Stable Partitions in Networks Under Symmetric Neighborhood-Composition Costs

We study changes in group membership on a fixed network. Link costs are symmetric, with the larger of the two neighborhood adjustments applied to both players. A stable partition is reachable through a finite sequence of best responses and allows no player to lower their cost by moving to another group. On a star with at least two group labels, the stable structure depends on a single threshold. Above it, all players belong to one group. Below it, the center is alone and all peripheral players belong to another group. At the threshold, both forms allow no profitable move, and we identify the initial partitions from which each is reachable. On a complete network with two group labels, explicit thresholds determine the stable difference between group sizes. At threshold values, this difference also depends on the initial partition. Compared with the asymmetric model, our rule permits stable separation on a star and larger differences in group sizes on a complete network under the same conditions.