[statnet_help] fragmented bipartite network...

[Carter T. Butts]

Hi, Daniel -

Most of the cases to which I believe you are referring deal with 
differential mixing; the "blocks" here are what are sometimes called 
"density" blocks, which are quantitative relaxations of the 
complete/null blocks.   I don't think anyone doubts that differential 
mixing exists, but that is very far from e.g. nontrivial global 
automorphism orbits or the like.  Indeed, John Boyd had a running bet 
for some years, in which he offered to pay a sum of money (I forget how 
much) to anyone who could show a statistically significant regular 
equivalence pattern (above and beyond SE - he also had some other 
boundary conditions that ruled out "easy" cases).  My vague recollection 
was that Steve Borgatti claimed to have one, and they then haggled over 
John's way of calculating "significance," but my memory on the subject 
is hazy and doubtless untrustworthy; I never did buy John's extreme 
conjecture, but it is true that he was not exactly overwhelmed with 
claimants.   At any rate, models for differential mixing with discrete 
group structure are well-trod.  As far as other kinds of generalized 
blocks (moving away from complete/null blocks), you can fit models with 
strict versions of e.g. regular, row/column dominant, and row/column 
functional blocks with clever use of constraints (in ergm, the bd() 
constraint term).  The most obvious path to soft versions of those block 
types is to create statistics that count violations of the block 
pattern.  Some can be implemented using the degrange() term, together 
with appropriate use of the optional attribute arguments.  (Obviously, 
these are all "confirmatory" models, in the sense that one has to 
specify the block structure one wants to impose/parameterize.  But that 
is not without its virtues.)

Vis a vis dependence, I'm not sure that it is very helpful to think in 
terms of "violating assumptions."  It is probably more useful to think 
of H-C and friends as giving you a "recipe" for the statistics you need 
to implement particular kinds of dependence conditions (should you want 
to do so).  So, e.g., if you want edges to depend on each other when 
they share endpoints, then you will want (in the unvalued case) 
indicators for each edge variable, and indicators for each mutual dyad.  
If you also want the corresponding effects to be homogeneous, then this 
reduces to the edge count and the count of mutuals.  Adding e.g. a 
2-outstar term to a model with edges and mutuals is not violating any 
particular assumption imposed by the latter - it's just that this new 
model will now belong to a different (and broader) dependence class than 
the original one.  (It will, in particular, have a form of Markov graph 
dependence.)  Nothing says that your model has to belong to /any/ 
particular dependence class - unless you want to impose such a 
condition.  Of course, if you /do /want to restrict your dependence to a 
particular class, then you will indeed need to ensure that your 
statistics are a subset of those admitted by that class (which, for H-C, 
can be determined from the cliques of the conditional dependence 
graph).  In my experience, this is rarely a useful way to proceed; 
however, it sometimes can be handy to know the type of dependence class 
to which your terms belong.  Likewise, it can sometimes be handy to 
start by positing a form of dependence that makes sense in a specific 
situation, and then deriving the statistics that result.   Pip, in 
particular, has done a great deal to elucidate these sorts of connections.

As far as long-range dependence, there's again nothing ruling it out.  
(Pip and Tom, IIRC, have a very nice typology working out statistics for 
dependence classes at different distances.)   For instance, k-cycles can 
be long-range, for large k.  The various component and bridging 
statistics can be arbitrarily long-range. The statistics that arise from 
density and dyad census mixtures do them one better by being completely 
global (i.e., they create conditional dependence between edge variables 
irrespective of whether there is even a path of any length between their 
endpoints).  All of these lead to well-defined models - those models 
just happen not to belong e.g. to the Markov graphs (or the social 
circuit graphs, the Bernoulli graphs, the u|man family, etc.).  If there 
is a reason that you need your model to belong to such a family, then 
you would not want to use terms that are not within the class specifying 
that family.  But otherwise, such restrictions are arbitrary, and may 
get in the way of specifying important mechanisms.

Hope that helps,

-Carter


On 12/7/23 11:02 PM, Gotthardt, Daniel wrote:

> Hello Carter,

>

> i agree that stricter types oft equivalence are very rare and I would 

> personally also look at either generalized blockmodeling or actually 

> just measures of structural or positional similarity - but indeed not 

> only local ones (which are already included in ergm of course). I did 

> mention them here because most results of the relevance of more global 

> equivalence structures I know have been found in especially kinship 

> research and organisational science (Krackhardt & Porter 1986 and e.g. 

> in insitutuional fields DiMaggio 1996 and Alsaas & Taamneh 2019). 

> There has also been some recent research in foreign trade and 

> political conflicts that indicate that block structures might matter 

> (Guler et al. 2002, Zhou & Park 2012, Olivella et al. 2022). I am 

> curious though which tools you are thinking about for implementing 

> aspects oft generalized block structures?

>

> Regarding hammersley-clifford I mostly wanted to be careful here, but 

> I did think that H-C and extensions like social circuit dependency 

> (which allows partial depensence) did matter to ensure some 

> (conditional) independence assumption with a few parameters (one for 

> each clique of the dependence graph) in ergms (see e.g. Koskinen & 

> Daraganova 2012 and Block er al. 2019). I thought dependencies (far) 

> beyond the local neigborhood might violate these properties. This is 

> probably beyond Harald's concerns but I would be happy if you could 

> indicate any literature to alleviate my misunderstanding.

>

> Best Regards

> Daniel

>

> --

> Daniel Gotthardt, M.A.

>

> Wissenschaftlicher Mitarbeiter / Research Associate

>

> Universität Hamburg

> Fakultät für Wirtschafts- und Sozialwissenschaften / Faculty of 

> Business, Economics and Social Sciences

> Fachbereich Sozialwissenschaften / Department of Social Sciences

> Soziologie, insb. Digitale Sozialwissenschaft / Sociology, esp. 

> Digital Social Science

>

> Max-Brauer-Allee 60

> 22765 Hamburg

> www.uni-hamburg.de 

> <https://urldefense.com/v3/__http://www.uni-hamburg.de__;!!CzAuKJ42GuquVTTmVmPViYEvSg!JdYBcV_E6DZQNgeXEHvFyB1XtKMZAxcMdc_AkjSbQU_HEoYKHjJfk8sROxBVNRM1pQa_K-uy5aEJXHEf_1N44xpY6bYmHv1j$>

>

> ------------------------------------------------------------------------

> *Von:* statnet_help <statnet_help-bounces at mailman13.u.washington.edu> 

> im Auftrag von Carter T. Butts <buttsc at uci.edu>

> *Gesendet:* Freitag, 8. Dezember 2023 07:06:46

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

> *Betreff:* Re: [statnet_help] fragmented bipartite network...

>

> Local automorphism orbits and their associations with covariates can 

> be modeled using graphlet statistics; see e.g. ergm.graphlets.  

> Nontrivial /global/ automorphisms are extremely rare in typical social 

> networks, so such terms would be unlikely to be useful - what one 

> might call the "strong algebraic paradigm" of network analysis (the 

> idea that we could explain most social network structure in terms of 

> small numbers of roles, as defined through algebraic equivalences) was 

> a very compelling idea that didn't really work out, and I don't think 

> many folks are pushing in that direction right now.  (See also 

> compositional factorization, as famously illustrated by the semigroup 

> on the cover of Wasserman and Faust (1994). Beautiful idea with some 

> lovely technical results, but one with few if any real-world success 

> stories.  Sometimes, things just don't work out.)  I think there could 

> be some potential uses for terms for adherence to (confirmatory) 

> generalized blockmodel structure (in the Doreian/Ferligoj/Batagelj 

> tradition), though some of this can already be emulated using existing 

> tools; there has also been a relative dearth of empirical cases in 

> which complex block types have been shown to be important for 

> capturing network structure.  If such cases were to become more often 

> encountered, this would naturally motivate more work to model them.

>

> With respect to your second comment, I am not sure what you mean by 

> "violating" Hammersley-Clifford.  H-C provides one way of establishing 

> an equivalence between sets of network statistics and associated 

> dependence conditions; Pip Pattison, Gary Robbins, and others have 

> obtained various refinements to the original result (allowing for more 

> subtle conditions to be treated).  H-C and friends simply say 

> (effectively) that certain classes of statistics implement certain 

> kinds of dependence.  These are important results for constructing and 

> interpreting statistics, but they are not rules that can be violated.

>

> Hope that clarifies things,

>

> -Carter

>

> On 12/7/23 8:52 PM, Gotthardt, Daniel wrote:

>> Dear Harald,

>>

>> after Martinas very insightful message and considering that you have 

>> kinship and business ties but not so many node covariates, I am 

>> wondering if you need or should think of structural equivalance as a 

>> driving factor. With White and others there is a strong tradition of 

>> focussing on this for kinship networks and DiMaggio and Burt have 

>> studied the importance oft business roles and structural position. In 

>> your case that probably means non-local forms of equivalence 

>> (automorphic, role, etc) that might matter directly in the network 

>> behavior or could represent unmeasured node attributes. Feature and 

>> embedding based measures are more scalable and now allow to measure 

>> those concepts better in larger networks.

>>

>> To the best of my knowledge this is not considered offen in 

>> generative network models and i don't think that we can include those 

>> less-localized mechanisms directly (yet). Plesae let me know if this 

>> is a direction that makes sense for you from a theoretical point of 

>> view and also something that could be identified in your data. I am 

>> currently working on this in the context oft actor-oriented models 

>> but am interested in the potential of ergms in this regard as well.  

>> At least as exogenous covariates this might be possible but otherwise 

>> we might violate conditional independence (Hammersley-Clifford 

>> theorem). I am curious to hear about the thoughts of experienced ergm 

>> modelers on this, though.

>>

>> Best Regards,

>> Daniel

>>

>> --

>> Daniel Gotthardt, M.A.

>>

>> Wissenschaftlicher Mitarbeiter / Research Associate

>>

>> Universität Hamburg

>> Fakultät für Wirtschafts- und Sozialwissenschaften / Faculty of 

>> Business, Economics and Social Sciences

>> Fachbereich Sozialwissenschaften / Department of Social Sciences

>> Soziologie, insb. Digitale Sozialwissenschaft / Sociology, esp. 

>> Digital Social Science

>>

>> Max-Brauer-Allee 60

>> 22765 Hamburg

>> www.uni-hamburg.de 

>> <https://urldefense.com/v3/__http://www.uni-hamburg.de__;!!CzAuKJ42GuquVTTmVmPViYEvSg!Jy0dmFtPSz9FGZILsxIzHWpAcAK5wDvLWuQ2s4hKJdX0uaJX7imnKxe9w1W52yrNrJRKiI-YzcF0M4kcXbfma0JgQ-N6zkZ-$>

>>

>> ------------------------------------------------------------------------

>> *Von:* Martina Morris <morrism at uw.edu>

>> *Gesendet:* Donnerstag, 7. Dezember 2023 23:45:59

>> *An:* Harald Waxenecker

>> *Cc:* Gotthardt, Daniel; statnet_help at u.washington.edu

>> *Betreff:* Re: [statnet_help] fragmented bipartite network...

>> Hi Harald,

>>

>> You do have a complicated analysis here, and I'm a bit under-equipped 

>> to help you Dx what is going on, as I don't have much experience with 

>> either bipartite or multi-level nets (let alone both together!).

>>

>> What I can say, though, is that factor and covariate effects on the 

>> nodes are, in the non-multilevel context, one of the most important 

>> brakes on the feedback effects caused by dyad-dependent terms, making 

>> them more well-behaved and more likely to produce the kinds of 

>> networks we actually observe (caveat: sometimes those dependent 

>> effects are needed, see Carter's work on amyloid fibrils).

>>

>> In this case, it seems like you don't have many attributes to work 

>> with -- indeed, only on one of the modes.  For gender, I would fit as 

>> a factor btw, not a quantitative covariate, tho if there are only 2 

>> levels this will not have much impact.  But when I think about the 

>> goals of board composition in non-profits (the closest I get to your 

>> world), it's clear that gender is not the only attribute that 

>> influences board member invitations -- and I would expect the same 

>> would be true here.  You might try adding  family name as a 

>> bxnodefactor (will pick up both family size and family activity level 

>> differentials), or sociality for either (or both) modes (to condition 

>> on the degree of each node).  Your additional terms can then be 

>> interpreted as effects operating beyond these differences in degree. 

>> Degree distributions definitely influence component size 

>> distributions, up to a point, so if your model is not getting these 

>> right, you can start there.

>>

>> Thinking about the orgs, it seems there must be org attributes that 

>> influence the size and composition of the board.  Org size, sector, 

>> geographic location, age, specialization, etc. -- I can imagine all 

>> of these would influence board memberships.  Properties these nodes 

>> show in the other nets you have might be able to be represented on 

>> the cheap here as nodal attributes in this network. If these effects 

>> are at work -- and if you're not including them in the model, it is a 

>> form of mis-specification that compromises all of the other model 

>> estimates.

>>

>> Then there's homophily, which works differently in bip nets -- for 

>> one, it's a dyad-dependent term.  But it's also more complicated to 

>> think about.  Perhaps families might choose to specialize in an org 

>> sector, or maybe the opposite, they aim to integrate across sectors.  

>> Orgs might want diversity (on some measure) for members, which would 

>> show up as anti-homophily in bip two-paths.  Again though, this would 

>> require more measured attributes for both orgs and persons.

>>

>> Adding model terms like components is different.  In my modeling 

>> world, we want our (parsimonious) models to represent the mechanistic 

>> effects that may actually generate the ties in the network.  For us, 

>> component size distributions are an *output* of a network formation 

>> process, not the generating mechanism (people aren't creating ties 

>> with the explicit intent of structuring the network component size 

>> distributions, with one key exception, and that we do model).  We 

>> instead use the component size distribution as a goodness-of-fit 

>> indicator -- to test whether the mechanistic terms we included in our 

>> model reproduce these higher order excluded network stats.

>>

>> But your context may be different.  When an org board is formed, if 

>> there is an explicit strategy to create specific component structures 

>> in the overall network then those intentions should be included as 

>> model terms.  I can imagine that bridging structural holes might be 

>> one of those strategies.  But again, not my area of expertise.

>>

>> I'm not sure how much any of this helps your specific issues.  But 

>> when models don't fit the data properly, it's worth thinking about 

>> specification from first principles.  So I hope this helps.

>>

>> best,

>> Martina

>>

>> On Mon, Dec 4, 2023 at 12:28 AM Harald Waxenecker 

>> <waxenecker at fss.muni.cz> wrote:

>>

>>     Dear Tom, Martina, Carter and Daniel

>>

>>     Thank you for your supportive answers. __

>>

>>     First, I will try to address some of your questions. The

>>     dependent network is a bipartite business network (6902 persons x

>>     5178 companies), based exclusively on interlocking directorates.

>>     This dependent bipartite network represents the business ties of

>>     elite members in their home country. We include two covariates

>>     for the first node set (persons): /traditional surname/ and

>>     /gender/. Isolates in this network represent elite members

>>     without any business ties. We belief that isolated nodes are

>>     meaningful in this network; e.g., women are often constrained to

>>     ‘reproduction’ rather than participating in ‘production’

>>     (businesses). However, in different network layers they

>>     contribute to elite cohesion.

>>

>>     Regarding these different layers: we have six more networks. The

>>     first is a one-mode kinship network (6902x6902), and the others

>>     are bipartite networks (based on interlocks), where persons form

>>     the first node set and entities the second. Hence, all matrices

>>     share a consistent number of rows (n = 6902), while the number of

>>     columns varies according to the number of entities in each

>>     network layer: offshore companies in Panama (n = 1537), business

>>     associations (n = 128), non-profit organizations (n = 236),

>>     political parties (n = 55), and public entities (n = 431).

>>

>>     We employ ‘bipartite homophily terms’, as proposed by Metz et al.

>>     (2018) https://doi.org/10.1017/S0143814X18000181

>>     <https://urldefense.com/v3/__https://doi.org/10.1017/S0143814X18000181__;!!K-Hz7m0Vt54!mZ6U-5ef-FwMtvk7aI512iZKTS20PMt72wzLingnjcBUoo1ETmzgxIYYk_qPcMmHbtcEowX7XXdRKk_R_lJbhAPBGYqXcg$>,

>>     to test whether a common property (‘homophily’) of the nodes in

>>     the first node set, such as a shared attribute (gender,

>>     traditional surname), a direct tie (kinship relation), or a

>>     mutual membership in other bipartite layers (offshore companies,

>>     business associations, etc.) contribute to the probability of two

>>     individuals forming ties with the same company in the dependent

>>     network.

>>

>>     Regarding the modeling process, it´s true that the model we

>>     shared relies only on dyad-dependent terms.We always ‘come back’

>>     to this model specification because all our attempts, which

>>     certainly were also based primarily on dyad-dependent terms, did

>>     not produce better results. We explored various options,

>>     including nodematch to control for component membership to split

>>     the network into smaller fragments. Then we incorporated

>>     component membership of the nodes as constraint to induce network

>>     fragmentation. While this partially improved network

>>     fragmentation, problems with goodness-of-fit persisted.

>>     Additionally, we encountered some computational limitations while

>>     running these options.

>>

>>     Now, we have incorporated several of your recommendations,

>>     introducing dyad-independent terms and utilizing components()

>>     from the ergm.components package. Please find the new outcomes

>>     (model 0) attached. We've also attached summary files and

>>     component distribution for a comparative analysis between the

>>     observed network and the simulated network.

>>

>>     We also tried to include the terms compsizesum() and dimers()

>>     into the model; however, we observe degeneracy issues. In

>>     addition, we still could not get results with bridges(), because

>>     it seems to be very time consuming and/or needs much

>>     computational capacity.

>>

>>     I think this bridges-term relates somehow to your question

>>     @Martina about cross-group ties in the simulated data. Or maybe I

>>     am wrong. Please, could you explain that in more detail? Thanks.

>>

>>     Thank you again for your support. Looking very forward to read

>>     your thoughts and advice.

>>

>>     Kind regards,

>>

>>     Harald

>>

>>     El 1/12/23, 21:53, "[NOMBRE]" <daniel.gotthardt at uni-hamburg.de>

>>     escribió:

>>

>>     Hello Harald,

>>

>>     if I understand you correctly you have a within-mode network as

>>     well as

>>

>>     a bipartite network. James Hollway et al. (2017) has described an

>>

>>     approach to handle these kinds of combined networks as multilevel

>>     social

>>

>>     spaces with stochastic actor-oriented models:

>>

>>     https://www.cambridge.org/core/journals/network-science/article/abs/multilevel-social-spaces-the-network-dynamics-of-organizational-fields/602BB810A44497EBDE2A111A6C2771A3

>>     <https://urldefense.com/v3/__https://www.cambridge.org/core/journals/network-science/article/abs/multilevel-social-spaces-the-network-dynamics-of-organizational-fields/602BB810A44497EBDE2A111A6C2771A3__;!!K-Hz7m0Vt54!mZ6U-5ef-FwMtvk7aI512iZKTS20PMt72wzLingnjcBUoo1ETmzgxIYYk_qPcMmHbtcEowX7XXdRKk_R_lJbhAOR74XZsg$>

>>

>>

>>     - There are also some tricks to transform these types of networks

>>     into

>>

>>     an extended multimodal network matrix, exemplified e.g. in Knoke

>>     et al.

>>

>>     (2021):

>>

>>     https://www.cambridge.org/core/books/abs/multimodal-political-networks/agency-influence-power/57CB185C6E9429B34A9DE181C37EADF3

>>     <https://urldefense.com/v3/__https://www.cambridge.org/core/books/abs/multimodal-political-networks/agency-influence-power/57CB185C6E9429B34A9DE181C37EADF3__;!!K-Hz7m0Vt54!mZ6U-5ef-FwMtvk7aI512iZKTS20PMt72wzLingnjcBUoo1ETmzgxIYYk_qPcMmHbtcEowX7XXdRKk_R_lJbhAMG16sRdw$>

>>

>>     I personally don't know of any ergm model that can handle this

>>     kind of

>>

>>     co-evolution of one-mode and two-mode networks but some kind of

>>

>>     multilevel ergms (see Wang et al. (2013)

>>

>>     https://www.sciencedirect.com/science/article/abs/pii/S0378873313000051

>>     <https://urldefense.com/v3/__https://www.sciencedirect.com/science/article/abs/pii/S0378873313000051__;!!K-Hz7m0Vt54!mZ6U-5ef-FwMtvk7aI512iZKTS20PMt72wzLingnjcBUoo1ETmzgxIYYk_qPcMmHbtcEowX7XXdRKk_R_lJbhAMb4-hbuA$>)

>>

>>

>>     might be the way to go: - I'm sure others here know more about the

>>

>>     capabilities of ergm.multi though.

>>

>>     If these kinship structures explain the fragmentation of the

>>     bipartite

>>

>>     network, you might need to include them either directly with the

>>

>>     approaches above or construct some corresponding dyadic or monadic

>>

>>     covariates to represent the kinship structure in your single

>>     level network.

>>

>>     Best Regards,

>>

>>     Daniel

>>

>>     Am 01.12.2023 um 02:13 schrieb Martina Morris:

>>

>>     >

>>

>>     > Hi Harald,

>>

>>     >

>>

>>     > I'm looking for some clarification here, which I think Tom

>>     Kraft might

>>

>>     > also have wondered about.

>>

>>     >

>>

>>     > You say:

>>

>>     >>

>>

>>     >> Our research focuses on tie formation and elite cohesion,

>>     specifically

>>

>>     >> examining interlocking directorates and kinship relations. The

>>

>>     >> dependent bipartite business network comprises 6,902

>>     individuals and

>>

>>     >> 5,178 companies, exhibiting sparsity (density = 0.00012) and

>>

>>     >> fragmentation with 4,455 components, including 3,850 isolates

>>     in the

>>

>>     >> first mode (persons)

>>

>>     >>

>>

>>     > For a bipartite network ties are allowed only between modes

>>     (persons,

>>

>>     > companies), not within.  It's clear how interlocking

>>     directorates would

>>

>>     > meet that criteria. But kinship relations would be among

>>     persons, so

>>

>>     > within-mode, not between, and this would not be a bipartite

>>     network.

>>

>>     >

>>

>>     > Is the model you've sent us for the interlocking directorships

>>     only?

>>

>>     > And by isolates in the person mode, do you mean persons who are

>>     not

>>

>>     > affiliated with any of the companies?  If so, then it's a bit

>>     odd to

>>

>>     > include them in the bipartite network.

>>

>>     >

>>

>>     > I'm wondering if this problem is better posed as a multilevel

>>     network

>>

>>     > (not my area of expertise).

>>

>>     >

>>

>>     > thanks,

>>

>>     > Martina

>>

>>     >

>>

>>     >

>>

>>     > On Thu, Nov 30, 2023 at 4:33 PM Carter T. Butts <buttsc at uci.edu

>>

>>     > <mailto:buttsc at uci.edu>> wrote:

>>

>>     >

>>

>>     >     __

>>

>>     >

>>

>>     >     Hi, Harald -

>>

>>     >

>>

>>     >     Coexistence of large complex components does not generally

>>     occur

>>

>>     >     unless something drives the fragmentation, and this is what

>>     your

>>

>>     >     models are telling you: the terms you are currently using

>>     do not

>>

>>     >     include the forces that are sufficient to reproduce your

>>     component

>>

>>     >     size distribution.  That means that you need to think about

>>     why your

>>

>>     >     network is split into fragments, and include terms that

>>     capture the

>>

>>     >     relevant social forces.  Thinking about likely mechanisms

>>     is step

>>

>>     >     zero, so do that before anything else!  Guided by your

>>     substantive

>>

>>     >     knowledge of what is likely going on, you will next (as

>>     others have

>>

>>     >     said) want to look at covariate effects relating to

>>     differential

>>

>>     >     mixing, since those are your most obvious and most

>>     important sources

>>

>>     >     of heterogeneity. If you find that there is still more

>>

>>     >     fragmentation that can be explained by other means, you may

>>     need to

>>

>>     >     consider model terms relating directly to component count

>>     or size.

>>

>>     >     These are still somewhat experimental, and are currently

>>     sequestered

>>

>>     >     in an add-on package called ergm.components

>>

>>     >     (https://github.com/statnet/ergm.components

>>     <https://urldefense.com/v3/__https://github.com/statnet/ergm.components__;!!K-Hz7m0Vt54!mZ6U-5ef-FwMtvk7aI512iZKTS20PMt72wzLingnjcBUoo1ETmzgxIYYk_qPcMmHbtcEowX7XXdRKk_R_lJbhAMQjqlvCA$>

>>

>>     >    

>>     <https://urldefense.com/v3/__https://github.com/statnet/ergm.components__;!!K-Hz7m0Vt54!iKts-XLv39sY0gvmpW6MWLIxNMCNKjKQKOhJszIbp3PIy_J5mdLCs0MytfHsBu-cjnQjk997tCRX0MMs6LDW$

>>     <https://urldefense.com/v3/__https:/github.com/statnet/ergm.components__;!!K-Hz7m0Vt54!iKts-XLv39sY0gvmpW6MWLIxNMCNKjKQKOhJszIbp3PIy_J5mdLCs0MytfHsBu-cjnQjk997tCRX0MMs6LDW$>>).

>>     However, this package can be installed from github (see the

>>     github page), and the terms will work automagically with ergm()

>>     and friends once the package is loaded.  Depending on your

>>     situation, you may need or want to examine the components() or

>>     compsizesum() terms, both of which are documented within the package.

>>

>>     >

>>

>>     >     Hope that helps,

>>

>>     >

>>

>>     >     -Carter

>>

>>     >

>>

>>     >     On 11/30/23 9:58 AM, Harald Waxenecker wrote:

>>

>>     >>

>>

>>     >>     Dear ‘statnet community’,____

>>

>>     >>

>>

>>     >>     __ __

>>

>>     >>

>>

>>     >>     Our research focuses on tie formation and elite cohesion,

>>

>>     >>     specifically examining interlocking directorates and kinship

>>

>>     >>     relations. The dependent bipartite business network comprises

>>

>>     >>     6,902 individuals and 5,178 companies, exhibiting sparsity

>>

>>     >>     (density = 0.00012) and fragmentation with 4,455 components,

>>

>>     >>     including 3,850 isolates in the first mode (persons). The

>>     attached

>>

>>     >>     documents contain descriptives and the component size

>>     distribution

>>

>>     >>     from the observed network.____

>>

>>     >>

>>

>>     >>     ____

>>

>>     >>

>>

>>     >>     The fragmented structure is important, as other network

>>     layers,

>>

>>     >>     like kinship relations, are expected to contribute to the

>>     cohesion

>>

>>     >>     of this business network. We apply ERGM to model these

>>     processes,

>>

>>     >>     but we struggle to capture the fragmented structure of the

>>

>>     >>     observed network. The component size distribution of

>>     the simulated

>>

>>     >> network differs significantly. In addition, the goodness-of-fit

>>

>>     >>     (GOF) for k-stars (in both modes) and geodesic distances (Inf)

>>

>>     >>     shows significant results. All these results are also

>>     attached.____

>>

>>     >>

>>

>>     >>     ____

>>

>>     >>

>>

>>     >>     We've explored various options, including constraints, MCMC

>>

>>     >>     propositions, and simulated annealing, but haven't achieved

>>

>>     >>     success. Please, we would like to ask for your help to

>>     improve our

>>

>>     >>     model. Thank you!____

>>

>>     >>

>>

>>     >>     __ __

>>

>>     >>

>>

>>     >>     Kind regards,____

>>

>>     >>

>>

>>     >>     Harald____

>>

>>     >>

>>

>>     >>     __ __

>>

>>     >>

>>

>>     >>     __ __

>>

>>     >>

>>

>>     >>     __ __

>>

>>     >>

>>

>>     >>     --- ____

>>

>>     >>

>>

>>     >>     *Harald Waxenecker

>>

>>     >>

>>

>>     >>     *____

>>

>>     >>

>>

>>     >>     *Masaryk University | Faculty of social studies*

>>

>>     >>     Department of Environment Studies

>>

>>     >>     A: Jostova 10 | 602 00 Brno | Czech Republic

>>

>>     >>     E: waxenecker at fss.muni.cz <mailto:waxenecker at fss.muni.cz>____

>>

>>     >>

>>

>>     >>     __ __

>>

>>     >>

>>

>>     >>

>>

>>     >> _______________________________________________

>>

>>     >>     statnet_help mailing list

>>

>>     >>

>>     statnet_help at u.washington.edu  <mailto:statnet_help at u.washington.edu>

>>

>>     >>

>>     https://urldefense.com/v3/__http://mailman13.u.washington.edu/mailman/listinfo/statnet_help__;!!CzAuKJ42GuquVTTmVmPViYEvSg!KK5UcPVRvb25ILHn7wJt4TEsP-Ic39L133WdzimKJv-378bLqah-hO8Gm9Yd_qoWgV_tbzbT6swweifmS5mRRQ$

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>>

>>

>>     > _______________________________________________

>>

>>     >     statnet_help mailing list

>>

>>     > statnet_help at u.washington.edu

>>     <mailto:statnet_help at u.washington.edu>

>>

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

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>>

>>     >    

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

>>     <https://urldefense.com/v3/__http://mailman13.u.washington.edu/mailman/listinfo/statnet_help*3e__;JQ!!CzAuKJ42GuquVTTmVmPViYEvSg!Jy0dmFtPSz9FGZILsxIzHWpAcAK5wDvLWuQ2s4hKJdX0uaJX7imnKxe9w1W52yrNrJRKiI-YzcF0M4kcXbfma0JgQ4LfDjoH$>

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>>     >

>>

>>     >

>>

>>     > _______________________________________________

>>

>>     > statnet_help mailing list

>>

>>     > statnet_help at u.washington.edu

>>

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

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>>

>>     -- 

>>

>>     Daniel Gotthardt, M.A.

>>

>>     Wissenschaftlicher Mitarbeiter / Research Associate

>>

>>     Universität Hamburg

>>

>>     Fakultät für Wirtschafts- und Sozialwissenschaften / Faculty of

>>

>>     Business, Economics and Social Sciences

>>

>>     Fachbereich Sozialwissenschaften / Department of Social Sciences

>>

>>     Soziologie, insb. Digitale Sozialwissenschaft / Sociology, esp.

>>     Digital

>>

>>     Social Science

>>

>>     Max-Brauer-Allee 60

>>

>>     22765 Hamburg

>>

>>     www.uni-hamburg.de

>>     <https://urldefense.com/v3/__http://www.uni-hamburg.de__;!!K-Hz7m0Vt54!mZ6U-5ef-FwMtvk7aI512iZKTS20PMt72wzLingnjcBUoo1ETmzgxIYYk_qPcMmHbtcEowX7XXdRKk_R_lJbhAPAULQvFQ$>

>>

>>     _______________________________________________

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>>

>>

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