Simulating Influence Propagation in a Vee Network using a Threshold Model

Setting the Stage: A Vee Network Scenario

Let's ground our simulation in a specific context relevant to your goals. Imagine a vee relationship consisting of a Hinge (H), Partner 1 (P1), and Partner 2 (P2). The "innovation" they are considering is a significant one: agreeing to adopt a shared digital calendar to coordinate schedules and major commitments. This is a behavior that requires buy-in; it's a perfect candidate for the Linear Threshold model, as a partner might only agree if they feel the others are sufficiently committed.

To build our model, we need to define its parameters: the network structure (who influences whom), the influence weights, and each person's individual adoption threshold.

1. Network Structure and Influence Weights ($w_{uv}$)

In a vee, influence isn't perfectly symmetrical. The hinge often has a strong influence on each partner, while the partners' primary channel of influence is back to the hinge. Direct influence between P1 and P2 is typically weak or non-existent, mediated entirely through H.

Let's represent this with a directed graph where an edge (u,v) with weight $w_{uv}$ signifies the influence u has on v. Based on the LT model's rules we reviewed, the sum of incoming weights for any node must be less than or equal to 1.

Let's assign the following weights:

• Hinge's influence on partners: Let's assume H has significant, equal influence on both partners.
  • $w_{H,P1}=0.6$
  • $w_{H,P2}=0.6$
• Partners' influence on Hinge: Let's assume the partners have some, but less, influence on the hinge. Perhaps P2 has been in the relationship longer or is more persuasive.
  • $w_{P1,H}=0.3$
  • $w_{P2,H}=0.4$
  • Check the constraint: The sum of incoming weights for H is $w_{P1,H}+w_{P2,H}=0.3+0.4=0.7$, which is $\le 1$. This is valid.

2. Individual Thresholds (${\theta }_v$)

Each person has a personal threshold (${\theta }_v$) representing their resistance to change or the amount of social proof they need to adopt the new calendar system. A low threshold means they are eager to adopt, while a high threshold indicates skepticism or caution.

We'll consider two cases to see how personality affects the outcome:

• Case A: Cautious Partners, Open-Minded Hinge
  • ${\theta }_H=0.25$ (Hinge is quite open to the idea).
  • ${\theta }_{P1}=0.5$ (P1 needs a moderate amount of buy-in).
  • ${\theta }_{P2}=0.7$ (P2 is very cautious and needs strong consensus).
• Case B: Enthusiastic Partners, Cautious Hinge
  • ${\theta }_H=0.8$ (Hinge is very reluctant and needs overwhelming support).
  • ${\theta }_{P1}=0.2$ (P1 is an early adopter).
  • ${\theta }_{P2}=0.4$ (P2 is also quite open to the idea).

With these parameters, we are ready to run the simulation. As a quick refresher on the activation rule, the video lecture from our previous lesson has a very clear example.

Running the Simulation: Step by Step

Your background in simulation will be very useful here. We are essentially running a discrete-time, deterministic simulation where the state of each node (active/inactive) is updated at each step based on the states in the previous step.

Simulation 1: Hinge Initiates (with Case A thresholds)

Let's assume the Hinge (H) proposes the shared calendar.

• Time t=0 (Initial State):

  • Set of active nodes: {H}
  • Inactive nodes: {P1, P2}

• Time t=1: We check if any inactive nodes become active.

  • Node P1: The only active neighbor is H. The influence on P1 is $w_{H,P1}=0.6$. P1's threshold is ${\theta }_{P1}=0.5$.
    • Is $0.6\ge 0.5$? Yes. P1 becomes active.
  • Node P2: The only active neighbor is H. The influence on P2 is $w_{H,P2}=0.6$. P2's threshold is ${\theta }_{P2}=0.7$.
    • Is $0.6\ge 0.7$? No. P2 remains inactive.

• Time t=2: We check again for new activations.

  • Set of active nodes: {H, P1}.
  • Node P2: The only active neighbor influencing P2 is still just H. P1 has no direct influence path to P2. The total influence on P2 from active neighbors is still 0.6.
    • Is $0.6\ge 0.7$? No. P2 remains inactive.
  • No other nodes can be activated. The process stops.

• Outcome 1: The final set of active nodes is {H, P1}. The innovation is only partially adopted. P2, the most cautious partner, does not join in.

Simulation 2: Partner Initiates (with Case A thresholds)

Now let's see what happens if the idea originates with P1.

• Time t=0 (Initial State):

  • Set of active nodes: {P1}
  • Inactive nodes: {H, P2}

• Time t=1:

  • Node H: The only active neighbor is P1. The influence on H is $w_{P1,H}=0.3$. H's threshold is ${\theta }_H=0.25$.
    • Is $0.3\ge 0.25$? Yes. H becomes active.
  • Node P2: Has no active neighbors (P1 cannot influence P2 directly). No change.

• Time t=2: Now H is active and can influence P2.

  • Set of active nodes: {P1, H}
  • Node P2: The only active neighbor influencing P2 is H. The influence is $w_{H,P2}=0.6$. P2's threshold is ${\theta }_{P2}=0.7$.
    • Is $0.6\ge 0.7$? No. P2 remains inactive.

• Outcome 2: The final set of active nodes is {P1, H}. This is the same outcome as before, but it took an extra step. It demonstrates the hinge's critical role as a bridge for influence. The idea had to "infect" the hinge before it even had a chance to reach P2.

Simulation 3: Hinge Initiates (with Case B thresholds)

Let's rerun the first simulation but with the "Enthusiastic Partners, Cautious Hinge" thresholds.

• Time t=0 (Initial State):

  • Set of active nodes: {H}
  • Inactive nodes: {P1, P2}

• Time t=1:

  • Node P1: Influence from H is $w_{H,P1}=0.6$. P1's threshold is ${\theta }_{P1}=0.2$.
    • Is $0.6\ge 0.2$? Yes. P1 becomes active.
  • Node P2: Influence from H is $w_{H,P2}=0.6$. P2's threshold is ${\theta }_{P2}=0.4$.
    • Is $0.6\ge 0.4$? Yes. P2 becomes active.

• Outcome 3: The final set of active nodes is {H, P1, P2}. Full adoption occurs in a single step because both partners had low thresholds.

These simple simulations reveal powerful insights: the success of a new idea depends not just on its merits, but on the interplay between network structure, influence dynamics, and individual personalities.

A Deeper Look at Thresholds and Beliefs

So far, we've treated thresholds as fixed numbers. However, in reality, a person's willingness to adopt a new idea is deeply tied to their existing web of beliefs and knowledge. An interesting paper by James Houghton models this by conceptualizing an individual's mind as a "semantic network." The cognitive effort required to adopt a new idea (and thus the threshold) depends on how well that idea "fits" with what they already believe.

The following reading details a formal model of this process, using a simulation between two individuals, Alice and Bob, to illustrate the core mechanism.

This perspective is powerful. It suggests that influencing someone in a relationship isn't just about applying pressure (increasing the sum of weights). It's also about framing the new idea in a way that connects to their existing values and beliefs, effectively lowering their adoption threshold. For instance, framing the "shared calendar" not as a control mechanism, but as a tool to "guarantee quality time" might make a skeptical partner more receptive.

Conclusion

In this lesson, we put the Linear Threshold model to work, moving from abstract rules to a concrete simulation within a vee network. This exercise demonstrates how a formal model can be used to explore complex interpersonal dynamics.

• Key Takeaways:
  • The LT model can be applied as a step-by-step algorithm to simulate influence in a small network like a vee.
  • The outcome of influence propagation is highly sensitive to the initial parameters: who initiates the idea, the weights of influence between partners, and each individual's personal threshold for adoption.
  • The hinge's position as a structural bridge is critical; influence between partners is typically mediated through them.
  • Thresholds can be viewed not just as random numbers, but as products of an individual's internal belief system, making "framing" an important influencing strategy.

Preview of the next lesson:
We have focused on the micro-dynamics of influence spreading from node to node. In the next lesson, we will zoom out to look at a different aspect of power derived from the network's overall structure. We will apply the Bonacich power index to assess structurally-derived power in extended polyamorous networks, moving beyond the simple vee to analyze more complex configurations.