[statnet_help] fragmented bipartite network...

[Carter T. Butts]

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

>

>     >>

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>

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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

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