ECC3841 Chap.2 Network Formation and Stability
Network Formation and Stability
Network formation links observed structure to a process. Block models let link probabilities depend on type, small-world processes combine local clustering with rare shortcuts, and preferential attachment gives already connected nodes a greater chance of receiving new links. Strategic formation instead asks whether agents add or delete links given benefits, costs and indirect connections.
Pairwise stability is a local incentive concept and can differ from efficiency because endpoints ignore effects on other nodes.
What this chapter covers
- 01
Type-dependent link probabilities in block models
- 02
Correct same-type and cross-type opportunity counts
- 03
Small-world rewiring and shortcut paths
- 04
Preferential attachment and degree inequality
- 05
Private benefits and costs of links
- 06
Nash link choices and bilateral consent
- 07
Pairwise stability for addition and deletion
- 08
Externalities that separate stability from efficiency
Worked example · free
Test an absent link
- +1A change is four minus five, equal to negative one.
- +1B change is three minus five, equal to negative two.
- +1Both endpoints must gain for an absent link to be added.
- +1The link stays absent because neither endpoint gains.
Key terms
- Block model
- A formation model with link probabilities that depend on node types or attributes.
- Small-world network
- A network combining strong local clustering with short global paths.
- Preferential attachment
- A growth process in which highly connected nodes attract new links more often.
- Pairwise stability
- A condition ruling out profitable unilateral deletion and mutually profitable addition.
- Link externality
- A benefit or cost of a link experienced by nodes outside its endpoints.
Network Formation and Stability FAQ
Does a block model prove preference for similarity?
No. Higher same-type link rates describe mixing after opportunity adjustment, but preferences, meeting opportunities and institutional sorting can create the same pattern. A complete response separates descriptive structure, private link incentives and the external effect on other nodes.
Why can stable networks be inefficient?
Endpoints consider their own payoff changes when adding or deleting a link. They may ignore shorter paths, congestion or information benefits experienced by other nodes. A complete response separates descriptive structure, private link incentives and the external effect on other nodes.
What does preferential attachment explain?
It explains how cumulative advantage can create a skewed degree distribution during network growth. It does not by itself explain type mixing or strategic consent. A complete response separates descriptive structure, private link incentives and the external effect on other nodes.
How should I test stability?
Check every existing link for profitable unilateral deletion and every absent link for a gain to both endpoints. Then calculate total welfare separately if efficiency is asked. A complete response separates descriptive structure, private link incentives and the external effect on other nodes.
Exam move
Practice possible-pair denominators until same-type and cross-type cases are automatic. Write n(n-1)/2 for an undirected within-type block and nAmB for a cross-type block before inserting observed links. Then state that the resulting probabilities describe mixing rather than proving preferences. Compare formation models by inputs, process and predicted statistic.
A block model needs types and link probabilities, a small-world model needs local structure and rewiring, and preferential attachment needs sequential growth with degree-dependent arrival. For each mechanism, list one statistic it is designed to reproduce and one feature it can miss. Use this to avoid explaining every observed graph with the same story.
In strategic problems, create a table with the current network, the proposed single-link change and each endpoint payoff difference. Existing links are tested for unilateral deletion; absent links require gains for both endpoints. Keep strict and weak inequalities visible. Do not jump from pairwise stability to efficiency. Recompute total welfare and name the third-party externality that creates a wedge.
When costs or indirect benefits change, predict the stable architecture before doing algebra: complete, star, path or empty. Check that prediction against every required link test. If a question supplies a final network and asks for parameters, derive inequalities rather than guessing a point value.
Conclude with the scope of the solution concept: pairwise stability checks one link at a time and does not rule out profitable coordinated rewiring. In final revision, take the same three-node example and vary link cost across several values. Track exactly where deletion, addition and welfare comparisons change. That parameter map is more useful than memorizing a single stable graph.
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