❚✐❛❣♦ ❋❡❧✐♣❡ ❆♥❞r❛❞❡
❆♥á❧✐s❡ ❞❡ ♣r♦♣r✐❡❞❛❞❡s t♦♣♦❧ó❣✐❝❛s ❞❛s
r❡❞❡s ❜✐♦❧ó❣✐❝❛s ✐♥t❡❣r❛❞❛s ❞❛
❊s❝❤❡r✐❝❤✐❛ ❝♦❧✐ ❡ ❞❛ ❙❛❝❝❤❛r♦♠②❝❡s
❝❡r❡✈✐s✐❛❡
❚✐❛❣♦ ❋❡❧✐♣❡ ❆♥❞r❛❞❡
❆♥á❧✐s❡ ❞❡ ♣r♦♣r✐❡❞❛❞❡s t♦♣♦❧ó❣✐❝❛s ❞❛s
r❡❞❡s ❜✐♦❧ó❣✐❝❛s ✐♥t❡❣r❛❞❛s ❞❛
❊s❝❤❡r✐❝❤✐❛ ❝♦❧✐ ❡ ❞❛ ❙❛❝❝❤❛r♦♠②❝❡s
❝❡r❡✈✐s✐❛❡
▼♦♥♦❣r❛✜❛ ❛♣r❡s❡♥t❛❞❛ ❛♦ ■♥st✐✲
t✉t♦ ❞❡ ❇✐♦❝✐ê♥❝✐❛s ❞❛ ❯♥✐✈❡rs✐✲
❞❛❞❡ ❊st❛❞✉❛❧ P❛✉❧✐st❛ ✧❏ú❧✐♦ ❞❡
▼❡sq✉✐t❛ ❋✐❧❤♦✧✱ ❈❛♠♣✉s ❞❡ ❇♦✲
t✉❝❛t✉✱ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡
❇❛❝❤❛r❡❧ ❡♠ ❋ís✐❝❛ ▼é❞✐❝❛✳
❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ◆❡② ▲❡♠❦❡
FICHA CATALOGRÁFICA ELABORADA PELA SEÇÃO TÉCNICA DE AQUISIÇÃO E TRATAMENTO DA INFORMAÇÃO
DIVISÃO DE BIBLIOTECA E DOCUMENTAÇÃO - CAMPUS DE BOTUCATU - UNESP BIBLIOTECÁRIA RESPONSÁVEL: SELMA MARIA DE JESUS
Andrade Tiago Felipe.
Análise de propriedades topológicas das redes biológicas integradas da
Escherichia coli e da Saccharomyces cerevisiae / Tiago Felipe Andrade. -
Botucatu [s.n], 2008.
Trabalho de conclusão (bacharelado – Física médica) – Universidade Estadual Paulista, Instituto de Biociências de Botucatu, 2008
Orientador: Ney Lemke
1. Física médica 2. Bioinformática
❆❣r❛❞❡❝✐♠❡♥t♦s
❘❡s✉♠♦
Pr♦❝❡ss♦s ❜✐♦❧ó❣✐❝♦s sã♦ ❝♦♠♣❧❡①♦s ❡ ♣♦ss✉❡♠ ♣r♦♣r✐❡❞❛❞❡s ❡♠❡r❣❡♥t❡s q✉❡ ♥ã♦ ♣♦✲ ❞❡♠ s❡r ❡①♣❧✐❝❛❞❛s ♦✉ ♣r❡✈✐st❛s ❛tr❛✈és ❞♦ r❡❞✉❝✐♦♥✐s♠♦✳ P❛r❛ s✉♣❧❛♥t❛r ❡ss❡s ❧✐♠✐t❡s✱ ♣❡sq✉✐s❛❞♦r❡s tê♠ ✉s❛❞♦ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ♠ét♦❞♦s ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ❜✐♦❧♦❣✐❛ s✐stê♠✐❝❛✱ ❝✉❥♦ ♦❜❥❡t✐✈♦ é ❛ ❝♦♠♣r❡❡♥sã♦ ❞❛s ✐♥t❡r❛çõ❡s ♥ã♦✲❧✐♥❡❛r❡s ❡♥tr❡ ♦s ♠ú❧t✐♣❧♦s ❝♦♠♣♦♥❡♥✲ t❡s ❞♦s ♣r♦❝❡ss♦s ❜✐♦❧ó❣✐❝♦s✳ ❊ss❛s ✐♥t❡r❛çõ❡s ♣♦❞❡♠ s❡r r❡♣r❡s❡♥t❛❞❛s ♣♦r ✉♠ ♦❜❥❡t♦ ♠❛t❡♠át✐❝♦ ❝❤❛♠❛❞♦ ❣r❛❢♦ ♦✉ r❡❞❡✱ ♦♥❞❡ ♦s ❡❧❡♠❡♥t♦s ✐♥t❡r❛❣❡♥t❡s sã♦ r❡♣r❡s❡♥t❛❞♦s ♣♦r ♥♦❞♦s ❡ ❛s ✐♥t❡r❛çõ❡s ♣♦r ❧✐❣❛çõ❡s q✉❡ ❝♦♥❡❝t❛♠ ♣❛r❡s ❞❡ ♥♦❞♦s✳ ❆s r❡❞❡s ♣♦❞❡♠ s❡r ❝❧❛ss✐✜❝❛❞❛s s❡❣✉♥❞♦ s✉❛ t♦♣♦❧♦❣✐❛✱ ♣♦❞❡♥❞♦ s❡r ❛❧❡❛tór✐❛s ♦✉ ❧✐✈r❡s ❞❡ ❡s❝❛❧❛✱ s❡♥❞♦ q✉❡ ♥❡ss❛ ú❧t✐♠❛ ♦ ♥ú♠❡r♦ ❞❡ ❝♦♥❡①õ❡s ❞❡ ✉♠ ♥♦❞♦ s❡❣✉❡ ❛ ❧❡✐ ❞❡ ♣♦tê♥❝✐❛✱ ❡①✐st✐♥❞♦ ♣♦✉❝♦s ♥♦❞♦s ❛❧t❛♠❡♥t❡ ❝♦♥❡❝t❛❞♦s ❡ ♠✉✐t♦s ♥♦❞♦s ❝♦♠ ♣♦✉❝❛s ❧✐❣❛çõ❡s✳ ❊❧❛s t❛♠❜é♠ sã♦ ❝❧❛ss✐✜❝❛❞❛s ❝♦♠♦ ❤✐❡rárq✉✐❝❛s ♦✉ ♥ã♦✱ ❡ s❡❣✉♥❞♦ ❡ss❛ ❝❧❛ss✐✜❝❛çã♦ r❡s♣❡✐t❛✲s❡ ✉♠❛ ♦r❞❡♠ ❞❡ ❝♦♥❡①ã♦ ❡♥tr❡ ♦s ♥♦❞♦s ❡ s❡✉s ❣r❛✉s ❞❡ ❝♦♥❡❝t✐✈✐❞❛❞❡✱ q✉❡ ❞❡✜♥❡ ❡ss❛ ❤✐❡r❛rq✉✐❛✳ ◆❡st❡ tr❛❜❛❧❤♦✱ ❛♥❛❧✐s❛♠♦s ❛s r❡❞❡s ♠♦❧❡❝✉❧❛r❡s ✐♥t❡❣r❛❞❛s ❞❛ ❜❛❝tér✐❛ ❊s❝❤❡r✐❝❤✐❛ ❝♦❧✐ ❡ ❞❛ ❧❡✈❡❞✉r❛ ❙❛❝❝❤❛r♦♠②❝❡s ❝❡r❡✈✐s✐❛❡✱ q✉❡ ✐♥❝❧✉❡♠ ✐♥t❡r❛çõ❡s ❢ís✐❝❛s ❡♥tr❡ ♣r♦t❡í♥❛s✱ ✐♥t❡r❛çõ❡s ♠❡t❛❜ó❧✐❝❛s ❡ ✐♥t❡r❛çõ❡s ❞❡ r❡❣✉❧❛çã♦ tr❛♥s❝r✐❝✐♦♥❛❧✳ ❆tr❛✈és ❞❡ ❢❡rr❛♠❡♥t❛s ❝♦♠♣✉t❛❝✐♦♥❛✐s✱ ❝♦♠♦ M athematica❘✱ ❡ ❞♦s ❞❛❞♦s ♦❜t✐❞♦s ❞❡ ❜❛♥❝♦s ❞❡ ❞❛❞♦s ♣ú❜❧✐❝♦s✱ r❡❛❧✐③❛♠♦s ❛ ♠❡❞✐çã♦ ❞❡ q✉❛tr♦ ♣❛râ♠❡tr♦s t♦♣♦❧ó❣✐❝♦s✿ ❛ ❞✐str✐❜✉✐çã♦ ❞❡ ❝♦♥❡❝t✐✈✐❞❛❞❡s
P(k)✱ ♦ ❝♦❡✜❝✐❡♥t❡ ❞❡ ❛❣r✉♣❛♠❡♥t♦ ♠é❞✐♦ ❞❡ t♦❞♦s ♦s ♥♦❞♦s ❝♦♠ k ❝♦♥❡①õ❡s C(k)✱ ♦
❣r❛✉ ❞❡ ♣r♦①✐♠✐❞❛❞❡ CC(k) ❡ ♦ ❣r❛✉ ❞❡ ✐♥t❡r♠❡❞✐❛çã♦ CB(k)✳ ❖P(k)é ✉♠❛ ❢✉♥çã♦ q✉❡ ❝❛❧❝✉❧❛ ♦ ♥ú♠❡r♦ t♦t❛❧ ❞❡ ♥♦❞♦s ❝♦♠ ❣r❛✉ k ❞❡ ❝♦♥❡①ã♦ ❡ s❡r✈❡ ♣❛r❛ ❝❧❛ss✐✜❝❛r ❛ r❡❞❡
❝♦♠♦ ❛❧❡❛tór✐❛ ♦✉ ❧✐✈r❡ ❞❡ ❡s❝❛❧❛✳ ❖ C(k) ♠♦str❛ s❡ ✉♠❛ r❡❞❡ é ❤✐❡rárq✉✐❝❛✱ ✐st♦ é✱ s❡ ♦
❛❣r✉♣❛♠❡♥t♦ ❞♦s ♥♦❞♦s ❞❡♣❡♥❞❡ ❞❛ ❝♦♥❡❝t✐✈✐❞❛❞❡ ❞♦s ♠❡s♠♦s✳ ❖ CC(k) ♠❡❞❡ q✉❛♥t♦ ✉♠ ♥♦❞♦ ♣❛rt✐❝✉❧❛r ❡stá ♣ró①✐♠♦ ❞❡ t♦❞♦s ♦s ♦✉tr♦s ♥♦❞♦s ❞❛ r❡❞❡ ❡ ♦ CB(k)♠❡❞❡ q✉❛♥t♦ ✉♠ ♥♦❞♦ ❡stá ♥♦s ♠❡♥♦r❡s ❝❛♠✐♥❤♦s ❡♥tr❡ ♦✉tr♦s ♥♦❞♦s ♥❛ r❡❞❡✳ ◆❛ ❛♥á❧✐s❡ ❝♦♠♣❛r❛t✐✈❛ ❞❛s r❡❞❡s ✈❡r✐✜❝❛♠♦s q✉❡ ❛s r❡❞❡s ❜✐♦❧ó❣✐❝❛s ✐♥t❡❣r❛❞❛s ❞♦s ♦r❣❛♥✐s♠♦s ❊s❝❤❡r✐❝❤✐❛ ❝♦❧✐ ❡ ❙❛❝❝❤❛r♦♠②❝❡s ❝❡r❡✈✐s✐❛❡ sã♦✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❧✐✈r❡ ❞❡ ❡s❝❛❧❛ ❡ ❡①♣♦♥❡♥❝✐❛❧✳ ❆❧é♠ ❞✐ss♦✱ ❛ ❞❡♣❡♥❞ê♥❝✐❛ ❞♦ C(k) ❡♠ r❡❧❛çã♦ ❛♦s ✈❛❧♦r❡s ❞❡ k ❞❡ ❛♠❜❛s r❡❞❡s ✐♥t❡❣r❛❞❛s s✉❣❡r❡ q✉❡
❡❧❛s sã♦ ❤✐❡rárq✉✐❝❛s✱ ♠❛s ❝♦♠ ✉♠❛ ❡str✉t✉r❛ ❤✐❡rárq✉✐❝❛ ❞✐❢❡r❡♥t❡ ❞❛s r❡❞❡s q✉❡ ❝♦♥té♠ ❛♣❡♥❛s ✉♠ t✐♣♦ ❞❡ ✐♥t❡r❛çã♦✳
❆❜str❛❝t
❇✐♦❧♦❣✐❝❛❧ ♣r♦❝❡ss❡s ❛r❡ ❝♦♠♣❧❡① ❛♥❞ ♣♦ss❡ss ❡♠❡r❣❡♥t ♣r♦♣❡rt✐❡s t❤❛t ❝❛♥ ♥♦t ❜❡ ❡①♣❧❛✐♥❡❞ ♦r ♣r❡❞✐❝t ❜② r❡❞✉❝t✐♦♥✐s♠ ♠❡t❤♦❞s✳ ❚♦ ♦✈❡r❝♦♠❡ t❤❡ ❧✐♠✐t❛t✐♦♥s ♦❢ r❡❞✉❝t✐♦✲ ♥✐s♠✱ r❡s❡❛r❝❤❡rs ❤❛✈❡ ❜❡❡♥ ✉s❡❞ ❛ ❣r♦✉♣ ♦❢ ♠❡t❤♦❞s ❦♥♦✇♥ ❛s s②st❡♠s ❜✐♦❧♦❣②✱ ❛ ♥❡✇ ✐♥t❡r❞✐s❝✐♣❧✐♥❛r② ✜❡❧❞ ♦❢ st✉❞② ❛✐♠✐♥❣ t♦ ✉♥❞❡rst❛♥❞ t❤❡ ♥♦♥✲❧✐♥❡❛r ✐♥t❡r❛❝t✐♦♥s ❛♠♦♥❣ ❝♦♠♣♦♥❡♥ts ❡♠❜❡❞❞❡❞ ✐♥ ❜✐♦❧♦❣✐❝❛❧ ♣r♦❝❡ss❡s✳ ❚❤❡s❡ ✐♥t❡r❛❝t✐♦♥s ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❜② ❛ ♠❛t❤❡♠❛t✐❝❛❧ ♦❜❥❡❝t ❝❛❧❧❡❞ ❣r❛♣❤ ♦r ♥❡t✇♦r❦✱ ✇❤❡r❡ t❤❡ ❡❧❡♠❡♥ts ❛r❡ r❡♣r❡s❡♥t❡❞ ❜② ♥♦❞❡s ❛♥❞ t❤❡ ✐♥t❡r❛❝t✐♦♥s ❜② ❡❞❣❡s t❤❛t ❧✐♥❦ ♣❛✐r ♦❢ ♥♦❞❡s✳ ❚❤❡ ♥❡t✇♦r❦s ❝❛♥ ❜❡ ❝❧❛ss✐✲ ✜❡❞ ❛❝❝♦r❞✐♥❣ t♦ t❤❡✐r t♦♣♦❧♦❣✐❡s✿ ✐❢ ♥♦❞❡ ❞❡❣r❡❡s ❢♦❧❧♦✇ ❛ P♦✐ss♦♥ ❞✐str✐❜✉t✐♦♥ ✐♥ ❛ ❣✐✈❡♥ ♥❡t✇♦r❦✱ ✐✳❡✳ ♠♦st ♥♦❞❡s ❤❛✈❡ ❛♣♣r♦①✐♠❛t❡❧② t❤❡ s❛♠❡ ♥✉♠❜❡r ♦❢ ❧✐♥❦s✱ t❤✐s ✐s ❛ r❛♥❞♦♠ ♥❡t✇♦r❦❀ ✐❢ ♥♦❞❡ ❞❡❣r❡❡s ❢♦❧❧♦✇ ❛ ♣♦✇❡r✲❧❛✇ ❞✐str✐❜✉t✐♦♥ ✐♥ ❛ ❣✐✈❡♥ ♥❡t✇♦r❦✱ ✐✳❡✳ s♠❛❧❧ ♥✉♠❜❡r ♦❢ ❤✐❣❤✲❞❡❣r❡❡ ♥♦❞❡s ❛♥❞ ❤✐❣❤ ♥✉♠❜❡r ♦❢ ❧♦✇✲❞❡❣r❡❡ ♥♦❞❡s✱ t❤✐s ✐s ❛ s❝❛❧❡✲❢r❡❡ ♥❡t✇♦r❦✳ ▼♦r❡♦✈❡r✱ ♥❡t✇♦r❦s ❝❛♥ ❜❡ ❝❧❛ss✐✜❡❞ ❛s ❤✐❡r❛r❝❤✐❝❛❧ ♦r ♥♦♥✲❤✐❡r❛r❝❤✐❝❛❧✳ ■♥ t❤✐s st✉❞②✱ ✇❡ ❛♥❛❧✐s❡❞ ❊s❝❤❡r✐❝❤✐❛ ❝♦❧✐ ❛♥❞ ❙❛❝❝❤❛r♦♠②❝❡s ❝❡r❡✈✐s✐❛❡ ✐♥t❡❣r❛t❡❞ ♠♦❧❡❝✉❧❛r ♥❡t✇♦r❦s✱ ✇❤✐❝❤ ❤❛✈❡ ♣r♦t❡✐♥✲♣r♦t❡✐♥ ✐♥t❡r❛❝t✐♦♥✱ ♠❡t❛❜♦❧✐❝ ❛♥❞ tr❛♥s❝r✐♣t✐♦♥❛❧ r❡❣✉❧❛✲ t✐♦♥ ✐♥t❡r❛❝t✐♦♥s✳ ❇② ✉s✐♥❣ ❝♦♠♣✉t❛t✐♦♥❛❧ ♠❡t❤♦❞s✱ s✉❝❤ ❛s M athematica❘✱ ❛♥❞ ❞❛t❛ ❝♦❧❧❡❝t❡❞ ❢r♦♠ ♣✉❜❧✐❝ ❞❛t❛❜❛s❡s✱ ✇❡ ❝❛❧❝✉❧❛t❡❞ ❢♦✉r t♦♣♦❧♦❣✐❝❛❧ ♣❛r❛♠❡t❡rs✿ t❤❡ ❞❡❣r❡❡ ❞✐str✐❜✉t✐♦♥ P(k)✱ t❤❡ ❝❧✉st❡r✐♥❣ ❝♦❡✣❝✐❡♥t C(k)✱ t❤❡ ❝❧♦s❡♥❡ss ❝❡♥tr❛❧✐t② CC(k) ❛♥❞ t❤❡ ❜❡t✇❡❡♥♥❡ss ❝❡♥tr❛❧✐t② CB(k)✳ P(k)✐s ❛ ❢✉♥❝t✐♦♥ t❤❛t ❝❛❧❝✉❧❛t❡s t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ ♥♦✲ ❞❡s ✇✐t❤ k ❞❡❣r❡❡ ❝♦♥♥❡❝t✐♦♥ ❛♥❞ ✐s ✉s❡❞ t♦ ❝❧❛ss✐❢② t❤❡ ♥❡t✇♦r❦ ❛s r❛♥❞♦♠ ♦r s❝❛❧❡✲❢r❡❡✳ C(k) s❤♦✇s ✐❢ ❛ ♥❡t✇♦r❦ ✐s ❤✐❡r❛r❝❤✐❝❛❧✱ ✐✳❡✳ ✐❢ t❤❡ ❝❧✉st❡r✐③❛t✐♦♥ ❝♦❡✣❝✐❡♥t ❞❡♣❡♥❞s ♦♥
♥♦❞❡ ❞❡❣r❡❡✳ CC(k) ✐s ❛♥ ✐♥❞✐❝❛t♦r ♦❢ ❤♦✇ ♠✉❝❤ ❛ ♥♦❞❡ ✐t ✐s ✐♥ t❤❡ ❧❡ss❡ ✇❛② ❛♠♦♥❣ ♦t❤❡rs s♦♠❡ ♥♦❞❡s ♦❢ t❤❡ ♥❡t✇♦r❦ ❛♥❞ t❤❡ CB(k) ✐s ❛ ♣♦✐♥t❡r ♦❢ ❤♦✇ ❛ ♣❛rt✐❝✉❧❛r ♥♦❞❡ ✐s ❛♠♦♥❣ s❡✈❡r❛❧ ♦t❤❡r ♥♦❞❡s ♦❢ t❤❡ ♥❡t✇♦r❦✳ ■♥ t❤❡ ❝♦♠♣❛r❛t✐✈❡ ❛♥❛❧②s✐s✱ ✇❡ ✈❡r✐✜❡❞ t❤❛t t❤❡ ✐♥t❡❣r❛t❡❞ ❜✐♦❧♦❣✐❝❛❧ ♥❡t✇♦r❦s ♦❢ ♦r❣❛♥✐s♠s ❊s❝❤❡r✐❝❤✐❛ ❝♦❧✐ ❛♥❞ ❙❛❝❝❤❛r♦♠②❝❡s ❝❡r❡✈✐s✐❛❡ ❛r❡✱ r❡s♣❡❝t✐✈❡❧②✱ s❝❛❧❡✲❢r❡❡ ❛♥❞ ❡①♣♦♥❡♥t✐❛❧✳ ▼♦r❡♦✈❡r✱ t❤❡ ❞❡♣❡♥❞❡♥❝❡ ♦❢ C(k)
♦♥ k ♦❜s❡r✈❡❞ ❢♦r ❜♦t❤ ♥❡t✇♦r❦s s✉❣❣❡sts t❤❡② ❛r❡ ❤✐❡r❛r❝❤✐❝❛❧✱ ❛❧t❤♦✉❣❤t t❤❡✐r str✉❝t✉r❡s
❛♣♣❡❛r t♦ ❜❡ ❞✐st✐♥❝t ❢r♦♠ t❤❡ ❤✐❡r❛r❝❤✐❝❛❧ str✉❝t✉r❡ ♦❢ ♥❡t✇♦r❦s ❝♦♥t❛✐♥✐♥❣ ♦♥❧② ♦♥❡ t②♣❡ ♦❢ ✐♥t❡r❛❝t✐♦♥✳
❙✉♠ár✐♦
Pá❣✐♥❛
❘❡s✉♠♦ ✹
❆❜str❛❝t ✺
✶ ■♥tr♦❞✉çã♦ ✼
✷ ❖❜❥❡t✐✈♦s ✶✶
✸ ▼❡t♦❞♦❧♦❣✐❛ ✶✷
✸✳✶ ❇❛♥❝♦s ❞❡ ❞❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✸✳✷ ❉❡✜♥✐çõ❡s ❞❛s ♣r♦♣r✐❡❞❛❞❡s t♦♣♦❧ó❣✐❝❛s ❞❡ r❡❞❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✸✳✸ ■♠♣❧❡♠❡♥t❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹
✹ ❘❡s✉❧t❛❞♦s ❡ ❉✐s❝✉ssã♦ ✶✻
✹✳✶ ❘❡❞❡ ✐♥t❡❣r❛❞❛ ❞❛ ❊✳ ❝♦❧✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✹✳✷ ❘❡❞❡ ✐♥t❡❣r❛❞❛ ❞❛ ❙✳ ❝❡r❡✈✐s✐❛❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✹✳✸ ❆♥á❧✐s❡ ❝♦♠♣❛r❛t✐✈❛ ❞♦ ❣r❛✉ ❞❡ ♣r♦①✐♠✐❞❛❞❡ ❡ ❞♦ ❣r❛✉ ❞❡ ✐♥t❡r♠❡❞✐❛çã♦
♣❛r❛ ❛s r❡❞❡s ✐♥t❡❣r❛❞❛s ❞❛ ❙✳ ❝❡r❡✈✐s✐❛❡ ❡ ❞❛ ❊✳ ❝♦❧✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶
✼
✶ ■♥tr♦❞✉çã♦
❖ ♠ét♦❞♦ r❡❞✉❝✐♦♥✐st❛ ❞❡ ❞✐ss❡❝❛çã♦ ❞♦s s✐st❡♠❛s ❜✐♦❧ó❣✐❝♦s ❡♠ s✉❛s ♣❛rt❡s ❝♦♥st✐✲ t✉✐♥t❡s t❡♠ ❛❥✉❞❛❞♦ ♥♦ ❡s❝❧❛r❡❝✐♠❡♥t♦ ❞♦ ❢✉♥❝✐♦♥❛♠❡♥t♦ ❞❡ ♠✉✐t♦s ♣r♦❝❡ss♦s ❜✐♦❧ó❣✐❝♦s✳ P♦ré♠✱ t❛✐s ♣r♦❝❡ss♦s sã♦ ❡①tr❡♠❛♠❡♥t❡ ❝♦♠♣❧❡①♦s ❡ ♣♦ss✉❡♠ ♣r♦♣r✐❡❞❛❞❡s ❡♠❡r❣❡♥t❡s q✉❡ ♥ã♦ ♣♦❞❡♠ s❡r ❡①♣❧✐❝❛❞❛s ♦✉ ♠❡s♠♦ ♣r❡✈✐st❛s ❛tr❛✈és ❞♦ ❡st✉❞♦ ❞❡ s✉❛s ♣❛rt❡s ✐♥❞✐✈✐✲ ❞✉❛✐s ❬✶❪✳ ❚❛❧ ❧✐♠✐t❛çã♦ ✐♠♣♦st❛ ♣❡❧♦ ♠ét♦❞♦ r❡❞✉❝✐♦♥✐st❛ é ✉♠ ❞♦s ❢❛t♦r❡s ♣r❡♣♦♥❞❡r❛♥t❡s ♥❛ ❢❛❧t❛ ❞❡ ✉♠❛ ♠❡❧❤♦r ❝♦♠♣r❡❡♥sã♦ ❡ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ t❡r❛♣✐❛s ❡✜❝❛③❡s ♣❛r❛ ❞♦❡♥ç❛s ❝♦♠♣❧❡①❛s ❬✷❪✳
P❛r❛ s✉♣❧❛♥t❛r ❡ss❡s ❧✐♠✐t❡s ❞♦ r❡❞✉❝✐♦♥✐s♠♦✱ ♣❡sq✉✐s❛❞♦r❡s tê♠ ✉s❛❞♦ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ♠ét♦❞♦s q✉❡ tr❛t❛♠ ♦s ♣r♦❝❡ss♦s ❜✐♦❧ó❣✐❝♦s ❞❡ ❢♦r♠❛ ✐♥t❡❣r❛❞❛✳ ❊st❛ ♥♦✈❛ ár❡❛ ❞❛ ❜✐♦❧♦❣✐❛ é ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ❜✐♦❧♦❣✐❛ s✐stê♠✐❝❛✳ ❆ ❜✐♦❧♦❣✐❛ s✐stê♠✐❝❛ ♦❜❥❡t✐✈❛ ❛ ❝♦♠♣r❡❡♥sã♦ ❞❛s ✐♥t❡r❛çõ❡s ♥ã♦✲❧✐♥❡❛r❡s ❡♥tr❡ ♦s ♠ú❧t✐♣❧♦s ❝♦♠♣♦♥❡♥t❡s ❞♦s ♣r♦❝❡ss♦s ❜✐♦❧ó❣✐❝♦s ❬✷❪✳
❚❛✐s ✐♥t❡r❛çõ❡s sã♦ ❣❡r❛❧♠❡♥t❡ r❡♣r❡s❡♥t❛❞❛s ♣♦r ✉♠ ♦❜❥❡t♦ ♠❛t❡♠át✐❝♦ ❝❤❛♠❛❞♦ ❣r❛❢♦ ♦✉ r❡❞❡ ❬✸❪✱ q✉❡ é ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ♥♦❞♦s G✭❝♦♠♣♦♥❡♥t❡s✮ ❡ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❛r❡st❛s A ✭✐♥t❡r❛çõ❡s✮ q✉❡ ❝♦♥❡❝t❛♠ ❝❛❞❛ ❞♦✐s ❝♦♠♣♦♥❡♥t❡s ♥♦ ❝♦♥❥✉♥t♦ G✳
❙ó r❡❝❡♥t❡♠❡♥t❡ ❝♦♠❡ç❛♠♦s ❛ ❞❡s✈❡♥❞❛r ❛ ❛rq✉✐t❡t✉r❛ ❞❛s r❡❞❡s ♥❛t✉r❛✐s q✉❡ ❝♦♥tr♦✲ ❧❛♠ ❛s ✐♥t❡r❛çõ❡s ❡♥tr❡ ♦s ❝♦♠♣♦♥❡♥t❡s ❞❡ ✉♠ s✐st❡♠❛ ❝♦♠♣❧❡①♦✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦✱ ❛ r❡❞❡ ❞❡ ❝♦♠♣✉t❛❞♦r❡s ♥❛ ✐♥t❡r♥❡t✱ ❛ r❡❞❡ ❞❡ r❡❛çõ❡s q✉í♠✐❝❛s ❡♠ ✉♠❛ ❝é❧✉❧❛ ✭♠❡t❛❜♦✲ ❧✐s♠♦✮✱ ❛s r❡❞❡s ❞❡ ❞✐str✐❜✉✐çã♦ ❡❧étr✐❝❛✱ ❛ r❡❞❡ ❞❡ ❝♦❧❛❜♦r❛çõ❡s ❡♥tr❡ ❝✐❡♥t✐st❛s✱ ❛t♦r❡s ❞❡ ❝✐♥❡♠❛✱ ♣❛r❝❡✐r♦s s❡①✉❛✐s ❡ ♦✉tr❛s ❬✹✱ ✺✱ ✻❪✳ ❆s r❡❞❡s ❢♦r♠❛♠ ♦ ❡sq✉❡❧❡t♦ ❞♦s s✐st❡✲ ♠❛s ❝♦♠♣❧❡①♦s q✉❡ ♣❡r♠❡✐❛♠ t♦❞♦s ♦s r❛♠♦s ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❤✉♠❛♥♦ ❡ sã♦ ✉♠ t❡♠❛ ❣❡♥✉✐♥❛♠❡♥t❡ tr❛♥s❞✐s❝✐♣❧✐♥❛r ❬✼❪✳
❖ ❡st✉❞♦ ❞❡ r❡❞❡s ♥❛t✉r❛✐s ♣❡rt❡♥❝❡ à t❡♦r✐❛ ❞❡ ❣r❛❢♦s ❛❧❡❛tór✐♦s✳ ❯♠ ❣r❛❢♦ ❛❧❡❛tór✐♦ é ❞❡✜♥✐❞♦ ❝♦♠♦ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ N ♥♦❞♦s ❝♦♥❡❝t❛❞♦s ♣♦r n ❛r❡st❛s q✉❡ sã♦ ❡s❝♦❧❤✐❞❛s
❛❧❡❛t♦r✐❛♠❡♥t❡ ❡♥tr❡ ❛s N(N −1)/2 ♣♦ssí✈❡✐s ❛r❡st❛s✳ ❊①✐st❡ ✉♠ t♦t❛❧ ❞❡✿
CNn(N−1)/2 = (N(N −1)/2)!
n!(N(N −1)/2−n)! ✭✶✮
❣r❛❢♦s ❞✐❢❡r❡♥t❡s q✉❡ ♣♦❞❡♠ s❡r ❝r✐❛❞♦s ❝♦♠ N ♥♦❞♦s ❡ n ❛r❡st❛s✱ t♦❞♦s ❡q✉✐♣r♦✈á✈❡✐s
❡st❛t✐st✐❝❛♠❡♥t❡✳ ❆ ❋✐❣✉r❛ ✶✭❆❛✮ ♠♦str❛ ❛ ✐❧✉str❛çã♦ ❞❡ ✉♠ ❣r❛❢♦ ❛❧❡❛tór✐♦ s✐♠♣❧❡s✳ ❯♠ ❣r❛❢♦ ❛❧❡❛tór✐♦ t❛♠❜é♠ ♣♦❞❡ s❡r ❞❡✜♥✐❞♦ ♣❡❧♦ ♠♦❞❡❧♦ ❜✐♥♦♠✐❛❧✱ ♥♦ q✉❛❧ ✉♠ ♣❛r ❞❡ ♥♦❞♦s q✉❛❧q✉❡r é ❝♦♥❡❝t❛❞♦ ❝♦♠ ✉♠❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ p✳ ❉❡st❛ ♠❛♥❡✐r❛✱ ♦ ♥ú♠❡r♦ ❡s♣❡r❛❞♦
❞❡ ❛r❡st❛s ♥♦ ❣r❛❢♦ s❡rá✿
E(n) =p[N(N −1)/2]. ✭✷✮
✽
♥♦♠✐❛❧✱ ♠❛s ♣❛r❛ N ❣r❛♥❞❡✱ ❡❧❛ é ❜❡♠ ❛♣r♦①✐♠❛❞❛ ♣♦r ✉♠❛ ❞✐str✐❜✉✐çã♦ ❞❡ P♦✐ss♦♥✿
P(k)≃e−kkk
k! . ✭✸✮
❖♥❞❡ k é ❛ ❝♦♥❡❝t✐✈✐❞❛❞❡ ♠é❞✐❛ ❞♦s ♥♦❞♦s ♥♦ ❣r❛❢♦✳ ❱❡r ❋✐❣✉r❛ ✶✭❆❜✮✳
❆❧❣✉♠❛s ❝❛r❛❝t❡ríst✐❝❛s ✐♠♣♦rt❛♥t❡s ❞❛s r❡❞❡s ♥❛t✉r❛✐s sã♦ ♠✉✐t♦ ❞✐❢❡r❡♥t❡s ❞❛s ♣r♦✲ ♣r✐❡❞❛❞❡s ❝♦♥❤❡❝✐❞❛s ♣❛r❛ ❣r❛❢♦s ❛❧❡❛tór✐♦s ❝♦♠♦ ♦ ♠✉♥❞♦ ♣❡q✉❡♥♦✱ ♦ ❛❣r✉♣❛♠❡♥t♦ ❡ ❛ ❞✐str✐❜✉✐çã♦ ❞❡ ❝♦♥❡❝t✐✈✐❞❛❞❡s✳ ❆ ❝❛r❛❝t❡ríst✐❝❛ ❞❡ ♠✉♥❞♦ ♣❡q✉❡♥♦ é ❝♦♥❤❡❝✐❞❛ ❞❡ ❣r❛❢♦s ❛❧❡❛tór✐♦s q✉❡✱ ❛♣❡s❛r ❞♦ ❣r❛♥❞❡ t❛♠❛♥❤♦ q✉❡ ✉♠❛ r❡❞❡ ❛❧❡❛tór✐❛ ♣♦ss❛ t❡r✱ ♦ ❝♦♠♣r✐✲ ♠❡♥t♦ ♠é❞✐♦ ❞❡ ❝❛♠✐♥❤♦ ❡♥tr❡ ❞♦✐s ♥♦❞♦s q✉❛✐sq✉❡r ❞♦ ❣r❛❢♦ ❝r❡s❝❡ ♠✉✐t♦ ❧❡♥t❛♠❡♥t❡ ✭❧♦❣❛r✐t♠✐❝❛♠❡♥t❡✮ ❝♦♠ ♦ s❡✉ ♥ú♠❡r♦ ❞❡ ♥♦❞♦s✱ ✐♠♣❧✐❝❛♥❞♦ q✉❡ s❡♠♣r❡ ❤á ❛❧❣✉♠❛ tr❛✲ ❥❡tór✐❛ ❝✉rt❛ ✉♥✐♥❞♦ ❞♦✐s ♥♦❞♦s q✉❛✐sq✉❡r ♥♦ ❣r❛❢♦✳ ❖ ❝♦❡✜❝✐❡♥t❡ ❞❡ ❛❣r✉♣❛♠❡♥t♦ ❡stá r❡❧❛❝✐♦♥❛❞♦ à ❢♦r♠❛çã♦ ❞❡ ♠ó❞✉❧♦s ❛❧t❛♠❡♥t❡ ❝♦♥❡❝t❛❞♦s ♥♦ ✐♥t❡r✐♦r ❞❡ ✉♠ ❣r❛❢♦✳ P❛r❛ r❡❞❡s ♥❛t✉r❛✐s✱ ❛ ❢♦r♠❛çã♦ ❞❡st❡s ♠ó❞✉❧♦s ♣❛r❡❝❡ s❡r ❢r❡qü❡♥t❡✱ ❡♥q✉❛♥t♦ q✉❡ ♣❛r❛ r❡❞❡s ❛❧❡❛tór✐❛s ❡❧❡s sã♦ r❛r♦s✳ ❏á ❛ ❞✐str✐❜✉✐çã♦ ❞❡ ❝♦♥❡❝t✐✈✐❞❛❞❡s ♣❛r❛ ✉♠ ❜♦♠ ♥ú♠❡r♦ ❞❡ r❡✲ ❞❡s ♥❛t✉r❛✐s t❡♥❞❡ ❛ s❡❣✉✐r ✉♠❛ ❧❡✐ ❞❡ ♣♦tê♥❝✐❛ q✉❡ ❝❛r❛❝t❡r✐③❛ ❛ ❛✉sê♥❝✐❛ ❞❡ ✉♠❛ ❡s❝❛❧❛ tí♣✐❝❛ ♥♦ s✐st❡♠❛✱ ❞✐❢❡r❡♥t❡♠❡♥t❡ ❞❛ ❞✐str✐❜✉✐çã♦ ♣❛r❛ ❣r❛❢♦s ❛❧❡❛tór✐♦s q✉❡ ♦❜❡❞❡❝❡ ✉♠❛ ❞✐str✐❜✉✐çã♦ ❞❡ P♦✐ss♦♥✳
❆ r❡♣r❡s❡♥t❛çã♦ ❞❡ s✐st❡♠❛s ❜✐♦❧ó❣✐❝♦s ♣♦r ❣r❛❢♦s ❛❧❡❛tór✐♦s ❞✐r❡❝✐♦♥❛❞♦s ❡ ♥ã♦ ❞✐r❡❝✐♦✲ ♥❛❞♦s ❡ ❛ ❛♣❧✐❝❛çã♦ ❞❛s ♣r♦♣r✐❡❞❛❞❡s t♦♣♦❧ó❣✐❝❛s ❞❡ss❡s ❣r❛❢♦s ♥♦ ❡st✉❞♦ ❞❡ss❡s s✐st❡♠❛s ❢♦r❛♠ ✐♥✐❝✐❛❞❛s ♣♦r ❏❡♦♥❣ ❡t ❛❧✳ ✭✷✸✸✸✮✳ ◆❡ss❡ tr❛❜❛❧❤♦ ♣✐♦♥❡✐r♦✱ ❛s ✈✐❛s ♠❡t❛❜ó❧✐❝❛s ❞❡ ✈ár✐♦s ♦r❣❛♥✐s♠♦s ❢♦r❛♠ ♦r❣❛♥✐③❛❞❛s ❡♠ ❢♦r♠❛ ❞❡ r❡❞❡ ❡ ❛s ♣r♦♣r✐❡❞❛❞❡s t♦♣♦❧ó❣✐❝❛s ❞❡ss❛s r❡❞❡s ♠❡t❛❜ó❧✐❝❛s ❢♦r❛♠ ❛♥❛❧✐s❛❞❛s ❡ ❜✐♦❧♦❣✐❝❛♠❡♥t❡ ✐♥t❡r♣r❡t❛❞❛s✳ ❆ ♣❛rt✐r ❞❡ss❡ tr❛❜❛❧❤♦✱ ❛ r❡♣r❡s❡♥t❛çã♦ ❞♦s s✐st❡♠❛s ❜✐♦❧ó❣✐❝♦s ❡♠ r❡❞❡s ❡ s✉❛s r❡s♣❡❝t✐✈❛s ❛♥á❧✐s❡s ✈ê♠ s❡♥❞♦ ❢❡✐t❛s ❝♦♠ ♠❛✐♦r ❢r❡qüê♥❝✐❛ ♣❛r❛ ❡❧✉❝✐❞❛çã♦ ❞❡ ♣r♦♣r✐❡❞❛❞❡s ❡♠❡r❣❡♥t❡s ❞❡ ♣r♦❝❡ss♦s ❜✐♦❧ó❣✐❝♦s ❝♦♠♣❧❡①♦s ❬✽✱ ✾✱ ✶✸❪✳
❊st✉❞♦s r❡❝❡♥t❡s tê♠ s✐❞♦ ❜❛s❡❛❞♦s ❡♠ ✉♠ ♥♦✈♦ ❝♦♥❝❡✐t♦ s♦❜r❡ r❡❞❡s ❜✐♦❧ó❣✐❝❛s✱ ♦ q✉❛❧ ❛s ❞❡✜♥✐ ❝♦♠♦ s❡♥❞♦ ✐♥t❡❣r❛❞❛s✱ ♦✉ s❡❥❛✱ r❡❞❡s ❜✐♦❧ó❣✐❝❛s q✉❡ ♣♦ss✉❡♠ ♠❛✐s ❞❡ ✉♠ t✐♣♦ ❞❡ ✐♥t❡r❛çã♦ ❢ís✐❝❛✳ ❊♠ ✉♠❛ r❡❞❡ ❜✐♦❧ó❣✐❝❛ ✐♥t❡❣r❛❞❛❬✶✶❪✱ ♦s ❣❡♥❡s r❡♣r❡s❡♥t❛♠ ♦s ♥♦❞♦s✱ ❡ ❝♦♥s✐❞❡r❛♥❞♦ q✉❡ ♦s ❣❡♥❡s g1 ❡g2 ❝♦❞✐✜❝❛♠ ❛s ♣r♦t❡í♥❛s p1❡ p2✱ ❡st❡s sã♦ ❝♦♥❡❝t❛❞♦s
♥❛ r❡❞❡✱ s❡ ♦❝♦rr❡✿
❛✮ ✐♥t❡r❛çã♦ ♣r♦t❡í♥❛✲♣r♦t❡í♥❛✿ p1❡ p2✐♥t❡r❛❣❡♠ ✜s✐❝❛♠❡♥t❡❀
❜✮ r❡❣✉❧❛çã♦ tr❛♥s❝r✐❝✐♦♥❛❧✿ g1 r❡❣✉❧❛ ❛ tr❛♥s❝r✐çã♦ ❞♦ ❣❡♥❡ g2 ♦✉
❝✮ ✐♥t❡r❛çã♦ ♠❡t❛❜ó❧✐❝❛✿ ✉♠ ♣r♦❞✉t♦ ❣❡r❛❞♦ ♣❡❧❛ r❡❛çã♦ ❝❛t❛❧✐s❛❞❛ ♣❡❧❛ ♣r♦t❡í♥❛ p1é
❝♦♥s✉♠✐❞♦♥❛ r❡❛çã♦ ❝❛t❛❧✐s❛❞❛ ♣❡❧❛ ♣r♦t❡í♥❛ p2✳
P(k) C(k) CC(k)
CB(k)
C(k) k
C(k) k
C(k) k
✶✵
P❛rt❡ ❞♦s r❡s✉❧t❛❞♦s ❞❡ss❡ tr❛❜❛❧❤♦ ❢♦✐ ❛♣r❡s❡♥t❛❞♦ ❡♦ ✹ ❡✈❡♥t♦s✱ s❡♥❞♦ ❡❧❡s✿
✶✳ ✼t❤ ■♥t❡r♥❛t✐♦♥❛❧ ❙②♦♣♦s✐✉♦ ♦♥ ▼❛t❤❡♦❛t✐❝❛❧ ❛♥❞ ❈♦♦♣✉t❛t✐♦♥❛❧ ❇✐♦❧♦❣②✱ ✷✵✵✼✱ ❇ú③✐♦s✲❘❏✱ ❝♦♦ tr❛❜❛❧❤♦ ✐♥t✐t✉❧❛❞♦✿ ◆❡t✇♦r❦ t♦♣♦❧♦❣②✲❜❛s❡❞ ✐♥ s✐❧✐❝♦ ❞✐s❝♦✈❡r② ♦❢ ❣❡♥❡ ❡ss❡♥t✐❛❧✐t② ✐♥ ❙❛❝❝❤❛r♦♠②❝❡s ❝❡r❡✈✐s✐❛❡✳ ❋♦✐ ❛♣r❡s❡♥t❛❞♦ s♦❜ ❛ ❢♦r♦❛ ❞❡ ♣ôst❡r ♣❡❧♦ ❛❧✉♥♦ ❞❡ ❞♦✉t♦r❛❞♦ ▼❛r❝✐♦ ▲✉✐s ❆❝❡♥❝✐♦✱ ❡ s❡✉ r❡s✉♦♦ ❝♦♥st❛ ♥♦s ❛♥❛✐s ❞❡ss❡ ❡✈❡♥t♦✳
✷✳ ❱■■ ❲♦r❦s❤♦♣ ❞❛ Pós✲●r❛❞✉❛çã♦✱ ✷✵✵✽✱ ❇♦t✉❝❛t✉✲❙P✱ ❝♦♦ ♦ tr❛❜❛❧❤♦ ✐♥t✐t✉❧❛❞♦✿ ❊st✉❞♦s ❝♦♦♣❛r❛t✐✈♦s ❞❛s ♣r♦♣r✐❡❞❛❞❡s ❞❛s r❡❞❡s ✐♥t❡❣r❛❞❛s ❞❛ ❊✳ ❝♦❧✐ ❡ ❞❛ ❙✳ ❝❡r❡✈✐s✐❛❡✳ ❆♣r❡s❡♥t❛❞♦ s♦❜ ❛ ❢♦r♦❛ ❞❡ ♣ôst❡r ♣❡❧♦ ❛❧✉♥♦ ❞❡ ✐♥✐❝✐❛çã♦ ❝✐❡♥tí✜❝❛ ❚✐❛❣♦ ❋❡❧✐♣❡ ❆♥❞r❛❞❡ ❡ ❝✉❥♦ r❡s✉♦♦ ❝♦♥st❛ ♥♦s ❛♥❛✐s ❞♦ ❡✈❡♥t♦✳
✸✳ ❳❳❳■ ❊♥❝♦♥tr♦ ◆❛❝✐♦♥❛❧ ❞❡ ❋ís✐❝❛ ❞❛ ▼❛tér✐❛ ❈♦♥❞❡♥s❛❞❛✱ ✷✵✵✽✱ ➪❣✉❛s ❞❡ ▲✐♥❞ó✐❛✲ ❙P✱ ❝♦♦ tr❛❜❛❧❤♦ ✐♥t✐t✉❧❛❞♦✿ ▼♦t✐✈♦s t♦♣♦❧ó❣✐❝♦s ♥❛ r❡❞❡ ✐♥t❡❣r❛❞❛ ❞❛ ❜❛❝tér✐❛ ❊✳ ❝♦❧✐ ❡ ❞❛ ❧❡✈❡❞✉r❛ ❙✳ ❝❡r❡✈✐s✐❛❡✱ ❛♣r❡s❡♥t❛❞♦ ❡♦ s❡ssã♦ ♦r❛❧ ♣❡❧♦ ♣r♦❢❡ss♦r ❞♦✉t♦r ◆❡② ▲❡♦❦❡✱ ❡ s❡✉ r❡s✉♦♦ ❡stá ✐♥❝❧✉✐❞♦ ♥♦s ❛♥❛✐s ❞❡ss❡ ❡✈❡♥t♦✳
✶✶
✷ ❖❜❥❡t✐✈♦s
✶✷
✸ ▼❡t♦❞♦❧♦❣✐❛
❖ tr❛❜❛❧❤♦ ❢♦✐ ❞❡s❡♥✈♦❧✈✐❞♦ ❡♠ ❡t❛♣❛s✱ ❞❡✈✐❞♦ ❛ ❣r❛♥❞❡ ❞❡♠❛♥❞❛ ❞❡ t❡♠♣♦✱ ❛s q✉❛✐s ❡stã♦ ❞❡s❝r✐t❛s ❞❡t❛❧❤❛❞❛♠❡♥t❡ ❛❜❛✐①♦✳
✸✳✶ ❇❛♥❝♦s ❞❡ ❞❛❞♦s
❖ tr❛❜❛❧❤♦ t❡✈❡ s❡✉ ✐♥í❝✐♦ ❝♦♠ ❛ ❧♦❝❛❧✐③❛çã♦ ❞♦s ❜❛♥❝♦s ❞❡ ❞❛❞♦s ❜✐♦❧ó❣✐❝♦s ♣ú❜❧✐❝♦s ♥❛ ✇❡❜✱ q✉❡ ❝♦♠♣♦rt❛♠ ♦s ❞❛❞♦s ❞❡ ✐♥t❡r❛çõ❡s ❢ís✐❝❛s ❡♥tr❡ ♣r♦t❡í♥❛s✱ ✐♥t❡r❛çõ❡s ♠❡t❛❜ó❧✐❝❛s ❡ ✐♥t❡r❛çõ❡s ❞❡ r❡❣✉❧❛çã♦ tr❛♥s❝r✐❝✐♦♥❛❧✱ r❡❢❡r❡♥t❡ ❛♦s ♦r❣❛♥✐s♠♦s ❡st✉❞❛❞♦s✳ ❖s ❞❛❞♦s ❡♠ ❢♦r♠❛ ❞❡ t❡①t♦ ❢♦r❛♠ ❛❞q✉✐r✐❞♦s ♣♦r ♠❡✐♦ ❞❡ ❞♦✇♥❧♦❛❞✱ ♦ q✉❡ ❞❡♠❛♥❞♦✉ ❣r❛♥❞❡ ♣❛rt❡ ❞♦ t❡♠♣♦ ❞❡ss❡ ♣r♦❥❡t♦✱ ♣♦✐s ♠✉✐t♦s ❞❛❞♦s ❡♥❝♦♥tr❛♠✲s❡ ❡s♣❛rs♦s✱ ♦ q✉❡ ❞✐✜❝✉❧t❛ ❛ ❝♦❧❡t❛ ❞♦s ♠❡s♠♦s✳ ❆ ❚❛❜❡❧❛ ✶ ❛♣r❡s❡♥t❛ ♦s ❜❛♥❝♦s ❞❡ ❞❛❞♦s ✉t✐❧✐③❛❞♦s ♣❛r❛ ❛ ❛q✉✐s✐çã♦ ❞♦s ❞❛❞♦s✳
❚❛❜❡❧❛ ✶✿ ▲✐st❛ ❞♦s ❜❛♥❝♦s ❞❡ ❞❛❞♦s ❜✐♦❧ó❣✐❝♦s ✉t✐❧✐③❛❞♦s ♣❛r❛ ♦ ♣r♦❝❡ss♦ ❞❡ ❛q✉✐s✐çã♦ ❞♦s ❞❛❞♦s✳
❇❛♥❝♦ ❞❡ ❞❛❞♦s ❖r❣❛♥✐s♠♦ ❚✐♣♦ ❞❡ ✐♥t❡r❛çã♦ ❘❡❢❡rê♥❝✐❛
❇■●● ❙✳ ❝❡r❡✈✐s✐❛❡ ▼❡t❛❜ó❧✐❝❛ ❤tt♣✿✴✴❜✐❣❣✳✉❝s❞✳❡❞✉✴❤♦♠❡✳♣❧
❨❊❆❙❚❘❆❈❚ ❙✳ ❝❡r❡✈✐s✐❛❡ ❘❡❣✉❧❛çã♦ tr❛♥s❝r✐❝✐♦♥❛❧ ❤tt♣✿✴✴✇✇✇✳②❡❛str❛❝t✳❝♦♠ ❇■❖●❘■❉ ❙✳ ❝❡r❡✈✐s✐❛❡ Pr♦t❡í♥❛✲♣r♦t❡í♥❛ ❤tt♣✿✴✴✇✇✇✳t❤❡❜✐♦❣r✐❞✳♦r❣
❑❊●● ❊✳ ❝♦❧✐ ▼❡t❛❜ó❧✐❝❛ ❤tt♣✿✴✴✇✇✇✳❣❡♥♦♠❡✳❥♣✴❦❡❣❣
❘❡❣✉❧♦♥❉❇ ❊✳ ❝♦❧✐ ❘❡❣✉❧❛çã♦ tr❛♥s❝r✐❝✐♦♥❛❧ ❤tt♣✿✴✴r❡❣✉❧♦♥❞❜✳❝❝❣✳✉♥❛♠✳♠① ❇❛❝t❡r✐♦♠❡✳♦r❣ ❊✳ ❝♦❧✐ Pr♦t❡í♥❛✲♣r♦t❡í♥❛ ❤tt♣✿✴✴✇✇✇✳❜❛❝t❡r✐♦♠❡✳♦r❣
❋❡✐t❛ ❛ ❝♦❧❡t❛ ❞♦s ❞❛❞♦s✱ ❛ s❡❣✉♥❞❛ ❡t❛♣❛ ❢♦✐ ♥♦r♠❛❧✐③❛r ❡ ✐♥t❡❣r❛r ♦s ❞❛❞♦s✳ ❖ ♦❜❥❡t✐✈♦ ❞❛ ♥♦r♠❛❧✐③❛çã♦ é ❝♦❧♦❝❛r ♦s ❞❛❞♦s ❡♠ ✉♠ ❢♦r♠❛t♦ ♣❛❞rã♦ ú♥✐❝♦ ❞❡ ❢♦r♠❛ q✉❡ ♦s ♠❡s♠♦s ♣♦ss❛♠ s❡r ✐♥t❡❣r❛❞♦s✳ ❊ss❡ ❡t❛♣❛ ❢♦✐ r❡❛❧✐③❛❞❛ ♠❛♥✉❛❧♠❡♥t❡✱ ♥♦r♠❛❧✐③❛♥❞♦ ♦s ♥♦♠❡s ❞♦s ❣❡♥❡s ♣❛r❛ ✉♠ ❢♦r♠❛t♦ ú♥✐❝♦✳
❈♦♠ ♦ ♣r♦❝❡ss♦ ❞❡ ♥♦r♠❛❧✐③❛çã♦ ❝♦♥❝❧✉✐❞♦✱ ❡①❡❝✉t♦✉✲s❡ ♦ ♣ró①✐♠♦ ♣❛ss♦ q✉❡ ❢♦✐ ❛ ✐♥t❡❣r❛çã♦ ❞♦s ❞❛❞♦s✳ P❛r❛ ❛ ❡①❡❝✉çã♦ ❞❡ss❛ ❡t❛♣❛ ❢❡③✲s❡ ♥❡❝❡ssár✐♦ ❛ ✉t✐❧✐③❛çã♦ ❞❡ ✉♠❛ ❛❜♦r❞❛❣❡♠ ❞❡ ✐♥t❡❣r❛çã♦✱ ❡ ❛ ❛❜♦r❞❛❣❡♠ ✉t✐❧✐③❛❞❛ ♥❡ss❡ tr❛❜❛❧❤♦ ❢♦✐ ❛ ❛❜♦r❞❛❣❡♠ ♠❛t❡✲ r✐❛❧✐③❛❞❛✱ ♦♥❞❡ ♦s ❞❛❞♦s sã♦ ❡①tr❛í❞♦s ❞♦s ❜❛♥❝♦s ❞❡ ❞❛❞♦s ❡ ♣♦st❡r✐♦r♠❡♥t❡ ❝❡♥tr❛❧✐③❛❞♦s ❡♠ ✉♠ ú♥✐❝♦ ❛rq✉✐✈♦✱ ❞❡ ❢♦r♠❛ ❛ ♣❡r♠✐t✐r ♦ ❡st✉❞♦ ❡ ❛♥á❧✐s❡ ❞❡st❡s ❞❛❞♦s ❡♠ ✉♠ ♣r♦❝❡ss♦ ♠✐♥✉❝✐♦s♦✳
✶✸
❜✐♦❧ó❣✐❝❛✳
✸✳✷ ❉❡✜♥✐çõ❡s ❞❛s ♣r♦♣r✐❡❞❛❞❡s t♦♣♦❧ó❣✐❝❛s ❞❡ r❡❞❡s
❛✮ ❉✐str✐❜✉✐çã♦ ❞❡ ❝♦♥❡❝t✐✈✐❞❛❞❡s
❆ ❞✐str✉✐❜✉✐çã♦ ❞❡ ❝♦♥❡❝t✐✈✐❞❛❞❡s✱ P(k)✱ é ✉♠❛ ❢✉♥çã♦ q✉❡ ❝❛❧❝✉❧❛ ♦ ♥ú♠❡r♦ t♦t❛❧
❞❡ ♥♦❞♦s ❝♦♠ ❞❡t❡r♠✐♥❛❞♦ ❣r❛✉ ❞❡ ❝♦♥❡①ã♦✱ ❡♠ ✉♠ ❞❛❞♦ ❣r❛❢♦✳ ❋♦r♠❛❧♠❡♥t❡✱ ❛ ❞✐str✐❜✉✐çã♦ ❞❡ ❝♦♥❡❝t✐✈✐❞❛❞❡s é✿
P(k) =
v∈V|deg(v)=k
1 ✭✹✮
♦♥❞❡v é ✉♠ ♥♦❞♦ ❞♦ ❣r✉♣♦ ❞❡ ♥♦❞♦s V ♣❡rt❡♥❝❡♥t❡s ❛♦ ❣r❛❢♦✱ ❡ deg(v)é ♦ ❣r❛✉ ❞♦
♥♦❞♦ v✳ ❊ss❛ ♠❡s♠❛ ✐♥❢♦r♠❛çã♦ é ❢r❡qü❡♥t❡♠❡♥t❡ r❡♣r❡s❡♥t❛❞❛ ❝♦♠♦ ✉♠❛ ❞✐str✐✲
❜✉✐çã♦ ❞❡ ❝♦♥❡❝t✐✈✐❞❛❞❡s ❝✉♠✉❧❛t✐✈❛✿
P(k) =
k′=k
p(k′). ✭✺✮
❊①✐st❡♠ ❞✐❢❡r❡♥t❡s t✐♣♦s ❞❡ r❡❞❡s ❝♦♠♣❧❡①❛s✱ ❡ ❝❛❞❛ ✉♠❛ ❞❡❧❛s ♣♦ss✉✐ ❞✐❢❡r❡♥t❡s ❝❛r❛❝t❡ríst✐❝❛s ❡♠ r❡❧❛çã♦ à ❞✐str✐❜✉✐çã♦ ❞❡ ❝♦♥❡❝t✐✈✐❞❛❞❡s✳ P♦r ❡①❡♠♣❧♦✱ ❛ r❡❞❡ ❛❧❡❛tór✐❛ ❞❡ ❊r❞ös✲❘é♥②✐ ♣♦ss✉✐ ✉♠❛ ❞✐str✐❜✉✐çã♦ ❜✐♥♦♠✐❛❧✱ ❡♥q✉❛♥t♦ r❡❞❡s ❧✐✈r❡s ❞❡ ❡s❝❛❧❛✱ ♣♦ss✉❡♠ ✉♠❛ ❞✐str✐❜✉✐çã♦ r❡♣r❡s❡♥t❛❞❛ ♣♦r ✉♠❛ ❧❡✐ ❞❡ ♣♦tê♥❝✐❛ ❬✹❪✿
P(k)∼k−γ ✭✻✮
♦ q✉❡ ✐♥❞✐❝❛ q✉❡ ❡①✐st❡♠ ♣♦✉❝♦s ♥♦❞♦s ❛❧t❛♠❡♥t❡ ❝♦♥❡❝t❛❞♦s✱ ❞❡♥♦♠✐♥❛❞♦s ✏hubs✑✳
❜✮ ❈♦❡✜❝✐❡♥t❡ ❞❡ ❛❣r✉♣❛♠❡♥t♦ ♠é❞✐♦
❖ ❝♦❡✜❝✐❡♥t❡ ❞❡ ❛❣r✉♣❛♠❡♥t♦ ♠é❞✐♦✱ C(k)✱ ❝❛r❛❝t❡r✐③❛ ❛ ❞❡♥s✐❞❛❞❡ ❞❡ ❝♦♥❡①õ❡s
♣ró①✐♠❛ ❛ ✉♠ ❞❡t❡r♠✐♥❛❞♦ ♥♦❞♦✳ ❙✉♣♦♥❞♦ ✉♠❛ r❡❞❡ ♥ã♦ ❞✐r❡❝✐♦♥❛❞❛✱ ❡ q✉❡ ✉♠ ❞❡ s❡✉s ♥♦❞♦s ♣♦ss✉❡♠ z ♥♦❞♦s ✈✐③✐♥❤♦s ♣ró①✐♠♦s✱ ❛ ❝♦♥❡①ã♦ ♠á①✐♠❛ ❞❡ss❡ ♥♦❞♦ é
❛♣r♦①✐♠❛❞❛ q✉❛♥❞♦ t♦❞❛sz(z+ 1)/2❧✐❣❛çõ❡s ♥♦ ♥♦❞♦ ❡stã♦ ♣r❡s❡♥t❡s✳ ■ss♦ t❛♠❜é♠
s✐❣♥✐✜❝❛ q✉❡ t♦❞❛s z(z+ 1)/2 ♣♦ssí✈❡✐s ❧✐❣❛çõ❡s ❡♥tr❡ ♦s ♥♦❞♦s ✈✐③✐♥❤♦s ♣ró①✐♠♦s
❡①✐st❡♠✳ ❈♦♥✈❡♥❝✐♦♥❛❧♠❡♥t❡ ❬✶✷✱ ✶✸✱ ✶✹❪✱ ♦ ❝♦❡✜❝✐❡♥t❡ ❞❡ ❛❣r✉♣❛♠❡♥t♦✱ C(k)✱ ❞❡ ✉♠
♥♦❞♦ é ❛ r❡❧❛çã♦ ❡♥tr❡ ♦ ♥ú♠❡r♦ t♦t❛❧ y ❞❡ ❧✐❣❛çõ❡s q✉❡ ❝♦♥❡❝t❛♠ ♥♦❞♦s ✈✐③✐♥❤♦s
✶✹
♣ró①✐♠♦s✱ r❡♣r❡s❡♥t❛❞♦ ♠❛t❡♠❛t✐❝❛♠❡♥t❡ ♣♦r✿
C = 2y
z(z−1). ✭✼✮
❆tr❛✈és ❞❛ ❛♥á❧✐s❡ ❞♦ ❝♦❡✜❝✐❡♥t❡ ❞❡ ❛❣r✉♣❛♠❡♥t♦✱ ♣♦❞❡✲s❡ ❞❡✜♥✐r s❡ ✉♠❛ r❡❞❡ ❜✐♦✲ ❧ó❣✐❝❛ ✐♥t❡❣r❛❞❛ é ❤✐❡rárq✉✐❝❛✱ ✐st♦ é✱ s❡ ♦ ❛❣r✉♣❛♠❡♥t♦ ❞♦s ♥♦❞♦s ❞❡♣❡♥❞❡ ❞❛ ❝♦✲ ♥❡❝t✐✈✐❞❛❞❡ ❞♦s ♠❡s♠♦s✳
❝✮ ●r❛✉ ❞❡ ♣r♦①✐♠✐❞❛❞❡
●r❛✉ ❞❡ ♣r♦①✐♠✐❞❛❞❡✱ CC(k)✱ ♠❡❞❡ q✉❛♥t♦ ✉♠ ♥♦❞♦ ♣❛rt✐❝✉❧❛r ❡stá ♣ró①✐♠♦ ❞❡ t♦❞♦s ♦s ♦✉tr♦s ♥♦❞♦s ❞❛ r❡❞❡✳ P❛r❛ ❝❛❧❝✉❧❛r ♦ ❣r❛✉ ❞❡ ♣r♦①✐♠✐❞❛❞❡ s♦♠❛✲s❡ ❛ ❞✐stâ♥❝✐❛ ❣❡♦❞és✐❝❛ ❞♦ ♥♦❞♦ ❡♠ r❡❧❛çã♦ ❛ t♦❞♦s ♦s ❞❡♠❛✐s ♥♦❞♦s ❞♦ ❣r❛❢♦ ❡ ❞❡♣♦✐s ✐♥✈❡rt❡✲s❡✱ ✉♠❛ ✈❡③ q✉❡ q✉❛♥t♦ ♠❛✐♦r ❛ ❞✐stâ♥❝✐❛ ♠❡♥♦r ❛ ♣r♦①✐♠✐❞❛❞❡✱ ❡ ❡ss❡ ✈❛❧♦r ❛✐♥❞❛ é ♥♦r♠❛❧✐③❛❞♦ ❡♠ r❡❧❛çã♦ ❛♦ ♥♦❞♦ ❞❡ ♠❡♥♦r ✈❛❧♦r✳ ▼❛t❡♠❛t✐❝❛♠❡♥t❡✱ ❡❧❛ é r❡♣r❡s❡♥t❛❞❛ ♣♦r✿
CC(v) =
1
t∈V /vdG(v, t)
✭✽✮
♦♥❞❡ dG(v, t)é ❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ v ❡ t✳
❞✮ ●r❛✉ ❞❡ ✐♥t❡r♠❡❞✐❛çã♦
●r❛✉ ❞❡ ✐♥t❡r♠❡❞✐❛çã♦✱ CB(k)✱ ♠❡❞❡ q✉❛♥t♦ ✉♠ ♥♦❞♦ ❡stá ♥♦s ♠❡♥♦r❡s ❝❛♠✐♥❤♦s ❡♥tr❡ ♦✉tr♦s ♥♦❞♦s ♥❛ r❡❞❡✱ ♦✉ s❡❥❛✱ ❛ ❢r❛çã♦ ❞♦s tr❛❥❡t♦s ♠❛✐s ❝✉rt♦s q✉❡ ✐♥❝❧✉❡♠ ♦ ♥♦❞♦i✳ ❚❛♠❜é♠ é ❝♦♥s✐❞❡r❛❞♦ ✉♠❛ ♠❡❞✐❞❛ ❞❡ r❡❧❡✈â♥❝✐❛ ❞♦ ♥♦❞♦✱ ❡ é r❡♣r❡s❡♥t❛❞♦
♠❛t❡♠❛t✐❝❛♠❡♥t❡ ♣♦r✿
CB(v) =
s=v=t
σst(v)
σst ✭✾✮
♦♥❞❡ σst(v) é ♦ ♥ú♠❡r♦ ❞❡ tr❛❥❡t♦s ♠❛✐s ❝✉rt♦s ❞❡ s ❛té t q✉❡ ✐♥❝❧✉✐ v ❡ σ(st) é ♦
♥ú♠❡r♦ ❞❡ tr❛❥❡t♦s ❞❡ s ❛té t✳
✸✳✸ ■♠♣❧❡♠❡♥t❛çã♦
✶✺
t♦r✐❝❛✱ ●r❛♣❤❯t✐❧✐t✐❡s ❡ ◆❡t✇♦r❦❳✳ ❖s ♥♦t❡❜♦♦❦s ❡stã♦ ❞✐s♣♦♥í✈❡✐s ♥♦ s✐t❡ ❬✶✺❪ ❡ ❞❡s❝r✐t♦s ❛❜❛✐①♦✿
❛✮ ❣❡r♥❡t✳♥❜✿ ❣❡r❛ ✉♠❛ r❡❞❡ ❛tr❛✈és ❞❛ ✐♠♣♦rt❛çã♦ ❞❡ ✉♠ ❛rq✉✐✈♦ ❝♦♠ ♦s ❞❛❞♦s❀
❜✮ ♣❦✳♥❜✿ ❛♣❧✐❝❛t✐✈♦ q✉❡ ❝❛❧❝✉❧❛ ♦s ❞❛❞♦s ❞❡ P(k)❡♠ ❢✉♥çã♦ ❞❡ k ♣❛r❛ ❛ ❊✳ ❝♦❧✐ ❡ ♣❛r❛
❛ ❙✳ ❝❡r❡✈✐s✐❛❡❀
❝✮ ❝❦✳♥❜✿ ❛♣❧✐❝❛t✐✈♦ q✉❡ ❝❛❧❝✉❧❛ ♦s ❞❛❞♦s ❞❡ C(k)❡♠ ❢✉♥çã♦ ❞❡k ♣❛r❛ ❛ ❊✳ ❝♦❧✐ ❡ ♣❛r❛
❛ ❙✳ ❝❡r❡✈✐s✐❛❡❀
❞✮ ❝❡♥t✳♥❜✿ ❛♣❧✐❝❛t✐✈♦ q✉❡ ❝❛❧❝✉❧❛ ♦s ❞❛❞♦s ❞❡ CC(k) ❡♠ ❢✉♥çã♦ ❞❡ k ♣❛r❛ ❛ ❊✳ ❝♦❧✐ ❡ ♣❛r❛ ❛ ❙✳ ❝❡r❡✈✐s✐❛❡❀
✶✻
✹ ❘❡s✉❧t❛❞♦s ❡ ❉✐s❝✉ssã♦
❆s r❡❞❡s ✐♥t❡❣r❛❞❛s ❝♦♥t❡♥❞♦ ✐♥t❡r❛çõ❡s ❢ís✐❝❛s ❡♥tr❡ ♣r♦t❡í♥❛s✱ ✐♥t❡r❛çõ❡s ♠❡t❛❜ó❧✐❝❛s ❡ ✐♥t❡r❛çõ❡s ❞❡ r❡❣✉❧❛çã♦ tr❛♥s❝r✐❝✐♦♥❛❧ ❞♦s ♦r❣❛♥✐s♠♦s ❊✳ ❝♦❧✐ ❡ ❙✳ ❝❡r❡✈✐s✐❛❡ ❝♦♥str✉í❞❛s ♣♦ss✉❡♠ ✷✳✸✹✾ ♥♦❞♦s ❡ ✶✾✳✾✵✽ ✐♥t❡r❛çõ❡s ❡ ✻✳✶✶✺ ♥♦❞♦s ❡ ✽✷✳✾✾✹ ✐♥t❡r❛çõ❡s✱ r❡s♣❡❝t✐✲ ✈❛♠❡♥t❡✳ ❈♦♠ ♦ ✐♥t✉✐t♦ ❞❡ ❝❛r❛❝t❡r✐③❛r ❛ ❡str✉t✉r❛ t♦♣♦❧ó❣✐❝❛ ❞❡ss❛s r❡❞❡s✱ ♦✉ s❡❥❛✱ ❝❧❛ss✐✜❝á✲❧❛s ❡♠ ❛❧❡❛tór✐❛s ♦✉ ❧✐✈r❡s ❞❡ ❡s❝❛❧❛ ❡ ❞❡t❡r♠✐♥❛r s❡ sã♦ ❤✐❡rárq✉✐❝❛s✱ ♥ós ❛♥❛❧✐✲ s❛♠♦s q✉❛tr♦ ♣❛râ♠❡tr♦s✿ ❛ ❞✐str✐❜✉✐çã♦ ❞❡ ❝♦♥❡❝t✐✈✐❞❛❞❡s✱ ♦ ❝♦❡✜❝✐❡♥t❡ ❞❡ ❛❣r✉♣❛♠❡♥t♦ ♠é❞✐♦✱ ♦ ❣r❛✉ ❞❡ ♣r♦①✐♠✐❞❛❞❡ ❡ ♦ ❣r❛✉ ❞❡ ✐♥t❡r♠❡❞✐❛çã♦✳
✹✳✶ ❘❡❞❡ ✐♥t❡❣r❛❞❛ ❞❛ ❊✳ ❝♦❧✐
✶✼
❖s ❛❥✉st❡s r❡❛❧✐③❛❞♦s ♥❛s ❞✐str✐❜✉✐çõ❡s ❞❡ ❝♦♥❡❝t✐✈✐❞❛❞❡s ❞❛ r❡❞❡ ✐♥t❡❣r❛❞❛ ❞❡ ✐♥t❡r❛çõ❡s ♠♦❧❡❝✉❧❛r❡s ❡♥tr❡ ❣❡♥❡s ❞❛ ❊✳ ❝♦❧✐ s❡❣✉❡♠ ✉♠❛ ❧❡✐ ❞❡ ♣♦tê♥❝✐❛ r❡♣r❡s❡♥t❛❞❛ ♣♦r✿
y=αx−β ✭✶✵✮
♦♥❞❡ α ❡ β sã♦ ❝♦♥st❛♥t❡s✳
❖s ❛❥✉st❡s ✉t✐❧✐③❛❞♦s ♥❛s ❞✐str✐❜✉✐çõ❡s ❞❡ ❝♦♥❡❝t✐✈✐❞❛❞❡s ❞❛ r❡❞❡ ✐♥t❡❣r❛❞❛ ❞❡ ✐♥t❡r❛✲ çõ❡s ♠♦❧❡❝✉❧❛r❡s ♠♦str❛❞♦s ♥❛ ❋✐❣✉r❛ ✷ ❢♦r❛♠✿
●r❛✉ ❞❡ ❡♥tr❛❞❛✿
P(k)in= 595.09k−1.24 ✭✶✶✮
●r❛✉ ❞❡ s❛í❞❛✿
P(k)out = 430.8k−1.15 ✭✶✷✮
✶✽
❋✐❣✉r❛ ✸✿ ❉❡♣❡♥❞ê♥❝✐❛ ❞♦ ❝♦❡✜❝✐❡♥t❡ ❞❡ ❛❣r✉♣❛♠❡♥t♦ C(k) ❡♠ r❡❧❛çã♦ à ❝♦♥❡❝t✐✈✐❞❛❞❡ k✳ P♦❞❡✲s❡ ♦❜s❡r✈❛r q✉❡ ♦C(k)é ❛❥✉st❛❞♦ ♣♦r ✉♠ ♠♦❞❡❧♦ ❧✐♥❡❛r q✉❛❞rát✐❝♦✱ ♦ q✉❡ ✐♥❞✐❝❛
q✉❡ ❛ r❡❞❡ ✐♥t❡❣r❛❞❛ ❞❡ ✐♥t❡r❛çõ❡s ♠♦❧❡❝✉❧❛r❡s ❡♥tr❡ ❣❡♥❡s ❞❛ ❊✳ ❝♦❧✐ ❝♦♥str✉í❞❛ ♥❡st❡ ♣r♦❥❡t♦ é ❧✐♥❡❛r q✉❛❞rát✐❝❛✳
❖ ❛❥✉st❡ r❡❛❧✐③❛❞♦ ♥❛ ❞❡♣❡♥❞ê♥❝✐❛ ❞♦ ❝♦❡✜❝✐❡♥t❡ ❞❡ ❛❣r✉♣❛♠❡♥t♦ C(k) ❡♠ r❡❧❛çã♦ à
❝♦♥❡❝t✐✈✐❞❛❞❡ k ❞♦s ❞❛❞♦s ❞❛ ❊✳ ❝♦❧✐ s❡❣✉❡ ✉♠ ♠♦❞❡❧♦ ❧✐♥❡❛r q✉❛❞rát✐❝♦✱ ❞❡s❝r✐t♦ ♣♦r✿
y=αeβx−γx2 ✭✶✸✮
♦♥❞❡ α✱ β✱ γ sã♦ ❝♦♥st❛♥t❡s✳
❖ ❛❥✉st❡ ✉t✐❧✐③❛❞♦ ♥❛ ❛♥á❧✐s❡ ❞❛ ❞❡♣❡♥❞ê♥❝✐❛ ❞♦ ❝♦❡✜❝✐❡♥t❡ ❞❡ ❛❣r✉♣❛♠❡♥t♦ C(k) ❡♠
r❡❧❛çã♦ à ❝♦♥❡❝t✐✈✐❞❛❞❡ k ♠♦str❛❞♦ ♥❛ ❋✐❣✉r❛ ✸ ❢♦✐✿
C(k) = 0.19e0.0058k−0.036k2 ✭✶✹✮
❊ss❡ ♠♦❞❡❧♦ ❧✐♥❡❛r q✉❛❞rát✐❝♦ ♣❛r❛ C(k) ✐♥❞✐❝❛ q✉❡ ❛ r❡❞❡ ✐♥t❡❣r❛❞❛ ❞❛ ❊✳ ❝♦❧✐ ♥ã♦
✶✾
❝♦♥❡❝t❛❞♦s ❞❡♥♦♠✐♥❛❞♦s ❤✉❜s ❬✺❪✳ ◆❡ss❛ r❡❞❡ ✐♥t❡❣r❛❞❛✱ ♣♦❞❡♠♦s ♦❜s❡r✈❛r q✉❡ ♣❛r❛ ❜❛✐✲ ①♦s ✈❛❧♦r❡s ❞❡k✱ ❡①✐st❡ ✉♠❛ ❢r❛❝❛ ❞❡♣ê♥❞❡♥❝✐❛ ❡♠ r❡❧❛çã♦ ❛♦ C(k)❡ t❛♠❜é♠ q✉❡ ♦ C(k)
❛✉♠❡♥t❛ ❣r❛❞❛t✐✈❛♠❡♥t❡ ❛ ♠❡❞✐❞❛ q✉❡ ♦ ❣r❛✉ ❞❡ ❝♦♥❡❝t✐✈✐❞❛❞❡ ❛✉♠❡♥t❛ ❛té ✉♠ ✈❛❧♦r ❞❡ ❛♣r♦①✐♠❛❞❛♠❡♥t❡ ✶✵✵ ❝♦♥❡①õ❡s✳ ❆❝✐♠❛ ❞❡ss❡ ✈❛❧♦r✱ ♦s ✈❛❧♦r❡s ❞❡ C(k) ❞❡❝❛❡♠ ❛❜r✉♣t❛✲
♠❡♥t❡ ❛ ♠❡❞✐❞❛ q✉❡ ♦ ❣r❛✉ ❞❡ ❝♦♥❡❝t✐✈✐❞❛❞❡ ❛✉♠❡♥t❛✳ ❈❧❛ss✐✜❝❛♠♦s ❡ss❡ ♠♦❞❡❧♦ ❝♦♠♦ ❧✐♥❡❛r q✉❛❞rát✐❝♦ ❞❡✈✐❞♦ ❛♦ ❛❥✉st❡ ✉t✐❧✐③❛❞♦ ♥❛ ❞❡♣❡♥❞ê♥❝✐❛ ❞♦ ❝♦❡✜❝✐❡♥t❡ ❞❡ ❛❣r✉♣❛♠❡♥t♦
C(k) ❡♠ r❡❧❛çã♦ à ❝♦♥❡❝t✐✈✐❞❛❞❡ k ❞♦s ❞❛❞♦s ❞❛ ❊✳ ❝♦❧✐✳
✹✳✷ ❘❡❞❡ ✐♥t❡❣r❛❞❛ ❞❛ ❙✳ ❝❡r❡✈✐s✐❛❡
❋✐❣✉r❛ ✹✿ ❉✐str✐❜✉✐çã♦ ❞❡ ❝♦♥❡❝t✐✈✐❞❛❞❡s ❞❛ r❡❞❡ ✐♥t❡❣r❛❞❛ ❞❡ ✐♥t❡r❛çõ❡s ♠♦❧❡❝✉❧❛r❡s ❡♥tr❡ ❣❡♥❡s ❞❛ ❙✳ ❝❡r❡✈✐s✐❛❡ ❝♦♥str✉í❞❛ ♥❡st❡ ♣r♦❥❡t♦✳ ❊ss❛ ❞✐str✐❜✉✐çã♦ é ❛❥✉st❛❞❛ ♣♦r ✉♠ ♠♦❞❡❧♦ ❧✐✈r❡ ❞❡ ❡s❝❛❧❛ ❞✐❢❡r❡♥❝✐❛❞♦✳
❖s ❛❥✉st❡s r❡❛❧✐③❛❞♦s ♥❛ ❞✐str✐❜✉✐çã♦ ❞❡ ❝♦♥❡❝t✐✈✐❞❛❞❡s ❞❛ r❡❞❡ ❞❡ ✐♥t❡r❛çõ❡s ♠♦❧❡❝✉❧❛r❡s ❡♥tr❡ ❣❡♥❡s ❞❛ ❙✳ ❝❡r❡✈✐s✐❛❡ s❡❣✉❡♠ ✉♠ ♠♦❞❡❧♦ ❞✐❢❡r❡♥❝✐❛❞♦ r❡♣r❡s❡♥t❛❞♦ ♣♦r✿
y=αxβe−γx ✭✶✺✮
✷✵
❖s ❛❥✉st❡s ✉t✐❧✐③❛❞♦s ♥❛ ❞✐str✐❜✉✐çã♦ ❞❡ ❝♦♥❡❝t✐✈✐❞❛❞❡s ❞❛ r❡❞❡ ✐♥t❡❣r❛❞❛ ❞❡ ✐♥t❡r❛çõ❡s ♠♦❧❡❝✉❧❛r❡s ♠♦str❛❞♦ ♥❛ ❋✐❣✉r❛ ✹ ❢♦r❛♠✿
●r❛✉ ❞❡ ❡♥tr❛❞❛✿
P(k)in = 1141.67k0.037e−0.57k ✭✶✻✮
●r❛✉ ❞❡ s❛í❞❛✿
P(k)out = 582.089k−0.43e−0.043k ✭✶✼✮
◆❛ ❛♥á❧✐s❡ ❞❡ss❡ r❡s✉❧t❛❞♦✱ ✈❡r✐✜❝❛♠♦s q✉❡ ❛ r❡❞❡ ❞❛ ❙✳ ❝❡r❡✈✐s✐❛❡ ♣♦ss✉✐ ✉♠ ❝♦♠♣♦r✲ t❛♠❡♥t♦ ❞✐st✐♥t♦ ❞♦ ♦❜s❡r✈❛❞♦ ❡♠ ♦✉tr♦s ❣r❛❢♦s✱ ♣♦✐s ❛ ❞✐str✐❜✉✐çã♦ ❞❡ ❝♦♥❡❝t✐✈✐❞❛❞❡s é r❡♣r❡s❡♥t❛❞❛ ♣♦r ✉♠❛ ❡①♣♦♥❡♥❝✐❛❧ ♠❛s ❝♦♠ ✉♠ ♣ré ❢❛t♦r ❧❡✐ ❞❡ ♣♦tê♥❝✐❛✳ ❙❡ ❡ss❡ ❛❥✉st❡ ❢♦ss❡ ♣✉r❛♠❡♥t❡ ❡①♣♦♥❡♥❝✐❛❧✱ ❞✐rí❛♠♦s q✉❡ ♦ ❣r❛❢♦ s❡r✐❛ ❝♦♠♣❛tí✈❡❧ ❝♦♠ ♦ ♠♦❞❡❧♦ ❞❡ ❊r❞ös✲❘é♥②✐ ❬✶✼❪✱ ♠❛s ♥❡st❡ ❝❛s♦ ❞✐③❡♠♦s q✉❡ ♦ ❣r❛❢♦ é ❝♦♠♣❛tí✈❡❧ ❝♦♠ ♦ ♠♦❞❡❧♦ ❧✐✈r❡ ❞❡ ❡s❝❛❧❛✱ ♥♦ ❡♥t❛♥t♦✱ ❝♦♠ ✉♠ ❝✉t♦✛ ❡①♣♦♥❡♥❝✐❛❧✳
❋✐❣✉r❛ ✺✿ ❉❡♣❡♥❞ê♥❝✐❛ ❞♦ ❝♦❡✜❝✐❡♥t❡ ❞❡ ❛❣r✉♣❛♠❡♥t♦ C(k) ❡♠ r❡❧❛çã♦ à ❝♦♥❡❝t✐✈✐❞❛❞❡ k✳ P♦❞❡✲s❡ ♦❜s❡r✈❛r q✉❡ ♦C(k)é ❛❥✉st❛❞♦ ♣♦r ✉♠ ♠♦❞❡❧♦ ❧✐♥❡❛r q✉❛❞rát✐❝♦✱ ♦ q✉❡ ✐♥❞✐❝❛
✷✶
❖ ❛❥✉st❡ r❡❛❧✐③❛❞♦ ♥❛ ❞❡♣❡♥❞ê♥❝✐❛ ❞♦ ❝♦❡✜❝✐❡♥t❡ ❞❡ ❛❣r✉♣❛♠❡♥t♦ C(k) ❡♠ r❡❧❛çã♦ à
❝♦♥❡❝t✐✈✐❞❛❞❡ k ❞♦s ❞❛❞♦s ❞❛ ❙✳ ❝❡r❡✈✐s✐❛❡ s❡❣✉❡ ✉♠ ♠♦❞❡❧♦ ❧✐♥❡❛r q✉❛❞rát✐❝♦✱ ❞❡s❝r✐t♦
♣♦r✿
y=αeβx−γx2 ✭✶✽✮
♦♥❞❡ α✱ β✱ γ sã♦ ❝♦♥st❛♥t❡s✳
❖ ❛❥✉st❡ ✉t✐❧✐③❛❞♦ ♥❛ ❛♥á❧✐s❡ ❞❛ ❞❡♣❡♥❞ê♥❝✐❛ ❞♦ ❝♦❡✜❝✐❡♥t❡ ❞❡ ❛❣r✉♣❛♠❡♥t♦ C(k) ❡♠
r❡❧❛çã♦ à ❝♦♥❡❝t✐✈✐❞❛❞❡ k ♠♦str❛❞♦ ♥❛ ❋✐❣✉r❛ ✺ ❢♦✐✿
C(k) = 0.16e0.012k−0.051k2 ✭✶✾✮
❊ss❡ ♠♦❞❡❧♦ ❧✐♥❡❛r q✉❛❞rát✐❝♦ ♣❛r❛ C(k)✐♥❞✐❝❛ q✉❡ ❛ r❡❞❡ ✐♥t❡❣r❛❞❛ ❞❛ ❙✳ ❝❡r❡✈✐s✐❛❡
♥ã♦ s❡❣✉❡ ♦ ♠♦❞❡❧♦ ❝❧áss✐❝♦ ❞❡ ❝❧❛ss✐✜❝❛çã♦ ♣r♦♣♦st♦ ♣♦r ❘❛✈❛s③ ❡t ❛❧✳ ❬✶✻❪✱ ♦♥❞❡ ♥♦❞♦s ❡s❝❛ss❛♠❡♥t❡ ❝♦♥❡❝t❛❞♦s ❢❛③❡♠ ♣❛rt❡ ❞❡ ár❡❛s ❛❧t❛♠❡♥t❡ ❝♦♥❡❝t❛❞❛s ❝♦♠ ❝♦♠✉♥✐❝❛çã♦ ❡♥tr❡ ❛s ❞✐❢❡r❡♥t❡s ár❡❛s ✈✐③✐♥❤❛s ❛❧t❛♠❡♥t❡ ❝♦♥❡❝t❛❞❛s ♠❛♥t✐❞❛s ♣♦r ❛❧❣✉♥s ♥♦❞♦s ❛❧t❛✲ ♠❡♥t❡ ❝♦♥❡❝t❛❞♦s ❞❡♥♦♠✐♥❛❞♦s ❤✉❜s ❬✺❪✳ ◆❡ss❛ r❡❞❡ ✐♥t❡❣r❛❞❛✱ ♣♦❞❡♠♦s ♦❜s❡r✈❛r q✉❡ ♣❛r❛ ❜❛✐①♦s ✈❛❧♦r❡s ❞❡ k✱ ❡①✐st❡ ✉♠❛ ❞❡♣ê♥❞❡♥❝✐❛ ❡♠ r❡❧❛çã♦ ❛♦ C(k) ❡ t❛♠❜é♠ q✉❡ ♦ C(k) ❛✉♠❡♥t❛ ❣r❛❞❛t✐✈❛♠❡♥t❡ ❛ ♠❡❞✐❞❛ q✉❡ ♦ ❣r❛✉ ❞❡ ❝♦♥❡❝t✐✈✐❞❛❞❡ ❛✉♠❡♥t❛ ❛té ✉♠
✈❛❧♦r ❞❡ ❛♣r♦①✐♠❛❞❛♠❡♥t❡ ✶✵✵ ❝♦♥❡①õ❡s✳ ❆❝✐♠❛ ❞❡ss❡ ✈❛❧♦r✱ ♦s ✈❛❧♦r❡s ❞❡ C(k) ❞❡❝❛❡♠
❛❜r✉♣t❛♠❡♥t❡ ❛ ♠❡❞✐❞❛ q✉❡ ♦ ❣r❛✉ ❞❡ ❝♦♥❡❝t✐✈✐❞❛❞❡ ❛✉♠❡♥t❛✳ ❈❧❛ss✐✜❝❛♠♦s ❡ss❡ ♠♦✲ ❞❡❧♦ ❝♦♠♦ ❧✐♥❡❛r q✉❛❞rát✐❝♦ ❞❡✈✐❞♦ ❛♦ ❛❥✉st❡ ✉t✐❧✐③❛❞♦ ♥❛ ❞❡♣❡♥❞ê♥❝✐❛ ❞♦ ❝♦❡✜❝✐❡♥t❡ ❞❡ ❛❣r✉♣❛♠❡♥t♦ C(k) ❡♠ r❡❧❛çã♦ à ❝♦♥❡❝t✐✈✐❞❛❞❡ k ❞♦s ❞❛❞♦s ❞❛ ❙✳ ❝❡r❡✈✐s✐❛❡✳
✷✷
❋✐❣✉r❛ ✻✿ ●r❛✉ ❞❡ ♣r♦①✐♠✐❞❛❞❡ ❞♦s ♥♦❞♦s ❡♠ ❢✉♥çã♦ ❞❡ k ♣❛r❛ ❛s r❡❞❡s ✐♥t❡❣r❛❞❛s ❞❛ ❊✳
❝♦❧✐ ❡ ❞❛ ❙✳ ❝❡r❡✈✐s✐❛❡✳ ❆♠❜♦s ♦r❣❛♥✐s♠♦s t✐✈❡r❛♠ s✉❛s ♠❡❞✐❞❛s ❛❥✉st❛❞❛s ♣♦r ✉♠❛ ❧❡✐ ❞❡ ♣♦tê♥❝✐❛✳
❖s ❛❥✉st❡s r❡❛❧✐③❛❞♦s ♥♦ ❣r❛✉ ❞❡ ♣r♦①✐♠✐❞❛❞❡ ❞❛ r❡❞❡ ✐♥t❡❣r❛❞❛ ❞❡ ✐♥t❡r❛çõ❡s ♠♦❧❡❝✉✲ ❧❛r❡s ❡♥tr❡ ❣❡♥❡s ❞❛ ❊✳ ❝♦❧✐ ❡ ❞❛ ❙✳ ❝❡r❡✈✐s✐❛❡ s❡❣✉❡♠ ✉♠❛ ❧❡✐ ❞❡ ♣♦tê♥❝✐❛ r❡♣r❡s❡♥t❛❞❛ ♣♦r✿
y=αxβ ✭✷✵✮
♦♥❞❡ α ❡ β sã♦ ❝♦♥st❛♥t❡s✳
❖s ❛❥✉st❡s ✉t✐❧✐③❛❞♦s ♥♦ ❣r❛✉ ❞❡ ♣r♦①✐♠✐❞❛❞❡ ❞❛ r❡❞❡ ✐♥t❡❣❛❞❛ ❞❡ ✐♥t❡r❛çõ❡s ♠♦❧❡❝✉✲ ❧❛r❡s ♠♦str❛❞♦s ♥❛ ❋✐❣✉r❛ ✻ ❢♦r❛♠✿
❊✳ ❝♦❧✐✿
CC(k) = 0.00011k0.079 ✭✷✶✮
❙✳ ❝❡r❡✈✐s✐❛❡✿
✷✸
❋✐❣✉r❛ ✼✿ ●r❛✉ ❞❡ ✐♥t❡r♠❡❞✐❛çã♦ ❞♦s ♥♦❞♦s ❡♠ ❢✉♥çã♦ ❞❡ k ♣❛r❛ ❛s r❡❞❡s ✐♥t❡❣r❛❞❛s ❞❛
❊✳ ❝♦❧✐ ❡ ❞❛ ❙✳ ❝❡r❡✈✐s✐❛❡✳ ❖ ❛❥✉st❡ ✉t✐❧✐③❛❞♦ ♣❛r❛ ❛♠❜♦s ♦r❣❛♥✐s♠♦s ❢♦✐ ❢❡✐t♦ ❛tr❛✈és ❞❡ ✉♠❛ ❧❡✐ ❞❡ ♣♦tê♥❝✐❛✳
❖s ❛❥✉st❡s r❡❛❧✐③❛❞♦s ♥♦ ❣r❛✉ ❞❡ ✐♥t❡r♠❡❞✐❛çã♦ ❞❛ r❡❞❡ ✐♥t❡❣r❛❞❛ ❞❡ ✐♥t❡r❛çõ❡s ♠♦❧❡✲ ❝✉❧❛r❡s ❡♥tr❡ ❣❡♥❡s ❞❛ ❊✳ ❝♦❧✐ ❡ ❞❛ ❙✳ ❝❡r❡✈✐s✐❛❡ s❡❣✉❡♠ ✉♠❛ ❧❡✐ ❞❡ ♣♦tê♥❝✐❛ r❡♣r❡s❡♥t❛❞❛ ♣♦r✿
y=αxβ ✭✷✸✮
♦♥❞❡ α ❡ β sã♦ ❝♦♥st❛♥t❡s✳
❖s ❛❥✉st❡s ✉t✐❧✐③❛❞♦s ♥♦ ❣r❛✉ ❞❡ ✐♥t❡r♠❡❞✐❛çã♦ ❞❛ r❡❞❡ ❞❡ ✐♥t❡r❛çõ❡s ♠♦❧❡❝✉❧❛r❡s ♠♦str❛❞♦s ♥❛ ❋✐❣✉r❛ ✼ ❢♦r❛♠✿
❊✳ ❝♦❧✐✿
CB(k) = 0,032k0.95 ✭✷✹✮
❙✳ ❝❡r❡✈✐s✐❛❡✿
CB(k) = 0,0047k1.64 ✭✷✺✮
✷✹
r❡❢❡r❡♥t❡ ❛♦ ❣r❛✉ ❞❡ ♣r♦①✐♠✐❞❛❞❡ ♥❡ss❛s r❡❞❡s é s❡♠❡❧❤❛♥t❡ ❛♦ q✉❡ ♦❝♦rr❡ ❡♠ ♦✉tr❛s r❡❞❡s ❝♦♠♣❧❡①❛s✱ ❜✐♦❧ó❣✐❝❛s ♦✉ ♥ã♦✱ ❡ s✐❣♥✐✜❝❛ q✉❡ ♦s ♠❡♥♦r❡s ❝❛♠✐♥❤♦s ❡♥tr❡ ♦s ♥♦❞♦s sã♦ ❝❛❞❛ ✈❡③ ♠❡♥♦r❡s ❛ ♠❡❞✐❞❛ q✉❡ ❛✉♠❡♥t❛ ♦ ❣r❛✉ ❞❡ ❝♦♥❡❝t✐✈✐❞❛❞❡ ❞♦s ♥♦❞♦s ♥❛s ❡①tr❡♠✐❞❛❞❡s ❞❡ss❡s ❝❛♠✐♥❤♦s✳ ❖ ❝♦♠♣♦rt❛♠❡♥t♦ ❞♦ ❣r❛✉ ❞❡ ✐♥t❡r♠❡❞✐❛çã♦ ♥❡ss❛s r❡❞❡s t❛♠❜❡♠ é s❡♠❡❧❤❛♥t❡ ❛♦ q✉❡ ♦❝♦rr❡ ❡♠ ♦✉tr❛s r❡❞❡s ❝♦♠♣❧❡①❛s ❡ s✐❣♥✐✜❝❛ q✉❡ q✉❛♥t♦ ♠❛✐s ❝♦♥❡①õ❡s ✉♠ ♥♦❞♦ ♣♦ss✉✐✱ ❡❧❡ ♣❛rt✐❝✐♣❛ ❞❡ ♠❛✐s ♠❡♥♦r❡s ❝❛♠✐♥❤♦s ❡♥tr❡ ♦✉tr♦s ♥♦❞♦s✳
❊♠ s✉♠❛✱ ♣♦❞❡♠♦s ♦❜s❡r✈❛r ♦s r❡s✉❧t❛❞♦s ♦❜t✐❞♦s ♥❛s ❚❛❜❡❧❛s ✷ ❡ ✸ ♣❛r❛ ❝❛❞❛ ♦r❣❛✲ ♥✐s♠♦ ❡st✉❞❛❞♦✳
❚❛❜❡❧❛ ✷✿ ❘❡s✉❧t❛❞♦s ♣❛r❛ ❛ r❡❞❡ ✐♥t❡❣r❛❞❛ ❞❛ ❜❛❝tér✐❛ ❊✳ ❝♦❧✐
P❛râ♠❡tr♦ ❊q✉❛çã♦ ❈♦❡✜❝✐❡♥t❡s
P(k) y =αx−β β
in =−0.89±0.15 ❀βout =−0.97±0.15
C(k) y =αeβx−γx2
β =−0.0071±0.005 ❀ γ =−9.27×10−6±3.45×10−5
CC(k) y =αxβ β =−0.081±0.007
CB(k) y =αxβ β = 1.6±0.4
❙❡❣✉♥❞♦ ❛ ❛♥á❧✐s❡ ❞♦s ♣❛râ♠❡tr♦s t♦♣♦❧ó❣✐❝♦s ❞❛ ❚❛❜❡❧❛ ✷✱ ♣♦❞❡♠♦s ✐♥❢❡r✐r q✉❡ ❛ r❡❞❡ ✐♥t❡❣r❛❞❛ ❞❛ ❊✳ ❝♦❧✐ é ❝❧❛ss✐✜❝❛❞❛ ❝♦♠♦ ❧✐✈r❡ ❞❡ ❡s❝❛❧❛ ❡ s❡❣✉❡ ♦ ♠♦❞❡❧♦ ❧✐♥❡❛r q✉❛❞rát✐❝♦✳
❚❛❜❡❧❛ ✸✿ ❘❡s✉❧t❛❞♦s ♣❛r❛ ❛ r❡❞❡ ✐♥t❡❣r❛❞❛ ❞❛ ❧❡✈❡❞✉r❛ ❙✳ ❝❡r❡✈✐s✐❛❡
P❛râ♠❡tr♦ ❊q✉❛çã♦ ❈♦❡✜❝✐❡♥t❡s
P(k) y=αx−βeγx β
in = 0.037±0.01❀ βout = 0.023±0.005
C(k) y =αeβx−γx2
β = 0.017±0.005 ❀ γ = 1.5×10−4±4×10−5
CC(k) y=αxβ β =−0.067±0.003
CB(k) y=αxβ β = 1.7±0.6
✷✺
✺ ❈♦♥❝❧✉sã♦
❖s r❡s✉❧t❛❞♦s ♦❜t✐❞♦s ✐♥❞✐❝❛♠ q✉❡ ❛ r❡❞❡ ❜✐♦❧ó❣✐❝❛ ✐♥t❡❣r❛❞❛ ❞❛ ❊✳ ❝♦❧✐ é ❧✐✈r❡ ❞❡ ❡s❝❛❧❛ ❡ ❛ ❞❡♣❡♥❞ê♥❝✐❛ ❞♦ C(k)♥♦s ✈❛❧♦r❡s ❞❡ k ✐♥❞✐❝❛ ✉♠ ♥♦✈♦ ♠♦❞❡❧♦ ❞❡ ❛❥✉st❡✱ ♦ q✉❡
♥♦s ❝♦♥❞✉③✐✉ ❛ ✉♠ ♠ét♦❞♦ ♥♦✈♦ ❞❡ ❝❧❛ss✐✜❝❛çã♦✱ ❞❡✜♥✐❞♦ ❝♦♠♦ ❧✐♥❡❛r q✉❛❞rát✐❝♦✳❏á ♣❛r❛ ❛ r❡❞❡ ✐♥t❡❣r❛❞❛ ❞❛ ❙✳ ❝❡r❡✈✐s✐❛❡ ♦s r❡s✉❧t❛❞♦s ✐♥❞✐❝❛♠ q✉❡✱ ❡❧❛ é ❝❧❛ss✐✜❝❛❞❛ ❝♦♠♦ ❧✐✈r❡ ❞❡ ❡s❝❛❧❛✱ ♣♦ré♠✱ ❞✐❢❡r❡♥❝✐❛❞❛✱ ✈✐st♦ q✉❡ ♦ ❛❥✉st❡ ✉t✐❧✐③❛❞♦ ♣♦ss✉✐ ✉♠❛ t❡♥❞ê♥❝✐❛ ♣❛r❛ ♦ ♠♦❞❡❧♦ ❡①♣♦♥❡♥❝✐❛❧✱ ❞❡✈✐❞♦ ❛♦ ❝✉t♦✛ q✉❡ ♦ ❛❥✉st❡ ❛♣r❡s❡♥t❛✳❆❧é♠ ❞✐ss♦ ❡❧❛ t❛♠❜é♠ é ❝❧❛ss✐✜❝❛❞❛ s❡❣✉♥❞♦ ♦ ♥♦✈♦ ♠ét♦❞♦ ❞❡ ❛❥✉st❡ q✉❡ ♣r♦♣✉s❡♠♦s✱ ❝♦♠♦ ❧✐♥❡❛r q✉❛❞rát✐❝♦✳
❖ ♥♦✈♦ ♠ét♦❞♦ ❞❡ ❝❧❛ss✐✜❝❛çã♦ ♣r♦♣♦st♦ ♣❛r❛ ♦s ❛❥✉st❡s ❞❛ ❞❡♣❡♥❞ê♥❝✐❛ ❞♦ C(k)
♥♦s ✈❛❧♦r❡s ❞❡ k ✈❛❧❡ ❛♣❡♥❛s ♣❛r❛ ❛ ❛✈❛❧✐❛çã♦ ❞❡ r❡❞❡s ❜✐♦❧ó❣✐❝❛s ✐♥t❡❣r❛❞❛s✱ ♦✉ s❡❥❛✱
q✉❡ ♣♦ss✉❡♠ ♠❛✐s ❞❡ ✉♠ t✐♣♦ ❞❡ ✐♥t❡r❛çã♦ ❢ís✐❝❛✳❖ ♠❡s♠♦ ❛✐♥❞❛ ❡stá ❡♠ ♣r♦❝❡ss♦ ❞❡ ❛✈❛❧✐❛çã♦ ♣❛r❛ s❡r ✈❛❧✐❞❛❞♦ ❝♦♠♦ ✉♠ ♥♦✈♦ ♠ét♦❞♦ ❞❡ ❝❧❛ss✐✜❝❛çã♦ ❡ ♣❛r❛ ✐ss♦ ❞❡✈❡✲s❡ ❝♦♠♣r♦✈❛r s✉❛ ✈❛❧✐❞❛❞❡ ♣❛r❛ ♦✉tr♦s ♦r❣❛♥✐s♠♦s✳❈♦♠♦ ♣♦❞❡♠♦s ♦❜s❡r✈❛r ♥❛s ❚❛❜❡❧❛s ✷ ❡ ✸✱ ♦s ✈❛❧♦r❡s ❞♦ ✐♥t❡r✈❛❧♦ ❞❡ ❝♦♥✜❛♥ç❛ sã♦ ✈á❧✐❞♦s ♣❛r❛ ❡ss❡s ❞❛❞♦s✳
◆❡st❡ tr❛❜❛❧❤♦ ♠❡❞✐♠♦s ❛ r❡❧❛çã♦ ❡♥tr❡ ❛ ♣r♦①✐♠✐❞❛❞❡✱ ✐♥s❡rçã♦ ❡ ❝♦♥❡❝t✐✈✐❞❛❞❡ ❞❡ ✉♠ ♥♦❞♦✳❖s r❡s✉❧t❛❞♦s ♦❜t✐❞♦s ✐♥❞✐❝❛♠ q✉❡ ❡①✐st❡ ✉♠❛ r❡❧❛çã♦ ❞♦ t✐♣♦ ❧❡✐ ❞❡ ♣♦tê♥❝✐❛ ❡♥tr❡ ❡st❛s q✉❛♥t✐❞❛❞❡s✳❖ s✐❣♥✐✜❝❛❞♦ ❜✐♦❧ó❣✐❝♦ ❞❡st❛ r❡❧❛çã♦ ❛✐♥❞❛ ♥ã♦ ❢♦✐ ❡st✉❞❛❞♦✱ ♠❛s ♥♦s ♣❛r❡❝❡ r❛③♦á✈❡❧ s✉♣♦r q✉❡ ❡st❛ r❡❧❛çã♦ ♣♦ss❛ ❡①♣❧✐❝❛r ♣♦r q✉❡ ♦s ♥♦❞♦s ❝♦♠ ✉♠ ♠❛✐♦r ♥ú♠❡r♦ ❞❡ ❝♦♥❡①õ❡s t❡♥❞❡♠ ❛ s❡r ♠❛✐s r❡❧❡✈❛♥t❡s ♣❛r❛ ✉♠ ♦r❣❛♥✐s♠♦ ❬✹❪✱ ♣♦✐s ❛ ✐♥s❡rçã♦ ❡ ❛ ♣r♦①✐♠✐❞❛❞❡ sã♦ ♣r♦♣r✐❡❞❛❞❡s q✉❡ ❞❡♣❡♥❞❡♠ ❞❡ t♦❞♦ ♦ ❣r❛❢♦ ❡ ♥ã♦ ❛♣❡♥❛s ❞❛ ✈✐③✐♥❤❛♥ç❛ ❞❡ ✉♠ ♥♦❞♦ ❡ ♣♦rt❛♥t♦ é ♥❛t✉r❛❧ ❛ss✉♠✐r q✉❡ ❡❧❛s ❡st❡❥❛♠ ♠❛✐s ❝♦rr❡❧❛❝✐♦♥❛❞❛s ❝♦♠ ❛ r❡❧❡✈â♥❝✐❛ ❞❡ ✉♠ ❞❛❞♦ ❣❡♥❡✳
✷✻
❘❡❢❡rê♥❝✐❛s
❬✶❪ ▼✳ ❍✳ ❱✳ ❱✳ ❘❡❣❡♥♠♦rt❡❧✱ ❊▼❇❖ ❘❡♣ ✺✱ ✶✵✶✻ ✭✷✵✵✹✮✳
❬✷❪ ❆✳ ❈✳ ❆❤♥✱ ▼✳ ❚❡✇❛r✐✱ ❈✳✲❙✳ P♦♦♥✱ ❛♥❞ ❘✳ ❙✳ P❤✐❧❧✐♣s✱ P▲♦❙ ▼❡❞ ✸✱ ❡✷✵✽ ✭✷✵✵✻✮✳
❬✸❪ ❇✳ ❇♦❧❧♦❜ás✱ ●r❛♣❤ ❚❤❛♦r②✿ ❛♥ ✐♥tr♦❞✉❝t♦r② ❝♦✉rs❛✱ ✶st ❡❞✳ ✭❙♣r✐♥❣❡r ❱❡r❧❛❣✱ ◆❡✇ ❨♦r❦✱ ✶✾✼✾✮✱ ❱♦❧✳ ✸✳
❬✹❪ ❘✳ ❆❧❜❡rt ❛♥❞ ❆✳✲▲✳ ❇❛r❛❜ás✐✱ ❘❡✈✐❡✇s ♦❢ ▼♦❞❡r♥ P❤②s✐❝s ✼✹✱ ✭✷✵✵✷✮✳
❬✺❪ ❆✳✲▲✳ ❇❛r❛❜ás✐ ❛♥❞ ❩✳ ◆✳ ❖❧t✈❛✐✱ ◆❛t ❘❡✈ ●❡♥❡t ✺✱ ✶✵✶ ✭✷✵✵✹✮✳
❬✻❪ ❆✳✲▲✳ ❇❛r❛❜ás✐✱ ▲✐♥❦❛❞✿ ❚❤❛ ◆❛✇ ❙❝✐❛♥❝❛ ♦❢ ◆❛t✇♦r❦s✱ ✶st ❡❞✳ ✭❇❛s✐❝ ❇♦♦❦s✱ ❈❛♠✲ ❜r✐❞❣❡✱ ✷✵✵✷✮✳
❬✼❪ ❊✳ ❈❛r✈❛❧❤♦ ❛♥❞ ❚✳ ▼❡♥❞♦♥ç❛✱ ❊♥s❛✐♦s ❞❛ ❝♦♠♣❧❛①✐❞❛❞❛ ✷✱ ✷✵✵✹✳
❬✽❪ ❇✳ P❡rr♦✉❞✱ ❏✳ ▲❡❡✱ ◆✳ ❱❛❧❦♦✈❛✱ ❆✳ ❉❤✐r❛♣♦♥❣✱ P✳✲❨✳ ▲✐♥✱ ❖✳ ❋✐❡❤♥✱ ❉✳ ❑ü❧t③✱ ❛♥❞ ❘✳ ❍✳ ❲❡✐ss✱ ▼♦❧ ❈❛♥❝❡r ✺✱ ✻✹ ✭✷✵✵✻✮✳
❬✾❪ ❆✳ ❊rt❡❧✱ ❆✳ ❱❡r❣❤❡s❡✱ ❙✳ ❲✳ ❇②❡rs✱ ▼✳ ❖❝❤s✱ ❛♥❞ ❆✳ ❚♦③❡r❡♥✱ ▼♦❧ ❈❛♥❝❡r ✺✱ ✺✺ ✭✷✵✵✻✮✳
❬✶✵❪ ❉✳ ❘✳ ❘❤♦❞❡s✱ ❙✳ ❆✳ ❚♦♠❧✐♥s✱ ❙✳ ❱❛r❛♠❜❛❧❧②✱ ❱✳ ▼❛❤❛✈✐s♥♦✱ ❚✳ ❇❛rr❡tt❡✱ ❙✳ ❑❛❧②❛♥❛✲ ❙✉♥❞❛r❛♠✱ ❉✳ ●❤♦s❤✱ ❆✳ P❛♥❞❡②✱ ❛♥❞ ❆✳ ▼✳ ❈❤✐♥♥❛✐②❛♥✱ ◆❛t ❇✐♦t❡❝❤♥♦❧ ✷✸✱ ✾✺✶ ✭✷✵✵✺✮✳
❬✶✶❪ ❏✳ P✳ ▼✳ ❞❛ ❙✐❧✈❛✱ ▼✳ ▲✳ ❆❝❡♥❝✐♦✱ ❏✳ ❈✳ ▼✳ ▼♦♠❜❛❝❤✱ ❘✳ ❱✐❡✐r❛✱ ❏✳ ❈✳ ❞❛ ❙✐❧✈❛✱ ◆✳ ▲❡♠❦❡✱ ❛♥❞ ▼✳ ❙✐♥✐❣❛❣❧✐❛✱ ■♥ s✐❧✐❝♦ ♥❛t✇♦r❦ t♦♣♦❧♦❣②✲❜❛s❛❞ ♣r❛❞✐❝t✐♦♥ ♦❢ ❣❛♥❛ ❛ss❛♥t✐❛❧✐t②✱ ✷✵✵✽✳
❬✶✷❪ ❉✳ ❏✳ ❲❛tts ❛♥❞ ❙✳ ❍✳ ❙tr♦❣❛t③✱ ◆❛t✉r❡ ✸✾✸✱ ✹✹✵ ✭✶✾✾✽✮✳
❬✶✸❪ ▼✳ ❊✳ ◆❡✇♠❛♥ ❛♥❞ ❉✳ ❏✳ ❲❛tts✱ P❤②s ❘❡✈ ❊ ❙t❛t P❤②s P❧❛s♠❛s ❋❧✉✐❞s ❘❡❧❛t ■♥t❡r✲ ❞✐s❝✐♣ ❚♦♣✐❝s ✻✵✱ ✼✸✸✷ ✭✶✾✾✾✮✳
❬✶✹❪ ▼✳ ❊✳ ◆❡✇♠❛♥✱ ❙✳ ❍✳ ❙tr♦❣❛t③✱ ❛♥❞ ❉✳ ❏✳ ❲❛tts✱ P❤②s ❘❡✈ ❊ ❙t❛t ◆♦♥❧✐♥ ❙♦❢t ▼❛tt❡r P❤②s ✻✹✱ ✵✷✻✶✶✽ ✭✷✵✵✶✮✳
❬✶✺❪ ❝✐s♣♦♥í✈❛❧ ❛♠✿ ❁❤tt♣✿✴✴✇✇✇✳❛s②✉r❧✳❝♦♠✴✺✼✹❃✳
❬✶✻❪ ❊✳ ❘❛✈❛s③✱ ❆✳ ▲✳ ❙♦♠❡r❛✱ ❉✳ ❆✳ ▼♦♥❣r✉✱ ❩✳ ◆✳ ❖❧t✈❛✐✱ ❛♥❞ ❆✳ ▲✳ ❇❛r❛❜ás✐✱ ❙❝✐❡♥❝❡ ✷✾✼✱ ✶✺✺✶ ✭✷✵✵✷✮✳
