Slides - SNAP - Stanford University

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How big do we have to make p before we are likely to have no isolated nodes? We know: P[v has degree 0] = e -c Event we are asking about is: I = some node is isolated v∈N I = I where Iv is the event that v is isolated We have: v ⎛ P ⎝ v∈N ( I ) = P⎜ I ⎟ ≤ P( I ) v ⎞ ⎠ ∑ v∈N 10/4/2011 Jure Leskovec, <strong>Stanford</strong> CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 14 v = ne −c Union bound i A i ∑ ≤ i A i A i
We just learned: P(I) = n e -c Let’s try: c = ln n then: n e -c = n e -ln n =n⋅1/n= 1 c = 2 ln n then: n e -2 ln n = n⋅1/n 2 = 1/n So if: p = ln n then: P(I) = 1 p = 2 ln n then: P(I) = 1/n → 0 as n→∞ 10/4/2011 Jure Leskovec, <strong>Stanford</strong> CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 15
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