1

My question is about the degree distribution of a special random graph. Suppose $n$ points are uniformly and independently chosen in a unit circle. Join two points iff they are within a distance of $r$, where $0<r<1$ is some fixed parameter. This generates a random graph on the unit circle. If $X$ is the r.v. corresponding to the degree of a node ,what is $P(X\mid \text{radial distance is } d)$?

Here is a rough idea: given a node at some distance from the center, the number of nodes within a vicinity of $r$ can be $0,1,2,3,....n-1$. So the number of nodes within a distance of $r$ of a particular node is a binomial r.v. with the parameter $p$ where $p$ is the area common to the unit circle and the circle of radius $r$ with centre at the particular node (whose degree distribution we are looking for). Will this approach work or do we need to try something else? What happens to degree distributio when $n$ goes to infinity?

madprob
  • 2,865
AgnostMystic
  • 1,654
  • Can somone please help me ,the statement below the problem is not showing the complete text that I have written.What can be wrong? – AgnostMystic Jan 03 '14 at 11:41
  • NOw the statement is ok .thanks fothe edit – AgnostMystic Jan 03 '14 at 11:45
  • Are the points picked on the unit circle or in the unit disk? Also could you explain what is $d$? is it related to $r$? – Siméon Jan 03 '14 at 11:46
  • 1
    As a general note, please make the title more informative in the future. – Lost1 Jan 03 '14 at 11:48
  • The quantity d is the radial distance of a node from the centre of the unit circle.As far as the points are cocerned they can be anywher in the disk or on the disk i.e., for a point d can be anything from 0 to 1 inclusive – AgnostMystic Jan 03 '14 at 12:08
  • 1
    I think you reasonning is correct, and you have to compute $p$ as a function of $d$ and $r$. For the limiting behaviour as $n\to+\infty$, check the De Moivre-Laplace central limit theorem. – Siméon Jan 03 '14 at 14:08

0 Answers0