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10/30/22, 5:18 PM L10: Linear Threshold Model: Network Science – CS-7280-O01
L10: Linear Threshold Model
Let us now see how we can model influence and social contagion with network models.
Consider a weighted (and potentially directed) network. The weight of the edge represents the strength of the relationship between nodes u and v. If the two

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nodes are not connected, we set .
The state of a node v can be either “inactive” or “active” .
In the context of social influence, for example, an inactive node may not be exposed to a certain behavior (e.g., smoking) or it may be that it has been exposed but it has not adopted that behavior.
Initially, the only active nodes are the sources of the cascade.
Each node v has a threshold . The Linear Threshold model assumes that a node v becomes active if the cumulative input from active neighbors of v is greater than the threshold :
Note that nodes can only switch from inactive to active once.
The Linear Threshold model is appropriate in diffusion phenomena when the ”critical mass” theory of social influence applies.
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10/30/22, 5:18 PM L10: Linear Threshold Model: Network Science – CS-7280-O01
An important question is: if we only activate a certain node , which are the nodes that will eventually become active? These nodes define the “activation cascade” of
. Note that this cascade may cover the whole network, may include only , or it may be somewhere in between. Of course, the cascade of includes nodes that are reachable from .
A common simplification of the linear threshold is the homogeneous case in which all nodes have the same threshold . In the visualization, the threshold is set to
for all nodes. Note that the nodes x and z will not be part of the cascade.
There are also variations of the model in which two behaviors A and B are spreading at the same time, meaning that the state of a node can take three different values (inactive, A and B). In that case, the state of a node can switch between states A and B over time.
Another common variation is the “Asynchronous Linear Threshold” model. In that case, each edge has a certain delay. This means that different nodes can become active at different times.
The edge delays can affect the temporal order in which nodes join the activation cascade – but they cannot change the size of the cascade. We will review an application of this model on brain networks at the end of this lesson.
Note: Depending on the literature, the activation threshold model can be greater than or greater than or equal to – this can vary.
Food For Thought
Explain why the size of the cascade does not depend on edge delays.
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