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1p 19 networks lecture 3 introduction proof continued

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Unformatted text preview: nvar(I1 ) + n (n − 1)cov(I1 , I2 ) � � nq (1 − q ) + n (n − 1) E[I1 I2 ] − E[I1 ]E[I2 ] , i j � =i i = where the second and third equalities follow since the Ii are identically distributed Bernoulli random variables with parameter q (dependent). We have E[I1 I2 ] = P(I1 = 1, I2 = 1) = P(both 1 and 2 are isolated) = (1 − p )2n−3 = q2 . (1 − p ) Combining the preceding two relations, we obtain � var(X ) = nq (1 − q ) + n (n − 1) = nq (1 − q ) + n (n − 1) � q2 − q2 (1 − p ) q2 p . 1−p 19 Networks: Lecture 3 Introduction Proof (Continued) For large n, we have q → 0 [cf. Eq. (3)], or 1 − q → 1. Also p → 0. Hence, var(X ) ∼ nq + n2 q 2 p ∼ nq + n2 q 2 p 1−p = nn−λ + λn log(n)n−2λ ∼ nn−λ = E[X ], a (n ) where a(n ) ∼ b (n ) denotes b (n) → 1 as n → ∞. This implies that E[X ] ∼ var(X ) ≥ (0 − E[X ])2 P(X = 0), and therefore, P(X = 0) ≤ E[X ] 1 = → 0. E[X ] E[X ]2 It follows that P(at least one isolated node) → 1 and therefore, P(disconnected) → 1 as n → ∞, completing the proof. 20 Networks: Lecture 3 Introduction Converse log(n) We next show claim (2), i.e., if p (n ) = λ n with λ > 1, then P(connectivity) → 1, or equivalently P(disconnectivity) → 0. From Eq. (4), we have E[X ] = n · n−λ → 0 for λ > 1. This implies probability of isolated nodes goes to 0. However, we need more to establish connectivity. The event “graph is disconnected” is equivalent to the existence of k nodes without an edge to the remaining nodes, for some k ≤ n /2. We have P({1, . . . , k } not connected to the rest) = (1 − p )k (n−k ) , and therefore, P(∃ k nodes not connected to the rest) = �� n ( 1 − p ) k (n −k ) . k 21 Networks: Lecture 3 Introduction Converse (Continued) Using the union bound [i.e. P(∪i Ai ) ≤ ∑i P(Ai )], we obtain P(disconnected graph) ≤ Using Stirling’s formula k ! ∼ � �k k e n /2 �� n ∑ k ( 1 − p ) k (n −k ) . k =1 k n , which implies (k ) ≤ n k in the k (e) preceding relation and some (ugly) algebra, we obtain P(disconnected graph) → 0, completing the proof. 22...
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