arXiv subjects

Bhupendra Gupta

Publications and source records attributed to Bhupendra Gupta.

6 recordsLinked to original sources

arXiv2009

Criticality of the Exponential Rate of Decay for the Largest Nearest Neighbor Link in Random Geometric Graph

Let n points be placed independently in d-dimensional space according to the densities $f(x) = A_d e^{-λ\|x\|^α}, λ> 0, x \in \Re^d, d \geq 2.$ Let $d_n$ be the longest edge length for the nearest neighbor graph on these points. We show that $(\log(n))^{1-1/α}d_n -b_n$ converges weakly to the Gumbel distribution where $b_n \sim \log \log n.$ We also show that the strong law result, % \lim_{n \to \infty} \frac{(λ^{-1}\log(n))^{1-1/α}d_n}{\sqrt{\log \log n}} \to \frac{d}{αλ}, a.s. % Thus, the exponential rate of decay i.e. $α= 1$ is critical, in the sense that for $α> 1, d_n \to 0,$ where as $α< 1, d_n \to \infty$ a.s. as $n \to \infty.$

math.PR↗

arXiv2009

Comment to "Coverage by Randomly Deployed Wireless Sensor Networks"

It is a correction paper on "P.J. Wan and C.W. Yi, "Coverage by Randomly Deployed Wireless Sensor Networks", IEEE Transaction On Information Theory, vol.52, No.6, June 2006." In the above paper, Lemma (4), on page 2659 play the key role for deriving the main results in the paper. The statement as well as the proof of Lemma (4), page $2659,$ is not correct. We have given the correct version of Lemma. This change in Lemma leads a drastic change in all the result derived in the above paper.

cs.IT↗

arXiv2008

Vertex Degree of Random Intersection Graph

A random intersection graph is constructed by independently assigning a subset of a given set of objects $W,$ to each vertex of the vertex set $V$ of a simple graph $G.$ There is an edge between two vertices of $V,$ iff their respective subsets(in $W$,) have at least one common element. The strong threshold for the connectivity between any two arbitrary vertices of vertex set $V,$ is derived. Also we determine the almost sure probability bounds for the vertex degree of a typical vertex of graph $G.$

math.PR↗

arXiv2008

A Strong threshold for the size of random caps to cover a sphere

In this article, we consider `$N$'spherical caps of area $4πp$ were uniformly distributed over the surface of a unit sphere. We are giving the strong threshold function for the size of random caps to cover the surface of a unit sphere. We have shown that for large $N,$ if $\frac{Np}{\log\:N} > 1/2$ the surface of sphere is completely covered by the $N$ caps almost surely, and if $\frac{Np}{\log\:N} \leq 1/2$ a partition of the surface of sphere is remains uncovered by the $N$ caps almost surely.

math.PR↗

arXiv2006

On the topological Properties of the One Dimensional Exponential Random Geometric Graph

In this paper we study the one dimensional random geometric graph when the location of the nodes are independent and exponentially distributed. We derive exact results and the limit theorems for the connectivity and other properties associated with this random graph. We show that the asymptotic properties of a graph with a truncated exponential distribution can be obtained using the exponential random geometric graph.

math.PR↗

arXiv2006

Vertex Degree of Random Geometric Graph on Exponentially Distributed Points

Let $X_1,X_2,...$ be an infinite sequence of i.i.d. random vectors distributed exponentially with parameter $\lam .$ For each $y$ and $n\geq 1,$ form a graph $G_n(y)$ with vertex set $V_n = \{X_1,...,X_n\},$ two vertices are connected if and only if edge distance between them is greater then $y$, i.e, $\|X_i-X_j\| \leq y.$ Almost-sure asymptotic rates of convergence/divergence are obtained for the minimum and maximum vertex degree of the random geometric graph, as the number of vertices becomes large $n,$ and the edge distance varies with the number of vertices.

math.PR↗