• Nature/people form Sj subnetworks of type j each independently with probability pj
  • May intersect and overlap
  • Weobserveresultingnetwork, infer the $p_j’s$

Estimation - Two Approaches

  • Sparse graphs: rare incidentals, Direct estimation is valid / consistent
  • Algorithm: corrects for small n, and provides estimates for non-sparse (see CJ paper)


  • Incidentals generated by combinations of other subgraphs
  • Sparsity definition relates rates of all subgraphs to each other (none grow too quickly)
  • Intutive example: links and triangles
    • $p_L = O(n^{-1/2})$, $p_T = O(n^{-3/2})$
    • Typical node involved in less than $n^{1/2}$ links, $n^{1/2}$ triangles

Links Form: Incidental Triangle


Estimation: Triangles

number of nodes = 42

number of triangles = 10

n choose 3 = 424140 / 321 = 11480

$\hat p_t = 10 / 11480 = .00087$

Theorem: Consistency and Distribution

Consider a sequence of sparse SUGMs

$S$: how many did you actually observe

$\bar s$: how many could there have been

The empirical frequency $\hat p^n_j = S^n_j / \bar S^n_j$ is (ratio) consistent:

$\hat p^n_j / p^n_j -> 1$ and $ D^{1/2}(\hat p^n - p^n) -> N(0, 1)$

Need for SERGMs/SUGMs:

  • Examine data from 75 Indian villages from BCDJ '13
  • How well do the model-recreated networks match real networks on non-modeled characteristics
  • Estimate SUGM based on covariates, allowing for triangle counts
  • Estimate standard link-based (block) model based on covariates
  • Does SUGM do better than block model at recreating networks?
  • Block model
    • prob of a link of both of the same category or similar to each other
    • prob of link if different
  • SUGM add in
    • prob of triangle if all similar
    • prob of triangle if some different

block model is a special case of a SUGM where we just look at links

  • Step 1: Estimate models
    • Block model, estimate $p_{LinkSame}$ $p_{LinkDiff}$
    • SUGM, estimate $p_{LinkSame}$ $p_{LinkDiff}$, $p_{TriadSame}$ $p_{TriadDiff}$
  • Step 2: randomly generate networks
    • Block model - randomly generate links
    • SUGM - randomly generate links, triangles...
  • Step 3: After generate networks, we can see whether or not these networks recreate the actual, original observations.


Dependencies are really important to pick up in social networks. Why? well that's the whole nature of social

  • "Social" by definition generates dependencies
  • Need tractable models to capture / test these
  • ERGMs are rich family, but not always accurately estimable
  • SERGMs and SUGMs offer easy and consistent estimation.

Network Models

  • Statistical models offer a medium, but also need models in context
  • Understand dependencies? Friends of friends, social enforcement...
  • What should we be testing for?
  • Example, see lecture 4.9

Strengths Random Networks:

  • Generate large networks with well identified properties
  • Mimic real networks (at least in some characteristics)
  • Tie specific properties to specify processes

Weaknesses of Random Network Models

  • Missing the "Why"
    • Why this process?(lattice, preferential attach...)
  • Missing implications of networks structure: context or relevance
    • welfare, efficiency?
  • Literature is missing careful empirical analysis of many "stylized facts" (small worlds, power laws, clustering...)
    • ERGMs have filling that niche, but need estimable models
    • New models are emerging