FAQ Policy. About this book This book provides an original graph theoretical approach to the fundamental properties of wireless mobile ad-hoc networks. Show all. Table of contents 11 chapters Table of contents 11 chapters Introduction to Ad-hoc Networks Pages Scope of the book Pages Modeling Ad-hoc Networks Pages Degree in Ad-hoc Networks Pages Hopcount in Ad-hoc Networks Pages Connectivity in Ad-hoc Networks Pages Interference in Ad-hoc Networks Pages Capacity of Ad-hoc Networks Pages Book Summary Pages Show next xx. Read this book on SpringerLink. Recommended for you. PAGE 1. The primary target of a topology control algorithm is to abandon long-distance communication links and instead route a message over several small energy-efficient hops.

### INTRODUCTION

In general there is a trade-off between networks connectivity and sparseness. There is a link u; v in E if and only if the two nodes u and v can communicate directly. The resulting topology Gtc should have the following properties: Property 1 Symmetry :The resulting topology Gtc should be symmetric, that is, node u is a neighbor of node v if and only if node v is a neighbor of node u. Asymmetric communication graphs are 3.

A simple ACK message confirming the receipt of a Message, for example, is already a nightmare in an asymmetric graph. Property 2 Connectivity :Two nodes u and v are connected if there is a path from u to v, potentially through multiple hops.

## Topology control in wireless ad hoc networks

If two nodes are connected in G, then they should still be connected in Gtc. Although a minimum spanning tree MST is a sparse connected subgraph, it is often not considered a good topology, since close-by nodes in the original graph G might end up being far away in Gtc G being a ring, for instance. If f c is bounded from above by a linear function in c, the graph Gtc is called a spanner. Researchers have studied a selection of cost metrics, the most popular being i Euclidean distance and ii various energy metrics.

The cost of a link in model i is the Euclidean distance of the link, in model ii the distance is raised to a predefined power.

In both models the cost of a path is commonly defined to be the sum of the costs of all links in the path. As mentioned, the primary target of a topology control algorithm is to abandon long-distance neighbors, or more formally. Property 3 Sparseness :The remaining graph Gtc should be sparse, that is, the number of links should be in the order of the number of nodes, i.

This reflects that not too many close-by nodes must be chosen, which reduces interference and thus saves energy.

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Since there still might be some nodes with many neighbors e. In particular the maximum degree in the graph 4. Gtc should be bounded from above by a constant. Since connectivity and sparseness run against each other, topology control has been a thriving research area. In addition to the properties 1, 2, and 3, one can often find secondary targets.

For instance, it is popular and often for free to ask the remaining graph to be planar in order to run a geometric a. When referring to a weighted graph we assume that the weights are symmetric:! The nodes of a Euclidean graph are assumed to be located in a Euclidean plane. Furthermore the edge weight of an edge u; v is defined to be! Note that the definition of Euclidean graphs does not contain a statement on the existence of certain edges. Unit Disk Graphs are often employed to model an ad-hoc network where all network nodes are placed in an unobstructed plane and have equal normalized transmission power and isotropic antennas that is antennas sending with identical power in every direction of the plane.

Strongly related to edge weights is the cost of an edge. The cost of an edge c u; v can be considered to represent the effort an algorithm is required to expend in order to send a message over u; v. A path p u; v from a node u to a node v being a sequence of consecutively contingent edges starting at u and ending at v, the cost of a path jp u; v j is accordingly defined to be the sum of the costs of all edges contained in the path.

Detail Explanation:In the first step each network node u computes a total order over all its neighbors in the network graph G. From an abstract point of view, this order is intended to reflect the quality of the links to the neighbors.

## Wireless ad hoc network

The neighbor order reflects a much more general notion of link quality, such as signal attenuation or packet arrival rate. In the second step the neighbor order information is exchanged among all neighbors. Typically a node u broadcasts its own neighbor order while receiving the orders established by all of its neighbors. During the third step, which does not require any further Communication, each node locally selects those neighboring nodes which will form its neighborhood in the resulting topology control graph, based on the previously exchanged neighbor order information.

## Ad-hoc networks - fundamental properties and network topologies - Semantic Scholar

Informally speaking, a node u only 6. Although the XTC algorithm is executed at all nodes, the detailed description as shown in the above box assumes the point of view of a node u. Lines 1 and 2 correspond to Steps I and II.

In the algorithm as described above, each node constructs in Step I a total order over all its neighbors in G. Property 1 is symmetry of the resulting graph, often has to be enforced by topology control algorithms.