Objective-Weighted Graph Partition Router

Integrate this at the MoE router initialization or at a periodic offline refresh, operating on a calibration batch of tokens rather than on every training token. (1) Construct V as token occurrences or token prototypes, and create a sparse graph E by connecting each token to its top-h cosine neighbors; set w_uv to cosine similarity clipped at zero. A candidate I is an integer vector z with z_v in {1,...,k}, where z_v is the selected expert. Define F using the fourth formula, with lambda_sim, lambda_load, and lambda_sep tuned so that load variance is strongly penalized while similar tokens prefer the same expert. (2) Maintain a population of P assignments. Select two parents by tournament or rank selection. For every token v, compute s_i=F(I^i) for maximization, set q_i=exp(beta*s_i) after subtracting max(s_1,s_2) for numerical stability, and set p_1=q_1/(q_1+q_2). Sample the child label from parent 1 with probability p_1 and parent 2 otherwise. (3) Apply local mutations: propose moving a token v from expert a to b, compute the exact or incremental Delta_M, accept the best positive-gain move, and additionally force moves from overloaded experts when N_a exceeds the capacity threshold. Run one-token and small j-token neighborhoods for a fixed budget. (4) Keep the best-scoring diverse assignments, where diversity is Hamming distance between z vectors, then export the best assignment as router-logit initialization: set the selected expert logit to +c and all others to -c, or use it as a temporary hard-routing mask for several warmup steps. The mathematical quantities are F, p_1, and Delta_M; estimate graph edges and weights from embeddings, and estimate objective coefficients on a held-out calibration set. First test on a 4-expert Switch-Transformer-style layer with a small language-model or CIFAR token benchmark, comparing random router initialization, k-means initialization, and learned noisy top-1 routing. Measure expert load coefficient of variation, auxiliary load-balancing loss, validation loss at equal training FLOPs, and the number of dropped or overflowed tokens. A successful result is lower initial routing imbalance and faster loss descent without reducing final perplexity or accuracy.