Bipartite networks

A bipartite (two-mode) network has two kinds of vertices, with ties only between kinds: people and the groups they belong to, firms and the boards their directors sit on, authors and papers. Davis, Gardner and Gardner’s (1941) Southern Women is the classic one: 18 women and the 14 social events each attended.

import ergmx
from ergmx import datasets

davis = datasets.load("davis")
print(datasets.describe("davis"))

Davis, Gardner and Gardner's (1941) Southern Women: the attendance of 18 women at 14 social events in Natchez, Mississippi, in the 1930s. Bipartite: the women are the first mode, the events the second. Vertex attributes: type (True for events, igraph's convention) and bipartite (0 or 1, networkx's). Fit it with bipartite=True. From their book Deep South (University of Chicago Press; reissued 2022, doi:10.7208/chicago/9780226817996.001.0001).

Declaring the modes

ergmx reads each vertex’s mode from a vertex attribute: false (or 0) for the first mode, true (or 1) for the second. Pass its name as bipartite=, or bipartite=True for igraph’s type or networkx’s bipartite attribute, which their bipartite functions create:

ergmx.summary_stats(davis, "edges + b1star(2) + b2star(2) + cycle(4)", bipartite=True)

{'edges': 89.0, 'b1star2': 214.0, 'b2star2': 322.0, 'cycle4': 341.0}

b1star(2) counts the pairs of events each woman attended, b2star(2) the pairs of women at each event, and cycle(4) the pairs of women who attended the same two events. In R’s ergm, the first mode is the first bipartite vertices of the network; in ergmx the vertices can be in any order.

Fitting

Only ties between the modes are modeled: the dyads within a mode are fixed, and the sample size of BIC is the 18 x 14 = 252 pairs of a woman and an event. Do women who attended many of the same events attend more together, beyond what their activity explains?

fit = ergmx.ergm(davis, "edges + gwb1dsp(0.5, fixed=TRUE)", bipartite=True, seed=1)
fit.summary()

Monte Carlo Maximum Likelihood Results:

                    Estimate  Std. Error  MCMC %  z value  Pr(>|z|)
edges                -1.2450      0.2585       0   -4.817    <1e-04 ***
gwb1dsp.fixed.0.5     0.3205      0.1239       0    2.587   0.00968 **
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Log-likelihood: -160.8087 (MC SE 0.027)   AIC: 325.6174   BIC: 332.6763
Converged after 3 iterations (4 chains, 1024 samples).

The positive gwb1dsp coefficient says yes: pairs of women tend to share several events. R’s ergm gives −1.25 and 0.33 for this model.

The terms of bipartite networks are in the term reference: stars, degrees, factors, covariates and homophily (b1nodematch) for each mode, and their shared partners. edgecov takes a first-mode by second-mode matrix, as in ergm.

Goodness of fit

For bipartite networks, gof() compares, as ergm does, the degrees of each mode, the dyadwise shared partners and the geodesic distances:

fit.gof(seed=1).plot();

Simulation, constraints, missing ties and the other features work as for other networks; pass bipartite= to ergmx.simulate() and ergmx.gof() when you give them a network rather than a fit.