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Learning Mean Field Control on Sparse Graphs
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Proposes local weak mean field control (LWMFC), a mean field control model for very sparse graphs with finite average degree (e.g., power-law networks with exponent above two), with a two-systems approximation and policy-gradient learning evaluated on synthetic and real networks.
Problem
Mean field approximations for multi-agent control on realistic sparse graphs are unavailable: graphon and graphex MF models require diverging average degree, excluding power-law networks with exponent above two.
Approach
Model agents via local weak convergence of graph sequences, using degree-dependent mean fields and a two-systems approximation (low-degree agents with explicit degree MFs, high-degree agents summarized into one MF with a neighbor-degree heuristic), then solve the resulting MFC MDP with single-agent PPO over high-level policies.
Key results
- On eight real-world KONECT networks, LWMFC reduces SIS total-variation error to about 1.6–5.0% versus 4.7–24% for LPGMFG and GXMFG in the stated table, and LWMFC outperforms both baselines on all networks and problems.
- On synthetic CL graphs with N=860 and 1598, MFMARL beats IPPO on all four problems (e.g., SIS -9.11 vs -19.70 at N=860).
- The extensive approximation LWMFC* moderately improves on LWMFC on most datasets but is computationally infeasible for Color and Rumor.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use MFMARL instead of IPPO | CL graph learning · SIS, N=1598 | objective: -22.42 -> -11.13 | Helped |
| Use MFMARL instead of IPPO | CL graph learning · SIR, N=1598 | objective: -7.73 -> -6.32 | Helped |
| Use MFMARL instead of IPPO | CL graph learning · Color, N=1598 | objective: -8.15 -> -6.94 | Helped |
| Use MFC instead of MFMARL | CL graph learning · Rumor, N=1598 | objective: 0.17 -> 1.33 | Helped |
| Use the extensive approximation instead of the two systems approximation (ablation) | LWMFC · CAIDA, SIS | total variation (%): 2.59 (1.14) -> 1.75 (0.90) | Helped |
| Use LWMFC instead of GXMFG | mean-field approximation · CAIDA, SIS | total variation (%): 9.07 (1.25) -> 2.59 (1.14) | Helped |
| Use the extensive approximation instead of the two systems approximation (ablation) | LWMFC · real-world networks, SIS and SIR | total variation | Mixed |
| Use LWMFC instead of LPGMFG or GXMFG | mean-field approximation · eight real-world networks, SIS, SIR, Color, and Rumor | total variation | Helped |
| Use MFMARL instead of IPPO | CL graph learning · synthetic CL graphs, N=860 and N=1598, all four problems | objective | Helped |
| Use MFMARL instead of IPPO | CL graph learning · synthetic CL graphs, N=167 and N=406, SIR | objective | Mixed |
Surprising findings
- On the two smaller graphs, IPPO is marginally better than LWMFC on SIR with N=167 and N=406.
- LWMFC is evaluated on the limiting model rather than the empirical system, so its objective may deviate from true system behavior.
Limitations
- Extensive approximation LWMFC* is computationally too expensive for Color and Rumor problems.
- LWMFC policies are evaluated on the limiting model, which may differ from the true finite-network behavior.
- Relies on a neighbor-degree heuristic and independence approximations in the two-systems approximation.
Experimental setup
Datasets: CAIDA Cities Digg Friends Enron Flixster Slashdot Yahoo YouTube synthetic Chung-Lu graphs
Benchmarks: CAIDA Cities Digg Friends Enron Flixster Slashdot Yahoo YouTube synthetic Chung-Lu graphs with N=167 to 1598
Metrics: average expected total variation between MF and empirical MF objective value after training
Baselines: LPGMFG GXMFG IPPO
Builds on: graphon mean field games LPGMFG GXMFG IPPO PPO
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