Network Topology

A network topology indicates the arrangement of communication links between nodes in a network. For traditional wired networks, the network topology is determined by the physical cables connecting network nodes, and it is generally fairly static. In contrast, most wireless networks use omnidirectional antennas to transmit their messages in all directions to all nodes within some maximum range. If nodes are mobile, the topology is dynamic and is implicitly determined by the environment, the transmission ranges (affected by the transmission power), and the node movement. Indirectional networks(networks that utilize steerable directional communication links, such as FSO links) a topology must be explicitly determined and managed as nodes move. Nodes have a finite number of links, and therefore a finite number directly connected neighbors. Each directional antenna or free-space optical terminal for these links must be commanded to point at its intended neighbor, who also must point one of its links back. More importantly, each node must determine which of its potentially many neighbors it should directly connect with.

This following sections explore some concepts relating to the network topology graph and controlling the topology.

2.1.4.1 Unit Disk Graph

The region of a plane bounded by a circle is called adisk. For theoretical analysis, the topology of a wireless network is often modeled as aunit disk graph(UDG), where an edge exists between any two nodes whose separation distance is less than 1 unit (the maximum range). A unit disk graph can be visualized

using disks with a radius of one unit, in which case the graph would include edges between a node and any node within its disk. Alternatively, the graph can be visualized using disks with a radius of half a unit (0.5), in which case the graph would include edges between any nodes whose disks intersect. Figure 2.3 shows a simple unit disk graph for a set of nodes that uses the latter visualization convention (half unit disks). Notice that the graph includes an edge between any two nodes whose half unit disks intersect (e.g. nodesxandy), and does not include an edge between nodes whose disks do not intersect (e.g. nodesyandz). Formally a unit disk graph can be defined as follows. GivenP, a set of points in a plane, the unit disk graphU(P) includes a vertex for each point and an edge(u,v) between a pair of vertices if and only ifdist(u,v)≤1, wheredist(u,v)is the Euclidean separation distance betweenuandv. Notice, that the unit disk graph places no constraint on the degree of each node. In a unit disk graph, a node is always connected to all other nodes within the maximum transmission range (that has been normalized to 1 unit), and is never connected to any node outside that range [KWZ03a]. The unit disk graph is an accurate model of a 2-dimensional broadcast wireless network if: (1) all nodes have the same transmission power and the same receive gain, yielding the same maximum transmission radius, (2) every node’s transmission pattern is a perfect circle (each node transmits and receives equally in every direction), and (3) there are no radio-opaque obstacles and no multipathing to interfere with this perfect transmission. While convenient for theoretical analysis, unit disk graphs rarely match real-world wireless network topologies [KWZ03a], [KGKS05b], [KGKS05a]. In this research, the Unit Disk Graph will be utilized to represent all of the connections that could potentially be formed among a set of nodes. Since not all of these connections can actually be formed simultaneously, an appropriate degree-constrained subgraph of the unit disk graph must be determined.

x _y

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Figure 2.3: Unit Disk Graph for the set of points given. Edges are included only for nodes whose half unit disks intersect.

2.1.4.2 Planar Graph

Some techniques (e.g. face routing) used in this research require graphs that are guaranteed to be planar. Aplanar graphis a graph that can be drawn in a plane without any of its edges crossing. If the positions of vertices of a non-planar graph are fixed in a 2-dimensional plane, a planar subgraph can be generated from the original graph by removing an edge for each pair of edges that intersect. There are many algorithms for generating a planar subgraph. One such algorithm is the Gabriel Graph (GG). Remember that a disk is the region of a plane bounded by a circle. Aclosed diskincludes points on the circle (includes all points≤the radius). Letdisk(u,v)be a closed disk that has the line segment uvas a diameter. Given a set of pointsP, the Gabriel Graph includes an edge(u,v)if and only ifdisk(u,v)contains no other points inP. Figure 2.4 displays the edge(u,v), for example, since nodew(and all other nodes inP) are located outside of the disk of which the line segment betweenuandvis a diameter. Ifwwere located inside the disk, the edge(u,v) would not be included in the Gabriel Graph.

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Figure 2.4: A Gabriel Graph Example:uandvare connected since no other vertex is within the disk. Delaunay Triangulation is another algorithm that can be used to generate a planar subgraph of an input graph. Delaunay Triangulation creates a triangulation for an input set of points. A circumcircle is a circle that passes through all the vertices of a given polygon. Given a set of pointsP, the Delaunay Triangulation includes a triangletuvif and only if the circumcircle of that triangle contains no other points inP. Figure 2.5 shows that the resulting triangulation would include triangletuv, sincew(and all other nodes inP) are located outside of the circumcircle. The Gabriel Graph is a subgraph of Delaunay Triangulation.

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Figure 2.5: A Delaunay Triangulation Example: the triangletuvis included since no other vertex is within the disk.

faces, that make up a graph, but always also one extra unbounded exterior face that encompasses all other space.

2.1.4.3 Topology Control

Topology Control in wireless ad hoc network and sensor networks has been studied extensively. How- ever, the main focus has been on networks using standard omnidirectional antennas. In these networks, topology control involves adjusting the transmission power of nodes to achieve a global topology that is connected and optimized for some metric, such as minimal energy use. In [San05] Santi gives a thorough overview of Topology Control research, dividing the field into Homogeneous and Nonhomogeneous meth- ods, where the homogeneity refers to the transmit power or range of the nodes. In homogeneous topology control, the Critical Transmitting Range (CTR) is the common minimum range at which all nodes will transmit, such that the network remains connected. Nonhomogeneous methods allow for each node to be assigned a unique transmit range and associated transmission power, enabling even more specific control of the topology. Santi further divides nonhomogeneous methods into Location-Based, Direction-based, and Neighbor-based distributed protocols. In Chapter 3 a distributed topology control method is introduced that explicitly controls the topology of a network of mobile nodes connected using directional links.