Shalizi and Thomas (2011) proved that in observational social-network studies, social contagion (influence of one person's past behavior on another's current behavior) and homophily (tie formation between similar people) are generically confounded when the traits driving tie formation are latent: no pattern of observable associations, including asymmetries in regression coefficients, distinguishes them without strong assumptions. They left open a coherent research program: determine what observational network data alone can establish about contagion when latent homophily is allowed — identification conditions or parametric families under which contagion is identified, sharp bounds on contagion effects and on the association and regression asymmetry that homophily alone can produce, and the conditions under which estimated network structure (communities or latent positions) suffices as an adjustment proxy. In the bare setting of one observational network plus longitudinal outcomes, generic identification and data-only sharp bounds remain open, and the specific thread of bounding homophily-generated regression asymmetry has no published follow-up.

Background

A recurring claim in observational social-network research is that behaviors and states — obesity, smoking, happiness, loneliness — spread through social ties by influence. The canonical example is Christakis and Fowler (2007), who reported that obesity spreads through the Framingham social network. The methodological obstacle is that the same associations arise without any influence. Homophily is the tendency of people to form social ties with others who resemble them; when the traits that drive tie formation are unobserved (latent homophily), two friends' behaviors are similar both under contagion (my friend's past behavior changes mine) and under homophily (we became friends because we were alike, and alike people behave alike). Shalizi and Thomas (2011) formalized this with directed acyclic graph (DAG) models — graphical representations in which arrows encode direct causal effects — in which an unobserved trait XiX_i influences both tie formation (the adjacency AijA_{ij}, so that AijA_{ij} is a collider: conditioning on it induces spurious association between the two friends' traits) and the outcome Yi(t)Y_i(t), while Yi(t)Y_i(t) may also depend on the friend's lagged outcome Yj(t−1)Y_j(t-1) (contagion). Their results: (i) contagion effects are generically unidentified — distinct causal structures (contagion, latent homophily, direct trait effects) generate exactly the same observable distribution of ties and outcomes; (ii) the popular 'argument from asymmetry' fails — with unreciprocated ties, homophily alone can produce asymmetric regression coefficients, as they demonstrate in a latent-homophily simulation with no direct influence between nodes' outcomes; (iii) in the linear-Gaussian version of their structure, any standardized regression coefficient of Yi(t)Y_i(t) on Yj(t−1)Y_j(t-1) can be reproduced by adjusting the path coefficients of the latent variables, so the regression coefficient alone cannot even bound the causal effect. As constructive responses they sketched three routes, each explicitly left open: (a) a random-halves procedure (repeatedly split nodes into two random halves and predict one half's outcomes from the other half's lagged outcomes), a proof of possibility with possibly very low power that also fails if another person's latent trait directly affects Yi(t)Y_i(t); (b) a bounding program in the spirit of Manski's partial identification — deriving the tightest bounds on a parameter that the observable data alone imply when the parameter cannot be pinned down exactly — bounds on the causal effect of Yj(t−1)Y_j(t-1) on Yi(t)Y_i(t), on the association generable by homophily alone, and on the regression asymmetry producible without influence (so that an observed asymmetry exceeding the bound would be evidence of contagion); (c) controlling for estimated communities or latent positions, which requires screening-off conditions — the estimates must render true community memberships conditionally independent of behaviors given the observed graph — and whose failure under misspecified block structure (e.g., when the truth follows a mixed-membership model, in which each node belongs to several communities with fractional weights, or a role model based on structural positions, rather than one community per node) was left unquantified. Later work built a substantial methodology literature around this challenge: sensitivity analysis, which asks how strong the unmeasured homophily would have to be to explain away an observed association (VanderWeele 2011); latent-position and embedding-based adjustment, i.e., controlling for estimates of each node's position in an unobserved social space recovered from the observed network (Xu 2018; McFowland and Shalizi 2023; Chen, Liu and Zhang 2022); negative-control and placebo-outcome identification, which bring in auxiliary variables that cannot carry the influence effect but do share the latent confounding, so that their observed associations reveal and help remove the confounding (Liu and Tchetgen Tchetgen 2022; Egami and Tchetgen Tchetgen 2024; Egami 2024); and mechanism tests in graphical models (Wu and Bhattacharya 2024). Each of these, however, buys identification with auxiliary variables or explicit modeling assumptions rather than solving the bare-setting question, which is the subject of this problem.

Problem Statement

In the latent-homophily causal structure of Shalizi and Thomas (2011) — an observational network in which ties AijA_{ij} form depending on unobserved, time-invariant traits XiX_i, XjX_j that also drive each node's behavior, while each outcome Yi(t)Y_i(t) may depend on its own past, on observed covariates, and on neighbors' lagged outcomes Yj(t−1)Y_j(t-1) — determine what the observable data alone can establish about social contagion in the bare setting: a single observational network plus longitudinal measurements of ties, covariates, and outcomes, with no randomized variation and no auxiliary variables such as negative controls or placebo outcomes. The single overarching objective is to replace the paper's generic non-identification result with a positive characterization on its three linked fronts: (1) Identification: exhibit observable-data conditions, or a parametric family of tie-formation and outcome processes, under which the causal effect of Yj(t−1)Y_j(t-1) on Yi(t)Y_i(t) is identified from observational network data without assuming away latent homophily: the conditions must neither make the components of XX that influence YY or AA observable (the covariate-sufficiency route of the source's Figures 3a–3b, which the source itself notes restores identifiability) nor reduce to the parametric network-growth and latent-structure families already established in the literature cited below (e.g., stochastic block and continuous latent-space models); (2) Bounding: when the effect is not identified, derive bounds in terms of the observable distribution on the causal effect of Yj(t−1)Y_j(t-1) on Yi(t)Y_i(t); on the magnitude of association that homophily alone can generate (for an explicitly specified association measure); and on the degree of regression asymmetry (the difference between the coefficient of Yi(t)Y_i(t) on Yj(t−1)Y_j(t-1) and that of Yj(t)Y_j(t) on Yi(t−1)Y_i(t-1), the two directions of the same tie) producible without direct influence, so that an observed association or asymmetry exceeding the bound would constitute evidence for contagion; (3) Proxy validity: characterize the classes of network models for which estimated latent structure (communities or latent positions) is a predictively sufficient statistic — i.e., conditioning on the estimates renders the outcomes conditionally independent of the latent traits, screening them off — and quantify the bias induced when the assumed block structure is misspecified (e.g., mixed-membership or role structures fitted as simple modules). These fronts jointly support one scientific objective: a defensible account of when, and how strongly, observational social-network data can support causal claims about contagion.

References

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