When Do Alternatives Threaten a Tie?
Opportunity Sets and Relational Persistence
Roberto Cantillan
Department of Sociology | Pontificia Universidad Católica de Chile
Homophily Explains Which Ties Form
Similarity structures social relations, through choice and through structured opportunity.
- We know a great deal about which kinds of ties occur
- We know much less about whether a compatible tie stays attached to the same person
Whether a tie survives depends not only on how well it fits, but on what else is available.
Feld 1981; McPherson, Smith-Lovin & Cook 2001; Zeng & Xie 2008; Kossinets & Watts 2009
Same Fit, Different Replaceability
- Two ties in which ego and alter share three characteristics
- In the first, few alternatives offer a comparable fit
- In the second, twenty classmates reproduce it
Homophily describes fit; relational redundancy describes how reproducible that fit is.
A dyadic analysis sees the same tie in both cases. The difference lies in the opportunity set outside the dyad.
Blau: Opportunity as Headcount or as Value?
Two premises
- Associations are more prevalent between persons in proximate positions
- Associations depend on opportunities for contact
Blau 1977, AJS 83:26–54
Two readings
- Read literally, the opportunity premise counts people
- Read with proximity, it weights them: an alter threatens an incumbent to the extent that it approximates the incumbent’s fit
Does persistence depend on how many alternatives a person has, or on how good they are?
Alternatives Matter — but How?
That alternatives destabilize relationships is not new.
- Dependence is conditional on alternatives
- Local supply of spousal alternatives raises divorce risk
- Actor-oriented network models treat tie maintenance as a choice among alternatives
What is missing is a tie-level estimand that separates the number of alternatives from their value, and a way to test which one matters.
Emerson 1962; Rusbult 1980; South & Lloyd 1995; Snijders 2001; Stadtfeld & Block 2017
What We Do, and What Is New
Not new: inclusive value (McFadden 1978), consideration sets (Manski 1977), the number of alternatives in the logit (Ackerberg & Rysman 2005), structural equivalence as competition (Burt 1987). New: asking this of ties, with an estimand built for it. Not Burt’s redundancy (next slides).
Three questions follow: does size matter, does compatibility protect a tie, and does the departed alter shape turnover?
I. Model
The Idea in Plain Words
Ego has a friend. What makes that friend easy to leave? Look at the people ego could befriend instead.
| Ego’s alternatives | How many | How good, relative to the friend | Threat to the tie |
|---|---|---|---|
| A crowd of poor matches | many | all far worse | low |
| A crowd with a few near-equals | many | a few as good | high |
| A handful, all near-equals | few | all as good | high |
- If number drives departures, ties with more alternatives end more often, even when the alternatives are poor (\(\alpha\) near 1)
- If only quality matters, number adds nothing once quality is held fixed (\(\alpha\) near 0)
The data decide: we regress whether a tie persists on its fit, on how many alternatives there are, and on how good they are.
Relational Redundancy
\[ C_{ij} = \underbrace{\log n_{ij}}_{\text{volume}} + \underbrace{\bar L_{ij}-U_{ij}}_{\text{competitive quality}} \]
\(\sum_q w_q M_{ijq}\) · multidimensional fit
opportunity as a headcount
\(\log\) mean of \(e^{U_{ik}}\) · value-weighted opportunity
- The log-sum of alternatives’ utility relative to the incumbent
- Splits exactly into the size of the opportunity set and its relational composition
- Distant alters add almost nothing; proximate alters add most
The decomposition separates how many alternatives there are from how good they are.
Inclusive value: McFadden 1978
Not Burt’s Redundancy
| Burt (1992): structural holes | Ours: relational redundancy | |
|---|---|---|
| Redundant is… | a contact who leads to the same people as ego’s other contacts | an incumbent whose compatibility feasible alternatives could reproduce |
| Compared with | ego’s existing contacts | people ego is not tied to |
| Computed from | ties among ego’s contacts (cohesion, structural equivalence) | compatibility on a choice scale, and the size of the roster |
| Consequence | the contact adds little new information or control | the tie is easier to give up |
- An ego whose contacts are all unconnected has no Burtian redundancy, yet any one of them can be highly redundant in our sense
- The closer antecedent is Burt’s (1987) structural equivalence as competition: equivalent actors are substitutes. Burt defines equivalence by shared ties; we define it by shared compatibility
- The two meet in friendship: classroom pressure comes from classmates who are already ego’s contacts
One Logit, Three Parameters
\[ \operatorname{logit}\Pr(\text{tie persists}) \approx \kappa\big[\rho+(1+\color{#175c50}{\gamma})\,U_{ij} -\color{#b33a3a}{\alpha}\log n_{ij}-\bar L_{ij}\big] \]
\(\alpha\) and \(\gamma\) are ratios: pooling protected and contested ties leaves them unchanged.
The Classical Hypotheses Are Nested
\(\alpha=1\)
Full-set evaluation
Every available alter competes with the incumbent. The default of actor-oriented models.
\(b_{\log n}=b_{\bar L}\)
\(\alpha=0\)
Bounded consideration
Only the value of alternatives matters, not their number.
\(b_{\log n}=0\)
\(\gamma=0\)
Pure relative choice
Compatibility protects a tie only through its value in choice.
\(b_U=-b_{\bar L}\)
Each hypothesis is one linear restriction; confidence sets follow by test inversion.
Consideration sets: Manski 1977; Abaluck & Adams-Prassl 2021. Number of alternatives in the logit: Ackerberg & Rysman 2005. Test inversion: Fieller 1954.
The Parameters Are Not Scale-Free
- \(\bar L\) is a log-sum: \(\bar L(cU)\neq c\,\bar L(U)\)
- Rescaling compatibility changes which alternatives dominate
- A rule-based score (e.g., rarity weights) mixes the parameters with the rule’s scale
Calibrate from partner choice
Conditional logit of observed nominations on match indicators, one stratum per ego.
Weights cross-fitted: each school’s weights come from the other schools.
Tests of mechanisms that operate through alternatives are tests of a utility scale as much as of a mechanism.
After an Exit: Memoryless Entry
Under relative choice, the entrant is drawn with probability \(\propto e^{U_{ik}}\)
What the model implies
- Who enters does not depend on who left
- Same-profile alternatives do not compete harder, given \(\bar L\)
What would break it
- Durable taste: ego recruits the kinds of alters it already has
- Positional demand: a vacated position attracts a similar entrant
Profiles can persist because positions are sought — or because the ecology keeps supplying them.
Vacancy chains: White 1970. Turnover with stable composition: Wellman et al. 1997.
II. Data and Research Design
Two Settings with Observed Opportunity Sets
Online discussion (DerStandard)
- A decade of threaded replies, 37 three-month windows
- 529,282 incumbent tie-periods (deleted accounts removed)
- Compatibility: topical similarity
- Opportunity set: prior co-presence in threads
- Ego fixed effects: opportunity varies within ego
School friendship (FIS)
- 2,928 pupils, 10 schools, 29 grade-cohorts, six waves
- 34,126 incumbent tie-periods
- Compatibility: sex, origin, generation, religiosity, lifestyle
- Opportunity set: grade roster; every nomination falls inside it
- Classes change within pupils between waves
Fraxanet et al. 2026; Leszczensky et al. 2022
From Observed Rosters to Parameters
01
Opportunity set
- Pretreatment \(\mathcal O_i\)
- Every tie must fall inside it
02
Choice scale
- Partner-choice conditional logit
- Cross-fitted weights
03
Decomposition
- \(U_{ij}\), \(\bar L_{ij}\), \(\log n_{ij}\)
- All on the choice scale
04
One logit
- \(\hat\kappa\), \(\hat\alpha\), \(\hat\gamma\) by ratios
- Test inversion; few-cluster wild bootstrap
The same compatibility scale feeds the persistence logit and the turnover diagnostics.
Wild restricted score bootstrap: Kline & Santos 2012
Checking the Method: What We Simulated
01
A truth we set
- How much of the pool ego weighs: \(\alpha\in\{0,.5,1\}\)
- Whether compatibility protects beyond choice: \(\gamma\in\{0,.3\}\)
02
Data that obey it
- 20,000 artificial egos, each with 15–180 alternatives described by four binary traits
- Ego holds \(K\) ties; each is contested with probability \(c\)
- Ego compares it with a random sample of \(12\,(n/60)^{\alpha}\) alternatives and keeps the best
03
An analyst who does not know it
- Sees the whole opportunity set and who stayed
- Runs our one persistence logit and reports \(\hat\alpha\), \(\hat\gamma\)
- Repeats with fresh random draws (Monte Carlo) to see the spread
Two ways to score compatibility. Choice scale: the true utility that generated the data. Rarity rule: each shared trait weighted by how rare it is, a common shortcut.
Because the truth is known, any miss belongs to the estimator or the scale, not to the world.
What the Estimators Return
Each panel is one design: \(K\) = ties ego holds; \(c\) = share of ties contested each period. Dashed line = truth; bars = 95% Monte Carlo intervals.
On the choice scale, \(\alpha\) is recovered and \(\gamma\) is conservative (attenuated, never inflated). On a rarity rule, \(\alpha\) is inflated and a premium appears where none exists.
III. Findings
Online: Value Counts Far More Than Number
Opportunity elasticity
.23
95% CI .11 to .36
.18 to .20 across audit specifications
Full-set evaluation (\(\alpha=1\))
rejected
\(p<10^{-10}\); also in each half of the decade
(first half \(p=.008\))
Compatibility premium
.33
95% CI .10 to .66
- Not constant over the decade: \(\hat\alpha\) is .42 (.21 to .76) in the first half and .09 (−.09 to .27) in the second; the premium falls from .88 to .07
- Established ties are less exposed (\(\hat\kappa\) .72 → .59) and their premium includes zero (.14; −.13 to .63)
Persistence responds far more to how good the alternatives are than to how many there are.
Friendship: Competition Comes From Ego’s Network
Pupils may nominate anyone in their grade, yet 81% of incumbent friends are classmates. Within pupils, with separate value and size for each group of alternatives:
\(b_{\bar L}\), \(p<.001\)
ratio of slopes, upper end of the 95% set
friends of friends \(-.09\), others \(-.05\), both n.s.
Within-class opportunity elasticity
.48
95% set .30 to .85 (grade-cohort clusters)
- Consideration that ignores class size: rejected (\(p<.001\))
- Full evaluation of the class: \(p=.024\) by cohort, \(p=.076\) by school
Alternatives threaten a friendship when they sit in the part of the setting ego draws on (the class, and within it ego’s own contacts); there, value counts about twice as much as number.
The Friendship Premium Was Who Pupils Are
Between pupils
1.85
95% set .30 to 41.5
Compatibility appears to protect friendships almost three times more than choice implies.
Within pupils
−.26
95% set −.55 to 1.10
With ego fixed effects the premium disappears.
Pupils whose friends are more compatible keep friends longer for stable reasons of their own. A premium must be estimated within actors.
Rule Scales Manufacture Mechanisms
| Apparent finding on a rarity-weighted scale | After calibrating to partner choice |
|---|---|
| Pure relative choice holds among weakly embedded ties | Not supported |
| Reciprocity insulates ties from competition | Not supported |
| Entrants resemble the departed alter (vacancy filling) | Vanishes |
| Alternative value lowers persistence within pupils | Detected only on the choice scale |
What pupils act on
Sex 1.75 · origin .47 · religiosity .19 · lifestyle .20 · generation .09
What the rarity rule assumes
Every shared category weighted by its local scarcity; utility dispersion overstated about threefold
Calibrate compatibility before testing anything about alternatives.
Who Replaces a Departed Friend?
Distance to the departed alter in entrant choice
| Specification | Rule weights | Estimated weights |
|---|---|---|
| Non-incumbent risk set | −.261 | −.066 |
| + ego’s portfolio taste | −.127 | −.036 |
| + network inheritance and foci | −.089 | −.009 |
Estimated weights, full model: bootstrap 95% interval [−.058, .061]; 11,784 exits, 878,087 candidates.
What predicts entry
- Compatibility with ego
- Ego’s durable taste (portfolio share .158)
- Ties to the departed alter and to ego’s other friends
- Sharing ego’s classroom
With these profiles and controls, who left adds no detectable information about who enters.
Profiles Persist Because the Ecology Supplies Them
What we built: an agent-based model. Artificial actors with four binary traits repeatedly pick \(K\) friends from a fixed set. A friend is worth the number of matching traits, the same bonus for every current friend, and a random shock. Nobody values a profile, scarcity or replacement.
What we read. Identity continuity: the same persons as at baseline. Profile continuity: the same trait combinations. A: hold one friendship fixed and add substitutes with the friend’s exact traits. B–C: vary how concentrated profiles are within each setting.
Persons churn (identity .39 → .18) while profiles hold (.45 → .83); group mixing stays near .99. In friendship, profile continuity (.618) is below the baseline the classroom supplies (.628).
Stable patterns need not be made of stable relationships. The ABM shows this is possible; it is not calibrated, so it cannot say how much occurs.
IV. Discussion
Takeaways
Size. The number of alternatives counts for less than proportionally: \(\hat\alpha\) is .23 online (.42 early, .09 late) and .48 in the class. Full evaluation is rejected online and, with some uncertainty, in the class.
Setting. Pressure comes from alternatives in the set ego draws on: in friendship, the class, and within it ego’s own contacts; other grade-mates add at most a tenth.
Protection. Compatibility protects online ties about a third more than choice implies; in friendship the apparent premium is ego heterogeneity.
Turnover. Profile continuity arises from ecological supply, without a detectable preference for replacing the previous occupant.
Relational opportunity: an opportunity structure bears on tie persistence mostly through how closely its alternatives approximate the incumbent’s value, and only inside the set ego draws on.
Implications
For Blau’s opportunity premise: for a particular tie, an opportunity is not another available person; its weight is its value relative to the incumbent
For actor-oriented models: the default treats every actor as an alternative (\(\alpha=1\)); the persistence logit is a diagnostic of that assumption before fitting large or multi-group networks
For measurement: compatibility scores must be put on the choice scale; rule-based scores produce spurious mechanisms
Settings in RSiena: Ripley et al. 2024. SAOMs as random-utility models: Pink, Kretschmer & Leszczensky 2020.
Limitations
Observational design: ego fixed effects remove stable, not time-varying, heterogeneity
What \(\alpha\) measures: an elasticity with respect to the opportunity set as we measure it. In simulations, proportional measurement error barely moves \(\hat\alpha\); but if ego draws on a neighbourhood whose size does not scale with the measured set, \(\hat\alpha\) is near .05–.08 even when the elasticity within the neighbourhood is 1 (Backup B6)
Precision and the cap: the classroom estimate excludes full evaluation by cohort but not by school; with the 10-nomination cap, \(\hat\alpha\) is .81 below the cap and .35 at it
Interpretation: \(\alpha\) is not a count of the people an actor considers; \(\gamma\) assumes a shared error scale for choice and maintenance
Thank You
Roberto Cantillan
Department of Sociology, PUC Chile
rcantillan@uc.cl
Backup Slides
B1: Proposition — Three Parameters
One slot; Type-I extreme-value errors; consideration set of size \(m_{ij}=\mu n_{ij}^{\alpha}\) drawn uniformly; incumbency premium \(\rho_{ij}=\rho_0+\gamma U_{ij}\); incumbent re-contested with probability \(c\).
\[ \Pr(\text{persist})=(1-c)+c\,\Lambda\big(\rho_0-\log\mu+(1+\gamma)U_{ij}-\alpha\log n_{ij}-\bar L_{ij}\big) \]
Local slopes of the logit:
\[ b_U=\kappa(1+\gamma),\qquad b_{\bar L}=-\kappa,\qquad b_{\log n}=-\kappa\alpha, \qquad \kappa=\frac{\Lambda}{1-c+c\Lambda}\in(0,1] \]
- Uses \(\mathbb E\big[\log\sum_{k\in S}e^{U_{ik}}\big]\approx\log m_{ij}+\bar L_{ij}\)
- Ratios are exact pointwise; with several slots \(\hat\kappa\) can exceed one (1.16–1.33 in simulations)
- Uniform consideration is one mechanism; other exposure or search processes give the same reduced form
B2: Online Discussion — Full Estimates
| Sample | \(b_U\) | \(b_{\bar L}\) | \(b_{\log n}\) | \(\hat\kappa\) | \(\hat\alpha\) [95% CI] | \(\hat\gamma\) [95% CI] | \(n\) |
|---|---|---|---|---|---|---|---|
| Ego active | .890 | −.671 | −.154 | .67 | .23 [.11, .36] | .33 [.10, .66] | 529,282 |
| Both active | .903 | −.658 | −.145 | .66 | .22 [.10, .36] | .37 [.14, .73] | 512,374 |
| New ties | .965 | −.719 | −.156 | .72 | .22 [.11, .34] | .34 [.11, .68] | 406,200 |
| Established ties | .673 | −.592 | −.216 | .59 | .37 [.18, .62] | .14 [−.13, .63] | 115,946 |
| Low reply intensity | 1.008 | −.740 | −.229 | .74 | .31 [.20, .44] | .36 [.15, .67] | 291,813 |
| High reply intensity | .782 | −.593 | −.131 | .59 | .22 [.05, .40] | .32 [.03, .82] | 225,710 |
| First 18 windows (2013–17) | .843 | −.449 | −.188 | .45 | .42 [.21, .76] | .88 [.26, 2.48] | 205,734 |
| Last 19 windows (2017–22) | .937 | −.877 | −.079 | .88 | .09 [−.09, .27] | .07 [−.13, .37] | 317,480 |
Ego and window fixed effects; SEs clustered by ego and window; utility scale from partner choice (\(\hat\lambda=1.747\)); Fieller intervals.
B3: School Friendship — Full Estimates
| Design | Scale | \(b_U\) | \(b_{\bar L}\) [\(p\)] | \(\hat\kappa\) | \(\hat\alpha\) [95% set] | \(\hat\gamma\) [95% set] |
|---|---|---|---|---|---|---|
| Between egos | Rule | .141 | −.072 [.016] | .07 | n.i. | .97 [−.05, 5.5] |
| Between egos | Calibrated rule | .437 | −.592 [.121] | .59 | n.i. | −.26 [−.65, 59] |
| Between egos | Estimated weights | .562 | −.197 [.033] | .20 | n.i. | 1.85 [.30, 41.5] |
| Within ego | Estimated weights | .606 | −.815 [.004] | .82 | .29 [−.15, 1.00] | −.26 [−.55, 1.10] |
| Within ego, classmates | Estimated weights | .589 | −1.103 [<.001] | 1.10 | .48 [.30, .85] | −.47 [−.60, −.25] |
| Within ego, other grade-mates | Estimated weights | .224 [.17] | — | — | — |
Wild restricted score bootstrap (10 schools, all \(2^{10}\) patterns; or 29 grade-cohorts, 9,999 draws). n.i. = not identified between egos.
B4: Diagnostic Signatures of Turnover Mechanisms
| Diagnostic | Relative choice | + Durable taste | + Positional demand |
|---|---|---|---|
| Same-profile alternatives, given \(\bar L\) | 0 | < 0 | < 0 |
| Portfolio taste in entrant choice | 0 | > 0 | ≤ 0 |
| Distance to departed alter, net of taste | 0 | small, sign-unstable | < 0 |
| Distance to departed alter, taste omitted | 0 | < 0 | < 0 |
A negative similarity-to-departed coefficient cannot by itself be read as vacancy filling.
Signatures established by known-truth simulations (1,500 egos, eight replications per regime and ecology).
B5: Parameters Across Settings
In both settings, the opportunity that bears on persistence is a bounded setting weighted by value, not a headcount.
B6: Known Truth with a Mismeasured Opportunity Set
| What the analyst measures, relative to what ego draws on | \(\hat\alpha\) if true \(\alpha=0\) | \(.5\) | \(1\) |
|---|---|---|---|
| The correct set | .02 | .44 | .90 |
| A random half of the alternatives | .03 | .53 | 1.01 |
| A random quarter | .05 | .59 | 1.22 |
| Correct set plus infeasible alters (+50% of set) | .00 | .44 | .92 |
| Correct set plus 100 infeasible alters, whatever \(n\) | .07 | 1.06 | 2.24 |
| Ego draws on a neighbourhood of 30% of the class; analyst measures the class | .40 | .46 | .85 |
| Neighbourhood size unrelated to class size; analyst measures the class | .02 | .05 | .08 |
| The same, analyst measures the neighbourhood | .12 | .40 | .80 |
Mean \(\hat\alpha\) over 6 replications of 20,000 egos; \(K=3\), \(c=.6\), \(\gamma=.3\).
Errors proportional to the set are benign; errors that do not scale with it are not. A low \(\hat\alpha\) is compatible with ego weighing a whole neighbourhood whose size does not grow with the measured set.
References
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Cantillan | When Do Alternatives Threaten a Tie? | 2026