Influence Maximization
Overview
Influence maximization selects k seed nodes in a network to maximize expected spread under a diffusion model (Independent Cascade or Linear Threshold). NP-hard, but the greedy algorithm achieves (1-1/e) ≈ 63% approximation guarantee due to submodularity. Practical for networks up to millions of nodes with CELF optimization.
When to Use
Trigger conditions:
- Selecting k influencers/users to seed a viral marketing campaign
- Maximizing information spread under a fixed budget (k seeds)
- Comparing seeding strategies (degree-based vs greedy vs random)
When NOT to use:
- When measuring existing influence (use centrality metrics)
- For community structure analysis (use community detection)
Algorithm
IRON LAW: Greedy With Lazy Evaluation (CELF) Is the Practical Standard
The naive greedy algorithm requires O(k × n × R) simulations where
R = Monte Carlo runs (10,000+). CELF exploits submodularity to skip
unnecessary evaluations, achieving 700x speedup. Always use CELF
over naive greedy. Simple heuristics (top-k by degree) are fast
but can perform 50%+ worse than greedy.Phase 1: Input Validation
Build network graph. Choose diffusion model: Independent Cascade (probability per edge) or Linear Threshold (threshold per node). Set k (number of seeds) and propagation probabilities. Gate: Graph loaded, diffusion model selected, k defined.
Phase 2: Core Algorithm
Greedy with CELF:
- Initialize: seed set S = ∅
- For each candidate node, estimate marginal gain: σ(S∪{v}) - σ(S) via Monte Carlo simulation (R=10,000 runs)
- Select node with highest marginal gain, add to S
- CELF optimization: reuse previous marginal gains, only re-evaluate when a node's upper bound exceeds current best
- Repeat until |S| = k
Phase 3: Verification
Compare greedy result against baselines: random seeds, top-k degree, top-k PageRank. Greedy should significantly outperform. Gate: Greedy spread > degree heuristic spread, difference is meaningful.
Phase 4: Output
Return seed set with expected spread and comparison.
Output Format
{
"seeds": [{"node": "user_42", "marginal_gain": 150, "selection_order": 1}],
"expected_spread": 2500,
"baselines": {"random": 800, "top_degree": 1900, "greedy": 2500},
"metadata": {"k": 10, "model": "independent_cascade", "mc_simulations": 10000, "nodes": 50000}
}Examples
Sample I/O
Input: Social network 10K nodes, k=5 seeds, IC model with p=0.1 per edge Expected: Greedy selects diverse, well-positioned seeds (not all high-degree), expected spread ~500-1000.
Edge Cases
| Input | Expected | Why |
|---|---|---|
| k=1 | Node with highest individual spread | Single seed, no overlap consideration |
| k > number of communities | One seed per community optimal | Diversity beats concentration |
| Very sparse graph (low p) | Small spread regardless of seeds | Network can't propagate with low probability |
Gotchas
- Monte Carlo variance: With R=1000, spread estimates have ~5% variance. Use R=10,000+ for stable results, especially when comparing close candidates.
- Diffusion model choice matters: IC and LT produce different optimal seed sets. IC favors high-degree nodes; LT favors nodes that can trigger cascades.
- Propagation probability estimation: Real-world edge probabilities are unknown. Common approaches: uniform (p=0.01-0.1), weighted inverse degree (1/in-degree), or learned from cascade data.
- Overlap penalty: Greedy naturally handles overlap (submodularity). Heuristics that independently select top nodes waste seeds on overlapping influence spheres.
- Scalability: Even with CELF, millions of nodes require further approximation (sketch-based methods like IMM or TIM+).
References
- For CELF and CELF++ implementation, see
references/celf-implementation.md - For scalable influence maximization (IMM), see
references/scalable-im.md