Typically clustering algorithms provide clustering solutions with prespecified number of clusters. The lack of a priori knowledge on the true number of underlying clusters in the dataset makes it important to have a metric to compare the clustering solutions with different number of clusters. This article quantifies a notion of persistence of clustering solutions that enables comparing solutions with different number of clusters. The persistence relates to the range of dataresolution scales over which a clustering solution persists; it is quantified in terms of the maximum over two-norms of all the associated cluster-covariance matrices. Thus we associate a persistence value for each element in a set of clustering solutions with different number of clusters. We show that the datasets where natural clusters are a priori known, the clustering solutions that identify the natural clusters are most persistent - in this way, this notion can be used to identify solutions with true number of clusters. Detailed experiments on a variety of standard and synthetic datasets demonstrate that the proposed persistence-based indicator outperforms the existing approaches, such as, gap-statistic method, X-means, Gmeans, PG-means, dip-means algorithms and informationtheoretic method, in accurately identifying the clustering solutions with true number of clusters. Interestingly, our method can be explained in terms of the phase-transition phenomenon in the deterministic annealing algorithm, where the number of distinct cluster centers changes (bifurcates) with respect to an annealing parameter.

This paper presents a framework that develops algorithms for solving combined locational and multihop routing optimization problems. The objective is to determine resource node locations in a multiagent network and to specify the multihop routes from each agent to a common destination through a network of resource nodes that minimize total communication cost. These problems are computationally complex (NP-hard) where the cost functions are riddled with multiple minima. Algorithms based on Maximum Entropy Principle, which guarantee local minima and are heuristically designed to seek the global minimum are presented. These algorithms accommodate practical constraints on resource nodes as well as on the routing network architectures. Simulation results show that the multihop routes and resource locations allocated by these algorithms achieve lower costs (as low as 47 percent) than those algorithms where resource locational optimization is done without multihop routing or where the locational and routing optimization objectives are separated. The enabling feature of these algorithms is accommodating problems with resource constraints which is demonstrated through simulations.