cs.AIOct 1, 2026
SaveNetwork 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
Figure 1: A coding agent writes algorithms that choose interventions on a network, and a Network World Model rolls out how each choice unfolds, returning a reward, feedback, and probe answers the agent uses to improve the algorithm (left). Across eight tasks on complex networks, the agent-designed algorithms outperform the strongest baselines, averaged over budgets (right).
| 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 | – / – | – / – | – / – | – / – |
Table 1: Performance of designed algorithms on four tasks and two datasets per task. Each cell reports results under two diffusion models: IC / LT, except for influence blocking, which reports IC / CLT. A dash indicates an OOM error or timeout.
Figure 2: Task score of the returned algorithm under three coding models, averaged over budgets.
Figure 3: Generalization across networks and scales. (a) Critical node detection transfer from PGP to Power Grid and Gnutella31, sign-corrected so higher is better. (b) Influence maximization performance across increasing network sizes under IC at the 10% budget.
Figure 4: Wall-clock time of one rollout sample under the Network World Model and Monte Carlo simulation, log scale.
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 |
Table 4: The probe operations. Every probe is metered as evaluator work, and a turn may ask at most six. frontier , robust(hidden) and gradient exist on the forward-model evaluators, and gradient only for seeding.
| 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” |
Table 5: The prompt of the write turn. The static part is sent once in the opening turn and the dynamic part is rebuilt every round.
| 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 |
Table 6: The six instructions. The two that need a partner are withheld while the population has one member; the instruction text for each is in Appendix D .
| 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 |
Table 7: The callable library by mechanism. Counts are callable members.
| 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 ( ) |
Table 8: The eight tasks on complex systems, summarized.
| 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 |
Table 9: The datasets used across the eight tasks, ordered by node count. For the three cascade-prediction corpora, the counts describe the full underlying network, while the runs replay a subsample restricted to the most active participants.
| 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 ) |
Table 10: The baseline methods we compare against, what each does, and where it comes from.
| 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 |
Table 11: Algorithm design repeated on GPT-5.6 Sol, Terra, and Luna.
| 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 |
Table 12: Comparison with LLM algorithm discovery systems on influence maximization on Network Science (spread, % of nodes activated, ) and critical node detection on Power Grid (remaining spread, % of nodes infected, ), under IC. All rows use GPT-5.6 Sol as the coding model.
Figure 5: Performance across intervention budgets under IC for influence maximization (IM), adaptive influence maximization (AIM), critical node detection (CND), and influence blocking (IB).
| 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 | – / – | – / – | – / – | – / – | – / – | – / – | – / – | – / – |
Table 13: Designed algorithms results for all eight tasks and every dataset. Every cell is the IC / LT pair (IC / CLT for influence blocking, SIR / SIS for epidemic control) obtained by replaying the returned solution on the shared oracle referee, so all methods are scored on the same referee. A dash in a cell indicates no result because of either an Out of Memory (OOM) error or a timeout that exceeds the 1,800 s wall-clock limit.
| 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 |
Table 13: Designed algorithms results for all eight tasks and every dataset (continued).
| 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) | |||||||
Table 14: Generalizability of the designed algorithms. The algorithm designed on one graph is replayed unmodified on the task’s other graphs. Designed on the target is that graph’s own returned program, as in Tables 1 and 13 . The NetHEPT adaptive program exceeded the 1,800 s wall-clock limit for one call on Digg at every budget.
| 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 |
Table 15: Frozen-checkpoint generalization across network size and topology. “Exact Brier” compares the predicted one-step probability with the analytic IC probability. The rollout column reports the model’s Brier score and the noise floor of the exact process at , and spread bias reports the same pair. The counterfactual effect is measured against the analytic intervention effect. Arrows give the better direction; spread bias is signed, so closer to zero is better.
| 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 |
Table 16: World-model encoders on Network Science. Each row changes only the encoder against the first row. Count bias is the relative bias of the final count of a free-running sampled rollout at the horizon; “ F1 drop” is the loss in one-step F1 when the action channels are zeroed at evaluation; “Effect corr.” is the Pearson correlation between the predicted and the measured effect of an action.
Figure 6: The four encoders on Network Science. Left: mean infected count of the free-running sampled rollout at each step, against the simulator (dashed). Right: the same rollouts as relative count bias by step; the bias settles within the first five steps on every backbone and does not grow with the horizon.
| Network | Step 1 | Step 2 | Step 5 | Step 10 |
|---|---|---|---|---|
| Network Science | ||||
| Power Grid |
Table 17: Relative count bias of the free-running sampled rollout against the Monte Carlo simulator, by rollout step, for the checkpoints used in the end-to-end runs, on the 50 held-out episodes of each run. Zero means no bias; a magnitude that grows with the step would mean compounding error.
Figure 7: Relative count bias of the free-running sampled rollout by step on the two benchmark networks, on the 50 held-out episodes of the main runs. The bias settles within the first few steps on both networks and does not grow.