The Montagues and the Capulets have never been friends and Juliet is quite aware of this. Even at her young age, she knows that her love for Romeo is an impossible dream her father will never accept. So, she designs a strategic plan. She will take a couple of sleeping pills, just enough to make her look like she is dead to trick everyone into thinking that she has passed away. Brilliant! If everything goes right, she will always be happy with Romeo… but if things go wrong… well, you never know.
She carefully designs each step of the plan. She gets the pills, she picks the right place to sleep her long nap and wait for her beloved one. There was just one thing that she was worried about: what if Romeo didn’t notice that she was sleeping and not dead? What is he is as silly as the rest of the characters in this made-up story and also assumes that she is dead? Would he run off with that pesky Rosaline or would he do something stupid, like, I don’t know… take his own life? Will this become the world’s greatest love tragedy or simply an advert for some really powerful sleeping pills?
Suddenly, a light bulb flashes in Juliet’s mind (ok, there were no light bulbs in the 15th century, but there were no sleeping pills either and you were fine with that). She would write a song with a hidden message and she would ask people to share it.
Juliet’s abilities as a songwriter are quite limited, as the reader will have noticed by now, but her abilities as a mathematician are not. She quickly starts modelling her situation and understands that after taking the pills, she only has a few hours to get Romeo listening to the song, otherwise, it’ll be too late for the loving couple. What if I sing this wonderful song to a person, say Mercutio, she thinks. Then, on the next time step unit (because yes, in the era before America was discovered, time was usually measured in discrete units and Juliet didn’t want to upset anyone with her maths), he might sing it to the next person, say that pesky Rosaline, with probability $p$, or he might wait for one step with probability $1-p$. Then, on the next time step, Mercutio might, again, sing the song to Rosaline or might decide to keep talking about the weather and Donald Trump. If each time step they repeat the same exercise, then eventually Mercutio will sing the song to Rosaline and then forget about it (the song is not that good). Then, Juliet assumes that Rosaline, even though she is kind of a witch, will do the same as Mercutio and eventually sing the song to the next person in line, Tybalt, and so on.
There are 36 characters in Shakespeare’s play, including Romeo and Juliet, so she imagines all the characters in a line, with her on the one extreme and her Romeo on the other, and she needs her song to be played 35 times to be 100% sure that Romeo gets the message. She cannot risk any ninety-something percent! And, since every character is assumed to behave the same, with the same probabilities, then Juliet just needs to wait for 35 successes (someone singing your own song is indeed a success) to make sure that Romeo listens to the song. She immediately notices that this is a Negative Binomial, with 35 successes and a certain probability $p$.
Bollocks! Her plan has a flaw! The expected number of time-steps that it will take for everyone to have heard her wonderful song is extremely large! Nothing positive may come from a negative binomial! If the probability of a person sharing her song, $p$, is $0.5$, then she expects 70 steps, but if it drops to 0.2, she would need to wait for 175 steps, or 350 steps if $p = 0.1$. It is far too much time! Also, she is aware that her song, pretty much like a Miley Cyrus song, sucks, so $p$ should be quite small.
If $p = 0.2$ then Juliet expects 175 steps for Romeo to find out.
If only there was another way to make my song goes viral, she ponders. Bingo! She says (even though bingo was invented a few years later). I shall share my song with more than one person! And ask them to do the same! Otherwise, my Romeo will end up with that fat Rosaline!
She imagines then each person is located in a different cell of a 6 by 6 grid, with herself on one of the corners and her Romeo on the opposite corner. Then, each time step, a person who already knows the song will share the song with probability $p$ with each of their neighbouring peers.
Juliet notices that this no longer is a Negative Binomial distribution. This is, instead, a percolation problem for which not many analytical solutions exist, so she needs to simulate thousands of times what will happen when she starts with her first “She is not dead, she is just having a nap…”.
Results are far more promising for the young couple with this percolating system! Firstly, she is no longer 35 songs away from her lover, but only 10 songs away (in the best case scenario). Also, her simulations show that with $p = 0.5$ she needs to wait 12.4 steps, with $p = 0.2$ only 24.8 steps, and even if her song is not that successful and $p=0.1$ she only has to wait 46.7 steps. It’s not too bad! Asking the ones who know the song to share it with more than one person improves the results considerably.
When Juliet asks Mercutio and the rest to share her song with their neighbours, her song goes viral! Even with $p= 0.2$, in fewer than 30 steps, Romeo will hear the song.
Her song goes viral! If she makes sure that her song is shared with more than one person then they will be safe! She sings
She is not dead, she is just having a nap,
Thy shall wait if you want her
dancing in your lapin your arms.
Attributions:
[Pictures: 1 – adapted from Flickr.com – shakespeare doll by Jimmie, CC-BY 2.0; 2 – adapted from clker.com – mapa-do-tesouro-md by Alexandre Moita (public domain); other pictures by Chalkdust]
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