Randomized and Distributed Self-Configuration of Wireless Networks: Two-Layer Markov Random Fields and Near-Optimality

This work studies the near-optimality versus the complexity of distributed configuration management for wireless networks. We first develop a global probabilistic graphical model for a network configuration which characterizes jointly the statistical spatial dependence of a physical- and a logical-configuration. The global model is a Gibbs distribution that results from the internal network properties on node positions, wireless channel and interference; and the external management constraints on physical connectivity and signal quality. A local model is a two-layer Markov Random Field (i.e., a random bond model) that approximates the global model with the local spatial dependence of neighbors. The complexity of the local model is defined through the communication range among nodes which corresponds to the number of neighbors in the two-layer Markov Random Field. The local model is near-optimal when the approximation error to the global model is within a given bound. We analyze the tradeoff between approximation error and complexity. We then derive sufficient conditions on the near-optimality of the local model. For a fast decaying wireless channel with power attenuation factor α > 4 , a node only needs to communicate with O(1) neighbors for a local model to be near optimal. For a slowly decaying channel with a power attenuation factor 2 ≤ α ≤ 4, a node may have to communicate with more than O(N^(4-α)/4) neighbors to result in a bounded approximation error. If the communication range is kept to be O(1), a bounded approximation error can also be achieved by reducing the density of active links to O(N^{(α-4)/(α+4)}) for α <; 4 and O(1) for α > 4 . The two-layer Markov Random Fields enable a class of randomized distributed algorithms such as the stochastic relaxation that allows a node to self-configure based on informat- - ion from neighbors. We validate the model, the analysis and the randomized distributed algorithms through simulation.