Understanding the dynamics of evolving social or infrastructure networks is a challenge in applied areas such as epidemiology, viral marketing, or urban planning. During the past decade, data has been collected on such networks but has yet to be fully analyzed. We propose to use information on the dynamics of the data to find stable partitions of the network into groups. For that purpose, we introduce a time-dependent, dynamic version of the facility location problem, that includes a switching cost when a client's assignment changes from one facility to another. This might provide a better representation of an evolving network, emphasizing the abrupt change of relationships between subjects rather than the continuous evolution of the underlying network. We show that in realistic examples this model yields indeed better fitting solutions than optimizing every snapshot independently. We present an O(log nT )-approximation algorithm and a matching hardness result, where n is the number of clients and T the number of time steps. We also give an other algorithms with approximation ratio O(log nT ) for the variant where one pays at each time step (leasing) for each open facility.

In this paper we propose a probabilistic analysis of the fully asynchronous behavior (i.e., two cells are never simultaneously updated, as in a continuous time process) of elementary finite cellular automata (i.e., {0, 1} states, radius 1 and unidimensional) for which both states are quiescent (i.e., (0, 0, 0) → 0 and (1, 1, 1) → 1). It has been experimentally shown in previous works that introducing asynchronism in the global function of a cellular automata was perturbing its behavior, but as far as we know, only few theoretical work exists on the subject. The cellular automata we consider live on a ring of size n and asynchronism is introduced as follows: at each time step one cell is selected uniformly at random and the transition is made on this cell while the others stay in the same state. Among the sixty-four cellular automata belonging to the class we consider, we show that nine of them diverge almost surely on all non-trivial configurations while the fifty-five other converge almost surely to a random fixed point. We show that the exact convergence time of these fifty-five automata can only take the following values: either 0, Θ(n ln n), Θ(n 2), Θ(n 3), or Θ(n2 n). Furthermore, the global behavior of each of these cellular automata is fully determined by reading its code.