A Universe of Sufficient Size

by Miriam Sved

  ‘Exactly,’ Josh says. ‘Eh. I’ve never understood why people go so nuts for dogs, especially dogs doing stupid human things.’

  ‘I like the cats,’ says Nagymama.

  ‘So anyway, there’s this video-sharing software called YouTube – it’s pretty amazing, it embeds Flash in, like, no time – and this guy I know, sort of an old school friend, put up a video of his husky who can say hello.’

  Nagymama’s eyebrows contract.

  ‘I mean, obviously he doesn’t actually say hello, the dog’s just barking, but he has this really weird bark that sounds exactly like someone saying hello.’ Lowering his chin and pursing his lips into gruff doggy acoustics. ‘Huwwow, huwwow … like that.’

  Nagymama shakes her head – whether in confusion or disbelief Josh isn’t sure.

  ‘It was pretty lame. This guy – his name’s Daniel – he and his dog just stand in front of the camera for a few seconds going huwwow to each other and shaking hands. And this video went nuts. Like, millions of views. It made the newspapers. Daniel was on telly, on one of those inane morning programs, with his huwwowing dog. He’s, like, a minor celebrity.’

  Nagymama is still frowning and looking at him through a semi-squint. It’s difficult to gauge her level of comprehension. He pushes on.

  ‘They call it going viral – when a video picks up so much momentum that it becomes unstoppable, self-perpetuating, like a stochastic system that creates its own noise. For, like, a fortnight you couldn’t leave the house without hearing about this dog that says hello. And I just started to wonder … how does this lame home movie go from my friend’s bedroom and end up all over the world? Because there must be a tipping point, a threshold between the pre- and post-doggy-hello worlds, and there must be a way to work out where that threshold is.’

  Josh is vaguely aware that he is sanitising this narrative somehow, making it sound like an abstract intellectual curiosity rather than what it was, or what it became, which was an obsession. The fact that Daniel-of-the-helloing-dog was in his physics class at school and was even nerdier than Josh himself – in the famous video the light from his glasses (so thick and heavy that when he bent over the desk in physics they sometimes slipped right off his face) bounced into the lens of his desktop camera, giving him a sightless robot look – acted on Josh as both a goad and a spur; an insult and a spec-reflected ray of hope. In early high school Daniel had weed himself multiple times, mostly in PE class. He had a pronounced lisp and his upper lip twitched when he talked. He was so profoundly unpopular throughout school that he’d sometimes drifted into the infernal realm set aside for nerds with whom other nerds didn’t want to be associated. And now he was on TV and had more than eight thousand Facebook friends. And all Josh wanted was an answer to the one question that the whole world – parents, teachers, exams – had always encouraged him to focus on: Josh wanted to know how? How had this happened to Daniel and, by extension, how could he make it happen to him?

  Nagymama says, ‘So you are working on … why people watch the dog which says hello?’

  ‘I’m working on … cascading networks. Small world theory. It’s about the ordering of information. You could think of it as any kind of movement through a network, like a viral epidemic. I wanted to find the contagion threshold, the point at which something spreads so effectively that it’s almost inevitable it will reach the whole population.’

  Nagymama is still looking at him with that head-tilted squint that could denote interest or scepticism. Or inebriation, or creeping senility. Josh has not told anyone else about the inner workings of his obsession, the mental corridors he has been stumbling down for months – almost a year – and the telling is itself proving to be a kind of cascading network, becoming more inevitable, more compulsive as he goes along. He finds he wants to get it all out there, now, quickly, before he thinks about it too much.

  He says, ‘I was seeing this girl, Bethany, and she was in a band.’

  Nagymama leans forward with a new kind of interest.

  ‘Don’t get too excited, Nagymama. We broke up.’

  She sits back again and makes a ‘tch’ sound.

  ‘She broke up with me. Sort of after I tried to be the manager for the band. Or the joint manager. And the keyboard player.’

  Nagymama looks confused. Josh pushes on with that same sense of desperate disclosure.

  ‘She plays the guitar. Bethany. She’s actually very, very good. She’s played since she was a kid – mostly classical, but she’s got this friend, Jasmine, who she’s known since she was a little kid, who’s a singer.’ (Calling Jasmine a singer is a bit like calling Muhammad Ali a famous poet.) ‘And one day we were all hanging out at Jasmine’s house and Jaz and Bethany started jamming, Bethany on the guitar and Jaz singing, and it was … amazing.’ It stank, musically speaking, although the song was an interestingly lopsided Jasmine original (part Bikini Kill, part Celine Dion). But Josh and Max, who were watching from the bong sofa, exchanged a look as Jasmine sashayed extravagantly around the room with the microphone, and another when she put the microphone back on its flimsy stand and draped her long hands around it and stared out at the ‘audience’ with those intensely special eyes. Of course the magical eye contact was meant for Max, with whom Jasmine was in a not-quite-relationship, both of them downplaying their mutual obsession by sleeping with lots of other people, both of them talking endlessly (Josh and Bethany had compared notes) about each other. But it didn’t matter: Josh felt it, those eyes conveying their magic right into his cerebral cortex, where the emotional centres of his brain quickly cascaded into the much more prolific strategic parts. He thought: I can do something with this.

  He says, ‘So I started playing the keyboard for them, and I convinced them that we should do a video.’ He also directed the video, Max holding the little camcorder, and oversaw its progress via a carefully selected network of YouTube nodes (kids he paid to share it on Facebook). Bethany could hardly accuse him of being an unsupportive boyfriend.

  Nagymama is staring at him blearily; he suspects he has lost her. He looks down at his hands and keeps going. ‘The video made a bit of headway for the band.’ Mostly driven by some fairly nasty public feedback about their musicality, Jasmine’s magic failing to translate to the screen with enough potency to overcome her voice. His next step was going to be a strategic series of small public appearances coupled with increasing online buzz. ‘But then Bethany broke up with me. I guess she thought there was something going on with me and Jasmine.’ She knew there was nothing going on with him and Jasmine. Josh knew she knew, and this was one of the most depressing parts of the whole scene with Bethany: Josh knew that Bethany knew that Jasmine wouldn’t touch him with the tip of one of her perfectly manicured nails. Bethany accused Josh of being obsessed with her oldest friend, knowing that Josh and Jasmine’s cool zones were so far apart that Jasmine saw him as some kind of anthropological oddity whose occasionally amusing presence she tolerated mostly for Bethany’s sake. Bethany, with her whip-crack verbal responses and unflappable affective facade, was a nerd who could cross over into that other world: Jasmine and Max’s world. Josh never could, not really.

  Nagymama says hesitantly, ‘So you do the mathematical theory about this band? To help the band with your girlfriend?’

  ‘Not about the band exactly. About the … buzz.’ Josh reaches into his pocket and holds on to the phone, feeling the chasm of time and age between him and his grandmother. ‘Think of it as a graph, edges and nodes, and the goal is to cover as much of the graph with these exponential nodes as you can. But you have a time limit, it’s like a wave, you have to cover the whole graph before the wave passes by. So obviously each node has a finite number of neighbours, and every one of them can either adopt a new behaviour – say, J – or stay the same. So you look at the cascading behaviour of this set, each node responding to its neighbours’ behaviour, and the question is: under what conditions can th
e set eventually convert all or most of the population to behaviour J? What’s the threshold for that to happen?’

  Nagymama is nodding slightly, one hand hovering in the air.

  ‘The interesting part is how large a threshold you can have and still see the new behaviour spreading to everyone, the maximum threshold for which there’s a finite contagious set.’ Josh takes a breath. ‘So think of the set as a two-way infinite path, with nodes –2, –1, 0, 1 and 2. And there’s this new behaviour, J. Say just a single node, 0, initially adopts J. But you have to remember the wave, and J is not constant, so 0 will convert 1 and –1 but will then switch back to non-J, and 1 and –1 will switch back as they convert 2 and –2, and it goes on with this blinking pattern so that the set can’t become contagious.’

  Nagymama says, ‘And J is this Jasmine?’

  ‘Yes, just for this example. But say you can start the infinite path with a bigger set, maybe 0 and the nodes around it, 1 and –1, all adopting J at the same time. Then in the first time-step those three nodes will stay with J, and 2 and –2 will switch to J. Under those conditions, where the first three nodes are following J when the next ones switch over, then you can convert the whole set, it becomes contagious. And when you generalise that, call it time-step B for band, the contagion threshold you come out with is B equals 1/2. It sounds simple, but the consistency of that result was actually incredibly interesting. Anything bigger than 1/2 and J will never get past the first trend transition.’

  In some peripheral corner of his brain Josh wonders what his mother is doing on the other side of the door, and when she might come crashing into the stairwell. He feels a mounting pressure to get through his explanation, as though something important is at stake.

  ‘I started trying it with non-progressive time processes as well, where the nodes don’t switch back from J after the first time-step. You’d think under those conditions the contagion threshold would be irrelevant, but it turns out to be the same: you always need this specific low contagion threshold to infect a population. Which, in the real world, means nothing can spread if it needs a majority of people around you to convince you to do it. It doesn’t happen. These results, they were pretty surprising when I started playing around with different formations; it was like watching some underlying human pattern play out on the graph. I could never get a consensus with a contagion threshold of greater than 1/2.

  ‘So that was the beginning of the theory. From there I started expanding the networks to take in more real-world parameters of cascading behaviour. Like, directed graphs where some nodes have more influence than others, and one-way influence – with two nodes beside each other the chances are one will have more influence on the other than vice versa.’

  And now he is so far down the rabbit hole that the old defensive heat rises in his chest – like trying to convince sluggish Sol Milos of his worth; trying to convince Jasmine to take him seriously, that he is a serious person.

  ‘The thing you have to understand,’ he says, ‘is that this stuff – cascading networks, contagion theory – is totally the new order of the world. Things used to be ordered from above, basically pre-designed by the people in charge, but now with the internet that model is on the way out, everyone will have a voice and the ordering of all those voices will happen down at ground level. Everyone’s going to want to be in on it and it’ll look like chaos, and the thing you have to figure out is how to get your voice out there, how to find a way through the chaos, a way to control it and harness it. And, well, I’ve done it. I’ve got this map for getting through the chaos quicker than anyone else. Or at least a rough guide. But what no-one seems to understand is how big this could actually be.’ Aware now that he is sounding petulant, immature; it’s unstoppable, like giving in to gravity, rolling down the verdant slope of whininess. ‘I’ve done all the hard work, and now I just need a bit of support to help me get it out there in the world, but all anyone seems interested in doing is putting obstacles in my path and making me prove myself by jumping over them.’ His mum: prove to me that you’re a responsible adult ; Sol Milos: prove that you’re collegiate and disciplined like a real mathematician; Bethany: prove that you can look past yourself, past this obsession with being a big shot.

  ‘And the thing is’ – still whining to the stairwell air above Nagymama’s head – ‘there’s actually a bit of a pressing time issue with this stuff. It’s not something many people are applying the way I’ve applied it, but there’s a whole world of people, really big players in Europe, the States, Silicon Valley, who could stumble onto the same territory any time. And I’m this kid in Australia who’s got there first, but do you think anyone will give a shit or even believe me if I don’t get it out there – and I mean get it out there big time? I need to get the attention of people who matter for this to have any hope of building into something for me, otherwise all the work will have been for nothing and I’ll have to start again on something else.’

  ‘What does the graph look like?’

  Nagymama has spoken abruptly but softly, disrupting Josh’s momentum.

  ‘Sorry, Nagymama, what?’

  ‘This graph of the contagion of Jasmine, you have drawn it, yes? The young people still do this with new theorems?’

  Josh feels his hand twitch for a pencil. ‘I drew it. I drew lots of them. I have some graphics software that was good for the early blinking graphs, and after that I did lots of them by hand, although I don’t really know why. They were sort of amazing, watching these patterns form, I wish I had some of them to show you. It was like … this underlying principle just waiting for the right coordinates to show itself. They were sort of beautiful.’ No need to go into how much time Josh spent colouring the graphs – like primary school pictures, different colours for the landmasses of J, the ordering contagion; swathes of darkness for the unreachable territory in the unsuccessful sets: here be dragons.

  ‘Beautiful,’ Nagymama says.

  They are quiet for a minute, and then, not quite looking at him, she says, ‘You have heard of this mathematician Pali Kalmar?’

  It is a question, although posed in the Hungarian grammatical contortions of a statement. A silly question.

  ‘Of course,’ Josh says. ‘Who hasn’t?’

  Nagymama says nothing but her lips are moving slightly in a way Josh finds a bit unnerving. Is she about to have a stroke or something?

  Josh fills the space with words. ‘He was the founder of Ramsey theory and went on to prove the prime number theorem – I guess that’s what he’s most famous for, although there’s almost nothing in twentieth-century maths that Kalmar didn’t have a hand in. Some of his stuff is quite fundamental to computer science.’ He gives way to the impulse he has been fighting all day and pulls the shiny phone out of his pocket. ‘They might not have invented this if it wasn’t for him.’ He waves it in Nagymama’s direction but, disappointingly, she doesn’t ask to see it. ‘Of course, he didn’t really know what he was working on at the time, what the implications were.’

  He has said this flippantly – it is slightly thrilling, to cut down giants with a flick of the wrist – but Nagymama seems to come suddenly back into focus; she looks at him sharply and says, ‘None of us. We did not know what we are doing.’

  There is a pause.

  ‘What do you mean, Nagymama?’

  Nothing. She is staring in front of her, eyes glassy again.

  ‘What do you mean none of you? Do you mean … did you actually work with Kalmar?’ The idea is exciting and absurd. His small, domestic, slightly nutty grandmother. Josh does a quick mental set of twentieth-century calculations. It is possible, chronologically, geographically.

  Nagymama looks up and suddenly she is flushed, glowing; she looks strangely fierce, and she reaches out a clawed hand to Josh. ‘We know nothing about what we are doing. We think everything is so clever, solving all the big problems of the universe, and all the time the s
ky is falling around us and we are blind to it.’

  She sounds almost angry, and the present tense is unnerving, although Josh is pretty sure she’s actually deep in the past, in some past he has never thought to find out about. He goes to speak but stops himself, waits.

  ‘Look where it brings me,’ Nagymama says eventually, gesturing outwards at … what? The stairwell, the funeral, the grandson? ‘Always I think somehow that there is a way to go back to this time and do this things again to be forgiven. To do this – how do you say, to get the forgiveness?’

  Josh shakes his head, lost in her drunk grammatical brambles.

  ‘Atone,’ she says. ‘This is it. Always I think there is time for atoning.’ She slumps down as if some febrile energy has run its course and drained away. ‘And I was wrong. Always I am wrong.’

  Josh begins to say something – he doesn’t know what it will be – but Nagymama waves him silent.

  ‘I will help you, darling,’ she says. ‘You tell me what you need for this theory, the cascading of Jasmine, and I will help with the money.’ She looks at him directly, a shrewd glint. ‘I assume it is about the money, what you need for this.’

  Josh clutches his phone and hopes he is not blushing – with excitement and embarrassment and avarice, and a little bit of guilt.

  ‘But I want for you to help me also with something. There is something I need to do. I have a … how do you say?’ Her face is suspended mid-sentence for a moment, then she conjures the word out of air with her hands. ‘A proposition.’

  ∴

  Budapest, 1938

  Ildiko came up with the upper limit problem herself, although she might tell you that we developed it together. I suppose we did develop it together, but it was her inspiration, her inception. I didn’t realise at the time what she intended it for, or I like to think I would have acted differently.

  It was on a day in April, after a meeting at the statue that was dominated, as usual, by talk of the trip to Vienna. Pali had been brought into the plan by then and even seemed quite enthusiastic about it, to our general surprise. There had been much discussion about how to present it to him. Ildiko was all for showing him the correspondence between Tibor and the professor and leaving it up to Pali to decide whether and under what conditions he would meet with the great man. But of course Pali couldn’t be depended upon to act rationally or in his own best interests, and we expected him to be oblivious to the professional opportunities the professor might confer. And there was such a series of logistical steps in order to get him, on the right day and in some respectable state, to the professor’s door. Tibor thought that the best way forward would be to try to interest Pali in the professor’s research, which led to long discussions about how to get our hands on it. Before last year we could have just strolled into the university library, knowing that if he was anyone at all he would have published in the Annals of Mathematics. As it was, we were limited to those publications we already owned, or what we could get hold of through friends, most of whom were in similar circumstances to us. Levi said something about seeing the name Voigt in one of the school mathematics journals, which got us excited – Tibor had kept every issue, so if the professor was one of the problem setters we would at least have a lead. It turned out to be a red herring. We went through dozens of issues before Levi decided that perhaps it was a Professor Vogel he had been thinking of and we gave up.