Giant component

Giant component - the effect that occurs in schemes for randomly placing particles in cells with an unlimited increase in the number of particles. The effect is that almost all particles (as a percentage) are collected in one cell.

Consider a generalized layout of n particles in N cells:

\eta _{1}+\dots +\eta _{N}=n,\qquad (1)

{\ displaystyle \ eta _ {1} + \ dots + \ eta _ {N} = n, \ qquad (1)}

{\ displaystyle \ eta _ {1} + \ dots + \ eta _ {N} = n, \ qquad (1)}

Denote by $\eta _{(1)}\leq \dots \leq \eta _{(N)}$ ${\ displaystyle \ eta _ {(1)} \ leq \ dots \ leq \ eta _ {(N)}}$ ${\ displaystyle \ eta _ {(1)} \ leq \ dots \ leq \ eta _ {(N)}}$ variational series of random variables $\eta _{1},\dots ,\eta _{N}$ ${\ displaystyle \ eta _ {1}, \ dots, \ eta _ {N}}$ $\ eta_1, \ dots, \ eta_N$ . In this way, $\;\eta _{(N)}$ ${\ displaystyle \; \ eta _ {(N)}}$ ${\ displaystyle \; \ eta _ {(N)}}$ - the maximum component of the circuit (or the maximum number of particles in one cell), and $\;\eta _{(N-1)}$ ${\ displaystyle \; \ eta _ {(N-1)}}$ ${\ displaystyle \; \ eta _ {(N-1)}}$ - the next largest component.

If at $n\to \infty$ ${\ displaystyle n \ to \ infty}$ $n \ to \ infty$ random value $\;\eta _{(N)}/n$ ${\ displaystyle \; \ eta _ {(N)} / n}$ ${\ displaystyle \; \ eta _ {(N)} / n}$ has a limit distribution that does not have accumulation at zero, and $\;\eta _{(N-1)}/n$ ${\ displaystyle \; \ eta _ {(N-1)} / n}$ ${\ displaystyle \; \ eta _ {(N-1)} / n}$ If it degenerates into zero, then they say that in the layout (1) a giant component arises. ^[one]

It is known, for example, that there is no giant component in the classical layout , and in the logarithmic scheme describing the lengths of the cycles in a random substitution , the giant component occurs when $n\to \infty$ ${\ displaystyle n \ to \ infty}$ $n \ to \ infty$ so that $\ln(n)/N\to \infty$ ${\ displaystyle \ ln (n) / N \ to \ infty}$ ${\ displaystyle \ ln (n) / N \ to \ infty}$ , i.e. provided that the parameter $N$ ${\ displaystyle N}$ $N$ growing slower than $\ln(n)$ ${\ displaystyle \ ln (n)}$ ${\ displaystyle \ ln (n)}$ . ^[2]

Literature

↑ Kolchin V.F. On the existence of a giant component in particle allocation schemes // Review of Applied and Industrial Mathematics. - 2000. - T. 7 , No. 1 . - S. 112-113 .
↑ Kazimirov N.I. Galton-Watson forests and random permutations . - Dis. for the competition step. Cand. Ph.D. - Petrozavodsk, 2003 .-- 127 p.

[1] Kolchin V.F. On the existence of a giant component in particle allocation schemes // Review of Applied and Industrial Mathematics. - 2000. - T. 7 , No. 1 . - S. 112-113 .

[2] Kazimirov N.I. Galton-Watson forests and random permutations . - Dis. for the competition step. Cand. Ph.D. - Petrozavodsk, 2003 .-- 127 p.

Giant component

Literature

More articles: