Draft
Cascading Failure in Weakly Connected Bernoulli Random Graph Ilya Gertsbakh,
[email protected]
Suppose, in a city live 900 blond girls, and 100 -readheads (call them reds). All these girls maintain mutual relations: probability that there is a telephone connection between any pair of girls is p. Suppose each evening all girls exchange calls. Denote by X the number of calls of a particular blonde with other blonds and by Y -the number of conversations this blond had with red girls. Obviously, X ∼ Bin(899, p), Y ∼ Bin(100, p). If it happens that X < Y , then the blond girl colors her hairs in red color. Suppose that telephone conversations take place each day, and are governed by some external random mechanism. Suppose that p = 0.002. On the average, each girl has avr = 999p = 1.99 conversations. Then a = P r(X < Y ) = 0.035, and on the average, 32 blond girls this evening color their hairs in red. Probability that nobody does it is equal α = (1 − a)899 = 7.5 · 10−5 . So, the probability that at least one blond girl changes hair color is 1. Obviously, next day the above telephone call exchange process will be repeated with a little ”acceleration” since there will be more reds and less blonds. Therefore, in some finite time, say 1-2 months all women population of the city will become redhead. Now consider the situation that p = 0.032, much larger. Here P (X < Y ) = approx = 7.7 · 10−8 , and the average number of blonds changing their color is near zero. The probability that no blond changes their hair color is 0.00007. So, now the process of spreading ”red-head” epidemic has stopped. Note that now each blond makes each conversation session, on the average avr = 999 · 0.032 ≈ 32 calls, Here exactly lies the explanation of this seemingly paradox situation. When a person makes more calls, she gets more accurate picture of the whole
1
population and becomes convinced that being blond is more popular. Now let us consider a little change in the problem and reduce the initial number of reds by 50. So the city has 950 blonds and 50 reds. Here the picture is similar but the threshold value of the cascade changes. Blondes change their color with probability 1 only for a ≤ 0.002. For p ranging from 0.004 to 0.018 the probability that at least one blond colors in red changes from 0.996 to 0.0012. It becomes practically zero for p exceeding 0.020. There is an ample literature on cascade failures. Let me mention two papers written by distinguished persons Duncan J. Watts, ”A simple Model of Global Cascades on Random Networks”, 2002, and a work of M. Gladwell , 2000. ” The Tipping Point: How Little Things Can Make a Big Difference”, Littic, Brown, New York.
2
