
So this tells us that, at most,
a threshold of three-eighths can spread in a square lattice.
So, now, we have to see if this contagion
with a threshold of exactly three-eighths will spread.
So this first node we considered will adopt
because three-eighths of their neighbors have adopted.
Now, the nodes on either side of them will adopt because, now, they're satisfied.
And, at this point, we can now see that entire new neighborhood has adopted,
which starts the entire process over again.
We can consider the nodes just outside
that neighborhood who share
three-eighths of their neighbors with the first node we considered.
They'll adopt, and then the nodes on either side of them will adopt,
and so forth. The pattern continues.
A new neighborhood becomes infected,
and the nodes just outside that neighborhood start to adopt.
And by this method,
the contagion cascades through the network.
What this tells us is that in a Moore lattice,
any threshold with a contagion less than or equal to three-eighths will spread easily,
or a contagion with a threshold greater than three-eighths won't spread at all.
Unlike ring lattices, which will always spread a contagion
of one half no matter how many neighbors the nodes have,
two-dimensional lattices see the critical threshold change
depending on the network properties.
As the neighborhood sizes get larger,
the neighborhoods actually become more overlapping,
and the maximum threshold increases from three-eighths and approaches one half.
And this tells us that,
in general, in two-dimensional lattices,
a three-eighths threshold contagion will always spread,
and sometimes with larger neighborhoods,
even higher threshold contagions can spread.
Both the ring lattice and the
two-dimensional lattices share this property of overlapping neighborhoods.
And we've seen that this property of overlapping neighborhoods is
crucial to the spread of social contagions that require reinforcement.
We can measure this quantity of
overlapping neighborhoods by using what's called a clustering coefficient.
In general, networks with the higher clustering coefficient will be more effective at
spreading contagions that require social reinforcement.
Finally, let's consider a random network.
Whereas the previous networks we considered had a very regular pattern to them,
where each node was connected to adjacent nodes,
in random networks, as we discussed in previous videos,
nodes are just randomly connected.
If you take a very large random network and
zoom in on it and look at a very specific neighborhood,
has what's called a tree-like property because
the nodes produce branches that look like trees.
Very quickly, you can see that none of the agents have overlapping neighbors.
So imagine that we start with a seed notice before and all of
their immediate neighbors adopt in what we called the seed neighborhood.
Now, you can immediately see that even though this entire neighborhood has adopted,
none of the nodes just outside that neighborhood will be
satisfied if the threshold is any greater than one node.
So if the contagion requires any social reinforcement whatsoever,
it won't spread in this random network because of the local tree-like properties.
We can look at any of the nodes that are just
outside the seed neighborhood and see that they have,
at most, one neighbor who has already adopted.
These diagrams help us to see how the existence of
overlapping neighborhoods is crucial
to allow complex contagions to spread through a network.
In the ring lattice and the square lattice,
we saw that overlapping connections allowed complex contagions to spread
easily because they provide for social reinforcement.
We also saw the different network structures,
each have a different type of critical threshold.
In the ring lattice, it was one half.
In the square lattice with eight neighbors, it was three-eighths.
However, in the random network,
we saw that because there are no overlapping neighborhoods,
and there's no opportunity for social reinforcement,
only simple contagions can spread.
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