In a load balancing network each processor has an initial collection of unit-size jobs, tokens, and in each round, pairs of processors connected by balancers split their load as evenly as possible. An excess token (if any) is placed according to some predefined rule. As it turns out, this rule crucially effects the performance of the network. In this work we propose a model that studies this effect. We suggest a model bridging the uniformly-random assignment rule, and the arbitrary one (in the spirit of smoothed-analysis) by starting from an arbitrary assignment of balancer directions, then flipping each assignment with probability α independently. For a large class of balancing networks our result implies that after O(logn)\mathcal O(\log n) rounds the discrepancy is whp O( (1/2-a) logn + loglogn)\mathcal O( (1/2-\alpha) \log n + \log \log n). This matches and generalizes the known bounds for α= 0 and α= 1/2.