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We investigate a probabilistic model for routeing in a multihop ad hoc communication network, where each user sends a message to the base station. Messages travel in hops via other users, used as relays. Their trajectories are chosen at random according to a Gibbs distribution, which favors trajectories with low interference, measured in terms of signal-to-interference ratio. This model was introduced in our earlier paper, where we expressed, in the limit of a high density of users, the typical distribution of the family of trajectories in terms of a law of large numbers. In this paper, we derive its qualitative properties. We analytically identify the emerging typical scenarios in three extreme regimes. We analyze the typical number of hops and the typical length of a hop and the deviation of the trajectory from the straight line, regime 1 in the limit of a large communication area and large distances and regime 2 in the limit of a strong interference weight. In both the regimes, the typical trajectory approaches a straight line quickly, in regime 1 with equal hop lengths. Interestingly, in regime 2, the typical length of a hop diverges logarithmically in the distance of the transmitter to the base station. We further analyze (regime 3) local and global repulsive effects of a densely populated subarea on the trajectories. [ABSTRACT FROM AUTHOR]

Maximization of the information divergence from any hierarchical log-linear model is studied. A new upper bound on the maximum is presented and its tightness analyzed. For the models given by the bases of a matroid, the latter is related to matroid representations by partitions or, equivalently, to ideal secret-sharing schemes. A new link between the divergence maximization, the maximum-likelihood principle, and secret sharing is established. [ABSTRACT FROM AUTHOR]