An open problem is to extend the results in the literature on unit disk graphs to  hypergraph models.  Motivated by recent results that the worst-case performance of the distributed maximal scheduling algorithm is characterized by the interference degree of the hypergraph, in the present work we investigate properties of the interference degree of the hypergraph and the structure of hypergraphs arising from physical constraints.  We show that the problem of computing the interference degree of a hypergraph is NP-hard and we prove some properties and results concerning this hypergraph invariant.  We then investigate which hypergraphs are realizable, i.e. which hypergraphs arise in practice, based on physical constraints, as the interference model of a wireless network.  In particular, given the results on the worst-case performance of the maximal scheduling algorithm, a question that arises naturally is: what is the maximal value of $r$ such that the hypergraph $K_{1,r}$ is realizable?  We show that this value is $r=4$.

If each node in a wireless network has information about only its $1$-hop neighborhood, then what are the limits to performance? This problem is considered for wireless networks where each communication link has a minimum bandwidth quality-of-service (QoS) requirement. Links in the same vicinity contend for the shared wireless medium. The conflict graph captures which pairs of links interfere with each other and depends on the MAC protocol. In IEEE 802.11 MAC protocol-based networks, when communication between nodes $i$ and $j$ takes place, the neighbors of both $i$ and $j$ remain silent. This model of interference is called the $2$-hop interference model because the distance in the network graph between any two links that can be simultaneously active is at least $2$. In the admission control problem studied in the present paper, the objective is to determine, using only localized information, whether a given set of flow rates is feasible. While distance-$d$ distributed algorithms have been analyzed for the $1$-hop interference model, an open problem in the literature is to extend these results to the $K$-hop interference model, and the present work initiates the generalization to the $K$-hop interference model. We show that the centralized version of the problem is NP-hard and then investigate distributed, low-complexity solutions for this problem. We propose a distributed algorithm for this problem where each node has information about only its $1$-hop neighborhood. The worst-case performance of the distributed algorithm, i.e. the largest factor by which the performance of this distributed algorithm is away from that of an optimal, centralized algorithm, is analyzed. Lower and upper bounds on the suboptimality of the distributed algorithm are obtained, and both bounds are shown to be tight. The exact worst-case performance is obtained for some ring topologies. The performance of the distance-$1$ distributed algorithm is compared with that of the row constraints, and these two distributed algorithms are shown to be incomparable.

This research is on performance analysis of networks and no data was used.