The [[Linear Threshold]] (LT) model is another approach to [[modeling]] influence spread in social networks. In this model, each node is assigned a threshold value, representing the fraction of its neighbors that need to be active before it becomes active itself. Each edge from a neighbor to a node has an associated weight, indicating the strength of influence. The sum of incoming edge weights to a node is typically constrained to be at most 1.
Mathematically, for a node $<math>v$</math>, let $<math>N(v)$</math> be the set of its neighbors, and $<math>w_{uv}$</math> be the weight of the edge from neighbor $<math>u$</math> to $<math>v$</math>, such that $<math>\sum_{u \in N(v)} w_{uv} \leq 1$</math>. Each node $<math>v$</math> is assigned a threshold $<math>\theta_v$</math>, drawn uniformly at random from the interval \[0,1].
The activation process proceeds as follows:
1.# At time $<math>t = 0$</math>, a set of seed nodes $<math>A_0$</math> are activated.
2.# For each subsequent time step $<math>t \geq 1$</math>, any inactive node $<math>v$</math> becomes active if the total weight of its active neighbors meets or exceeds its threshold:▼
▲2. For each subsequent time step $t \geq 1$, any inactive node $v$ becomes active if the total weight of its active neighbors meets or exceeds its threshold:
3.# The process continues until no new activations occur.▼
▲3. The process continues until no new activations occur.
The LT model reflects situations where individuals adopt behaviors based on the collective influence of their peers, such as adopting a new technology or participating in social movements.
