Consider a graph having n = 8vertices labeled 1 , 2 , . . . , 8 . Suppose that each edge is independently present with probability p. The degree of vertex i, designated as D i , is the number of edges that have vertex i as one of its vertices. Find C o r r ( D i , D j ) , the correlation between D i and D j .

Question

Consider a graph having   n  =  8vertices labeled   1  ,  2  ,  .  .  .  ,  8 . Suppose that each edge is independently present with probability p. The degree of vertex i, designated as       D    i   , is the number of edges that have vertex i as one of its vertices. Find   C  o  r  r  (      D    i    ,      D    j    ) , the correlation between       D    i   and       D    j   .

emerhelienapj · Accepted Answer

Step 1                    ρ        (                  D          i                ,                  D          j                )                            =                              C            o            v            (                          D              i                        ,                          D              j                        )                                V            a            r            (                          D              i                        )            V            a            r            (                          D              j                        )                                                            =                              V            a            r            (                          I                              i                j                                      )                                V            a            r            (                          D              i                        )            V            a            r            (                          D              j                        )                                                            =                              p            (            1            −            p            )                                7            p            (            1            −            p            )            ⋅            7            p            (            1            −            p            )                                                            =                  1          7                    The second equality comes from the fact that the covariance between edges is 0 unless the edge is the same. The third:       I          i      j       is bernoulli with parameter p, and       D    i   and       D    j   are binomial with parameters 7 and p.

makaunawal5 · Accepted Answer

Step 1Here&#039;s a sketch; if I have time, I&#039;ll complete this later.I denote   I  =      I          i      j      .Let       E          i      j       be the edge from vertex i to j. We know that       P    (      E          i      j         is present  )  =      P    (      I          i      j        =  1  )  =  p. In fact, we know that there are   7  +  6  +  ⋯  +  1  =  28 possible edges, so that

Find the correlation between D_{i} and D_{j}

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