Network World Models as Environments for Algorithm Design on Complex Systems
Organizations: Emory University · University of Illinois Urbana-Champaign
Abstract
World models, which simulate an environment and predict how it changes under actions, are increasingly used in real-world applications such as robotics. Complex systems call for the same tool because the effect of an action is not immediate. Seeding nodes for a campaign, or immunizing nodes against an epidemic, changes little on its own; what matters is the outcome that unfolds over the steps that follow. Designing an algorithm that selects such actions to maximize expected performance on a task is inherently iterative, and every candidate must be scored by the outcome it produces. Obtaining that outcome has relied on simulation, whose cost becomes a bottleneck when candidates are evaluated over many sampled trajectories. We propose an action-conditioned Network World Model that learns a network's diffusion dynamics under interventions over time, applies each action to the network, and predicts the outcome that follows. It serves as a fast evaluator inside an algorithm design loop in which a coding agent designs and refines executable algorithms using feedback from full rollouts, action-level credit, and counterfactual probes over alternative interventions. Across eight network tasks and five diffusion models, the designed algorithms match or exceed the strongest reported baseline in 138 of 141 settings while enabling up to 14.5 times faster rollouts than Monte Carlo simulation. Code will be released upon acceptance.
Figures & tables
| Influence maximization (IC / LT): spread, % of nodes activated ( ) | ||||||||
|---|---|---|---|---|---|---|---|---|
| Network Science (1,589 nodes) | Digg (116,893 nodes) | |||||||
| Method | 1% | 5% | 10% | 20% | 1% | 5% | 10% | 20% |
| IMM | 8.7 / 10.9 | 24.8 / 30.2 | 38.0 / 45.1 | 59.2 / 68.5 | 25.9 / 47.0 | 36.0 / 60.9 | 44.2 / 69.7 | 55.7 / 79.6 |
| OPIM | 8.8 / 10.8 | 24.2 / 29.5 | 37.7 / 45.3 | 58.3 / 67.4 | 27.4 / 50.5 | 41.6 / 68.4 | 51.8 / 77.6 | 62.7 / 85.5 |
| SubSIM | 8.7 / 10.7 | 24.3 / 29.1 | 37.3 / 45.0 | 57.8 / 68.5 | 27.3 / 50.1 | 41.7 / 68.4 | 51.8 / 77.7 | 62.7 / 85.4 |
| DeepIM | 4.8 / 5.3 | 15.3 / 20.0 | 27.7 / 32.9 | 45.8 / 53.0 | – / – | – / – | – / – | – / – |
Appendix figures & tables18 assets
Supplementary material from the paper’s appendix.
Appendix
| Probe | Question it answers about the plan |
|---|---|
| drop | The reward with and without ’s actions: ’s marginal contribution |
| swap | The reward if replaces |
| add | The marginal gain of one more budgeted action on an unselected , at : where the headroom is |
| best_swap | The best replacement for among the 200 highest-degree unselected nodes, with its delta and the runner-up: local search handed over as information, for the model to turn into a rule |
| overlap | The expected number of nodes the cascades of and both reach, : whether two seeds are redundant, which drop cannot say |
| horizon | The spread curve runs out to : a plan that wins early and loses late against one that keeps growing |
| Component | Content |
|---|---|
| Static (the opening turn) | |
| task statement | family, objective and its direction, the diffusion model, (absolute and as a share of ), , the allowed operations, the exogenous context (outbreak or rumor with its size, observation block, rounds of an adaptive task) |
| network profile | , arc count, directedness, density, degree quantiles, -core size, reciprocity, clustering, assortativity, communities and their sizes; no adjacency |
| contract and rules | the entry point and return shape, the two structured comment lines (mechanism and forecast), the import whitelist, the budget and validity rules, two worked exemplars at the level the search should start from |
| library | the callable API of the library without its simulation-based members, with the source of the classical members, and the withheld members named with the reason |
| anchor leaderboard | every anchor with its reward and standard error on this instance, the best first, “beating the top row is the bar” |
| Operator | Category | Purpose |
|---|---|---|
| crossover | Explore | Combine parts from a parent algorithm’s mechanism with a partner algorithm’s |
| synthesize | Explore | Write one new strategy taking the best-supported idea from each prior algorithm |
| from_scratch | Explore | Implement a completely novel idea, chosen by the idea search below, as a new strategy (must not re-implement a library algorithm or something already in the population) |
| refine | Exploit | One small targeted change: adjust one term or fix one weakness the diagnostics expose |
| parameters | Exploit | Only change numerical constants; keep the same functions and control flow |
| simplify | Exploit | Remove parts of the algorithm that the diagnostics do not justify; a shorter child algorithm is the goal |
| Group | Members |
|---|---|
| Network primitives ( primitives , 16) | |
| degree and centrality (6) | compute_degree , compute_out_degree , compute_weighted_degree , get_top_degree_nodes , compute_pagerank , compute_centrality |
| communities and budget (2) | detect_communities (label propagation), allocate_budget (largest-remainder quota per community) |
| path influence (1) | path_influence_scores (truncated path products) |
| reverse influence sampling (3) | batch_reverse_sample , ris_select , estimate_sample_size |
| live-edge realizations (2) | sample_live_edge_graph , reachable_count |
| Task | Dynamics | Task output | Reported metric |
|---|---|---|---|
| Influence maximization | IC, LT | Seed set | Final spread ( ) |
| Adaptive influence maximization | IC, LT | Per-round seeding policy | Final spread ( ) |
| Critical node detection | IC, LT | Node removals against an outbreak | Final infected ( ) |
| Influence blocking | IC, CLT | Counter-seeds, node or arc blocks, or weight cuts against a rumor | Final rumor size ( ) |
| Epidemic control | SIR, SIS | Vaccinations, quarantines, arc cuts, or contact reductions | Attack rate ( ) |
| Source localization | IC, LT | Source set | Consistency with the observation ( ) |
| Dataset | Nodes | Edges | Tasks |
|---|---|---|---|
| Email-EU ( Yin et al., 2017 ; Leskovec et al., 2007 ) | 1,005 | 24,929 | Influence Blocking |
| UCI Students ( Opsahl and Panzarasa, 2009 ) | 1,266 | 6,451 | Cascade Reconstruction |
| Network Science ( Newman, 2006 ) | 1,589 | 2,742 | Influence Maximization, Adaptive Influence Maximization |
| Cora-ML ( McCallum et al., 2000 ; Bojchevski and Günnemann, 2018 ) | 2,810 | 7,981 | Source Localization |
| Power Grid ( Watts and Strogatz, 1998 ) | 4,941 | 6,594 | Critical Node Detection, Source Localization |
| CA-GrQc ( Leskovec et al., 2007 ) | 5,242 | 14,484 | Cascade Reconstruction |
| Task | Method | What it does | Reference |
|---|---|---|---|
| Influence maximization | IMM | Reverse influence sampling with a martingale stopping rule; the standard RIS reference (library) | ( Tang et al., 2015 ) |
| OPIM | Online-processing RIS that tightens its bound as samples arrive (published code) | ( Tang et al., 2018 ) | |
| SubSIM | Sublinear-time reverse reachable set generation with tightened bounds (published code) | ( Guo et al., 2020 ) | |
| DeepIM | Learned seed-set generator with a graph encoder and a spread predictor, trained per network (published code) | ( Ling et al., 2023 ) | |
| DegreeDiscount | Degree ranking with a discount for neighbors already seeded; the anchor (library) | ( Chen et al., 2009 ) | |
| Adaptive influence maximization | EPIC | AdaptGreedy instantiated with reverse influence sampling per round; the anchor (library) | ( Han et al., 2018 ) |
| Network Science (1,589 nodes) ( ) | Power Grid (4,941 nodes) ( ) | |||||||
| Model | 1% | 5% | 10% | 20% | 1% | 5% | 10% | 20% |
| GPT-5.6 Sol | 8.92 | 25.27 | 39.06 | 60.06 | 26.18 | 22.04 | 17.26 | 10.24 |
| GPT-5.6 Terra | 8.89 | 25.23 | 38.98 | 60.01 | 26.19 | 22.05 | 17.28 | 10.25 |
| GPT-5.6 Luna | 8.81 | 25.21 | 38.95 | 60.00 | 26.32 | 22.34 | 17.82 | 10.49 |
| Influence maximization ( ) | Critical node detection ( ) | |||||||
| Network Science (1,589 nodes) | Power Grid (4,941 nodes) | |||||||
| Method | 1% | 5% | 10% | 20% | 1% | 5% | 10% | 20% |
| EoH | 5.5 | 19.5 | 32.5 | 54.6 | 26.6 | 23.4 | 19.4 | 14.3 |
| OpenEvolve | 5.1 | 21.9 | 36.8 | 50.8 | 26.6 | 23.8 | 21.4 | 17.7 |
| LLaMEA | 6.1 | 21.1 | 34.9 | 56.2 | 26.6 | 23.4 | 19.5 | 14.3 |
| ReEvo | 6.6 | 22.3 | 35.5 | 57.0 | 26.6 | 23.5 | 19.6 | 15.0 |
| Influence maximization (IC / LT): spread, % of nodes activated ( ) | ||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Network Science (1,589 nodes) | Digg (116,893 nodes) | NetHEPT (15,229 nodes) | ||||||||||
| Method | 1% | 5% | 10% | 20% | 1% | 5% | 10% | 20% | 1% | 5% | 10% | 20% |
| IMM | 8.7 / 10.9 | 24.8 / 30.2 | 38.0 / 45.1 | 59.2 / 68.5 | 25.9 / 47.0 | 36.0 / 60.9 | 44.2 / 69.7 | 55.7 / 79.6 | 11.6 / 14.8 | 28.9 / 36.2 | 42.4 / 51.6 | 61.7 / 72.5 |
| OPIM | 8.8 / 10.8 | 24.2 / 29.5 | 37.7 / 45.3 | 58.3 / 67.4 | 27.4 / 50.5 | 41.6 / 68.4 | 51.8 / 77.6 | 62.7 / 85.5 | 12.1 / 15.5 | 31.4 / 39.2 | 44.7 / 54.3 | 64.5 / 75.7 |
| SubSIM | 8.7 / 10.7 | 24.3 / 29.1 | 37.3 / 45.0 | 57.8 / 68.5 | 27.3 / 50.1 | 41.7 / 68.4 | 51.8 / 77.7 | 62.7 / 85.4 | 12.6 / 16.2 | 31.3 / 39.0 | 44.8 / 54.5 | 64.3 / 75.4 |
| DeepIM | 4.8 / 5.3 | 15.3 / 20.0 | 27.7 / 32.9 | 45.8 / 53.0 | – / – | – / – | – / – | – / – | – / – | – / – | – / – | – / – |
| Epidemic control (SIR / SIS): attack rate, % of nodes ever infected ( ) | ||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Infectious SocioPatterns (10,972 nodes) | Oregon1 (10,670 nodes) | Brightkite (58,228 nodes) | ||||||||||
| Method | 1% | 5% | 10% | 20% | 1% | 5% | 10% | 20% | 1% | 5% | 10% | 20% |
| DAVA | 19.9 / 23.0 | 7.4 / 8.3 | 1.0 / 1.0 | 1.0 / 1.0 | 7.9 / 9.6 | 1.0 / 1.0 | 1.0 / 1.0 | 1.0 / 1.0 | 40.9 / 47.1 | – / – | – / – | – / – |
| NetShield+ | 21.2 / 24.9 | 17.3 / 20.6 | 12.8 / 15.6 | 7.4 / 8.9 | 4.3 / 5.4 | 2.1 / 2.3 | 1.8 / 1.9 | 1.6 / 1.7 | 24.1 / 29.4 | 8.5 / 10.5 | 4.5 / 5.4 | 2.7 / 3.0 |
| GreedyWalk | 20.9 / 24.6 | 16.5 / 19.8 | 12.3 / 14.9 | 7.0 / 8.5 | 4.8 / 5.9 | 2.5 / 2.8 | 1.9 / 2.1 | 1.6 / 1.7 | 23.2 / 28.4 | 8.4 / 10.5 | 4.6 / 5.5 | 2.8 / 3.1 |
| EI | 20.7 / 24.4 | 16.3 / 19.3 | 11.5 / 13.6 | 8.1 / 9.5 | 4.2 / 5.2 | 2.0 / 2.1 | 1.7 / 1.8 | 1.6 / 1.6 | 23.5 / 28.7 | 8.7 / 10.9 | 4.5 / 5.3 | 2.8 / 3.1 |
| Influence maximization (IC), program designed on Digg ( ) | ||||||||
|---|---|---|---|---|---|---|---|---|
| Network Science (1,589) | NetHEPT (15,229) | |||||||
| Algorithm | 1% | 5% | 10% | 20% | 1% | 5% | 10% | 20% |
| Designed on the target | 8.9 | 25.3 | 39.1 | 60.1 | 12.9 | 33.0 | 47.8 | 68.5 |
| Transferred from Digg | 8.9 | 25.3 | 39.1 | 60.0 | 12.9 | 33.0 | 47.8 | 68.6 |
| Adaptive influence maximization (IC), program designed on NetHEPT ( ) | ||||||||
| Network Science (1,589) | Digg (116,893) | |||||||
| Shift | Target | KL ( ) | Exact Brier ( ) | AUROC ( ) | Brier ( ) | Spread bias ( ) | Effect MAE norm ( ) / Pearson ( ) |
|---|---|---|---|---|---|---|---|
| None | BA-100 | 0.0059 | 0.00011 | 0.923 | 0.0083 / 0.0078 | / | 0.0095 / 1.0000 |
| Size | BA-200 | 0.0048 | 0.00002 | 0.934 | 0.0076 / 0.0072 | / | 0.0103 / 0.9999 |
| Size | BA-500 | 0.0043 | 0.00002 | 0.943 | 0.0079 / 0.0064 | / | 0.0098 / 0.9999 |
| Size | BA-1000 | 0.0041 | 0.00003 | 0.948 | 0.0075 / 0.0066 | / | 0.0129 / 0.9999 |
| Topology | WS-100 | 0.0036 | 0.00000 | 0.932 | 0.0074 / 0.0073 | / | 0.0088 / 1.0000 |
| Topology | ER-100 | 0.0051 | 0.00023 | 0.930 | 0.0093 / 0.0081 | / | 0.0131 / 0.9995 |
| Variant | F1 ( ) | Brier ( ) | Count bias | F1 drop | Effect corr. ( ) | Train (min) |
|---|---|---|---|---|---|---|
| GraphSAGE | 0.8765 | 0.0003 | 0.792 | 0.849 | 6.5 | |
| GCN | 0.8750 | 0.0003 | 0.810 | 0.849 | 8.8 | |
| GATv2 | 0.8756 | 0.0004 | 0.812 | 0.849 | 11.4 | |
| GCNII | 0.8745 | 0.0004 | 0.796 | 0.848 | 20.8 |
| Network | Step 1 | Step 2 | Step 5 | Step 10 |
|---|---|---|---|---|
| Network Science | ||||
| Power Grid |