[statnet_help] statnet_help Digest, Vol 167, Issue 7

Carter T. Butts buttsc at uci.edu
Sun Jul 19 16:10:04 PDT 2020


Hi, Adam -

Ah, my fault, I misread your email - I thought you were referring to Pr(
i=>j | j=>i), not Pr(i=>j | i=>k).  That is a two-star effect. 
(Specifically, a 2-outstar effect.)  More generally, the directed
k-stars correspond to that type of dependence; those terms can be
accessed within ergm using ostar() and istar().

However, it should be noted that those terms are usually (not always!)
ill-behaved unless some extra structure is employed to keep dependence
in check.  It is hence more common to work with their degree statistic
representation (the star statistics and degree statistics are
equivalent), and to use curved terms such as gwdegree() in order to
ensure vanishing marginal effects of tie addition.  There are many
papers on this - the classic Snijders et al. (2006) paper is a good
place to start.

I will note in passing that there are exceptions to the "no free star
terms" heuristic.  In particular, in some systems (especially physical
systems) one has negative dependence among edges sharing a vertex: each
current tie limits the ability to form new ties, e.g. because it
reflects reduced surface area available for binding, steric hindrance,
etc.  In that case, negative star effects can be reasonable, and do not
necessarily lead to degenerate behavior.  We also see this in some
social systems, e.g. in the context of inhomogeneous 2-star suppression
associated with ostracism.  Your system may or may not have these
properties.

In general, I think you will find it useful to go read some of the
literature on ERGMs if you wish to build effective models, particularly
given your goal of producing models that are physically credible.  You
can get a certain distance by imitating other papers and using
trial-and-error, but at some point you will need to do the math and
figure out what terms produce the types of dependence structure (and
heterogeneity) that are sensible for your application.  (You may also
need to consider your choice of reference measure, e.g. if you need to
extrapolate to new cases or if your system has support constraints that
must be respected.) Reading the fine print is especially important when
applying the approach to systems where it has not been extensively used,
because some of the heuristics that you will see in the literature are
chosen for things like friendship networks, and do not always generalize
to new settings.  One can do very cool things with ERGMs, but in terms
of subtlety they are like e.g. compartment models or agent based models,
and not like OLS.  A certain amount of attention to detail is necessary
for optimal performance.

Hope that helps,

-Carter

On 7/19/20 12:43 PM, Adam Haber wrote:

> Hi Carter,

>

> Thanks (again!) for the detailed response. I’m probably missing

> something, but I’m not sure I understand why mutuality would be the

> relevant term here. Let’s say that my nodes are embedded in 2D, and

> node i is at the origin. I also have two other nodes - node j at (1,0)

> and node k at (1.001, 0) - such that j and k are very close, relative

> to i. I’m trying to build a model in which the edges (i->j) and (i->k)

> are /dependent/ - if one of them exists, the other is more likely to

> exists as well. Mathematically, I think a good way to denote what I’m

> trying to achieve is P(i->k | i->j, j and k are very close) > P(i->k),

> where P(i->k) is the marginal probability that an edge from i to k

> exists (already taking into account their distance and potentially the

> effects of other terms).

>

> If I understand correctly, mutuality would affect P(j->i | i->j), or

> P(i->k | k-> i), but not P(i->k | i->j). But again, maybe I’m missing

> something - I’m quite new to ERGMs.

>

> Regarding interaction - I thought one possible way to do this is to

> “discretise” the close/far into categories (clusters of nodes), and

> then add an interaction term between this new “cluster factor” and

> relevant triad-related terms, as these seem to capture this dependence

> I’m after. Like you said, I’m not sure ERGM exposes such functionality.

>

> Thanks again!

> Best,

> Adam

>

>> On 19 Jul 2020, at 22:01,

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>> Today's Topics:

>>

>>   1. Interactions in a distance-dependent model (Adam Haber)

>>   2. Re: Interactions in a distance-dependent model (Carter T. Butts)

>>

>> *From: *Adam Haber <adamhaber at gmail.com>

>> *Subject: **[statnet_help] Interactions in a distance-dependent model*

>> *Date: *19 July 2020 at 13:07:23 GMT+3

>> *To: *statnet_help at u.washington.edu

>>

>>

>> Hello,

>>

>> Following a recent question I’ve posted here (and got very helpful

>> responses - thank you!), I’m trying to add more “domain knowledge”

>> into the model. Specifically, the nodes in the network we're studying

>> are embedded in 3D, and we’ve seen that (as expected) adding

>> distance-dependence to the model (via an edgecov(x) term such that x

>> is the distance matrix) indeed improves the GOF.

>>

>> I want to take this one step further: I know that if there’s a

>> (directed) edge i->j, and j and k are “close” (spatially), it should

>> increase the probability that there’s an edge i->k. Another option

>> would be to “discretize” the distances and group nodes into groups of

>> “spatial loci", and add a term that if there’s an edge i->j and j and

>> k are in the same spatial cluster (a binary indicator function), than

>> this should increase the probability of the edge i->k.

>>

>> Is there a way to incorporate this sort of reasoning into an ergm

>> model using any of the available terms? I went over the examples I

>> could find and did not encounter anything similar...

>>

>> Thanks again,

>> Respectfully,

>> Adam Haber

>>

>>

>>

>>

>>

>> *From: *"Carter T. Butts" <buttsc at uci.edu>

>> *Subject: **Re: [statnet_help] Interactions in a distance-dependent

>> model*

>> *Date: *19 July 2020 at 14:15:20 GMT+3

>> *To: *statnet_help at u.washington.edu

>>

>>

>> Hi, Adam -

>>

>> If you have a mutuality term in the model, it already accounts for

>> that effect.  Do you intend to say that you expect proximate vertices

>> to reciprocate at higher rates - /above and beyond the propinquity

>> effect/ - than distant vertices? (Again, if you have a propinquity

>> effect, it will already be the case that, ceteris paribus,

>> reciprocation will be more likely for proximate vertices.)

>>

>> If you want this type of effect, it can be realized with the dyadcov

>> term, or by creating an interaction term between mutual and edgecov. 

>> I don't think the user-level functionality for the latter option is

>> yet exposed, though it's pretty easy to code it using ergm.userterms;

>> however, dyadcov will probably suit your purposes.

>>

>> Hope that helps,

>>

>> -Carter

>>

>> On 7/19/20 3:07 AM, Adam Haber wrote:

>>> Hello,

>>>

>>> Following a recent question I’ve posted here (and got very helpful responses - thank you!), I’m trying to add more “domain knowledge” into the model. Specifically, the nodes in the network we're studying are embedded in 3D, and we’ve seen that (as expected) adding distance-dependence to the model (via an edgecov(x) term such that x is the distance matrix) indeed improves the GOF.

>>>

>>> I want to take this one step further: I know that if there’s a (directed) edge i->j, and j and k are “close” (spatially), it should increase the probability that there’s an edge i->k. Another option would be to “discretize” the distances and group nodes into groups of “spatial loci", and add a term that if there’s an edge i->j and j and k are in the same spatial cluster (a binary indicator function), than this should increase the probability of the edge i->k.

>>>

>>> Is there a way to incorporate this sort of reasoning into an ergm model using any of the available terms? I went over the examples I could find and did not encounter anything similar...

>>>

>>> Thanks again,

>>> Respectfully,

>>> Adam Haber

>>>

>>> _______________________________________________

>>> statnet_help mailing list

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>>> http://mailman13.u.washington.edu/mailman/listinfo/statnet_help

>>>

>>

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