Routing is widely used to scale large language models, from Mixture-of-Experts gating to multi-model/tool selection. A common belief is that routing to a task ``expert'' activates sparser internal computation and thus yields more certain and stable outputs (the Sparsity--Certainty Hypothesis). We test this belief by injecting routing-style meta prompts as a textual proxy for routing signals in front of frozen instruction-tuned LLMs. We quantify (C1) internal density via activation sparsity, (C2) domain-keyword attention, and (C3) output stability via predictive entropy and semantic variation. On a RouterEval subset with three instruction-tuned models (Qwen3-8B, Llama-3.1-8B-Instruct, and Mistral-7B-Instruct-v0.2), meta prompts consistently densify early/middle-layer representations rather than increasing sparsity; natural-language expert instructions are often stronger than structured tags. Attention responses are heterogeneous: Qwen/Llama reduce keyword attention, while Mistral reinforces it. Finally, the densification--stability link is weak and appears only in Qwen, with near-zero correlations in Llama and Mistral. We present RIDE as a diagnostic probe for calibrating routing design and uncertainty estimation.
Figures & tables
Figure 1: Multi-model overview of ΔHoyer across Early/Middle/Late segments.
Figure 2: Multi-model overview of ΔAttn (mean ± SEM). Negative values indicate reduced attention share to domain keywords.
Figure 3: Instance-level correlations between ΔC1 (prompt Hoyer) and ΔC3 across models and contrasts. Positive values indicate that instances with larger decreases in Hoyer tend to also exhibit larger decreases in the corresponding instability measure (entropy or semantic variation).
Appendix figures & tables8 assets
Supplementary material from the paper’s appendix.
Appendix
Model
ΔHoyer Tag–Ctrl
ΔHoyer Instr–Ctrl
ΔHoyer Tag–Instr
Llama-3.1-8B-Instruct
−0.015±0.001
−0.016±0.001
+0.001±0.000
Mistral-7B-Instruct-v0.2
−0.016±0.001
−0.017±0.001
+0.000±0.000
Qwen3-8B
−0.005±0.000
−0.003±0.000
−0.002±0.000
Appendix
Table 1: Early segment: mean ± SEM of ΔHoyer under three contrasts.
Model
ΔHoyer Tag–Ctrl
ΔHoyer Instr–Ctrl
ΔHoyer Tag–Instr
Llama-3.1-8B-Instruct
−0.014±0.001
−0.015±0.001
+0.000±0.000
Mistral-7B-Instruct-v0.2
−0.012±0.001
−0.015±0.001
+0.003±0.000
Qwen3-8B
−0.007±0.001
−0.009±0.001
+0.002±0.000
Appendix
Table 2: Middle segment: mean ± SEM of ΔHoyer under three contrasts.
Model
ΔHoyer Tag–Ctrl
ΔHoyer Instr–Ctrl
ΔHoyer Tag–Instr
Llama-3.1-8B-Instruct
−0.008±0.000
−0.012±0.000
+0.004±0.000
Mistral-7B-Instruct-v0.2
−0.002±0.000
−0.015±0.000
+0.013±0.000
Qwen3-8B
−0.001±0.000
−0.007±0.001
+0.006±0.000
Appendix
Table 3: Global segment: mean ± SEM of instance-level paired differences in ΔHoyer under three contrasts.
Model
ΔAttn Tag–Ctrl (Mean ± SEM)
ΔAttn Instr–Ctrl (Mean ± SEM)
Llama-3.1-8B-Instruct
−0.0389±0.0031
−0.0371±0.0030
Mistral-7B-Instruct-v0.2
+0.0154±0.0005
+0.0164±0.0005
Qwen3-8B
−0.0409±0.0032
−0.0396±0.0031
Appendix
Table 4: Mean ± SEM of ΔAttn under Tag–Ctrl and Instr–Ctrl across three models (Prompt-last view).
Model
ΔAttn Tag–Ctrl (Mean ± SEM)
ΔAttn Instr–Ctrl (Mean ± SEM)
Llama-3.1-8B-Instruct
−0.0592±0.0045
−0.0599±0.0046
Mistral-7B-Instruct-v0.2
+0.0158±0.0004
+0.0175±0.0005
Qwen3-8B
−0.0493±0.0039
−0.0480±0.0038
Appendix
Table 5: Mean ± SEM of ΔAttn under Tag–Ctrl and Instr–Ctrl across three models (First-gen view). Negative values indicate reduced domain-keyword attention share relative to control.
Model
r(ΔAttn,ΔEntropy)
p -value
Llama-3.1-8B-Instruct
0.117
1.3×10−10
Mistral-7B-Instruct-v0.2
−0.072
7.8×10−5
Qwen3-8B
0.229
5.4×10−37
Appendix
Table 6: Pearson correlation between ΔAttn and ΔEntropy (instance-level), with corresponding p -values.
Model
r (Instr vs. Ctrl)
r (Tag vs. Ctrl)
r (Tag vs. Instr)
Llama-3.1-8B-Instruct
0.141∗∗∗
0.013ns
0.105∗∗∗
Mistral-7B-Instruct-v0.2
0.191∗∗∗
0.043∗
0.150∗∗∗
Qwen3-8B
0.309∗∗∗
0.204∗∗∗
0.139∗∗∗
Appendix
Table 7: Pearson correlations between C1 ( ΔHoyer ) and C3-Entropy ( ΔEntropy ) under three contrasts. Superscripts indicate significance: ∗∗∗p<0.001 , ∗∗p<0.01 , ∗p<0.05 , and ns not significant.
Model
Instr–Ctrl
Tag–Ctrl
Tag–Instr
Llama-3.1-8B-Instruct
0.067∗∗∗
−0.027ns
0.037∗
Mistral-7B-Instruct-v0.2
0.142∗∗∗
0.128∗∗∗
0.177∗∗∗
Qwen3-8B
0.023ns
0.072∗∗∗
0.017ns
Appendix
Table 8: Pearson r between C1 ( ΔHoyer ) and C3-SemVar ( ΔVar ). Superscripts: ∗∗∗p<0.001 , ∗∗p<0.01 , ∗p<0.05 , ns not significant.
Large language model routing selects models, tools, experts, or inference strategies to balance answer quality, cost, latency, and deployment constraints.
Sparse mixture-of-experts (MoE) language models activate only a small subset of parameters for each token, making router behavior a central part of model computation. This paper studies routing behavior of Mixtral 8x7B-Instruct under benign and harmful prompts using two complementary signals: activation-based routing scores derived from expert selection frequencies and gradient-based scores derived from router-gate sensitivities. We analyze expert- and layer-level routing behavior and conduct expert-suppression interventions. The results show that activation-based expert usage is broad and long-tailed, whereas gradient-based importance is concentrated. At expert level, benign and harmful prompt groups remain close under both signals with modest separation. At layer level, activation-based routing is most selective around layers 8-15, while gradient-based importance is concentrated in final layers. Expert classification shows most experts are shared across benign and harmful prompts, though a limited subset shows clear group preference. Top-ranked expert sets show stronger benign-malicious overlap under gradient scores than activation scores, suggesting concentration on a common late-layer expert set. In intervention experiments, suppressing top five benign-dominant experts from activation scores reduces restricted responses from 24 to 14 over 100 prompts, while suppressing gradient-derived experts reduces them from 34 to 22 with fewer unintended reversals. Overall, safety-relevant routing in Mixtral is subtle, depth-dependent, and distributed rather than dominated by a fixed set of experts.
Md Nurul Absar Siddiky
Department of Electrical and Computer Engineering University of Hawai’i at M¯anoa Honolulu, HI, USA
Dynamic LLM routers promise to cut inference costs by sending each query to the cheapest model that can answer it correctly. We analyze six commercial routers across 14 settings on a diverse benchmark spanning eight task categories, finding that none of them outperforms a router that randomly selects between two well-chosen models at matched cost. Some underperform by more than 10 percentage points. We trace this gap to four patterns prevalent across routers: difficulty blindness, length reversal, semantic matching, and roster suboptimality. We show that the first three are what the standard objective rewards: cost-accuracy Pareto efficiency on realized costs favors escalating moderately hard queries over the hardest ones, shorter queries over longer ones, and routing by a query's source over its difficulty. We also argue that the two assumptions that would justify large rosters, model granularity and model specialization, do not hold empirically. We propose an alternative evaluation methodology that does not reward these patterns, and as a proof of concept, we design a simple two-model router that avoids all four. Nevertheless, its gain over random routing is limited, because a well-chosen roster leaves little to route.
Reward models are a standard tool to score responses from LLMs. Reward models are built to rank responses to a fixed prompt sampled from a single model, for example to choose the best of n sampled responses. In this paper, we study whether scores from response-level reward models lifted to score a model's suitability for a prompt, prior to seeing responses from that model. Specifically, we show that it is straightforward to predict the expected reward that an LLM would earn from the reward model under repeated sampling. Further, we show that these expected reward predictions are precise and discriminative enough to support an application to a model routing protocol that routes prompts to models at inference time to maximize reward while controlling computational cost. We demonstrate the performance of this routing procedure on the open-perfectblend dataset, using a model pool composed of Llama3.1-Instruct 8B/70B, Gemma2-IT 9B/27B, and Gemma1-IT 7B models. Our simple expected reward prediction--based routing (ERP) outperforms baselines that route prompts to models with the best average performance within each prompt's category, and explains the success of more complex routing protocols that implicitly estimate an expected reward. Our approach has the added advantage of being trivially extensible as new models are added to the pool.
Kenan Hasanaliyev, Silas Alberti, Jenny Hamer +5
Stanford University · Inception Labs · Cognition Labs +2