Tutorial 5: Constraints, two-mode networks and operators

Goal. Three tools beyond the basic model: constraints that restrict the networks the model is defined on, two-mode terms for networks between two kinds of nodes (people and clubs), and operators that evaluate terms on a derived network.

  • Builds on Tutorial 2 to Tutorial 4.
  • Needs the core nwcommands and nwergm (see Install).
  • The operators in the last section need nwergm 1.1.0 or later. The other sections work with 1.0.1 as well.

Data. The Florentine marriage network and the Lazega friendship network (both shipped with nwcommands), and a small two-mode network of 30 people and 12 clubs that is simulated in the script (so that you can recreate it exactly; it is not a real data set). Run with nwergm 1.3.1 and the core 3.10.0; the script and log are in example.do and example.log. Output is shortened to the coefficient table and the log-likelihood line.

Constraints

A constraint changes the set of networks that the model is defined on. constraints(edges) holds the number of ties fixed: the model then describes how a fixed number of ties is arranged, and the density parameter drops out. Compare the unconstrained and the constrained fit:

. nwwebuse florentine, nwclear

. nwergm flomarriage, edges nodecov(wealth) gwesp(0.5) seed(1)
...
--------------------------------------------------------------------------------
   flomarriage | Coefficient  Std. err.      z    P>|z|     [95% conf. interval]
---------------+----------------------------------------------------------------
         edges |  -2.599772   .5426303    -4.79   0.000    -3.663308   -1.536237
nodecov_wealth |   .0107953   .0051785     2.08   0.037     .0006457    .0209449
     gwesp_0.5 |  -.0170638   .2676109    -0.06   0.949    -.5415716    .5074439
--------------------------------------------------------------------------------

MCMLE converged after 1 iteration(s).
...
Log-likelihood:   -51.5528 (df = 3)   AIC:     109.1   BIC:     117.5   (MC std. err. =  .00149)

. nwergm flomarriage, edges nodecov(wealth) gwesp(0.5) constraints(edges) seed(1)
...
--------------------------------------------------------------------------------
   flomarriage | Coefficient  Std. err.      z    P>|z|     [95% conf. interval]
---------------+----------------------------------------------------------------
         edges |          0  (omitted)
nodecov_wealth |   .0100932   .0048998     2.06   0.039     .0004898    .0196966
     gwesp_0.5 |   .0213663   .2638801     0.08   0.935    -.4958292    .5385619
--------------------------------------------------------------------------------

MCMLE converged after 1 iteration(s).
...
Log-likelihood:   -55.3402 (df = 2)   AIC:     114.7   BIC:     120.3   (MC std. err. =  .16122)

nwergm prints a note that the constraint holds edges constant, and the coefficient of edges is shown as omitted. The other coefficients are close to those of the unconstrained model. Other constraints written as in R’s constraints= include degrees, odegrees, idegrees, bd(), blocks(), blockdiag(), fixedas(), fixallbut() and observed; see help nwergm_constraints. The related obsconstraints() describes how the network was observed.

Two-mode networks

In a two-mode (bipartite) network the ties go between two kinds of nodes, here people (mode 1) and clubs (mode 2). We build the network from a 12 x 30 matrix of memberships with nwset ..., bipartite; a person’s group (kind, 1 or 2) and a club’s type (also 1 or 2) are stored in one variable, the persons first. The simulation makes ties more likely when person and club are of the same kind (probability 0.30, otherwise 0.08), so we know what the model should find.

. nwset, mat(B) bipartite name(clubs)
. set obs 42
. generate byte kind = .
. mata: st_store(., "kind", (g1 \ g2))

. nwsummarize
--------------------------------------------------
   Network name:  clubs
   Network id:  1
   Directed: false
   Valued: false
   Two-mode: true
   Nodes: 42
      Level 1: 30 ()
      Level 2: 12 ()
   Selfloop: false
   Edges: 71
   Minimum value:  0
   Maximum value:  1
   Density:  .197
   Temporal: false

Terms that name a mode follow R’s b1/b2 terms, with the digit moved to the end (bdegree1() is R’s b1degree, bgwdegree2() is gwb2degree). nodemix() on a two-mode network gives one coefficient per combination of the person’s and the club’s level; the first cell (1,1) is the reference.

. nwergm clubs, edges nodemix(kind)
...
----------------------------------------------------------------------------------
           clubs | Coefficient  Std. err.      z    P>|z|     [95% conf. interval]
-----------------+----------------------------------------------------------------
           edges |  -.7563261   .1787266    -4.23   0.000    -1.106624   -.4060283
nodemix_kind_2_1 |  -1.512357   .3931267    -3.85   0.000    -2.282872   -.7418434
nodemix_kind_1_2 |  -2.379168   .6162542    -3.86   0.000    -3.587004   -1.171332
nodemix_kind_2_2 |  -.2340726   .3707259    -0.63   0.528    -.9606821    .4925369
----------------------------------------------------------------------------------
...
Log-likelihood:  -160.5841 (df = 4)   AIC:     329.2   BIC:     344.7   (MC std. err. =       0)

The cell (1,1) is the reference. Ties between persons of kind 2 and clubs of kind 1 (2_1) and between persons of kind 1 and clubs of kind 2 (1_2) are much less likely than same-kind ties (coefficients -1.51 and -2.38), whereas (2,2) does not differ from (1,1) (-0.23, p = 0.53). This is exactly the pattern put into the simulated data. The model is dyad-independent, so it is fitted by pseudolikelihood. A degree term for the clubs makes it dyad-dependent:

. nwergm clubs, edges nodemix(kind) bgwdegree2(0.5) seed(1)
...
----------------------------------------------------------------------------------
           clubs | Coefficient  Std. err.      z    P>|z|     [95% conf. interval]
-----------------+----------------------------------------------------------------
           edges |  -.7414179   .1930383    -3.84   0.000    -1.119766   -.3630697
nodemix_kind_2_1 |  -1.506756   .4068944    -3.70   0.000    -2.304254   -.7092575
nodemix_kind_1_2 |  -2.212356   .5936993    -3.73   0.000    -3.375985   -1.048727
nodemix_kind_2_2 |  -.0559743   .4063925    -0.14   0.890    -.8524889    .7405403
  bgwdegree2_0.5 |  -1.690794   1.394711    -1.21   0.225    -4.424378     1.04279
----------------------------------------------------------------------------------

MCMLE converged after 2 iteration(s).
...
Log-likelihood:  -160.1196 (df = 5)   AIC:     330.2   BIC:     349.7   (MC std. err. =   .0943)

The geometrically weighted degree of the clubs, with decay 0.5, is not significant here (-1.69, s.e. 1.39): there is no evidence that the popularity of the clubs is unevenly distributed. The other terms for two-mode networks are listed in help nwergm_bipartite: concurrency, minimum degree and degree ranges, stars, covariate ranges, shared-partner terms (dsp, gwdsp, coincidence) and the curved versions with an estimated decay.

Operators (version 1.1.0)

An operator evaluates terms on a derived network, or combines and transforms terms: Sum, Prod, S (the subgraph induced by a set of nodes), F (filtered ties), NodematchFilter, Symmetrize, Project, I, For, Label, Log, Exp, Offset and Curve. They are written like R’s operator terms inside operators(); the terms inside an operator are options of nwergm. Coefficient names follow R’s, with . replaced by _. Two examples.

Proj1() applies terms to the one-mode projection of the people: two people are tied if they share a club. edges and triangle of the projection ask whether people who belong to common clubs are unusually often connected, over and above the model for the two-mode ties.

. nwergm clubs, edges nodemix(kind) operators(Proj1(edges triangle)) seed(1)
...
----------------------------------------------------------------------------------
           clubs | Coefficient  Std. err.      z    P>|z|     [95% conf. interval]
-----------------+----------------------------------------------------------------
           edges |  -1.103858   .3445076    -3.20   0.001     -1.77908   -.4286353
nodemix_kind_2_1 |  -1.775325   .5405127    -3.28   0.001    -2.834711   -.7159399
nodemix_kind_1_2 |  -2.355423   .6577301    -3.58   0.000     -3.64455   -1.066296
nodemix_kind_2_2 |  -.3295673   .4577906    -0.72   0.472     -1.22682    .5676857
     Proj1~edges |   .1847007    .198093     0.93   0.351    -.2035545    .5729559
  Proj1~triangle |   -.012532   .0275408    -0.46   0.649    -.0665108    .0414469
----------------------------------------------------------------------------------

MCMLE converged after 3 iteration(s).
...
Log-likelihood:  -158.9038 (df = 6)   AIC:     329.8   BIC:     353.1   (MC std. err. =  .09301)

Neither projected term is significant (Proj1~edges 0.18, s.e. 0.20), as the data were generated without any such structure.

Symmetrize() evaluates a term on the undirected network that is obtained from a directed one. With rule(strong), two lawyers are tied if the friendship is reciprocated, so Symmetrize(strong)~triangle counts triangles among mutual friendships:

. nwergm lazega_fr, edges mutual nodematch(prac) operators(Symmetrize(triangle, rule(strong))) seed(1)
...
---------------------------------------------------------------------------------------------
                  lazega_fr | Coefficient  Std. err.      z    P>|z|     [95% conf. interval]
----------------------------+----------------------------------------------------------------
                      edges |  -2.736101   .0829356   -32.99   0.000    -2.898652    -2.57355
                     mutual |    .900128   .1705302     5.28   0.000     .5658949    1.234361
             nodematch_prac |   .8942052   .1007393     8.88   0.000     .6967598    1.091651
Symmetrize(strong)~triangle |   .7278742   .4769188     1.53   0.127    -.2068695    1.662618
---------------------------------------------------------------------------------------------

MCMLE converged after 2 iteration(s).
...
Log-likelihood: -1702.0947 (df = 4)   AIC:    3412.2   BIC:    3438.2   (MC std. err. =  .08466)

Closure among reciprocated friendships is positive but not significant (0.73, s.e. 0.48, p = 0.13), next to the reciprocity (mutual) and same-practice effects. The AIC of this model (3412.2) can be compared with the two Lazega models of Tutorial 4, which have the same other terms and the same N: 3409.9 without a closure term and 3349.9 with gwesp(0.5). Closure among reciprocated friendships alone hardly improves the model (3412.2 against 3409.9), while the gwesp term does: the closure in this network is not restricted to mutual ties.

Operators cannot be combined with constraints(), obsconstraints(), freedyads(), blockdiag() or fixdensity, and a curved (free-decay) term cannot be used inside an operator. The complete list and arguments are in help nwergm_operators.

Things to try

  • Change the decay of bgwdegree2() from 0.5 to 1 and compare the log-likelihoods.
  • Use Label() to give a term your own coefficient name, and Offset() to hold a coefficient at a fixed value.
  • Replace rule(strong) by rule(weak) in the Symmetrize example.
  • Constrain the degrees of the people in the club model with constraints(b1degrees) and read the warnings: edges is then fixed by the constraint and the other terms may become collinear.

Where next

nwergm and R’s ergm lists what is the same and what is missing; Terms is the map of all term families.

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nwcommands is free software under the GNU General Public License, version 3, as are the model packages nwsaom, nwergm, nwdynam and nwrem. Citing and licenses · Get help · GitHub. Issues with the core go to the nwcommands repository; issues with a model package go to that package's own repository.

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