We present a stabilizing algorithm for finding clustering of path (line) networks on a distributed model of computation. Clustering is defined as covering of nodes of a network by subpaths (sublines) such that the intersection of any two subpaths (sublines) is at most a single node and the difference between the sizes of the largest and the smallest clusters is minimal. The proposed algorithm evenly partitions the network into nearly the same size clusters and places resources and services for each cluster at its center to minimize the cost of sharing resources and using the services within the cluster. Due to being stabilizing, the algorithm can withstand transient faults and does not require initialization. We expect that this stabilizing algorithm will shed light on stabilizing solutions to the problem for other topologies such as grids, hypercubes, and so on.