Incorporating variable structure in SD

[George Richardson]

On Wed, 20 Nov 1996, Kostis Christodoulou wrote:

[...]
> In SD the decision rules and structure that describe a system
> remain fixed over time. As a result, SD models might not be able to
> offer a lot of valid insight in problems where the underlying reality
> is constant and fundamental change (e.g, the telecommunications
> industry) and where one simulates for a long period of time
> ignoring the issue of variable structure.
[...]

In one sense, the claim here is correct: a system dynamics model does
not change its equations as it runs and therefore has what most would
call "fixed structure."

In another sense, which is very important to all of us trying to apply a
dynamic feedback perspective to real problems, the *active* structure of
a well-formulated system dynamics model changes over time as the
simulation runs. Nonlinearities are the source of this endogenous
change in active structure. Indeed, most of us believe that a linear
view, implying no change in active or dominant feedback structure, is an
inadequate perspective guaranteed to distort or misrepresent the slice of
dynamic reality we hope to simulate. The history of efforts to apply a
feedback perspective show pretty conclusively that a feedback view
without the notion of changing dominant or active structure is generally
regarded as weak or inappropriate. Thats really why we build nonlinear
models.

Thus, a well-formulated model capturing a full range of nonlinear
structural effects expected to be potentially involved over a 30-year or
300-year time frame could be run with some confidence over such long
time frames. It would endogenously change active structure, built into
the model equations, as the simulation evolved. The problem, of course, is
doing a good job uncovering and incorporating the necessary endogenous
nonlinear structures. But frankly, thats the problem that system
dynamicists are trying to solve everyday with simulations of whatever
time frame.

I have concluded for myself that the notion of active system structure and
endogenous changes in dominant structure in nonlinear models is the best
way to handle the wish for formal models that change structure. I am
skeptical of exogenous changes, except as scenario tests.

This issue is a good one to talk about, however, because it gets at some
of the deepest fundamentals in our field.

...GPR

----------------------------------------------------------------------
George P Richardson G.P.Richardson@Albany.edu
Associate professor of public adm., public policy, and info science
Rockefeller College of Public Affairs and Policy Phone: 518-442-3859
University at Albany - SUNY, Albany, NY 12222 FAX: 518-442-3398
----------------------------------------------------------------------

["Kostis Christodoulou"]

Dear friends,

A few weeks ago, the issue of sustainability came up on the mailing list.
Bob Eberlein argued that one could simulate for as long one cares
about (300 years for a country or 30 years for a company) and look at
the behaviour over the entire period.

I feel rather uncomfortable with such an approach and would question
whether its practical or meaningful because of the changing structure
problem. In SD the decision rules and structure that describe a system
remain fixed over time. As a result, SD models might not be able to
offer a lot of valid insight in problems where the underlying reality
is constant and fundamental change (e.g, the telecommunications
industry) and where one simulates for a long period of time
ignoring the issue of variable structure. Do people think that such
an approach is a valid one or should we try to extend SD so that
we can incorporate changing structure in our models?

Regards,


Kostis Christodoulou
PhD Programme
London Business School
London NW1 4SA, UK
Tel: +44-171-262 5050, ext. 3260
Fax: +44-171-724 7875
-------------------------------------------------------------------
The original emails follow:
Bob Eberlain (BE)
Why not just simulate for as long as you care about (say 300 years for
a country or 30 years for a company) and look at the behavior over
the entire period.
System dynamics models can easily capture this time frame and thus
move the whole discussion to much more concrete issues around specific
policies and their long term impacts.

Kostis Christodoulou (KC)
Yes, but to implicitly assume that the structure of the country or
industry will not change _at all_ over the course of 300 or 30 years
is a bit far fetched. How can we have faith in such a simulation?

BE
If it were true that there were nothing that could be said about the
nature of a country after 100 years you would be right in marking such
simulations as without merit. However, the same argument would
suggest that we do not need to worry about the future beyond a couple
of decades (or perhaps years). Many people truly believe this. I do
not.

KC
I also believe we need to worry about the future beyond a couple of
decades and thats why simulation is a powerful tool to analyze
"would-be worlds". However, the nagging feeling I have about SD is
that the decision rules and structure that describe a system remain
fixed over time. Dont you think it would make sense to take SD a step
further, and allow variable structure, where the system can
endogenously change its decision rules and underlying entities
structure. The experience gained from such simulation experiments
could be more enlightening or do you reckon that itll be a complete
mess?

BE
The issue you raise of variable structure is an interesting one. If
you look at great models (world dynamics, urban dynamics - things that
Jay wrote) you will see that they are often pretty abstract. Jay
feels, and I think he is right, that the abstraction allows the
incorporation of what is often thought of as changing structure. This
raises a lot of very hard questions about validity and lots of other
good stuff.

I dont have answers, but these are issues worth pondering.

[wakeland@ccgate.leupstv.com]

I have long been intrigued by the concept of fixed vs. variable model
structure. A few years back, I wrote a "think piece" that addresses
this subject. The paper, called "Metamodeling Aspects of Model
Conceputualization," was prepared for the 1990 ISSS Conference. It is
available on the web at:

http://sysc.pdx.edu/Faculty/Wakeland/papmetam.html

As a very brief synopsis, the idea is that one can build logic into a
dynamic model that sort of "sits on top of" the primary model logic
and attempts to monitor whether or not the model is still internally
consistent. If not, it might intercede to change parameters or even
trigger a switch to an alternative equation (structure).

Wayne Wakeland
Adjunct Assoc. Prof.
Systems Science Ph.D. Program
Portland State University

[as7748@cnsvax.albany.edu]

On Wed, 20 Nov 1996, Kostis Christodoulou wrote:
> I feel rather uncomfortable with such an approach and would question
> whether its practical or meaningful because of the changing structure
> problem. In SD the decision rules and structure that describe a system
> remain fixed over time ... should we try to extend SD so that
> we can incorporate changing structure in our models?

Previous and current work in the field would indicate that the answer is
YES. No doubt, this is a challenging task which requires a great deal
of wisdom and acumen, both in terms of having a good theory about what is
happening in the real world, and of how to model it.

Forresters National Model of the US Economy is an excellent example of
this practice. As he described it in 1974:

"The structure of the socio-economic model is intended to be general and
to apply to any country ... The model will treat all major aspects of the
socio-economic system as internal variables to be generated by the
interplay of mutual influences within the model structure."

The meaning of a "changing structure" in a system dynamics model should
perhaps be clarified. It is not the elements of the model (level, rates
and feedback loops) that are changing over time. As Kostis indicated, in
SD the "decision rules" that are built into a model are fixed. What is
changing is loop polarity and loop dominance as a result of
nonlinearities built into the model. These changes ought to reflect
"structural" changes in society.

The need to build models that reproduce perceived changes in system
structure is necessary especially when one is interested in the dynamics
of the transition stage, rather than of the equilibrium. For example,
Forrester argues that in the transition stage of the national model, as
the economy reaches limits to growth and changes its mode of behavior,
profound changes in society are occuring in terms of "laws, attitudes,
management methods, traditions, values, expectations, and religions."

Aldo Zagonel dos Santos
Doctoral student of
Simulation & Modeling in Public Administration
University at Albany, SUNY
e-mail: AS7748@CNSVAX.ALBANY.EDU

[saeed@ait.ac.th (Professor Khali]

I have talked about this elsewhere in detail, but one way to assure a model
has in it possibilities for structural change is to incorporate into the
reference mode multiple case histories with contradicting patterns. This
way, latent structure for re-enacting those patterns, and many more, is
created in the model.

A model, how so ever complex, must always be seen as a slice of reality.
Its tendency towards a robust equilibrium might have nothing to do with
reality, yet such a hypothetical equilibrium is necessary for creating a
decision space in which policies for sustainability can be searched.

Khalid
Khalid Saeed
Professor and Program Coordinator
Infrastructure Planning & Management Program
School of Civil Engineering
ASIAN INSTITUTE OF TECHNOLOGY
P.O. Box 4, Klongluang, Pathumthani, THAILAND 12120
phones: (66-2) 524-5681, (66-2) 524-5785; fax: (66-2) 524-5776
email saeed@ait.ac.th
Visit our program website at: http://www.ipm.ait.ac.th/

["Kostis Christodoulou"]

Many thanks to Tom for an excellent reply.

It was a thoughtful reply, addressing all the points I had in mind.
After working with SD and the mechanical paradigm, I think its about
time to adopt some ideas from the evolutionary paradigm and the field
of complexity theory.

I thus believe that even though SD has many useful insights to offer, I
find two aspects of it rather limiting. As I indicated, I dont believe that
SD provides the tools to endogenously deal with evolving structure
(meaning, say, entry, exit, mergers of firms in an industry or change in the
firms decision rules). In addition, there is no explicit behavioural
representation: by taking the "view from above" and, as a result,
by usually focusing on simply one single value (an "average" of some
sort), for an entire population, we miss the important characteristic and
only true reality which is variability and diversity. I would thus
disagree with Tom that Brian Arthurs Bar Problem is not "any
different from traditional SD models" as preferences are explicitly
represented and the introduction of agents increases the overall
variance.

Tom has a valid point that the difficult issue with evolutionary models
is their parametrization. On the other hand, one is able to create
more "would-be worlds" and get a feeling for the overall variability.
I am not trying to put SD down. I believe that it offers excellent insights,
but some phenomena cannot be explained with the existing SD armoury.
So, dont you agree that its about time we took SD a step further?

Regards,


Kostis Christodoulou
KChristodoulou@lbs.lon.ac.uk
PhD Programme
London Business School
London NW1 4SA, UK
Tel: +44-171-262 5050, ext. 3260
Fax: +44-171-724 7875

[marco@idsia.ch (Marco Wiering)]

Thomas Fiddaman raises a good point. Like I have
already said some weeks ago, machine learning
can be very useful for SD-modeling. Thomas
mentions GAs, GP, CA, and EP, but I am more
thinking in terms of structured recurrent
neural networks (but this is only a personal
preference). It is possible to implement
a SD model in a recurrent network, after
which the network can improve the weights
the modeler gave it. When enough data is
available, and the nets are designed well,
Im sure this will give a better endproduct in
99% of the cases!

Marco Wiering
PhD candidate Artificial Intelligence
marco@idsia.ch
http://www.idsia.ch/~marco

[Jim Hines]

Tom Fiddaman as usual made a thoughtful contribution to the discussion
on variable structure as have a number of others. I would like to
comment on Toms suggestion that incorporating genetic programming is
not different in principle from regular SD modeling.

There are two ways that GPs can be used in system dynamics modeling.
These two ways are different from one another and different from
"traditional" system dynamics models.

What I have in mind is a genetic program that will actually create new
policies and new stock and flow structure. There could be two purposes.
One would be to find good policies or good structure. The second would
be to understand how people actually create new policies and new stock
and flow structure.

The first purpose -- finding good policies and structure -- is not a new
purpose, but the method is new. The second purpose -- understanding how
people create new structure -- is actually new and the method is new,
too.

I personally think that exploring how people create new policies and S&F
structure is of great potential importance, allowing us to design
organizations that evolve more efficiently in directions that
participants desire. It requires recasting the genetic program (or the
genetic algorithm) so that it represents not simply a "problem solver",
but represents the way that people create policies and structure. The
task is to recast GPs and GAs in terms of the way people actually
behave.

The task of representing how people really make decisions in order to
better design organizations is key to system dynamics modeling. And, in
this sense I strongly agree when Tom says the use of GPs is "not
different in principle" from system dynamics. However, it is also true
that the decisions being represented and the aspects of the organization
being designed are new.

Jim Hines
LeapTec and M.I.T.
jimhines@interserv.com

["Christian E. Kampmann"]

Kostis question about variable structure in SD models touches some
important points about methodology. Much of what I was going to say was
anticipated by Toms comments, which I wholeheartedly agree with, but
some elaboration might be useful.

First, a more academic point: it is very easy to get confused about
what is meant by "structure". As far as I can see, theres a hierarchy
of structure:

At the highest (shall we call it fundamental) level, structure is
whatever is invariant during the simulation. For an SD model, that is
clearly the entire model equation structure. For a genetic-algorithm
model (and similar techniques--GA for short), it would be the "rules of
the survivival game", such as selection procedures among competing
"genes" and, not least, the way the coding between the genes and their
phenotypes (the expression of that gene in the actions / properties of
the organism).

At the next level is what George R. calls active structure, i.e. the
dominant feedback mechanisms at any point in time during a simulation.
In a GA model, that might be the current ensemble of genotypes and
phenotypes and (if the model includes an endogenous environment) their
interaction in the ecological system.

Depending on the particular situation, I could imagine several layers in
the hierarchy. For instance, Khalids switches that activate latent
structure or Waynes logical superstructure lie somewhere between the
two levels just described.

Another example that came up in my dissertation work was how to measure
learning in a set of market experiments with human subjects. (Bent
Bakkens thesis talks extensively about these issues.) A simplistic
measure of learning as improvement in performance is inadequate because
performance may improve simply as a dynamic consequence of a fixed
decision or judgment procedure. If, for instance, the dynamic task
facing the subject is much like a simple search for an optimum price,
with a fairly stable and dynamically simple environment, performance can
improve over time for a wide variety of simple error-correction or
hill-climbing procedures. But real learning would imply a change in
the mental model of the subject, i.e., a change in the decision/judgment
rule itself. In my case, it was possible to incorporate and measure
this as a change in the coefficients or weights in a single rule. I
would say my learning measure falls somewhere between the fundamental
level and GPRs notion of active structure.

All this said, I think these considerations are not nearly as important
as a more basic methodological point:

Tom points out that in principle SD models could be used to set up a
universal computer and hence do anything a computer can do, adding right
away that this is of course ludicrous. Yes indeed. The "in-principle
principle" has done much to dull the mind. When you assess a method
like SD, you have to look, not at what it could do in principle but how
it tends to shape the kinds of problems you look at and the kinds of
hypotheses / theories you form.

An analogy from economics might serve: Analytical economists tend to
assume that consumer preferences are stable and exogenous. "In
principle" one could incorporate theories of endogenous preferences, but
first, it takes a lot of the power out of statements about welfare,
efficiency, etc. and second, it is cumbersome to treat analytically. As
a result, you rarely (never?) see mathematical theories in economics
that incorporate the idea that preferences can be directly manipulated
by e.g. marketing. (On the economic-history and political-economy side,
there are of course plenty of examples, e.g. Torstein Veblen, but thats
a different story.)

SD has a tendency to view problems (patterns of behavior) as created
endogenously by feedback loops. Have you ever tried, for instance, to
look at a random walk. Try it! I bet youll see cycles. There is
nothing wrong with that. SD has a set of biases and propensities for
theory building that adds a unique perspective, and it has many
particular strengths, not least the ability to communicate and explain
the model results. The key, I believe, is to be wise about the
strengths and weaknesses of the inherent biases of the approach. Thats
why Im also skeptical about grand schemes of integrating SD with other
modeling techniques (artificial intelligence, GAs etc.) to create a
bigger better SD. In principle this can be done, but in the
process, I think youd lose the methodology.

Still, I dont want to be too black and white about it. For instance,
the de Vries - Janssen study that Tom cites sounds fascinating.
Interfacing with other methods is a good idea because it helps reveal
limitations and qualify the results, not because a combined grand method
is better or truer.

Back again to the issue of endogenous structure, in the context of
evolutionary models:

John Sterman and Mike Radzicki wrote a paper for an Amer. Econ. Assoc.
conference a couple of years ago where they argued that SD was well
suited to address questions of evolutionary economics, using a simple
model of technological lock-in as a demonstration. The thrust of the
paper, as I recall it, is that the SD perspective provides a simple and
clear tool for understanding evolutionary processes. In contrast to the
black-box nature of GA models, SD can explain the outcomes in a way
that makes it more appropriate for policy making. Indeed, some GA-style
models (e.g., some of Nelson and Winters early models of innovation and
immitation among firms) can be distilled into a simpler feedback
explanation.

OK. But I would have added a word about the limitations of our
approach. By an a priori specification of the kinds of decision rules
(structures) that can be activated, our models (by definition?) cannot
be truly evolutionary. And the element of surprise and discovery, while
probably there, is much less than in GA models. It may be that in
princple a richer diversity of structure is possible in SD, but to
incorporate it, you first have to be convinced it matters and why.
Where do you get that inkling from? In my view, you can better use SD
ex post to distill and explain evolutionary outcomes. And recognize
that this is not always possible: The technological lock-in example is
simple enough that you can do it, but we have a hard time explaining
emergent properties of the system that depend upon the diversity of
populations.

--
Christian E. Kampmann
cek@chaos.fys.dtu.dk
Postdoctoral Fellow
Dept. of Physics, Bldg. 309
Tech. Univ. of Denmark (DTU)
2800 Lyngby
DENMARK
Phone: (+45) 45 25 31 05
Fax: (+45) 45 93 16 69
WWW: http://chaos.fys.dtu.dk/~cek (under construction)

[Alan Pengelly]

This is an interesting discussion, and relevant to some work we are doing.
Our aim is to develop a prototype adaptive (telecommunications) network, but
at the moment we are concentrating on building models. Our first choice
was to use discrete event simulation. This was fine up to a point, but I
had used SD on previous projects (organisational structures) and felt that
it might be useful. The issue of evolving structure is important for us - the
network we envisage will change its topology with respect to usage and
performance criteria. As has been mentioned previously, approaches based
on cellular automata are promising. However, just how you would map this onto
SD is by no means certain. Likewise we are using evolutionary game theory
as the primary `learning engine. I still feel that SD should provide us
with some valuable insights into the behaviour of an adaptive network, but
evolving structure would seem to be a problem. If anyone is researching
(or would like to) in this area might want to drop me a line.

regards

Alan
apengell@srd.bt.co.uk
Dr Alan Pengelly,
Project Manager,
Software Engineering Laboratory,
BT Labs,
Martlesham Heath,
Ipswich,
Suffolk IP5 7RE,
U.K.
tel +44 1473 646652
fax +44 1473 642299

[tomfid@MIT.EDU (Tom Fiddaman)]

Kostis raised a great question and I cant wait to see all the other replies.

I think theres already been a great deal of work aimed at creating models
with variable structure. It generally falls into the complexity theory
realm - genetic algorithms and genetic programming, classifiers, cellular
automata, evolutionary economics, etc.

In principle, these models arent any different from traditional SD models.
They just push structure to a higher level. Consider Brian Arthurs Bar
Problem, in which 100 Santa Fe residents try to decide which night to go to
the local watering hole. If either too many or too few show up, its not
fun to be there. Each individual uses a forecasting model (basically a
vector of weights on attendance in the past few nights) to form an
expectation of whether tonight will be a good time to show up. Individuals
use a genetic algorithm to refine their forecasting model. The resulting
behavior is very complex and interesting.

In one sense, there is shifting structure; as individuals add or remove
terms from their forecasting models, feedback loops are created and
destroyed. In another sense, though, all the loops are potentially there at
the outset; they just arent active. This is no different from shifting
loop dominance in a simple model. Its just that the modeler specifies the
loops in a different, less explicit manner. Because of this, the behavior
may be more surprising than that of a conventional SD model, and also
harder to interpret - though it sounds difficult to get much harder than
the model George Backus just described.

If you really wanted to abuse yourself, you could build an SD model of a
microprocessor and then use the model to run any arbitrary code. Of course,
this is ludicrous and we have to acknowledge methodological difference at
some point - SD is not a superset of all disciplines.

Traditional SD has several ways of dealing with the changing structure
issue. Sensitivity analysis changes the relative gains of feedback loops,
and thus changes the active structure of a model. I often build in several
alternative decision rules into key areas of the model, so that I can test
its behavior under alternative conditions. At some point, though, my
ability to speculate about what loops are possible breaks down, and it
would probably be more productive to create some sort of evolutionary
simulation, in which a huge number of loops were possible, but simple rules
(e.g. a GA) guided which actually became active.

I recently spent quite a bit of time debating whether to build a
traditional SD model or an evolutionary model for my dissertation on
climate policy - a problem with a 300 year time horizon. I ultimately chose
to build a traditional model, because I wanted to use the model to
communicate some fairly simple points about embodiment of energy
requirements in capital, endogenous technology, etc. to other modelers and
policymakers. Traditional SD models are easy to explain and its usually
easy to attribute behavior to structure. Existing climate/economy models
dont even have any positive loops (other than capital accumulation), so
you dont need variable structure to improve them a lot.

Had I built an evolutionary model, I would have had to restrict my scope a
lot in order to make it computationally feasible, and it would be extremely
difficult to parameterize the model in a way that made sense to
policymakers. The evolutionary aspects of the model would introduce a whole
new set of structures that would be tough to validate.

Still, I think there are some very valuable insights to be gained -
particularly in the domain of social preferences, which are generally
assumed to be static. Bert de Vries and Marco Janssen at RIVM in the
Netherlands added a GA-based evolutionary layer to a traditional SD
climate/economy model to do just this - I can dig up the citation for
anyone whos interested. A major insight from this is just as Kostis
suggests - that the true scope of uncertainty is a lot greater than one
would anticipate from a model with fixed decision rules. Their approach
reminds me of Jay Forresters heuristic for including discrete and
stochastic elements in a model - build the continuous, deterministic model
first, then add other elements where necessary.

I think SD has a lot to offer evolutionary modeling. Building good models
to serve as the infrastructure for interacting agents is one area where we
could add a lot to the state of the art. Several evolutionary models Ive
looked into are filled with discrete time and IF..THEN logic in places
where continuous formulations would be much more appropriate. At the same
time, I think we have a lot to learn - perhaps especially about humility &
taking results with a big grain of salt.

---------------------

BTW, I think Bobs original point was a good one - that looking at a single
summary number cant capture everything going on in a model with many state
variables (unless youre a neoclassical economist, in which case you
probably wouldnt be reading this anyway).

At the same time, if one can do a bunch of runs and then rank them by
attractiveness, one ought to be able to figure out what criteria make them
so, and make those explicit. Any time the model is about more than one
person, theres likely to be a plurality of objective functions. Making
them clear makes it easier to identify common ground.

Im going to ignore Bobs admonition and cite a paper:
P. Gardiner and A. Ford, Which Policy Run is Best, and Who Says So? in A.A.
Legasto Jr. et al, System Dynamics. New York: North Holland. 1980.
Hartmut Bossels work on system orientors is also interesting in this
context, though I dont have the citation at my fingertips.

______________________________________________________________
Thomas Fiddaman, PhD Candidate http://web.mit.edu/tomfid/www
MIT Sloan School of Management, System Dynamics Group
E60-355, 30 Memorial Drive, Cambridge, MA 02142
MIT: 617-253-3958 home: 603-497-2273 email: tomfid@mit.edu
______________________________________________________________

[tomfid@MIT.EDU (Tom Fiddaman)]

Several people have asked me for citations to Hartmut Bossels work on
orientors and Bert de Vries et. al. on layering a GA onto an SD model.
Thanks go to Joel Rahn for finding this one:

>Hartmuts work came to mind as I read previous posts. Your item
>inspired me to find one of the original references. I dont know if
>there is more recent work by Bossel on this. Here is the reference. It
>would be nice if you could post it on the list; my office e-mail
>system only allow for a reply directly to you. Thanks.
>
>Hartmut Bossel (ed.), Concepts and Tools of Computer-assisted Policy
>Analysis, vol. 1 to 3, Birkhauser, Basel, 1977, ISBN 3-7643-0921-0,
>3-7643-0922-9 and 3-7643-0923-7. Also known as ISR (Interdisciplinary
>Systems Research series) nos. 36, 37 and 38.

There are also several others:

Hartmut Bossel, Viability and Sustainability: Matching Development Goals to
Resource Constraints. Futures v.19 no. 2, 1987, pgs 114-128.

______________, Modeling and Simulation. Wellesley, Mass: AK Peters. 1994.

______________, Deriving Indicators for Sustainable Development. Report
#A9601. Wissenschaftliches Zentrum fur Umweltsystemforschung, Universitat
Gesamthochschule Kassel. 1996.

Contact: bossel@usf.uni-kassel.de

Marjolein van Asselt and Jan Rotmans. Uncertainty in Integrated Assessment
Modeling - A Cultural Perspective-based Approach. RIVM Report #461502009,
Globo report #9. RIVM, Netherlands.

A similar (and perhaps better) paper appeared in the first issue of
Environmental Modeling and Assessment, a journal which just started this
year.

The work is based on ideas by Bert de Vries and the PhD thesis of Marco
Janssen, both at RIVM:

Marco Janssen. Meeting Targets: Tools to Support Integrated Assessment
Modeling of Global Change. PhD Dissertation, University of Maastricht,
Netherlands. 1996.

______________________________________________________________
Thomas Fiddaman, PhD Candidate http://web.mit.edu/tomfid/www
MIT Sloan School of Management, System Dynamics Group
E60-355, 30 Memorial Drive, Cambridge, MA 02142
MIT: 617-253-3958 home: 603-497-2273 email: tomfid@mit.edu
______________________________________________________________