RELATIONAL ADVANTAGE · DIFFUSION

Does Relational Advantage Travel Across Tasks?

Evidence from Microcredit Entry and Adoption

Roberto Cantillan

Department of Sociology · Pontificia Universidad Católica de Chile

Who is selected—and who adopts?

The question

Entry

An organization selects households as entry points.

Adoption

Households decide whether to commit to a costly practice.

Does the relational advantage that helps at entry carry into adoption?

Katz & Lazarsfeld 1955 · Banerjee et al. 2013, 2019 · Beaman et al. 2021

An opening matters through what follows

Diffusion as a sequence of accomplishments

An event acquires consequences through its connections to subsequent action.

Entry

An opportunity arrives.

Information

People become aware.

Adoption

People decide to commit.

What connects one accomplishment to the next need not be the same relation.

Conceptual adaptation of Rule, Sabetta & Bearman (2026), “Pathways and Chance,” Sociologica, §4.

Fixed networks. Successive tasks.

Two coexisting fixed relational domains connect the same six actors. Actor 1 is central in A and actor 2 in B. Successive tasks place more relevance on A and then B, illustrating a possible reversal in their relative advantage.

Conceptual illustration, not a simulation result. Both domains coexist; only task relevance changes.

Relational advantage across tasks

The concept, its antecedents, and the conditions for reordering.

Portability is continuity of relational advantage

The concept

An advantage is portable when the position that helps an actor at one task remains advantageous at the next.

A relation is a domain of ties, such as advice or visiting.

A task is an accomplishment needed for the process to advance. Organizations can perform selection; households can receive information or adopt.

We compare successive tasks within a process, with the actors’ positions held fixed.

The premises we inherit—and the question we add

Tasks can require different resources. Learning, gaining access and committing need not draw on the same relations.

Different networks can work together. Their overlap and interaction shape diffusion; combining them can hide important differences.

Network advantage depends on the task. Hansen and colleagues show that a network beneficial for one task can hinder another.

We make continuity of relative advantage across successive tasks the object of explanation.

Coleman 1988 · Gould 1991 · Hansen et al. 2001 · Becker et al. 2020 · Chandrasekhar et al. 2026 · Distinctions: A1.

El punto de partida es conocido: distintas tareas pueden necesitar recursos diferentes, y distintas relaciones pueden actuar juntas. Gould muestra cómo las redes organizacionales y de vecindad sostienen conjuntamente la movilización. Becker y sus colegas conectan influencia personal y difusión por rutas comerciales. Chandrasekhar y sus colegas estudian cómo compartir contactos entre capas modifica la difusión. Hansen es el antecedente más cercano: una configuración ventajosa para una tarea puede perjudicar otra.

La pregunta que organizo aquí es qué ocurre con los mismos actores al pasar a la siguiente tarea: cuándo conservan su ventaja relativa y cuándo cambian de lugar. Dentro del modelo aditivo, cambiar la tarea puede invertir el orden si desplaza suficientemente el peso hacia dominios que favorecen a actores diferentes. A1 distingue compartir contactos, ocupar posiciones semejantes y conservar ventaja. También reconoce los antecedentes sobre secuencias y cambio de canal.

Advantage depends on what the task values

Theory · A minimal formalization

For a task, add position × relevance across domains. The positions stay fixed; their relevance can change.

$R_{ir}=\sum_l\omega_{lr}x_{il}.$

$x_{il}$ : actor $i$ ’s position in domain $l$ . $\omega_{lr}$ : how much task $r$ values that domain’s capacities.

$R_{ir}$ is a theoretical score. This minimal model adds fixed, comparable positions with nonnegative weights.

Hold the positions fixed. Change what the task values.

Fixed positions; a different ordering

Theory · A minimal formalization

Actor 1 leads in A; actor 2 leads in B. Fixed positions can preserve or reverse the ordering, depending on the weight shift.

Actor	Position A	Position B	Task 1	Small shift	Large shift
Actor 1	.8	.2	.68	.50	.32
Actor 2	.5	.4	.48	.45	.42

A’s weight: .8 initially → .5 (small shift) or .2 (large shift). B receives the remainder.

The pair ties when A’s weight is .4. Crossing that threshold reverses the order.

When does advantage persist—and when can it reverse?

Similar task weights preserve the basis of advantage. Exactly unchanged weights preserve the additive ordering.

Ahead in every relevant domain: an actor cannot fall strictly behind the same rival under nonnegative weights.

Opposing domain advantages: a task-weight shift can reverse the pair, if the shift is large enough to reverse their initial gap.

Changing tasks alone is insufficient: the change must matter for the actors’ relative profiles.

Additive benchmark with fixed, comparable positions · Exact pairwise criterion: A4.

What would challenge this explanation?

Evidence needed · Distinguish a possibility from an explanation

Measure task demands and domain resources independently of the outcomes; follow the same actors across stages.

If relevance shifts toward domains where actors hold opposing advantages, ask whether those differences explain who loses or gains relative advantage.

Repeated conservation despite large, independently established shifts would challenge the proposed explanation. Reordering without a shift would call for another mechanism.

A change in domain associations motivates the explanation; a direct test must connect demands, positions, and individual outcomes.

Pooling can hide a task-specific advantage

Conditional measurement implication

Domain-specific position preserves distinctions between relational infrastructures.

Union-network position pools ties before asking what the task requires.

When an adoption association is concentrated in a particular domain, pooling other relations can dilute that signal.

The value of aggregation depends on the task and on how actors’ positions correspond.

Empirical design

Two datasets, distinct tasks, and comparable household networks.

Different studies observe different stages

Study	Unit and outcomes	Role
BSS · 49 villages, 10,618 households	Household designation and own adoption	Within-setting stage comparison
Information RCT · 68 villages	Reach generated by randomized sets of 3 or 5 seeds	Information benchmark compared with BSS adoption
Behavioral simulation	Individual information and adoption probabilities	Sequential-process illustration

Designation is not information receipt. BSS does not observe information; the simulation illustrates the general process rather than reconstructing the BSS sequence.

Networks precede the arrival of microcredit

Setting · Karnataka, India

10,618

Households in the adoption analysis

49 / 75

Adoption villages / network villages

17.3%

Households that took up a BSS loan

Networks measured in 2006, about six months before entry.
BSS designated 12% of households using locally influential roles.
Designation is organizational selection; adoption is the household’s own outcome.

Banerjee, Chandrasekhar, Duflo & Jackson 2013

Nominations become comparable household networks

Network construction · BSS

Nodes: households on a common village roster.

Ties: combine each pair of name generators. A nomination in either direction creates one binary, undirected tie; self-ties are removed.

Positions: compute diffusion centrality separately in each domain, then scale it to $[0,1]$ within village and domain.

The same households and position measure allow comparisons across relations.

Diffusion centrality: Banerjee et al. (2013) · Intuition, example and implementation: Appendix A2–A2a.

Centrality includes what your contacts can reach

Measurement · Diffusion centrality

Start a message at one household. Count transmission opportunities to its contacts, then through their contacts, for a finite number of rounds.

Weight each step by the transmission parameter $q$ . A household with well-connected contacts can score higher than another with the same number of ties.

Compute this separately in each domain. Repeated arrivals count.

Degree counts immediate contacts. Diffusion centrality also counts onward transmission opportunities.

Banerjee et al. (2013), Eq. 5 · Appendix A2–A2a: intuition, worked example and implementation.

The relations we observe

Measurement · The main comparison

Domain	Reported ties
Advice/Decision	Advice and decision support
Exchange	Money lending and borrowing
Visiting	Visits to each other’s homes

These are the focal domains. The SI also examines every remaining BSS relation.

Paired name generators; undirected layers · Alternative generators and domain composition: A7.

Entry and adoption use identical comparisons

Design · Primary evidence

Entry designation

Was the household in the lender’s designated entry pool?

Own adoption

Did the household take up a BSS loan?

The three domain positions enter together in each model.
Compare households within village, with identical controls.
Estimate the two equations jointly; cluster uncertainty by village.

Controls: union degree, wealth, and network-survey participation. Linear probability models. · Appendix A3: joint equations and contrast.

Empirical evidence

Which domain is associated with entry—and which with adoption?

Households rank differently across domains

Results · Positional differentiation

Cross-domain positional differentiation in the 75 Karnataka villages.

.742

Median rank correlation

50%

Top-decile overlap in the median village

21.8%

Move at least 25 percentile points

75 villages · Advice/Decision versus Visiting · Correlation IQR: .686–.796

Observed pattern · Entry and adoption differ

Results · Same households, two outcomes

Coefficients for entry designation and adoption, with 95% confidence intervals.

Entry: Advice/Decision $b=0.219$ , $p<.001$

Adoption: Visiting $b=0.173$ , $p=.012$

10,618 households · 49 villages · All three domains in each model · Village-clustered 95% CIs

The domain contrast changes sign

Results · Beyond a difference in significance

Visiting minus Advice/Decision

At entry: $-0.246$
At adoption: $+0.206$

Θ = 0.451

Change in the domain association contrast
SE $0.103$ · $p<.001$

Moving from the 25th to the 75th percentile of Visiting position corresponds to 4.3 pp more adoption.

Adoption base rate: 17.3%. Rounded domain gaps may differ slightly from the unrounded joint contrast. · Appendix A3: definition of the contrast.

The Visiting association survives key checks

Results · Robustness

Check	Result
Leave one village out	Visiting stays positive and significant in all 49 estimates.
Exclude the entry pool	Visiting $b=0.168$ , $p=.025$ .
Small-cluster inference	Visiting wild-cluster bootstrap $p=.021$ .
Alternative measurement	All 8 reversal contrasts and 16 adoption domain contrasts are positive.
Additional relational layers	Visiting remains positive: $b=.148$ – $.179$ , all $p<.05$ .

Seven of eight reversal tests have p < .05; 15 of 16 adoption contrast intervals exclude zero.
A5: rival explanations · A11: medical, temple, kin/nonkin and rice/kerosene layers (separate specifications).

Visiting exceeds the union association

Results · Same sample and controls; separate one-measure models

Union centrality

−0.025

Association with adoption
SE $0.070$ · $p=.72$

Visiting centrality

+0.156

Association with adoption
$p=.030$

Visiting exceeds the union coefficient by 0.181 (SE $0.058$ , $p=.003$ ).

Two single-measure models estimated jointly. Visiting .156 here; .173 in the joint three-domain model.

The Visiting signal is concentrated in nearby ties

Results · Main qualification

Nearby Visiting ties

+0.080

$p=.017$

Other Visiting ties

+0.019

$p=.71$

Learning from neighbors and neighborhood differences in exposure remain competing explanations.

Near = within five positions in census enumeration order; 32.9% of Visiting ties. Location proxy, not coordinates. · Appendix A6: composition and secondary test.

Benchmark · Information and adoption

Information RCT: 68 villages, randomized sets of 3 or 5 seeds.
BSS adoption: 10,618 households in 49 other villages.

Relational domain	Information · RCT	Own adoption · BSS
Advice	.362 (.242)	.039 (.024)
Decision Support	−.214 (.123)	−.059 (.019)
Visiting	−.322 (.196)	.069 (.037)
Material exchange	.401 (.264)	.009 (.024)
Visiting − Advice	−.684 (.325)	+.029 (.043)

Information favors Advice over Visiting. The adoption contrast is small and imprecise.

Joint four-layer models; predictors and outcomes in SD units; SEs in parentheses. RCT contrast: randomization p = .030. Different studies and units; no observed individual sequence across both stages.

Este benchmark compara información y adopción utilizando cuatro capas separadas y coeficientes estandarizados. La información proviene del RCT de 68 aldeas: las identidades de tres o cinco iniciadores se asignan al azar y el alcance se registra mediante llamadas. La adopción proviene de los 10.618 hogares de BSS en 49 aldeas. El RCT no observa aquí adopción de microcrédito.

En información, la diferencia entre Visiting y Advice es −0,684 desviaciones estándar; la prueba por aleatorización da p igual a 0,030. En adopción, esa diferencia es positiva pero pequeña: 0,029, con error estándar de 0,043. No permite afirmar que Visiting supera a Advice con precisión en esta especificación armonizada. Esto no equivale al contraste principal, que utiliza el dominio combinado Advice/Decision.

Su valor como benchmark es mostrar qué perfil relacional acompaña al alcance informativo en un diseño con iniciadores aleatorios y compararlo con el perfil de adopción. Son estudios y unidades diferentes: conjuntos de iniciadores por aldea frente a hogares. La estandarización facilita presentar los perfiles, pero no convierte la comparación en una secuencia individual observada ni en una estimación de un efecto común. El resultado es evidencia complementaria; no una segunda demostración concluyente de inversión.

A sequential behavioral model

Fixed actors and layers; changing demands, access, and reinforcement.

What is the model designed to explain?

Computational model · Purpose

Question: can changing the relevant domain reorder the same actors in a sequential information–adoption process?

Comparison: change adoption weights; keep the networks, initiators and informed population fixed.

Mechanism: task relevance changes contacts; information access and reinforcement generate individual outcomes.

Aim: examine its consequences and limits under explicit assumptions.

Follow individual advantage through the sequence, and separate task change from the baseline stage difference.

Buscamos entender qué le ocurre a la ventaja individual cuando el proceso cambia de tarea. La formalización muestra que invertir el orden es posible; la simulación pregunta si esa posibilidad se mantiene cuando hay caminos de transmisión, acceso a información y necesidad de apoyo para adoptar.

Para estudiarlo, cambiamos la relevancia de los dominios en adopción y conservamos las redes, los iniciadores y la población informada. Comparamos con un escenario donde esa relevancia permanece igual. Así podemos distinguir lo que añade el cambio de tarea de las diferencias que ya existen entre informarse y adoptar. El modelo especifica un mecanismo de comportamiento: los pesos cambian los contactos, la información determina elegibilidad y el apoyo de vecinos permite adoptar. Estudiamos sus consecuencias y sus límites. Identificar si ese mecanismo explica causalmente lo observado en las aldeas exige evidencia adicional sobre recursos, demandas y acciones; la simulación no proporciona esa identificación empírica.

Build two layers before changing the task

Computational model · Network construction

Generate 40 independent pairs, each with 40 actors, mean degree 6, four groups and heterogeneous activity. Within-group pairs have four times the sampling weight.

Initially align the degree rankings in A and B. Relabel 0%, 50% or 100% of B’s actors while preserving B’s topology.

Relabeling changes who occupies its positions and which cross-layer neighborhoods overlap. Aligned degrees do not make the layers identical.

Fixed ties; changing task relevance

Model · Individual portability

Apply task weights to ties, rather than actor scores. The cascade then computes behavioral outcomes.

$P_r=\beta_r[\omega_{Ar}A+(1-\omega_{Ar})B].$

The weights come from section 01. $\beta_r$ sets contact intensity: .4 for information; .8 for adoption.

An A-only tie gets probability $\beta_r\omega_{Ar}$ ; a B-only tie gets $\beta_r(1-\omega_{Ar})$ . A tie in both gets $\beta_r$ .

The matching rule changes contact opportunities. Cascade outcomes are computed from the resulting paths and response rules.

Information first; adoption second

Computational model · The process

01 · Information

News travels

Three fixed initiators start the cascade. Newly informed actors can pass the news onward.

02 · Eligibility

Keep who heard

Information finishes spreading. Hold that informed population fixed across adoption contrasts.

03 · Adoption

Support accumulates

Reuse the initiators. Informed actors adopt after support from h distinct active neighbors; new adopters become sources.

Continue adoption until no actor changes state. Exclude the three forced initiators from individual rankings.

Repeat the process; estimate each actor’s chances

Computational model · From cascades to individual rankings

Repeat the sequence to estimate each actor’s own chances of information, adoption and adoption given information.

Illustrative actor, over 10 repetitions	Frequency
Receives information	8 / 10
Adopts through the full sequence	4 / 10
Adopts among occasions informed	4 / 8

Since adoption requires information: $P(A_i)=P(I_i)\,P(A_i\mid I_i)$ .

Illustrative counts only. Actual design: 64 information populations × 32 adoption contact worlds.

Three comparisons answer different questions

Computational model · The process

Comparison	What it asks
Information vs adoption	Do the same actors exchange places between stages?
Changed vs unchanged adoption weights	Do task weights change the adoption ordering, with the same eligibility and response-rule specification?
Difference in between-stage reversal rates	Does changing weights add more reversals, net of the unchanged-weight baseline?

The net difference can be negative. A tie is recorded separately from a strict reversal.

The experiments vary three dimensions

Model · Individual portability

Dimension	Levels
Functional differentiation $s$	0 · 0.5 · 1
Adoption task demand $\tau_2$	Unchanged +1 · Partial 0 · Complete shift −1
Layer correspondence	Aligned degree ranks · Partial relabeling · Full relabeling

At $s=0$ , a task shift leaves the mixture unchanged. At $s=1$ , information uses A; adoption uses A, an equal mixture, or B. At $s=.5$ , these shifts are moderated.

Paired task contrasts share layers, initiators, informed actors and contact uniforms. Design and exact controls: A8.

Response rules answer different questions

Computational model · Response experiments

Absolute: require $h=1,2,3$ distinct supporting neighbors. The illustrative benchmark is $h=2$ .

Proportional: require $\max(1,\lceil q\,k_{i,2}\rceil)$ , where $k_{i,2}$ is weighted degree in the adoption layer. $q=.3,.4,.5$ .

Anchored: use the same fractions of actor $i$ ’s degree in A; requirements remain fixed across adoption task contrasts.

Proportional rules change both contacts and requirements. Anchored and absolute contrasts hold requirements fixed.

Simulation results

Individual reordering, response rules, and the limits of interpretation.

Task shift adds reordering with two supports

Simulation results · Individual level

+24.6 pp

Net increase in information–adoption reversals.

39 / 40 networks show an increase.

Network-pair bootstrap interval: 21.9–26.8 pp.

40 network pairs · h = 2 · Complete shift · Bands: network-pair bootstrap.
Ranking stability with more Monte Carlo draws remains to be checked; full rules: A9.

Adoption rankings change for the same actors

Simulation results · Individual level

29.0%

Pairs reverse between unchanged and changed adoption weights.

Positions and initiators remain fixed.

40 synthetic network pairs · h = 2 · Complete task shift · Bands: bootstrap over network pairs.

Reordering persists after information access

Simulation results · Individual level

+37.8 pp

Net increase in reversals using adoption given information.

Access remains necessary; it need not preserve relative advantage.

40 synthetic network pairs · h = 2 · Complete task shift · Bands: bootstrap over network pairs.

The domain associated with advantage changes

Simulation results · Individual level

Information tracks position in A.

Own adoption tracks B more strongly.

A also remains associated with own adoption in this sequential process.

Mean within-network associations · Same nonforced actors · h = 2, s = 1, full relabeling.

High requirements can flatten the ranking

Simulation results · Individual level

h = 3: almost no adoption after the shift.

About 97% of pairs are tied, also at proportional 40%–50% requirements.

Fewer reversals can mean a collapsed ranking.

All nine response rules · Fixed requirements (anchored) and task-relative requirements (proportional). · Appendix A9: adoption, ties and all nine rules.

Implications and scope

What follows from the evidence—and what remains to be established.

What the evidence establishes—and leaves open

Interpretation

Observed empirical pattern

Entry associates with Advice/Decision; own adoption with Visiting.

Union centrality has a weaker adoption association.

Open explanation

Tie capacities are not directly measured.

Consultation may help interpret organizational selection. Observation, coordination or support may help interpret adoption. These are possible, nonexclusive mechanisms.

BSS compares designation and adoption associations. The model illustrates own probabilities; intervention reselection is a separate question (A10).

Central in which relation—and for which task?

Discussion

Relational advantage can change
as the process moves to its next task.

Designation association
Advice/Decision

Own-adoption association
Visiting

Roberto Cantillan
Pontificia Universidad Católica de Chile

Appendix

A1 · Shared contacts, similar positions, portable advantage

Question	What it describes
Do domains connect actors to the same people?	Tie overlap
Are the same actors prominent across domains?	Positional correspondence
Do actors retain their advantage at the next task?	Portability of advantage

These are distinct questions. Changing what a task requires can reorder actors whose advantages lie in different domains.

Overlap and diffusion: Chandrasekhar et al. 2026 · Layer-switching: Min et al. 2016 · Targeting and network structure: Hsiao & Christakis 2026.

Compartir contactos entre capas no equivale a ocupar posiciones semejantes, ni asegura que una ventaja se conserve entre tareas. La tercera pregunta requiere considerar qué demanda cada tarea y cómo esas demandas valoran el perfil del actor. La inversión exige un cambio suficiente: una diferencia pequeña puede conservar el orden.

Hay antecedentes explícitos sobre secuencias. Min y sus colegas estudian cómo la transmisión depende del canal por el que llegó la información y de aquel por el que se transmite después. Eso es dependencia de la trayectoria dentro de una difusión; aquí seguimos la ventaja de los actores entre logros sucesivos. Hansen también propone estudiar los rendimientos de las redes a través del tiempo. El aporte debe situarse en esta conversación, sin reclamar prioridad sobre toda forma de secuencia.

Hsiao y Christakis preguntan cuándo seleccionar actores muy conectados genera difusión amplia. Su resultado depende de cómo se organizan las conexiones del conjunto. Esa distinción ayuda a separar nuestra pregunta individual de las consecuencias colectivas: reordenar oportunidades individuales no garantiza una diferencia grande en el resultado agregado. La regla de refuerzo también importa. En Chandrasekhar, varias interacciones a través de capas con un mismo contacto pueden contribuir al umbral; no debemos equiparar ese supuesto con cualquier regla que exija fuentes distintas.

A2 · Same degree; different onward opportunities

Consider a five-household chain: 1 — 2 — 3 — 4 — 5.

Households 2 and 3 each have two direct contacts. Their contacts offer different numbers of two-step walks.

Contribution, using illustrative $q=.5$ , $T=2$	Household 2	Household 3
One-step walks: $q \times 2$	1.00	1.00
Two-step walks: $q^2 \times$ walk count	$.25\times3=.75$	$.25\times4=1.00$
Total, before scaling and without $t=0$	1.75	2.00

Illustrative arithmetic, not the parameter choice used in estimation. Walks can return to their origin.

A2a · Translate the intuition into the calculation

$\mathrm{DC}(i)=\left[\sum_{t=0}^{T}(qA)^t\mathbf{1}\right]_i$

$A^t\mathbf{1}$ counts $t$ -step walks from each household. Multiplying by $q^t$ weights their transmission opportunities; the sum combines rounds.

Our parameters: $q=1/\lambda_{\max}(A)$ ; $T=$ layer diameter.

Our scale: minimum 0, maximum 1, within each village and domain.

Measure: Banerjee et al. (2013), Eq. 5 · Layer parameters: Chandrasekhar et al. (2026).
The $t=0$ term adds a common constant, removed by min–max scaling.

Ahora la ecuación resume los pasos del ejemplo. A describe los vínculos; elevarla a t cuenta recorridos de t pasos. El vector de unos suma sus destinos. Como nuestras capas son simétricas, esta suma por fila describe recorridos desde el hogar. q elevado a t pondera cada recorrido, y sumamos hasta T.

La medida viene de Banerjee y sus colaboradores. Para los parámetros por capa seguimos el procedimiento de Chandrasekhar y sus colaboradores: q es la inversa del mayor valor propio y T el diámetro. q sitúa el proceso en el umbral espectral entre decaimiento y crecimiento de las contribuciones de recorridos largos. T fija un horizonte basado en la red; los recorridos contados pueden repetirse y no se restringen a caminos más cortos. En capas desconectadas tomamos la mayor distancia finita dentro de sus componentes.

Por último, restamos el mínimo y dividimos por el rango dentro de cada aldea y dominio. Eso permite leer la posición entre cero y uno; no convierte las capacidades en unidades equivalentes. La definición original empieza en t igual a uno. Nuestro término cero añade uno a todos los hogares y desaparece al normalizar. La medida comparte un criterio de cálculo entre dominios; no estima por sí misma recursos ni probabilidades de adopción.

A3 · Joint outcome models and the reversal contrast

$Y_{iv}^{k}=\alpha_v^k+\sum_{l\in\{AD,E,V\}}\beta_l^k\,\mathrm{DC}_{liv}+\mathbf{X}_{iv}'\gamma^k+\varepsilon_{iv}^k$

$\Theta=(\beta_V^U-\beta_{AD}^U)-(\beta_V^G-\beta_{AD}^G)$

$G$ : designation; $U$ : own uptake. Same sample and controls.
$\alpha_v^k$ : village fixed effects. Controls: union degree, wealth, survey participation.
Joint estimation retains covariance across outcomes; uncertainty clustered by village.

A4 · When do two actors exchange places?

$(R_{i1}-R_{j1})(R_{i2}-R_{j2})<0.$

Writing $R_{ir}=m_i+s\tau_r d_i$ gives:

$\left(\Delta m_{ij}+s\tau_1\Delta d_{ij}\right)\left(\Delta m_{ij}+s\tau_2\Delta d_{ij}\right)<0$

Common position: $m_i=(x_{i1}+x_{i2})/2$ .
Relational profile: $d_i=(x_{i1}-x_{i2})/2$ .
$\Delta m_{ij}$ and $\Delta d_{ij}$ compare actors $i$ and $j$ .
A negative product means opposite orderings at the two tasks.

Different task scores do not necessarily imply different rankings.

A4a · Matching weights

Theory · A minimal formalization

In two domains, advantage is a weighted average of fixed, comparable positions. This is the additive assumption.

$\omega_{Ar}=\frac{1+s\tau_r}{2},\qquad \omega_{Br}=1-\omega_{Ar}.$

Domain differentiation $s$ : at $s=0$ , both weights are always .5. Task demand cannot favor one domain over the other.

Task demand $\tau_r$ : positive favors A, negative favors B. At $s=1$ , demand can shift all the weight from A to B.

A4b · The score-change identity

Theory · A minimal formalization

Subtract the same actor’s score at the two tasks:

$R_{i2}-R_{i1}=\underbrace{\frac{s}{2}}_{\text{domain difference}}\;\underbrace{(\tau_2-\tau_1)}_{\text{task change}}\;\underbrace{(x_{iA}-x_{iB})}_{\text{position profile}}.$

The change depends jointly on different domain capacities, changed task demand, and where this actor stands across domains.

If any factor is zero, this actor’s score stays the same. A changed score still need not change the ordering.

A5 · Checks on rival explanations

Threat	What could produce the pattern	What we do
General prominence	Visiting just marks wealthier or more active households	Control union degree and wealth; compare opposite relational profiles after balancing on prominence; drop the entry pool
Composition, homophily	Caste structures both relations and access to credit	Sub-caste fixed effects; residualize on caste, religion, housing; estimate by caste
Spatial proximity	Visiting is local; nearby households may share exposure	Census enumeration order as a proxy; split Visiting ties into near and far
Measurement	Sampled respondents; composites; the pool is not the households contacted	Each constituent name generator; four scales; four sample restrictions

A generic prominence account predicts the same positions have similar associations across tasks, not that different domains order them for different tasks.

A6 · Composition and the secondary test

Caste heterogeneity: the adoption association is concentrated among lower-caste households (interaction $b=.159$ , $p=.008$ ).

Balanced relational profiles: Visiting-oriented households adopt 2.4 pp more; interval $[-1.2,5.9]$ .

Cross-village moderation: entry-pool advantage × domain divergence is imprecise ( $b=-.008$ , SE $.182$ ; $p=.96$ ).

These secondary comparisons locate the pattern; they do not independently establish the mechanism.

A7 · Decision support versus advice: exploratory

Separating the generators behind Advice/Decision: Decision Support centrality is negatively associated with adoption (standardized $b=-0.059$ , $p=.003$ ).
The relative profile matters: Decision Support minus Advice predicts lower adoption (⁠ $b=-0.099$ , $p=.005$ ); their sum does not (⁠ $b=-0.020$ , $p=.48$ ).
Sharpest among entry-pool households (interaction $b=-0.074$ , $p<.001$ ).
The reversal is larger when Advice/Decision is represented by the decision generator (0.495 to 0.538) than by the advice generator (0.192 to 0.287).

Deliberative prominence as such does not predict lower adoption; ranking higher as a source of decision support than of advice does. Outside the main comparisons and exploratory.

A8 · Sequential cascade design

Appendix · Computational specification

Component	Fixed design
Network pairs / actors / initiators	40 / 40 / 3
Information / adoption contacts	64 information populations / 32 adoption worlds each
Transmission	Information 0.4 / Adoption 0.8
Response rules	h = 1, 2, 3; proportional and anchored q = .3, .4, .5
Controls	Unchanged task; s = 0; identical layers
Validation	6,480 exact null checks; reconstruction error 0

A9 · All response rules retain their signed results

Appendix · s = 1, complete shift, full relabeling

Rule	Adoption: no shift → shift	Tied pairs after shift	Extra reversals
absolute 1	82.6% → 79.4%	5.0%	+8.2 pp
absolute 2	76.0% → 44.8%	10.3%	+24.6 pp
absolute 3	7.9% → 0.4%	97.0%	-6.7 pp
anchored 0.3	64.7% → 47.4%	6.1%	+29.1 pp
anchored 0.4	10.6% → 21.4%	18.6%	+22.6 pp
anchored 0.5	4.1% → 11.6%	29.6%	+35.4 pp
proportional 0.3	64.7% → 5.7%	68.1%	-0.4 pp
proportional 0.4	10.6% → 1.0%	97.1%	-25.9 pp
proportional 0.5	4.1% → 1.0%	97.2%	-12.2 pp

A10 · Collective gains depend on the specification

Appendix · Illustrative interventions on observed networks

Observed-network scenario	Reselect gain	Gain relative to no-shift contrast
5% seeds · q=.3 · proportional	+8.26 pp	+14.71 pp
5% seeds · q=.3 · anchored	−0.18 pp	+6.28 pp
12% seeds · q=.4 · proportional	+6.16 pp	+10.85 pp
12% seeds · q=.4 · anchored	−2.57 pp	+2.12 pp

A positive task contrast can coexist with a negative reselection gain.

A11 · Additional relations in the adoption models

Supplementary evidence

Added relation	Added $b$	$p$	Visiting $b$
Medical consultation	.004	.898	.173
Temple co-presence	.030	.285	.175
Temple + nonkin (nonkin term)	−.016	.655	.179
Kinship	.058	.097	.169
Rice/kerosene inflow	.107	.048	.148
Rice/kerosene outflow	−.029	.614	.148

Every Visiting coefficient has p < .05. Inflow and outflow enter together; the other rows refer to separate specifications.

Selected references · 1

Banerjee, A., Chandrasekhar, A. G., Duflo, E., & Jackson, M. O. (2013). “The Diffusion of Microfinance.” Science 341(6144). doi:10.1126/science.1236498.
Banerjee, A., Chandrasekhar, A. G., Duflo, E., & Jackson, M. O. (2019). “Using Gossips to Spread Information: Theory and Evidence from Two Randomized Controlled Trials.” The Review of Economic Studies 86(6):2453–2490. doi:10.1093/restud/rdz008.
Beaman, L., BenYishay, A., Magruder, J., & Mobarak, A. M. (2021). “Can Network Theory-Based Targeting Increase Technology Adoption?” American Economic Review 111(6):1918–1943. doi:10.1257/aer.20200295.
Becker, S. O., Hsiao, Y., Pfaff, S., & Rubin, J. (2020). “Multiplex Network Ties and the Spatial Diffusion of Radical Innovations: Martin Luther’s Leadership in the Early Reformation.” American Sociological Review 85(5):857–894. doi:10.1177/0003122420948059.

Selected references · 2

Centola, D. & Macy, M. (2007). “Complex Contagions and the Weakness of Long Ties.” American Journal of Sociology 113(3):702–734. doi:10.1086/521848.
Chandrasekhar, A. G., Chaudhary, V., Golub, B., & Jackson, M. O. (2026). “Multiplexing in Networks and Diffusion.” Proceedings of the National Academy of Sciences 123(28). doi:10.1073/pnas.2534923123.
Hsiao, Y. & Christakis, N. A. (2026). “The Influence “Paradox”: When More Network Ties Lead to Less Change.” American Sociological Review 91(3):464–489. doi:10.1177/00031224261438845.
Larson, J. M. & Rodríguez, P. L. (2023). “The Risk of Aggregating Networks When Diffusion Is Tie-Specific.” Applied Network Science 8:21. doi:10.1007/s41109-023-00546-7.

Selected references · 3

Omodei, E. & Arenas, A. (2016). “Untangling the Role of Diverse Social Dimensions in the Diffusion of Microfinance.” Applied Network Science 1(1):14. doi:10.1007/s41109-016-0016-x.
Rule, A., Sabetta, L., & Bearman, P. (2026). “Pathways and Chance.” Sociologica 20(1). doi:10.60923/issn.1971-8853/23641.
Min, B., Gwak, S.-H., Lee, N., & Goh, K.-I. (2016). “Layer-switching cost and optimality in information spreading on multiplex networks.” Scientific Reports 6:21392. doi:10.1038/srep21392.
Valente, T. W. & Davis, R. L. (1999). “Accelerating the Diffusion of Innovations Using Opinion Leaders.” The ANNALS of the American Academy of Political and Social Science 566(1):55–67. doi:10.1177/000271629956600105.

Selected references · 4

Katz, E. & Lazarsfeld, P. F. (1955). Personal Influence: The Part Played by People in the Flow of Mass Communications. Free Press.
Coleman, J. S. (1988). “Social Capital in the Creation of Human Capital.” American Journal of Sociology 94:S95–S120. doi:10.1086/228943.
Gould, R. V. (1991). “Multiple Networks and Mobilization in the Paris Commune, 1871.” American Sociological Review 56(6):716–729. doi:10.2307/2096251.

Selected references · 5

White, H. C. (2008). Identity and Control: How Social Formations Emerge. 2nd ed. Princeton University Press.
McAdam, D. (1986). “Recruitment to High-Risk Activism: The Case of Freedom Summer.” American Journal of Sociology 92(1):64–90. doi:10.1086/228463.
Cai, J., de Janvry, A., & Sadoulet, E. (2015). “Social Networks and the Decision to Insure.” American Economic Journal: Applied Economics 7(2):81–108. doi:10.1257/app.20130442.