@MISC{_thestrongly, author = {}, title = {The Strongly Connected Components of}, year = {} }

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Abstract

We consider a random digraph Dn.uu, ln \ on vert ices l,..., n, where, lor each vertex r., we choose at random one of the n possible arcs with head l and one of the ri possible arcs with tail r'. We show that the expected size of the largesl component of D;,,.,,,,, is, ,- 2 ( * ) = + o t l t 1. The results By Di, , (n) we mean the random digraph on the ver t ices 1,....n obta ined by choosing for each ver tex i. a parent p ( i) uni formly f iom 11,..., n). Thus, p ( l) i wi l l be the unique arc of D;,, with head L Dou,fu) is the random digraph obtained by choosing, independently for each i, a son s(i) in the same fashion. In this paper, we are interested in the random digraph Di,.o,,(n) obtained by choosing independently a parent and a son lor each vertex. or, equivalently, taking the union of independent copies of Di,,(n) and D,,,,,(n). In what follows we drop the (n) from D1n, Do,, and Din.os,&id assume that n is sufficiently large. In [1], Cooper and Frieze show that the probabil ity that Di,,.ou1 is strongly connected is-, 1 I- ( l- ) )- +ol l). In th is paper, we g ive an a l ternat ive proofof thei r resul t and show that the expected size of the largest component of Di,,.,,,,, is u-2 ( ' ) t * o,1 t. \ e- l / n