## At a rate of what?

The latest quarterly Pentagon report on Iraq was released today. (NYTimes story) It probably won’t surprise anyone to hear that things are bad; Iraqi casualties went up 50% relative to last quarter. One number that particularly struck me is that total Iraqi casualties have reached 120 per day.

Think about this for a moment. If a terrorist action, or set of terrorist actions, were to kill 120 people in the United States, consider what the news would be like, what the inquests would be like, how long it would be remembered for. This has now reached the level of daily occurrence.

Technical aside: When trying to interpret the impact of this, we really need to scale things to the size of the population. The real number that affects the public as a whole in a mass casualty event is the average number of degrees of separation between a random person and a person affected. Simply scaling the number of people affected linearly — the US has ten times the population of Iraq, it’s as if 1200 people were killed here — is incorrect, since as groups get smaller you’re more likely to know someone else in it. Does someone know a good result on mean distance in very large social networks?

Published in: on September 1, 2006 at 16:03  Comments (6)
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1. Isn’t it like 6-7 or so? (The whole 6 degrees to kevin bacon thing?)

2. Isn’t it like 6-7 or so? (The whole 6 degrees to kevin bacon thing?)

3. Mean distance is usually pretty much on a logarithmic scale.
Basically, a simplified formula is:
N (Network breadth) = (average number of people each individual knows)
Mean distance in a population = log (base N) of total population
Now, it turns out that in social networks N actually operates under a power law, so an arithmetic mean gives weird results. Some people are just way, way more well-connected than others.
There’s also the fact that people tend to cluster into richly-interconnected clusterss, and that those clusters are usually linked together by folks with very high N values.
But, anyway, it turns out that whatever the value is, the distance is usually quite low, as Kevin Bacon can tell you.
(One of my favorite books talks a lot about social networks – “Linked” by Barabasi. I’m happy to lend if you’re willing to forgive me for redacting my last name out of your very fine poem.)

4. Mean distance is usually pretty much on a logarithmic scale.
Basically, a simplified formula is:
N (Network breadth) = (average number of people each individual knows)
Mean distance in a population = log (base N) of total population
Now, it turns out that in social networks N actually operates under a power law, so an arithmetic mean gives weird results. Some people are just way, way more well-connected than others.
There’s also the fact that people tend to cluster into richly-interconnected clusterss, and that those clusters are usually linked together by folks with very high N values.
But, anyway, it turns out that whatever the value is, the distance is usually quite low, as Kevin Bacon can tell you.
(One of my favorite books talks a lot about social networks – “Linked” by Barabasi. I’m happy to lend if you’re willing to forgive me for redacting my last name out of your very fine poem.)

5. Good point. I just realized that my earlier post was too hasty: to get the “emotional scaling” right, one needs more than just how mean distance scales as a function of population size. Let’s say you have a population of N people, and you choose a subset K of k people who are directly affected by the event, and a random person x in N. The distance of the person from this event is the min (over y in K) of the distance from x to y. The average emotional distance of the population from the event is therefore the average of this quantity over x —
D(K, N) ≡ avgx∈N miny∈K |x, y|
We don’t really know about how the k people were chosen to die in these events; for a simple model, we could just assume that they were randomly chosen from the population. Then we can average over all ways to choose k people, and get a notion of the mean separation of people from sorrow in a population:
D(k, N) = avgK⊆N, |K|=k D(K, N)
D(1, N) is simply the mean “Kevin Bacon” distance between two people in a population of size N. D(k, N) is the mean distance between an arbitrary person and an arbitrary group of k people. I think that this is a good measure of the radius of sorrow in any catastrophic event: if you live in a population of N people that are bound in the sort of social network we’re talking about, and k are in some way affected by this, then you are so many degrees away from someone who was hit.
I don’t know if anyone has calculated a function like this for large populations, where the network is basically a country. I suppose it would be straightforward enough to do simulations to compute it.
It seems like a useful thing, though: it would be a kind of language for expressing the impact of a distant catastrophe. Instead of simply being told, “2,500 people died in this earthquake” you could learn something like “the average person was within three degrees of separation of someone who was killed.”
(It occurs to me that the question of which average to use is important: means are probably too sensitive to outliers, and in power-law networks that’s a major issue. A median would be useful, or of course a full histogram)

6. Good point. I just realized that my earlier post was too hasty: to get the “emotional scaling” right, one needs more than just how mean distance scales as a function of population size. Let’s say you have a population of N people, and you choose a subset K of k people who are directly affected by the event, and a random person x in N. The distance of the person from this event is the min (over y in K) of the distance from x to y. The average emotional distance of the population from the event is therefore the average of this quantity over x —
D(K, N) ≡ avgx∈N miny∈K |x, y|
We don’t really know about how the k people were chosen to die in these events; for a simple model, we could just assume that they were randomly chosen from the population. Then we can average over all ways to choose k people, and get a notion of the mean separation of people from sorrow in a population:
D(k, N) = avgK⊆N, |K|=k D(K, N)
D(1, N) is simply the mean “Kevin Bacon” distance between two people in a population of size N. D(k, N) is the mean distance between an arbitrary person and an arbitrary group of k people. I think that this is a good measure of the radius of sorrow in any catastrophic event: if you live in a population of N people that are bound in the sort of social network we’re talking about, and k are in some way affected by this, then you are so many degrees away from someone who was hit.
I don’t know if anyone has calculated a function like this for large populations, where the network is basically a country. I suppose it would be straightforward enough to do simulations to compute it.
It seems like a useful thing, though: it would be a kind of language for expressing the impact of a distant catastrophe. Instead of simply being told, “2,500 people died in this earthquake” you could learn something like “the average person was within three degrees of separation of someone who was killed.”
(It occurs to me that the question of which average to use is important: means are probably too sensitive to outliers, and in power-law networks that’s a major issue. A median would be useful, or of course a full histogram)