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This title is an effort to elicit a simplified explanation as to how come the transitiveties term causes less numerical instability than ttriple when fitting an Exponential Random Graph Model (ERGM).

From the ERGM R Package manual:

Transitive ties: This term adds one statistic, equal to the number of ties i–>j such that there exists a two-path from i to j

Transitive triples: By default, this term adds one statistic to the model, equal to the number of transitive triples in the network, defined as a set of edges i→j, j→k, i→k.

From Novel Approaches to Degeneracy in Network Models by Timothy Blackburn,

In an effort to maximize entropy, the ERGM can be thought of as "spreading out" mass across the graph space as much as possible while still maintaining the mean constraints. This sometimes leads to a large amount of mass being placed on extremal configurations (such as the empty and complete graphs) and very little mass being placed on the observed graph. This problem is referred to as degeneracy, and when a model is degenerate predictions from it will be invalid.

So, we can fire up R and run an ERGM model with transitiveties and an ERGM model with ttriple; even if each model has a nicely prepared network as input, we can expect the former to converge easily while the latter will make us wait; it will probably never fit. The computer will probably get stuck into an empty graph black hole.

But how does transitiveties do that?

I imagine the question also applies to the cyclicalties term.

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The answer is in the definitions of the terms. Essentially with the transitiveties term you count ties for which adjacent nodes have at least one partner in common (that's the two-path part in the definition). In contrast, the ttriple term counts triangles. What leads to degeneracy is the fact that a single flip of a dyad (one MCMC step) can create a big change in graph ttriple statistic:

  • adding one tie to the network can create a whole lot of triangles
  • removing one tie can destroy a lot of triangles

All of this leads to a bimodal distribution that is concentrated on networks which are almost complete (full) or almost empty. The transitiveties does not suffer from that problem that much.

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  • $\begingroup$ That's very useful, thank you very much! $\endgroup$ Sep 6, 2022 at 0:49

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