Random graph models for wireless communication networks

This thesis concerns mathematical models of wireless communication networks, in particular ad-hoc networks and 802.11 WLANs. In ad-hoc mode each of these devices may function as a sender, a relay or a receiver. Each device may only communicate with other devices within its transmission range. We use graph models for the relationship between any two devices: a node stands for a device, and an edge for a communication link, or sometimes an interference relationship. The number of edges incident on a node is the degree of this node. When considering geometric graphs, the coordinates of a node give the geographical position of a node. One of the important properties of a communication graph is its connectedness — whether all nodes can reach all other nodes. We use the term connectivity, the probability of graphs being connected given the number of nodes and the transmission range to measure the connectedness of a wireless network. Connectedness is an important prerequisite for all communication networks which communication between nodes. This is especially true for wireless ad-hoc networks, where communication relies on the contact among nodes and their neighbours. Another important property of an interference graph is its chromatic number — the minimum number of colours needed so that no adjacent nodes are assigned the same colour. Here adjacent nodes share an edge; adjacent edges share at least one node; and colours are used to identify different frequencies. This gives the minimum number of frequencies a network needs in order to attain zero interference. This problem can be solved as an optimization problem deterministically, but is algorithmically NP-hard. Hence, finding good asymptotic approximations for this value becomes important. Random geometric graphs describe an ensemble of graphs which share common features. In this thesis, node positions follow a Poisson point process or a binomial point process. We use probability theory to study the connectedness of random graphs and random geometric graphs, which is the fraction of connected graphs among many graph samples. This probability is closely related to the property of minimum node degree being at least unity. The chromatic number is closely related to the maximum degree as n→∞; the chromatic number converges to maximum degree when graph is sparse. We test existing theorems and improve the existing ones when possible. These motivated me to study the degree of random (geometric) graph models. We study using deterministic methods some degree-related problems for Erdaős-Rényi random graphs G(n, p) and random geometric graphs G(n, r). I provide both theoretical analysis and accurate simulation results. The results lead to a study of dependence or non-dependence in the joint distribution of the degrees of neighbouring nodes. We study the probability of no node being isolated in G(n, p), that is, minimum node degree being at least unity. By making the assumption of non-dependence of node degree, we derive two asymptotics for this probability. The probability of no node being isolated is an approximation to the probability of the graph being connected. By making an analogy to G(n, p), we study this problem for G(n, r), which is a more realistic model for wireless networks. Experiment shows that this asymptotic result also works well for small graphs. We wish to find the relationship between these basic features the above two important problems of wireless networks: the probability of a network being connected and the minimum number of channels a network needs in order to minimize interference. Inspired by the problem of maximum degree in random graphs, we study the problem of the maximum of a set of Poisson random variables and binomial random variables, which leads to two accurate formulae for the mode of the maximum for general random geometric graphs and for sparse random graphs. To our knowledge, these are the best results for sparse random geometric graphs in the literature so far. By approximating the node degrees as independent Poisson or binomial variables, we apply the result to the problem of maximum degree in general and sparse G(n, r), and derived much more accurate results than in the existing literature. Combining the limit theorem from Penrose and our work, we provide good approximations for the mode of the clique number and chromatic number in sparse G(n, r). Again these results are much more accurate than existing ones. This has implications for the interference minimization of WLANs. Finally, we apply our asymptotic result based on Poisson distribution for the chromatic number of random geometric graph to the interference minimization problem in IEEE 802.11b/g WLAN. Experiments based on the real planned position of the APs in WLANs show that our asymptotic results estimate the minimum number of channels needed accurately. This also means that sparse random geometric graphs are good models for interference minimization problem of WLANs. We discuss the interference minimization problem in single radio and multi-radio wireless networking scenarios. We study branchand-bound algorithms for these scenarios by selecting different constraint functions and objective functions.