Metcalfe's law is clearly wrong after a network get's overly large. I could in theory call a random person in China but if I have no way of communicating with them it's practically worthless to do so.
But seriously: I've had people on conference calls when they couldn't understand the language, but someone wanted them to see slides. The fact they could join the call (in that case on Skype) definitely made the network more valuable.
You might argue that the fact you can't speak to some of them drops the value per user, and that is true to some extent. But the fact they are on there and attract other people (some of whom you can speak to) adds value for you, too.
Orders of magnitude. Sure, adding someone else in china is in theory worth something. However adding one more friend that I regularly communicate with on a network is worth more to me than every person in China.
PS: For large networks X Log X is probably much closer to reality than X ^2. Just compare the amount of internet bandwidth between NY to California vs the bandwidth between the US and China.
Metcalf's law is not suggesting "The value of the network to any random node within the network." Because that measure is very subjective in the ways you say.
Instead, it's "The total value of the network to all nodes". Because the more people in the network, the higher the probability that the nodes you DO care about are also connected.
I'd extend the rule further and suggest that the value of a network scales as some Odlyzko and Tilly suggest, but with an additional negative function subtracting value (v) from the network:
v = n(log(n) - f(n)
That is: as the network grows, the added value of each additional member is reduced (log(n)). Further, each additional member of the network exacts a cost to the network as a whole as well. Doing some simple modeling, I suspect that this isn't a strictly linear factor, but itself grows with n, quite possibly as the original Metcalfe's law suggestion. That is: any given member is increasingly less likely to be a positive contribution to the network, but might well present an equal opportunity to be a net negative to the group as a whole:
v = n(log(n)) - kn^2
Where 0 < k < 1 and n > 1.
Moreover, let's look at some group sizes which might allow us to estimate for k in various contexts.
For software team size, it's very typical that a core engineering group has a size of 5-10 members, more or less. This suggests that k is about 0.2 for software development: every added team member exacts a cost, within a single group, of about 20%. This sees value grow for 1 < n < 8, then fall with larger n, hitting negative values at about n=14.
For an elementary school classroom where the ideal class size seems to be around 22-25 students, k would be around 0.08.
For Dunbar's Number, the number of relationships people can manage (100 - 300, typically set at 150), k is between 0.028 (100), 0.2 (150), and 0.113 (300).
For city sizes, it's likely that different cities offer different matches of positive and negative factors. k of 0.0005 gives value max at n ~ 10,000, k of 0.0006 is ~ 100,000, and 0.000007 is around 1 million.
To scale to 1 billion users with net positive value means you have to keep k to less than 0.00000001. That is: any one member can have only a 1 in 10 million chance of being annoying to other members.
> To scale to 1 billion users with net positive value means you have to keep k to less than 0.00000001. That is: any one member can have only a 1 in 10 million chance of being annoying to other members.
Or you could just design a network where new users don't annoy existing users, and reduce it down all the way to zero.
In the case of cities, physical distribution means that even within a given city, the interactions of its citizens, while potentially very high, are generally reasonably low. It's less my direct contacts (likely within a fair approximation of Dunbar's Number) that are high, than my 2nd and 3rd order possible contacts which are high.
In a small town, those 2nd order connections are inherently constrained to the size of the population: my 300 direct contacts may expand to the 3,000 or 30,000 of a small town, but not the million or more of a large metropolis.
Similarly, for more complex organisms, you also have more complex immune systems. An interesting (and staggering) recent fact I ran across is that the individual cell mortality rate among ocean lifeforms is about 20%. Per day. If you're a cell in the ocean, you've got 1:5 odds of not being here tomorrow, because of the viral load:
> The value to huge communications networks isn't in making connections but in avoiding making them at all.
The social networking startup I'm at is focused entirely on that problem. It's taken a lot of work to have the scope of possible interactions be the entire network, while at the same time, limiting the interactions to any given user to just those with a positive value.
Since the algorithms are not omniscient, we also have a ruthlessly efficient feedback mechanism whereby users can indicate when they have received a negative value communication with just a single swipe.
There's tons of interesting problems in this space. :)
we also have a ruthlessly efficient feedback mechanism whereby users can indicate when they have received a negative value communication with just a single swipe.
That's helpful, though you've really got to recognize that a huge part of the problem is assessing indirect feedback.
If you've seen Derek "Veritasium" Muller's "The Problem with Facebook" videos, one of the challenges is that interactions with FB content are hard to gauge. Dating sites have a similar challenge, in that feedback on interactions ("how did the date go") are rarely collected. As opposed to YouTube where a huge signal is "did the user stay on the page for the duration of the entire video". If I watch 5 seconds of a 3 minute video (or 30 seconds of a 60 minute vid), odds are I wasn't very impressed.
There's also the challenge of sorting out abuse of moderation systems, particularly those trying to get legitimate (but unpopular) voices banned or restricted.
The good news is that there are some people whose interactions are so widely negative (spammers and trolls) that you can attack them head on and reduce the cost coefficient significantly (recognizing that the cost constant is constructed of both a specific value and the number of connections). Spam is as annoying as it is because a single spammer affects so many other users.
The flipside is that you can increase the network value by finding people others really want to connect with. Here I see G+ as being horribly naive (also YouTube) in repeatedly making recommendations that I'm absolutely not interested in, without offering me an opportunity to say "don't show me this person" or "don't show me this product / video / category" ever, ever again. One of my long-standing challenges to the "deep data" (snooping) perspective is: rather than compile a massive dossier on my and attempt to bother me by way of it, when you do find me in an intentional mood to find something, get really good at figuring out whether you're offering me what I want or not. Why Google should know my location to within 2 feet every 60 seconds of the past five years ... but not be able to tell me the dot pitch of the monitor I'm shopping for, strikes me as a stunningly obtuse mismatch of data focus.
That said: yes, the ability to dismiss stuff I don't want to see and be bothered with is absolutely useful. As I'd repeatedly said at G+: let me say "not now", "not this hour", "not today / this week / month" (essentially: timeouts). And of course "not ever". G+'s blocking feature is also grossly inadequate. Some people, yes, I simply don't want to deal with. For others, I just don't want to see their insipid posts, or deal with their insipid comments on my own posts.
