PopCo

  ‘Cheers,’ he says, walking away. I watch him go, half expecting him to fade into invisibility, cackle loudly, or spontaneously combust. He does none of these things, of course; he simply walks away like he’s on his way to the most normal place ever. I stop watching him and keep walking in the direction I was going.

  So Hiro is involved in whatever this is? Where was he going when I bumped into him? To find me? To break into my room to see if I had left a response there? Again, I find myself hoping that this is not just some silly game, endorsed by the PopCo Board. Or perhaps I do hope it’s a silly game. The idea of it being something else isn’t actually that appealing, now I think about it. As I walk over to the seminar room I realise that I am tired, and wish that I’d gone back to bed again this morning after all. Not one particle in my body wants to go to this seminar but there will be big trouble if I don’t. Sighing heavily, I approach the room and quietly open the door, pretending to be invisible as I slip into the seat Esther has saved for me.

  ‘Glad you could join us, Alice,’ says Mac, from the front of the room. This really is just like being at school.

  There is a man at the front of the room with Mac. I am assuming his name is Mark Blackman, as this is what is written on the board behind him. He is older than the other speakers have been, with slicked-back grey hair and black-framed spectacles. He is dressed rather eccentrically in a tweed jacket, yellow cravat and jeans.

  ‘Hello,’ he says, now, getting up to speak. ‘As Mr MacDonald has already done such a good job of introducing me I shall not waste time re-introducing myself. The only extra thing I can tell you is that I have an Erdös number of 3. Does anyone know who Paul Erdös was?’ He pronounces the name correctly: air-dish.

  I put my hand up. ‘A Hungarian mathematician,’ I say when he nods at me.

  ‘Thank you. And my Erdös number means?’

  ‘That you have written a paper with someone who has written a paper with someone who wrote a paper with Paul Erdös,’ I say.

  ‘Very good. Are you mathematically inclined, yourself?’

  I can hear Dan groan in the seat behind me.

  ‘My grandmother was,’ I say. ‘She had an Erdös number of 2.’

  ‘Wow!’ He looks impressed. ‘Does anyone else know what we’re talking about?’

  I look around the room. Kieran has his hand up, as do Grace, Richard and the blonde girl I’ve seen hanging around with Kieran who either works in videogames or in Kieran’s strange team.

  ‘OK, thanks,’ Blackman says. ‘You can put your hands down now. All right. What we are talking about here is networks. If you want to take over the world with some undoubtedly pointless plastic moulded product, as Mr MacDonald assures me you do, then you need to understand network theory. You need to understand why your toy, or even your disease, or your idea – it’s the same principle – can be relatively unknown one day and then, overnight, be the thing everyone has. Or not. OK. Forget Erdös for a moment. Who has heard of Kevin Bacon, the movie actor?’ Most people put up their hands. ‘Good. You, boy, with the long, odd-coloured hair.’

  Kieran looks up. ‘Me?’ he says, grinning.

  ‘Tell us about the game Six Degrees of Kevin Bacon.’

  He starts talking in a lazy voice. ‘Six Degrees of Kevin Bacon, or “The Kevin Bacon Game” appeared in 1997, I think. Originally invented by a group of fraternity brothers, it posited the theory that Kevin Bacon was the centre of the movie universe, as it was possible to connect any other actor in the history of films to him in, on average, less than four steps. The way it works is this: if you have acted in a film with Kevin Bacon, you have a Bacon number of one. If you have acted in a film with someone who acted in a film with Bacon, your Bacon number is two and so on. The idea was supposed to prove Stanley Milgram’s thesis about “Six Degrees of Separation” – that everyone in the world can connect to anyone else in six steps or less. However, in a closed network like movie actors, the number was found to be much less than that …’

  Blackman is almost laughing. He looks at Mac. ‘Looks like you could have saved my fee and paid this young man instead.’

  Mac smiles. ‘Kieran has an alarming mind,’ he says. ‘Anyway, please go on. This is fascinating.’

  ‘As I’m sure you could all guess, an Erdös number is the same as a Kevin Bacon number, but it relates to degrees of separation in a network of mathematicians rather than movie stars. Scientists have found that these networks, whatever – or whoever – they are comprised of, will display the same structure and properties. We call this the “Small World Phenomenon”. This is interesting when we come to consider how disease, or products, or ideas, can infect these kinds of networks.

  ‘A small-world network is itself defined as a group which displays a level of interconnectedness whereby each “node” on the network can be reached from any other node in six steps or less. Your company, PopCo, will, no doubt, display properties of a small-world network. Perhaps the whole of the toy industry would. Other small-world networks have been found to exist in the US power-grid system and the neurological system of the microscopic worm C. elegans. They have also been found to exist in the World Wide Web, the metabolic network of E. coli and in the network of boards of directors in US Fortune 1000 companies. Kevin Bacon and, particularly, Erdös numbers occur when the group is aware of its own interconnectedness. Erdös, incidentally, for those of you who don’t know, was a very famous mathematician who created a model for random graphs – which, coincidentally, happen to form the basis for much of the initial mathematical work in this area of study.

  ‘Small-world networks have to display a particular combination of “clusters” – say, groups of friends in the same town, or the people in your actual office – and “random links” – you and your most remote friend, or you and the person from accounts with whom you smoke, probably in the rain, outside the office building. These random links provide “short cuts” through the network. Mr MacDonald is an important node in the PopCo network, as so many people connect to him. He is like a big airport, in that sense, linking many smaller places. So, out in the real world, you might not know anyone who lives in Australia, but I do. If you know me, then you are connected to my Australian friends by only two degrees. What is surprising about scientists’ findings is that only a few random links seem to have the effect of connecting the network in such a way that it will suddenly display small-world properties. Not only does this happen, seemingly ‘naturally’, in many different types of network, the ‘critical point’ where this type of connectedness occurs is also familiar: it is the same, mathematically, as the moment a polymer solidifies or water freezes. It looks like a phase transition. It is a natural value.’

  Mark Blackman pauses dramatically and then turns to the whiteboard. For the next ten minutes or so, he attempts to draw examples of what he means. I am thinking of cobwebs while he does this, I don’t know why. Just as I am at the point of drifting away, I click myself back in and look at the board. This is an exciting idea, I realise, that some property that you find in nature – the point where one thing becomes another, where water becomes ice or steam – is replicated in this odd network theory. I don’t understand the physics of it but I know that there is a mathematical reason for these things. How can that apply to people too?

  Other people are completely lost.

  ‘What is a phase transition again?’ asks Esther, rocking back in her chair.

  ‘It is the transformation of a thermodynamic system from one phase to another.’ Blackman scratches his head. ‘You don’t really need to understand it to understand network theory; it’s just very interesting to note that a network changes from being unconnected to being connected in the same mathematical way that, say, water turns into ice.’

  Esther frowns. ‘I still don’t get it.’

  Kieran speaks up. ‘You’ve heard of “critical mass”, surely? Or a “critical point”?’

  ‘Yeah,’ she says.

  ‘When something reaches i
ts critical point, it changes from one thing into another, or, you know, explodes. What Mark is saying is that it’s the same with how we all connect. Say you move to a new town, yeah, and you don’t know anyone? For a few weeks you don’t know anyone, then maybe you start a college course. On the first day, you talk to one person but this person isn’t very well connected. You now know one person, and you aren’t connected to many more. Slowly, you make some more friends until, say, you come across some really popular person who seems to know everyone. Because this person knows everyone, she invites you to a party. You meet more people. You start going out with some guy you met at this party. His dad, say, is the mayor of the town. Suddenly you have connected with two important hubs in the network and you find you are only a couple of steps away from anyone in the town. The moment you tip from being a no one into being really connected – that’s the phase transition.’

  ‘OK,’ says Esther. ‘I get it.’

  Kieran slumps back in his chair, a little as if he was a coin-operated information machine that has just been deactivated and is waiting for someone to insert more coins. Blackman puts down his chalk and addresses us all again.

  ‘What this young man has just explained,’ he says, pointing at Kieran, ‘is how “short cuts” appear in the network. These short cuts are what turn ordinary networks into small-world phenomena. However, although we all have access to these “short cuts”, it seems that the problem is how we find them. However well-connected we are in theory, it is very hard for us to evaluate our links beyond the local level. I may know that I know you, but I do not know everyone you know. I certainly don’t know anything about who your friend Simon (whom I have never met) knows. Yet I am connected to his friends by only three steps. The problem isn’t the connectivity but our ability to navigate it. Stanley Milgram found this with his small-world study, where he asked randomly selected people in the US Midwest to try to get letters to a stockbroker in Boston, on the east coast. The letters could not be posted. Instead, those taking part were asked to hand them to people they knew who might have social ties that might take the message closer to the target. Although Milgram did prove that the letter would get to the target eventually, he didn’t show people using their contacts very well.

  ‘So, if we apply this to your desire to spread a plastic product among the world’s children – a commendable aim, I am sure – then you may have to navigate the network on behalf of the children. Build something into your product that means it spreads itself. Disease works like this. You may never come into contact with the person on the fourteenth floor of your office block, but you may spread your germs to them none the less. This is called “unconscious” or “automatic” transmission, and I believe this is what marketing departments want built into products they are responsible for selling. Of course they do! Automatic transmission does their job for them. Lazy bastards!

  ‘Consider Hotmail. Every time you send a message using Hotmail, it comes with an advertisement for Hotmail itself. The message is the ad. The product spreads itself. Or, perhaps more pertinently, consider MSN Messenger, used, I believe, by teenagers everywhere – certainly by my son, who doesn’t allow me access to my own computer in the evenings. In order to talk to other people on MSN, you have to have the MSN technology yourself. This piece of software promotes itself!

  ‘All this mathematical network theory relates also to disease epidemics, the “madness of crowds” and the breakdown and recovery of network systems. Why did so many people go crazy buying tulips in Holland in the seventeenth century, even selling their houses to do so? Why did seemingly rational people get carried along by the 1990s dot-com hysteria? It seems that we, in our networks, are wired up in such a way that we are both resistant and vulnerable to ‘infection’ from diseases and ideas; system failure and so on. However, one very important rule holds true: you are more likely to get a disease or buy a particular book if everyone else has it too. This is called a power law, or is sometimes referred to as the Matthew Principle. What is the Matthew Principle? It is, of course: “for whomsoever hath, to him shall be given”, otherwise known as “the rich get richer while the poor get poorer”. The more people that have a product, the more people will buy it. Now. Kieran, I think it was?’

  Kieran seems to wake with a jolt. ‘Huh?’

  ‘Who was Stanley Milgram’s mentor?’

  ‘Um …’

  ‘Ha! Thank God you don’t know everything. Milgram’s mentor was Solomon Asche, who demonstrated that people would give the wrong answer to a simple question if enough other people in the room had given that answer first. He conducted a very fascinating series of experiments, whereby, for example, the subject would enter a room in which he or she believed the other people also to be subjects in an experiment to do with reasoning or some such. The other people in the room were, however, actors. The people in the room would be shown a series of shapes, like this …’ He approaches the whiteboard, rubs out his diagrams from before and draws a circle, a square and a triangle, in a horizontal line, starting from the far left-hand side of the board, as if they were letters in a word. Circle, square, triangle, from left to right. Then, on the far right-hand side of the board, he draws a vertical line, like an ‘I’ or a ‘l’.

  ‘Now,’ he says. ‘Asche asked the people in the room to say whether they thought the square was closer to the triangle, or closer to the line. Obviously, it is closer to the triangle. However, he found that when he asked the actors first – who were primed to give the wrong answer, and say instead that it was closer to the line – the subject would often give the wrong answer too. Such is the madness of crowds. We may not like to think we follow the crowd but, to differing extents, we all do. We do odd things – eating meat, praying to an invisible, non-existent bearded man – because everyone else does. Crazes in your own industry often involve children wanting the toy that everyone else has got, particularly at Christmas, as I should well know having bought my son Transformers, Mutant Turtles, Power Rangers and goodness knows what else before he grew out of them.

  ‘Once you have got your odious plastic object to the stage that everyone wants it, the craze is set. But to get it there in the first place, you have to build in a way for the product to navigate networks and proliferate itself as widely as possible. How is your product going to be spread not just between friendship groups but beyond them? It only took one infected rat on a ship to spread the bubonic plague to a whole new country, remember, because once the plague was there, among a network of local rats, it could look after itself. Perhaps television is the way that an idea for a toy can spread to “remote” nodes on the network. Or perhaps you will do as other toy companies have done – or so I read in my weekend newspaper – and give the toy away to the most popular child in every school. Once you have infected the most connected hubs, the other nodes will soon follow! In this way, you would be the connecting point between various networks. But remember, of course, that any toy or product which involves communicating, swapping, giving, comparing and so on, has built-in automatic transmission. Think of your vile plastic objects as dandelion clocks strewn by the wind. The dandelion seeds wouldn’t fly in the wind had they not been designed that way. Any questions?’

  Kieran puts his hand up immediately. ‘Who designed the dandelions?’ he asks. ‘You can’t mean God, because you already said …’

  ‘The creators,’ Blackman says dreamily. ‘The original mathematicians.’

  ‘Cool! So …’

  Mac stands up. ‘I think we have taken enough of Mr Blackman’s time now. What a fascinating discussion. Thank you.’ He looks at his watch. ‘OK, now I think you are expected for your sailing classes soon. Can you please refer to the times on the noticeboard for each group please. Mark, I’ll see you out.’

  * * *

  The daytime is all right, and if I have to be alone in the light, I can cope. At night, however, I have to be with people. To find myself alone at night is just too hideous a proposition. What would I do if it actua
lly happened, and I did find myself alone in the dark? I think I would probably scream until I passed out, or until someone came. I understand that, in many ways, this is an irrational fear. By definition I cannot be hurt if I am alone, because being alone means that no one is there. Yet in my most panic-stricken moments I would even wish the enemy there in the dark with me. The enemy. The enemy is a two-headed man, or sometimes two heads on the body of an insect or a spider. I could not ever describe this to anyone. I could not describe the way that, since the incident at the bus stop, those men have become super-real in my mind. Only I understand the way that one whisker has become four, blue eyes have become red and a sneer has become a grimace with fangs; fangs dripping with blood and gristly bits of small schoolchildren. But I’d still rather they came than that I was left alone in the dark. Perhaps it’s because I would rather just get on with dying than wait for the inevitable, alone and unseeing. It is because of this that I have been sleeping in my grandparents’ room.

  Interesting things have happened as a result of my fear (which my grandparents say is actually a ‘phobia’: like fear, but more important). Because the nights are so potentially horrific, so mind-curdlingly, heart-stoppingly terrifying, everything outside of this set of ‘things of which I am afraid’ (we have been doing sets in maths) seems just brilliant. For example, the daytime is brilliant. If, during the day, I ever feel sad about my father being lost, or I feel bored at school, or upset because of a fight at playtime or anything like that, I simply have to remind myself that it is not night and it is not dark and I am not alone. Other people seem moderately happy, or just the same kind of happy/sad, all the time. If the way they felt was a graph, it would be a single flat line, with maybe a blip for Christmas and another for their birthday, perhaps. But the graph of my moods has mountains and valleys galore. Dawn, for example, can make me soar on its own. A whole nine, or maybe ten, hours until it starts to get dark again! Hooray! In the summer it will be brilliant, because there will be a good deal less darkness than there is now. In some ways, fear is a good thing. It makes you appreciate life more: the bits it hasn’t spat all over seem all the more delicious for it.