Information Diffusion — Cascade Models, Influence Maximization

🎯 Learning Objectives

• Explain the Independent Cascade and Linear Threshold models
• Model information spread using SIR/SIS epidemic models
• Understand influence maximization as a submodular optimization problem
• Apply greedy algorithm for seed selection

1. Cascade Models

1.1 Intuition

1.2 Independent Cascade (IC) Model

1. Start with set of active (infected) seeds at time 0
2. At each time step t, every node that became active at t-1 has one chance to activate each inactive neighbor with probability p(u,v)
3. Process continues until no new activations Properties:

• Activation attempts are independent across edges
• Each edge has its own probability
• Once a node activates, it stays active

1.3 Linear Threshold (LT) Model

2. Influence Maximization

2.1 Problem

2.2 Greedy Algorithm

python
Run Inline
def greedy_influence_maximization(G, k):
    S = set()
    for i in range(k):
        best_node = None
        best_gain = 0
        for v in V \ S:
            gain = expected_influence(S ∪ {v}) - expected_influence(S)
            if gain > best_gain:
                best_gain = gain
                best_node = v
        S.add(best_node)
    return S

3. Epidemic Models

3.1 SIR Model

• S → I: rate β (infection rate)
• I → R: rate γ (recovery rate) Basic reproduction number: $R_0 = \frac{\beta}{\gamma}$
• $R_0 > 1$: epidemic grows
• $R_0 < 1$: epidemic dies out

3.2 SIS Model

• S → I: rate β
• I → S: rate γ (no immunity — can be reinfected)

4. 📝 Practice Questions

Q1: In the IC model, what is the expected number of activations if a seed has one neighbor and p=0.5?

Answer: Expected = 1 (seed itself) + 0.5 (neighbor activation) = 1.5. The neighbor activates with probability 0.5. Q2: Why is the influence function in greedy seed selection submodular?

Answer: Submodularity means diminishing returns: adding a node to a larger set gives less marginal gain than adding it to a smaller set. The influence function is submodular because each new seed may activate nodes that are already activated by existing seeds — the overlap reduces marginal gain. Q3: Compare the IC model with the LT model.

Answer: IC is probabilistic (each neighbor has independent chance to activate). LT is threshold-based (cumulative influence must exceed threshold). IC spreads like a branching process; LT captures peer pressure effects where multiple friends collectively influence. Q4: What does R₀ > 1 mean in the SIR model?

Answer: R₀ > 1 means each infected person infects more than one other person on average — the epidemic grows exponentially. R₀ < 1 means each infection causes less than one secondary infection — the epidemic dies out. Q5: What approximation guarantee does the greedy algorithm provide for influence maximization?

Answer: Due to submodularity and monotonicity, the greedy algorithm achieves at least (1-1/e) ≈ 63% of the optimal influence spread. This is the best polynomial-time approximation unless P = NP.

5. 🔗 Cross-References

• Week 1 - SNA: Network structure affects diffusion
• Week 6 - Fake News: Information diffusion and misinformation
• BSCS4021 (Advanced Algorithms): Submodular optimization, greedy Join Discord PreviousFake News DetectionNextPrivacy Mechanisms