5 research outputs found
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
The Cover Time of Random Walks on Graphs
A simple random walk on a graph is a sequence of movements from one vertex to
another where at each step an edge is chosen uniformly at random from the set
of edges incident on the current vertex, and then transitioned to next vertex.
Central to this thesis is the cover time of the walk, that is, the expectation
of the number of steps required to visit every vertex, maximised over all
starting vertices. In our first contribution, we establish a relation between
the cover times of a pair of graphs, and the cover time of their Cartesian
product. This extends previous work on special cases of the Cartesian product,
in particular, the square of a graph. We show that when one of the factors is
in some sense larger than the other, its cover time dominates, and can become
within a logarithmic factor of the cover time of the product as a whole. Our
main theorem effectively gives conditions for when this holds. The techniques
and lemmas we introduce may be of independent interest. In our second
contribution, we determine the precise asymptotic value of the cover time of a
random graph with given degree sequence. This is a graph picked uniformly at
random from all simple graphs with that degree sequence. We also show that with
high probability, a structural property of the graph called conductance, is
bounded below by a constant. This is of independent interest. Finally, we
explore random walks with weighted random edge choices. We present a weighting
scheme that has a smaller worst case cover time than a simple random walk. We
give an upper bound for a random graph of given degree sequence weighted
according to our scheme. We demonstrate that the speed-up (that is, the ratio
of cover times) over a simple random walk can be unboundedComment: 179 pages, PhD thesi
Connectivity threshold for Bluetooth graphs
International audienceWe study the connectivity properties of random Bluetooth graphs that model certain "ad hoc" wireless networks. The graphs are obtained as "irrigation subgraphs" of the well-known random geometric graph model. There are two parameters that control the model: the radius that determines the "visible neighbors" of each node and the number of edges that each node is allowed to send to these. The randomness comes from the underlying distribution of data points in space and from the choices of each vertex. We prove that no connectivity can take place with high probability for a range of parameters and completely characterize the connectivity threshold (in ) for values of close the critical value for connectivity in the underlying random geometric graph