>> Nikhil Devanur: It is my great pleasure to welcome Bobby Kleinberg. Bobby is one
of Microsoft's new faculty fellows. He has done great work on design analysis of
algorithms and applications to e-commerce, networking and many other things. And he's
going to talk about network formation.
>> Robert Kleinberg: Okay. Thanks Nikhil for inviting me to come give this talk. This
presentation will have sort of a Coals to Newcastle character to it in that it is some work
applying random graph theory and edge percolation to analyze network formation in the
presence of contagious risk. Much of the technical content of what I am going to present
while non-trivial for us might be sort of an exercise for some of the people in this room.
So I encourage you to think of this talk as an introduction to a provocative branch of
applied math about which very little is known and which, you know, there is perhaps no
more distinguished group of people all in one place to study these sorts of questions that I
have right here in this room.
>>: [inaudible] [laughter]
>> Robert Kleinberg: Hmm. As I was thinking about the types of questions that I am
going to get during this talk, that is not one that occurred to me. [laughter] A lot of us are
fascinated by the subject of complex networks and when researchers have looked at why
complex networks look the way they do, people take very different approaches toward
answering that question depending on whether they come from mathematics/physics or
from social sciences. So those who like to address complex networks from probabilistic
perspective will come up with models of network formation that say something like links
form according to the following simple rules. Nodes show up one at a time and link to
other nodes with probability proportional to varying degrees. Or with some probability
depending on your distance from other nodes in some post metric that is independent of
the network topology.
On the other hand, in the social sciences, you see models that don't have any explicit
probability built into them at all. So nodes form a network based on an endogenous
optimization process in which each of them constructs links strategically so as to
optimize some payoff that depends on the global structure of the network that they form.
An example of such a model just to introduce you to the aspect of a theory that may be
less familiar is this one from Jackson and Wolinsky. In this model there are two
parameters that define people's utilities for the network that gets formed. There is an
Alpha which is the cost of forming the link and a delta which is some decay parameter
that specifies how much of a preference you have for being close to nodes as opposed to
far away from them. And then your payoff is linear in the number of links you form so
you experience a negative payoff, a cost of Alpha every time you form a link. And then
for every other node in the network you have positive payoff which diminishes
geometrically with your distance from that node.
So for example, in a network where B and C are already linked together and A is
strategically deciding which subset of two possible links to form to them, it could either
do the empty set with a payoff of zero. One link with this payoff or two links with that
payoff and for different choices of Delta and Alpha you can get each of these three
behaviors to be utility optimizing for node A so that you can sort of draw a phase space,
you know, in the two-dimensional space of Alpha Delta settings and plot the graphs that
arise as optimum graphs at each point of the phase space.
There is an interesting issue involved in defining what it means for a graph to be
equilibrium of this game. I trust that you are all familiar with Nash equilibrium, but
generally Nash is not the right equilibrium concept for looking at network formation
games, because Nash considers what happens when individuals make unilateral
deviations, but the process of forming a link between two individuals should really be
thought of as a bilateral process where both of them must willingly opt into forming the
link. And accordingly game theorists analyze these games using stability conditions that
seem similar to Nash but are not exactly Nash. The stability condition that will be
relevant to this talk is the following one. We will assume that any node can unilaterally
secede from the entire network area and it can delete all of its links. And that any two
nodes can jointly decide to create a link between them if it benefits both of them and
benefits at least one strictly.
So the network is stable if neither of those two moves--if it is a local optimum under
those two types of moves. So there is no individual that can strictly improve its payoff by
seceding from the network and there is no pair that can each weekly improve their payoff
and at least one strictly by forming a link between them.
>>: [inaudible] individuals to cut off [inaudible]
>> Robert Kleinberg: We are not allowing individuals to cut off a subset of their nodes.
Good. So any network that is stable under that stronger definition is stable under our
definition. And so when we prove impossibility results for stable networks under our
definition, they automatically carry over to impossibilities for stable ones under the other
definition. But an existence theorem for stable networks under ours does not imply
existence for the other and so that is just something to bear in mind. Actually I claimed
that the more interesting result about stability of this talk would be the impossibility one
which does carry over but it is obviously a self-serving claim so you are free to form their
own opinion about which of the two is more interesting.
>>: But the reason for doing it that way is because of what you can do [inaudible]
because that is an interesting [inaudible]. I guess that is also a matter of [inaudible]
>> Robert Kleinberg: Speaking as a participant in the research process, the reason we
did this, we knew that a question of primary interest to us was going to be a price of
anarchy type question that was comparing the social optimum against the set of all Nash
equilibria. And so if we adopted the most permissive definition of stability and still got a
strong price of anarchy separation, that would imply the same for a less permissive
definition of stability. So given that in any such talk when I got to this slide there was
going to be controversy over what stability means, we wanted to make an initial choice of
stability definition that would resolve the controversy in a way that is most favorable to
saying the price of anarchy result is a strong one. That is what we did.
All right. Moving on. The Jackson-Wolinsky style payoff function that I just described
is--there are many other papers in the literature that propose different payoff functions
but nearly all of them have this mindset that there is a cost to link formation and the
benefit that you get indirectly from forming those costly links is that you make this
connected world and then we all benefit from being connected together in this network.
But there is a different type of network formation process that occurs quite often in
reality where links are beneficial and we pay a cost for all being connected together.
That will be the subject of this talk. So we originally got interested in this subject in
2zerozero9 because of financial contagion where these are banks making loans to each
other and at least in the popular consciousness about what happened in the fall of
2zerozero8, there is this sense that a tidal wave of failure propagated through the network
and the economy paid a large cost for being too densely connected in the presence of this
possibility of risk. So that although each individual business arrangement was beneficial
to the two parties involved, the overall outcome of constructing a densely interconnected
web got us trapped in this risky situation.
Okay, so here I have a picture of the loan network of the US federal banking structure.
This is taken from Beck and Atalay. The models that I am going to be talking about in
this paper are undirected graphs, whereas this one is it a directed graph whose edges
represent A makes a loan to B. At some level--the reason that failure propagates through
this network is because of something called counterparty risk. So these banks make
overnight loans to each other. It is literally the end of one day a bank that doesn't have
enough cash on hand to fulfill its minimum reporting requirement mandated by the Fed,
will ask to borrow a small amount from another bank so that in the morning when the Fed
has the right to inspect them and see if they have enough cash on hand, they will pass that
inspection. And then they will immediately give it back to the bank that loaned it to
them, unless they default. Which because these are giant banks there is almost no chance
that that will happen, but there is a small amount of risk any time money is given from
one party to another that it won't be given back.
So these risks of defaults were each individually viewed correctly as being
extraordinarily unlikely events. It is just that there were so many of them that when a lot
of them started happening at once unexpected results came out of that. Now that I have
told you about the general scheme of links benefit people, but being overly connected can
be to the detriment of the individuals, it is possible to find examples of that cropping up
all over the place. For example sexual contacts are usually treated by the participating
parties as being beneficial, but an excessive amount of linking carries attendance risks.
In some sense this one actually hues more closely to the story of my talk because this is
an undirected graph. You know the probability spread or failure is symmetric when a
link is formed between two individuals.
Another example would come from covert organizations, things like networks of
terrorists. The organization benefits from individuals being linked together because it
allows them to coordinate plans requiring more than one actor to execute the plan. And
the danger is that if one of these people is captured and coercively interrogated they may
divulge the identities of others and so the organization is at risk if it engages in an
excessive amount of linking and then one of the individuals is compromised.
There is another example that is not in the slides for some reason. Exchanging private
data could be another example of this. When I share my cell phone number with
somebody, that benefits me because that person, presumably it is somebody that I wanted
to share my phone number with, and it benefits me that they now have my personal
information and can call me easily. But if that person shares their address book with
someone else, who shares their address book with someone else and eventually one of the
people in this web turns out to be a spammer, then it was not beneficial to me that we
formed that highly interconnected network.
So now I am ready to present the basic framework that we use to qualitatively analyze all
of these situations although in each case it would be quite easy to find at least one
qualitative feature that differentiates this abstract model from the banking application or
the STD transmission application or which ever one you want to pick apart. So we have
nodes that are forming a network of max degree capital delta among each other. It is an
undirected graph. And there are two important parameters of the model. I think I should
write them down on the white board since they will only be on this slide for a minute but
they will be referenced throughout the talk. So Q is the probability that a node fails
spontaneously. And spontaneous failures are independent events, each with the
probability of Q.
In addition each edge makes an independent decision whether to be live or blocked. The
live edges are capable of transmitting failure and the blocked ones are not. And P is the
probability of an edge being live. So the way this determines the set of failed nodes in
the network is that first of all every node that fails spontaneously fails, and secondly
everybody that did not fail spontaneously also fails if they belong to a connected
component containing a failed node in the sub graph of live edges.
>>: [inaudible] time?
>> Robert Kleinberg: There is no time to mention. Now there is a notion that first the
network is formed and the failure is revealed, but those are the only two stages of the
game. So, yeah, if your application is STDs then you would want to have a time
dimension which is--that's the primary way that the STD application differs from the
abstract model, but that just is what it is.
Throughout the talk we will think of both spontaneous failures and transmission of long
edges as being rare events, so Q and P are both positive but they are less than 1,
significantly less than 1. So if a node has degree desivy and failure probability thesivy,
then its payoff is linear in the number of links it forms, so that is ADI; let me see if the
formula is on my next--good. So if you can see the slide then you can see the payoff
formula and if you can see this panel of the board then you will see the payoff formula
even when the slide is gone. You get a payoff of DI for every link you form, but only if
you don't fail. If you do fail you get a negative payoff of B, regardless of how many links
you formed.
>>: So the liveness otherwise on the edge only influences [inaudible] in failure?
>> Robert Kleinberg: Not the payoff, yeah, that is right. So you get benefit--if you don't
fail regardless of whether your edges were live or blocked you reap the benefits of it. We
are going to be comparing graphs now according to a parameter which is the minimum
welfare of any node in the network. This is another one of those points in the talk where
someone could interrupt me and say well why did you decide to--actually a philosophical
discussion I would love to have is why computer scientists are so bent on taking a graph
or an algorithm, summarizing it in a single number and then just saying mine is hundred
times better yours because on this one dimensional continuum mine came out 1zerozero
times higher than yours. But, okay, it is effective CS research that people seem bent on
taking a single number that measures, you know optimality and then ranking things
according to that number. And we choose min welfare. Now there are many other
natural choices we could have made. Average welfare would have been a very natural
one. We chose min welfare because at one point in this talk I will present an argument
that only works using min welfare and not the average welfare and I think I will
remember to tell you when that happens.
>>: Welfare in the top 1% [inaudible].
>> Robert Kleinberg: Yeah, that's right, that's right. So that I think would actually be a
pretty uninteresting parameter for analyzing this model. And as I hinted earlier, a
motivating question, not the only question that we will address but a motivating question
throughout the talk has to do with the price of anarchy. That is comparing this parameter
in a socially optimal graph, a graph that maximizes the min welfare versus the same
parameter in a stable graph under the stability definition that I showed you before. So
now it is useful to do some thought experiments to map out a region of parameter space
in which the question is interesting and distinguish it from regions in which the question
is completely uninteresting.
If A is less than BQP, what it means is that for any node A the risk of forming a link to B,
which is--okay, if B fails with probability Q and transmits that failure with probability P.
So anytime I form a link I get a negative BQP payoff from having formed the link. If that
is greater than the benefit of forming the link then we would just get a big independent
set and so A less than BQP is an uninteresting part of the parameter range. If there is
someone who I know for sure is going to fail, and A is greater than BP, then I would link
to them anyway despite knowing for certain that they are going to fail. So that is an
uninteresting part of the parameter range. Similarly if there is an edge that I know for
certain is going to transmit failure and A were greater than BQ, then I would still choose
to form this edge despite the fact that it is guaranteed to transmit failure. So based on
these thought experiments we want to be focusing on a range of parameters where A lies
between BQP and the min of BP and BQ.
>>: [inaudible] even the first one yet because [inaudible] attitudes or maybe because I
have some fellow [inaudible] so the cost of [inaudible] isn't as much of [inaudible]
because it doesn't increase the probability by that much.
>> Robert Kleinberg: Yes, I understand what you are saying.
>>: Something to do something that you can also disconnect [inaudible] isolated just
completely [inaudible]
>> Robert Kleinberg: It sort of--okay. So there is another--I forgot to write this down.
Another aspect of the model that I told you about is that people form a maximum of
capital Delta links. And we think of capital Delta as a constant that does not scale with
the network size. So I am going to get to your question but let me first talk about the
bounded max degree. You have unlimited max degree then when the number of
individuals in the world becomes astronomically large, you get into ridiculous solutions
where people are--that is not true. They lose their payoff if they fail. In an early version
of the model we didn't have the, you lose your payoff if you fail. Then you have the
pathology that people form tons of links because they know they are almost guaranteed to
fail but they are still getting a payoff of A for every link that they are forming.
I don't know how to answer your question, so definitely in a network of two nodes if A
were less BQP, they wouldn't form the link, but you are making a valid point that the
same argument does…
>>: I mean you've always got the option of disconnecting yourself [inaudible] BQP, that
is one thing that you would do. I mean given the [inaudible] you might just…
>> Robert Kleinberg: Well, that's true but if I have 1zerozero links and the first one
increases my failure probability by BQP but the second one by a little less than that and
so it is at least possible that for a very large degrees…
>>: But this assumes that Delta is bounded [inaudible] probably will [inaudible]
>> Robert Kleinberg: Yes. But the justification for Delta bounded was also not
completely airtight, okay. The paper makes a lot of assumptions to get into an interesting
range of parameters where I think an interesting phenomenon is observed. And I have
argued to forcefully and thereby gotten into a bit of trouble claiming that every other part
of the parameter’s face is uninteresting. I haven't even gotten to this last line which is an
important one actually and is entirely unjustified. So rather than merely assuming A is
between these two trivial upper bounds and lower bounds, I am going to assume that it
has a lot of room on either side. So it differs from the min BP and BQ by a factor Delta
which is small and exceeds the other one by Delta inverse. So the important one here
actually is the A less than Delta min of BP and BQ and it is important enough that I am
going to include it on my cheat sheet here and I think the cheat sheet is now complete,
which is good because I think if I wrote down here nobody could actually see it.
So now it is possible for me to tell you about our main results. It turns out that payoff A
over P which is the payoff you would get from forming 1 over P links if you are
guaranteed not to fail. It is a range in payoff space to where a critical transition takes
place. And we will see that as Delta tends to zero the optimal, the min welfare of a
socially optimal graph is A over P times a constant which is strictly greater than one by
an amount that is tending to zero with Delta. All of this suppresses the dependence on N,
the network size, because it turns out that, you know, the limit of the optimal min welfare
as N goes to infinity a constant not depending on N.
In qualitative terms what this is saying is that socially optimal graphs actually push up to
and slightly beyond a phase transition point in payoff space and one thing that we will see
is that in order to get payoffs beyond the space transition, you need to be able to know the
identities of the people you are linking to so in an anonymous model where each
individual just chooses their degree and then a random graph with that degree sequences
generated, it is not possible for them to achieve payoffs exceeding A over P by any
constant factor. We will also see that in stable graphs the largest min welfare achievable
is little o of A over P as Delta goes to zero. So it is bounded by G of Delta times A over
P where G of Delta tends to zero with Delta.
And in qualitative terms what is going on is compared to socially optimal graphs. Stable
graphs have higher average degree, which exposes their nodes to a greater amount of risk,
so much so that it wipes away all but a vanishingly small fraction of the benefit achieved
by doing that linking. And this is a form of tragedy of the commons where because of
the individuals do not feel most of the negative impacts that they generate on others, they
continue engaging in behaviors that are detrimental to society as a whole, while making
small increases in their own payoff until the general payoff shared by everybody is close
to zero. So I would like to talk now about how some of these results have proved.
So recall that forming a graph nodes spontaneously fail with probability Q and edges are
live with probability P. So this is edge percolation in a graph which is in endogenous to
the model. So in particular when people happen to form a complete graph, then the live
edge sub graph is a GNP. And as is well known in GNP where P is, you know, less than
1 over N by a constant factor all of the components are small so the probability of our
component size exceeding S is exponentially small NS and on the other hand when P is 1
plus X over N there is a giant component whose size is a constant fraction of N. The
constant depending on this X in enumerator and if you plot the dependence, you know, at
zero All the way up until the numerator is equal to 1 and then it immediately takes off
with a positive derivative, this continuity in the derivative at 1.So that will be important
in the talk also. The derivative here is positive but not infinite.
In analyzing the, you know, min welfare of arbitrary networks, we need to consider not
just GNP but edge percolation in general graphs of Max degree capital Delta. And I want
to start out by showing you that the min welfare cannot exceed A over P by very much. So
to do so let me remind you of a little bit of the existence proof of the giant component. So
if you just want to prove that there is a constant probability of forming a giant component
in GNP when P is 1+ X over N, you imagine performing breadth-first search starting from
any root node and in any step of this breadth-first search, you expose all of its adjacent
edges. Some number of additional nodes are found and placed into the back of the queue
and so the expected change in the queue size is 1+ X for the new nodes that show up -1 for
the node that was just inspected. So there is a positive drift in this random walk and for our
purposes the only thing that is necessary for our analysis is that there is a constant
probability that this random walk process continues beyond 1 over P steps. This is just a
picture of the edge exposure process of the random walk. So then if we have a graph with
minimum degree D and we do edge percolation where the min expected degree of vertices
is one plus X for some constant X greater than zero, what this argument establishes is that
for any node I there is a constant probability that it's live edge component contains at least
Absalom times D nodes.
If you think about the consequences in terms of the payoffs to nodes, that means that if
you want everybody to get payoffs 1+ X times A over P, you need to have minimum
degree at least 1 plus X over P, is there a…
>>: [inaudible] P times D-1 [inaudible]
>> Robert Kleinberg: We are thinking…
>>: We are thinking of 1 and then adding D-1 [inaudible]
>> Robert Kleinberg: Yeah, we are in the large D regime so…
>>: [inaudible] cheat sheet…
>> Robert Kleinberg: I should make a note for the next version of these slides. Yeah,
because that is right, you only expose D-1, the edge you came in on does not get reexposed. It will be fixed, in the next version of these slides. If everybody is getting a
payoff of 1+ X times A over P, they get a payoff of A per link so they must be forming 1
plus X over P links so they are getting expected degree 1 plus X. So that means that
everybody has a constant probability of being in a live edge component with this many
nodes. When that happens, their probability of failing is, let's see I am out of space to write
on my cheat sheet. I can just do this in the air, oh, there is, okay. The probability of
surviving is 1-Q to the power, Absalom D. And, okay we are taking D to be roughly A
over P and by A--okay, D is roughly A over P which is like Delta B, so my--I don't know
how to do this in this amount of space.
People have such a high failure probability that it washes away all of the benefit of
linking and they get a negative payoff. They get a positive payoff of A over P if they
survive. If they fail, they get a negative payoff of B which is like A over Delta P. so the
only way that this is beneficial for them is that their survival probability is like 1-Delta.
But it is much worse than that. So that shows that if X is fixed and I take Delta P and Q to
be small enough with respect to that fixed X then people cannot achieve a payoff of 1 plus
XA over P. On the other hand if we treat Delta as fixed and we allow X to be sufficiently
small, then I will show you that you can get payoffs that are strictly greater than A over P.
It is useful to think of two natural ways of achieving this. One is a bunch of disjoint
cliques, and in that case to get up to this level of payoff we are going to need to have the
degree be slightly more than 1 over P so the clique size has to be a little more than 1 over
P. And then what is going to happen is we are going to form a bunch of disjoint GNPs, for
this value of N. And the other way would be to make a gigantic high-growth graph which
basically looks like a, locally looks like a D regular tree in the realm of a very slightly
supercritical branching process.
Okay. So in the paper we do general P and Q sufficiently small in this next set of slides
focus on case P equals Q. And let's think about which of these two types of networks is
better. Starting from degree 1 over P should you increase your degree to 1 plus X over P
for a very small X? If you do you will gain AX over P performing those links. And you
lose B times the increase in your failure probability. And we know that your failure
probability depends in the GNP case on the giant component size and in the branching
process case on the size of the component of the, you know, branching process containing
the root.
So in the--here I have drawn the anonymous case, so that the random D regular graph on
N nodes, D is slightly more than A over P, so it has high girth and locally in the
neighborhood of the root the live edge sub graph is going to look like a super critical
branching process. So the size of the giant live edge component is going to be Tao of X
times N where Tao is another one of these functions that zero up until X hits 1 and then-sorry the way that I have parameterized it, is zero until X hits zero and then it takes off with
positive derivative.
So as you increase your degree from A over P to 1 plus X times A over P, what is
happening is your failure probability is increasing at a rate which is the derivative of Tao at
zero. So you are getting a benefit of AX over P from the links and your failure probability
is increasing by X Tao prime of zero, so this benefits you if A over P is greater than B Tao
prime of zero. But I am assuming that A over P is much less than B; A over P is less than
Delta B and Delta is sufficiently small parameter. So in an anonymous market it does not
benefit you to increase your degree at all beyond the thresholds of forming, of criticality.
Okay, that is what this hand wavy argument meant to convince you of. So now let's look at
the opposite case where you are forming cliques each of which is going to sample a live
edge component which is a GNP of expected degree slightly larger than 1. And let's do the
same thought experiment. You start with clique size 1 over P and you consider is it
beneficial to grow up to 1+ X times P.
So now what happens is the clique that node I belongs to is going to have a giant
component of size theta of X times 1 over P and in order for I to fail, the most likely cause
of failure is that it belongs to the giant component and somebody in the giant component
spontaneously fails. There are these other events that have truly tiny probability where I
belongs to one of the other components and somebody else in that tiny component fails.
These contribute negligibly. So, you know, in each of these three cases I does not fail in
this picture because it is in the giant component but the only failures happen outside. I
does not even belong to the giant component. The only case that is bad for I is that both it
ends the failed node. I am making the assumption by the way that there is only one failed
node here. It is a reasonable assumption because spontaneous failures happen with
probability P so the expected number of failures is 1 plus X, so the actual number of
failures is Poisson with expectation nearly 1, so maybe there is not one but a couple of
nodes that fail. A constant number of nodes are failing. I is in trouble if it belongs to the
giant component and at least one of the failures is also in there.
And so as you increase from 1 over P to 1+X over P, the benefit to I is again a gain in
payoff of AX over P and the detriment to I is a negative B times the derivative of theta
squared. So once again we are comparing AX over P versus 2BX theta of X theta prime of
zero. And growing beyond the face transition is beneficial if A over P is greater than 2B
theta of X theta prime of zero. And what happens now since we are thinking now of BEQ
Delta as being fixed and X is allowed to be as small as we want, we can always set X small
enough that A over P is greater than 2B theta of X theta prime of zero. And it means that if
we push past the phase transition by a sufficiently small amounts people's payoffs are still
growing.
So again the qualitative reason for this is that in that branching process if you are in a
giant component, it is so large that you are virtually guaranteed to fail. And if you are in a
bunch of small cliques then you need to events to happen, you need to be in giant
component and the failed guide needs to be in the giant component. And so it is really the
difference between theta itself which has positive derivative at zero and theta squared
whose derivative is zero. That is what determines the qualitatively different behavior
between growing cliques past the phase transition versus growing the degree of the
branching process past the phase transition.
So this slide is just summarizing what I have talked about so you can achieve
supercritical payoffs by forming disjoint cliques and if you are doing anonymous linking,
that is impossible, because giant components in this case are much larger than giant
components in this case and so as soon as you get past the phase transition the penalty of
going beyond that transition is too steep.
For the final part of this talk I am going to talk a little about stable graphs. Remember I
advertised this price of anarchy results that says there is a tragedy of commons where
additional linking washes out almost all of the positive benefits that people get from
forming these links. So in particular we just saw that a union of cliques of size just barely
beyond A over P gets a payoff which is actually reasonably high. So why is that graph not
stable already in and of itself? I mean after all…
>>: [inaudible]
>> Robert Kleinberg: I'm sorry, thanks. Not A over P, 1 plus X over P. So these cliques
are already getting pretty close to, you know, they are past the phase transition so if they
grow a little more the giant component is getting bigger reasonably fast so you might think
that this graph is already stable because there is not much room to grow the cliques, but if
these two guys form a single link between them, then they get a benefit of A. The cost of
forming that link is really just the probability that the node I linked to belongs to the giant
component in its graph and someone in that giant component failed. And that is small
enough that if these are optimally sized cliques these two nodes are going to benefit by
forming that link. So what is going on here is forming that link does not expose INJ to
very much failure, but because each of them is so highly connected to its own community,
that probability of failure multiplies over everybody that is linked over INJ so the total
amount of negative utility that they create for the world by forming that link is much
greater than the positive utility that they get. This is the form of the tragedy of the
commons where--Okay so the original tragedy of the commons story is there is a pasture,
everybody puts their cows out to graze on the pasture. The farmers have an infinite supply
of cows and they are just going to keep putting cows on the pasture as long as the farmer
gets additional benefit from that. And the model of the pasture is that it produces a net
benefit which is a function of number of cows grazing, a decreasing function of the number
of cows grazing it that eventually decreases to zero. And that net benefit is shared equally
among the cows.
Then from the standpoint of any individual farmer, if the benefit has not yet reached zero
they are going to put one additional cow on, because that cow will still get a positive
benefit from grazing and the farmers total share of the benefit is going up because the
farmer is increasing his number of cows. So you end up with this phenomenon that they
keep populating the pasture until the total benefit of everybody is zero. The same thing is
going to happen in our model so people will just keep linking and linking until they have
eaten up essentially all of the utility that they got from the linking. And that is the only
time that they will stop. So how do you actually analyze it? Let's consider two nodes I
and J in a graph where they have decided not to link to each other. If they decide not to
link, that means that each of them--there are two reasons for them to decide not to link.
One is that either I or J is already at its degree threshold, Capital Delta and the rules of
the game don't allow for it to form another link.
The other possibility is that one of them has a failure probability that is high enough that
the link does not benefit the other party. So the benefit to I if it forms the link is A and
the cost of forming the link is at most B times P times the failure probability of J. It
might be strictly less than this because maybe there is already another path from J to I in
the graph and so maybe I is already absorbing some fraction of the probability of J
failing. All we are going to need is that the cost is less than or equal to BPCJ. So if I
doesn't want to link that means that the benefit is less than or equal to the cost and so the
failure probability of J is already at least A over BP.
So now let's think about three types of nodes. Those are already at their max degree and
that is why they are not linking, those who have failure probability greater than A over
BP and that is why nobody is linking to them anymore. And the third type of nodes have
low degree and low risk. So if we are in a stable graph then every node must be one of
these three types. And furthermore the nodes of type III must form a clique because I
have just argued that if I and J both have degree below Capital Delta and they both have
risk less than A over BP then they are going to link to each other. So the nodes of type I
already have high degree. The nodes of type II…
>>: [inaudible] trying to form a clique and Delta blocks them, no?
>> Robert Kleinberg: Nodes of type III.
>>: Yes.
>> Robert Kleinberg: Because they are type III that means that they don't have type I so
their degree is not Capital Delta.
>>: [inaudible] many of them. They try to form a clique and [inaudible].
>> Robert Kleinberg: Then I would not call them type III. So consider a graph which is
already stable. Its nodes have types I, II and III. The nodes of type III by definition don't
have degree Delta and by the arguments on the preceding slide they are already linked
together in a clique, because by our assumption that the graph is stable there could not be
any missing edges between them. That also means there are not very many of them
because in a clique of Max degree Capital Delta, you know that you have at most Capital
Delta +1 nodes.
Suppose now for contradiction that everybody achieves a payoff greater than or equal to
epsilon A over P. It means that the type II nodes have a payoff which is at least epsilon
A over P and on the other hand type II means that your [inaudible] J is reasonably large A
over BP. These nodes are getting a payoff of ADJ minus A over P, which is at least
epsilon A over P. It implies that DJ is greater than or equal to 1 plus epsilon over P.
Good. So types I and II both have degrees greater than or equal to 1 plus epsilon over P.
I am silently here making the assumption that the Capital Delta is greater than 1+ epsilon
over P. But again that is the interesting part of the parameter space because we just saw
the sort of optimal cliques have size 1+ X over P, so if you're Max degree bound is
significantly less than 1 over P then the world is not even able to get up to the size of
GNPs where giant components start to emerge.
So if there were no nodes of type III we would be in a graph of min degree 1+ epsilon
over P. We would be doing edge percolation with probability P and already saw that that
process produces for every node a constant probability of belonging to, you know, a
component of size at least constant over P and that that eats up all of the positive benefit
from linking and gives them a negative payoff. So if nodes of type III did not exist, we
would already be done at this point. We would have reached a contradiction because we
assumed that everybody's payoff is at least epsilon A over P, and deduced that their
payoffs were negative. Since we have the type III nodes we have to do a little finesse to
finish the argument. So assume now that the number of nodes in the world is very, very
big, bigger than Delta to the power of Delta. We have this clique of type III nodes which
as I said has at most Delta +1 members in it. And that means within a hop distance of
Capital Delta you can only encounter about Delta to the power of Delta nodes. If N is
bigger than that, then there must be somebody who is far from this clique and within
Delta hops of this node, you see nothing but high degree nodes. So now run the same
breadth-first search argument starting with this node and you know that in the first Delta
steps of that breadth-first search you will never expose one of these type III nodes, so it is
as if they did not exist. And the first Capital Delta steps are already enough to get the
constant probability of being in a component of constant over P many nodes. So when N
is that large, we take one of these for nodes and deduce that it has a negative payoff
contradicting our assumption then that everybody's payoff is at least epsilon A over P.
So that is the story of what happens in stable graphs. This proves that, you know, in this
parameter range that I delineated people will do so much linking that they eat up almost
all of the utility gained from forming links.
So just a final slide of reflections and open questions before I end the talk, I guess the two
main qualitative conclusions of this research were first of all when we first started
thinking about this contagion model a natural guess was optimal graphs would consist of
cliques that go right up to but not beyond the point where the giant component emerges.
It is a natural guess because as soon as the giant component emerges, then there is a sharp
transition in the behavior of failure propagation and it is natural to think well, that is
eating up so much utility that you would push right up to the phase transition but not past
it. We have seen that that is actually not true. You push past the phase transition by a
small amount. You are able to get just barely supercritical payoffs but not, you know, a
factor that tends to zero with this Delta perimeter.
And the other qualitative insight that we saw is that when you form Nash stable graphs as
opposed to socially optimal ones they are much less benefit for their nodes as measured
by their min welfare function. So one of the open questions we would like to answer is is
it also the case that the average welfare in stable graphs is bad? And that translates pretty
crisply into a question about the expected sum of squares of the sizes of percolation
clusters in a graph of average degree 1 plus epsilon over P when you are doing edge
percolation with probability P. And it is possible actually that the answer to this question
is well known and we just don't know it.
>>: [inaudible]
>> Robert Kleinberg: And this is known for irregular graphs, and not just for regular
graphs?
>>: [inaudible]
>> Robert Kleinberg: Yes. The question is if I have a graph whose average degree is 1+
epsilon over P but it is not a D regular graph. It could have highly irregular degree
distribution. Is it still the case that the expected sum of percolation cluster sizes squared
diverges the way that it does [inaudible].
>>: [inaudible] you are saying that some of the degree is high?
>> Robert Kleinberg: Yes. We are assuming the average degree is high. So we don't
want to assume that the min degree is high.
>>: [inaudible]
>> Robert Kleinberg: So in some of our [inaudible] we assumed that the min degree is
large but then, you know, when I had to analyze stable graphs, the way I honestly
achieved that was I showed that stability implies that all of the low degree nodes are in a
clique over here and then I went over here. So I had to work for that min degree. And
the way that I was able to obtain it was--so this is why we are stuck on min welfare
instead of average welfare, right? The way I obtained this picture was by assuming that
everybody's payoff is at least epsilon A over P. So now you see why this question about
average welfare becomes a question about graphs of average degree D instead of min
degree D. It is possible that there is an easy answer to that question. We thought about it
a bit and did not see how to do it ourselves, but I would love it if one of you would
enlighten us.
We also don't know what the structure of socially optimal graphs is. So I showed you
that a union of cliques can get beyond A over P, and that nothing can get beyond A over
P by a factor much bigger than 1. But, you know, we have no strong reason to believe
that a union of cliques is literally the optimal graph and it might be interesting to prove or
disprove that conjecture, similar structural question about stable graphs. So in our paper
we show that there exists a clique size such that the union of cliques that size is stable.
But there might be sort of many other stable graphs. And an interesting style of question
is whether, you know, all nearly optimal graphs or all stable graphs must look like a
union of cliques with just a small number of edges added and deleted.
And finally for those in the room who are interested in modeling type questions as
opposed to this is a math problem that we couldn't solve that comes directly out of our
paper. So, you know, can you extend the model to, you know, change the payoff
structure of the game in some way that makes the users internalize the cost of linking so
that the outcome of playing the game to a stable outcome looks more like the social
optimum and less like the stable graphs where the tragedy of commons has set in and we
have eaten up all of the payoff. Okay, so thank you for your time and I am happy to stay
and take questions. [applause]