Connectivity of sparse Bluetooth networks

Consider a random geometric graph defined on n vertices uniformly distributed in the d-dimensional unit torus. Two vertices are connected if their distance is less than a “visibility radius ” rn. We consider Bluetooth networks that are locally sparsified random geometric graphs. Each vertex selects c of its neighbors in the random geometric graph at random and connects only to the selected points. We show that if the visibility radius is at least of the order of n−(1−δ)/d for some δ> 0, then a constant value of c is sufficient for the graph to be connected, with high probability. It suffices to take c ≥ √ (1 + ɛ)/δ + K for any positive ɛ where K is a constant depending on d only. On the other hand, with c ≤ √ (1 − ɛ)/δ, the graph is disconnected, with high probability. 1 Introduction an

On the Expansion and Diameter of Bluetooth-Like Topologies

The routing capabilities of an interconnection network are strictly related to its bandwidth and latency characteristics, which are in turn quantifiable through the graph-theoretic concepts of expansion and diameter. This paper studies expansion and diameter of a family of subgraphs of the random geometric graph, which closely model the topology induced by the device discovery phase of Bluetooth-based ad hoc networks. The main feature modeled by any such graph, denoted as BT (r(n), c(n)), is the small number c(n) of links that each of the n devices (vertices) may establish with those located within its communi- cation range r(n). First, tight bounds are proved on the expansion of BT (r(n), c(n)) for the whole set of functions r(n) and c(n) for which connectivity has been established in previous works. Then, by leveraging on the expansion result, tight (up to a logarithmic additive term) upper and lower bounds on the diameter of BT (r(n), c(n)) are derived

Huge networks, tiny faulty nodes

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2007.Includes bibliographical references (p. 87-91).Can one build, and efficiently use, networks of arbitrary size and topology using a "standard" node whose resources, in terms of memory and reliability, do not need to scale up with the complexity and size of the network? This thesis addresses two important aspects of this question. The first is whether one can achieve efficient connectivity despite the presence of a constant probability of faults per node/link. Efficient connectivity means (informally) having every pair of regions connected by a constant fraction of the independent, entirely non-faulty paths that would be present if the entire network were fault free - even at distances where each path has only a vanishingly small probability of being fault-free. The answer is yes, as long as some very mild topological conditions on the high level structure of the network are met - informally, if the network is not too "thin" and if it does not contain too many large "holes". The results go against some established "empyrical wisdom" in the networking community. The second issue addressed by this thesis is whether one can route efficiently on a network of arbitrary size and topology using only a constant number c of bits/node (even if c is less than the logarithm of the network's size!). Routing efficiently means (informally) that message delivery should only stretch the delivery path by a constant factor. The answer again is yes, as long as the volume of the network grows only polynomially with its radius (otherwise, we run into established lower bounds). This effectively captures every network one may build in a universe (like our own) with finite dimensionality using links of a fixed, maximum length and nodes with a fixed, minimum volume. The results extend the current results for compact routing, allowing one to route efficiently on a much larger class of networks than had previously been known, with many fewer bits.by Enoch Peserico.Ph.D

Irrigating ad hoc networks in constant time

We propose very simple randomized algorithms to compute sparse overlay networks for geometric random graphs modelling wireless communication networks. The algorithms generate in constant time a sparse overlay network that, with high probability, is connected and spans the whole network. Moreover, by making use of the "power of choice" paradigm, the maximum degree can be made as small as O(log log n), where n is the size of the network. We show the usefulness of this kind of overlays by giving a new protocol for the classical broadcast problem, where a source is to send a message to the whole network. Our experimental evaluation shows that our approach outperforms the well-known gossiping approach in all situations where the cost of a message can be charged to the pair (sender, receiver), i.e. to the edge connecting the two. This includes sensor networks. Copyright 2005 ACM