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Causal Models for Growing Networks

Gecia Bravo-Hermsdorff, Lee M. Gunderson, Kayvan Sadeghi

TL;DR

The paper addresses causal modeling of growing networks by shifting focus from node-exchangeable distributions to invariances of the causal mechanisms generating dyadic edges. It constructs a taxonomy of 96 deletion-invariant causal meta-DAGs over dyad variables and reduces them to 21 transitively closed classes, enabling distributed and asynchronous evaluation. As a canonical example, it introduces the Distributed Affine Preferential Attachment (DAPA) model, where $x_{ij}\sim\text{Bernoulli}(p_{ij})$ with $p_{ij}=\frac{\alpha+\theta_{in}d_i^{in}+\theta_{out}d_i^{out}}{j-2+\alpha+\beta}$, revealing three sparsity regimes and a flexible power-law degree distribution with exponents determined by $\theta_{in}$ and $\theta_{out}$. The framework yields natural baselines for causal inference in relational data and supports generalization, interventions, and counterfactual analyses in distributed settings, with practical implications for understanding growth, phase transitions, and network resilience in real-world systems.

Abstract

Real-world networks grow over time; statistical models based on node exchangeability are not appropriate. Instead of constraining the structure of the \textit{distribution} of edges, we propose that the relevant symmetries refer to the \textit{causal structure} between them. We first enumerate the 96 causal directed acyclic graph (DAG) models over pairs of nodes (dyad variables) in a growing network with finite ancestral sets that are invariant to node deletion. We then partition them into 21 classes with ancestral sets that are closed under node marginalization. Several of these classes are remarkably amenable to distributed and asynchronous evaluation. As an example, we highlight a simple model that exhibits flexible power-law degree distributions and emergent phase transitions in sparsity, which we characterize analytically. With few parameters and much conditional independence, our proposed framework provides natural baseline models for causal inference in relational data.

Causal Models for Growing Networks

TL;DR

The paper addresses causal modeling of growing networks by shifting focus from node-exchangeable distributions to invariances of the causal mechanisms generating dyadic edges. It constructs a taxonomy of 96 deletion-invariant causal meta-DAGs over dyad variables and reduces them to 21 transitively closed classes, enabling distributed and asynchronous evaluation. As a canonical example, it introduces the Distributed Affine Preferential Attachment (DAPA) model, where with , revealing three sparsity regimes and a flexible power-law degree distribution with exponents determined by and . The framework yields natural baselines for causal inference in relational data and supports generalization, interventions, and counterfactual analyses in distributed settings, with practical implications for understanding growth, phase transitions, and network resilience in real-world systems.

Abstract

Real-world networks grow over time; statistical models based on node exchangeability are not appropriate. Instead of constraining the structure of the \textit{distribution} of edges, we propose that the relevant symmetries refer to the \textit{causal structure} between them. We first enumerate the 96 causal directed acyclic graph (DAG) models over pairs of nodes (dyad variables) in a growing network with finite ancestral sets that are invariant to node deletion. We then partition them into 21 classes with ancestral sets that are closed under node marginalization. Several of these classes are remarkably amenable to distributed and asynchronous evaluation. As an example, we highlight a simple model that exhibits flexible power-law degree distributions and emergent phase transitions in sparsity, which we characterize analytically. With few parameters and much conditional independence, our proposed framework provides natural baseline models for causal inference in relational data.

Paper Structure

This paper contains 36 sections, 4 theorems, 36 equations, 8 figures, 1 table.

Table of Contents

  1. Causality and Networks
  2. Outline (see Figure \ref{['fig:FirstNatureFigure']}).
  3. Our framework
  4. The Growing Network
  5. The Causal Meta-DAG
  6. Invariances of the Causal Meta-DAGs
  7. Seven Types of Causal Arrows
  8. Arrows Between Dyads Sharing a Node
  9. Arrows between Dyads Not Sharing a Node
  10. The Arrow from a Dyad to Itself
  11. Invariance to Node Deletion
  12. Invariance to Node Marginalization
  13. Computing the Transitive Closure
  14. Streamlining Network Models
  15. Basic Preferential Attachment
  16. ...and 21 more sections

Key Result

Theorem 1

There are $2^5 3 = 96$ deletion-invariant causal meta-DAGs with finite ancestral sets, given by the subsets of $\{\textcolor{NonLocalExteriorColor}{Far},\textcolor{TransitiveColor}{Path},\textcolor{NonLocalInterfaceColor}{Mid},\textcolor{HubColor}{Hub},\textcolor{NonLocalInteriorColor}{Near},\textco

Figures (8)

  • Figure 1: The narrative arc of the paper at a glance. Left column: The nodes of the growing network, represented as circles, have a total ordering. The variables in the model are indexed by the dyads (pairs of nodes), represented as squares (\ref{['sec:GrowingNetworkDescription', 'ref:TheMetaDAGDescription']}). Middle column: Causal relationships between these dyad variables are represented as arrows in a causal DAG describing the generative process of the growing network. We classify the relevant types of such causal arrows, represented by colors (\ref{['ref:InvarianceCausalModel', 'sec:typesofcausalarrows', 'sec:deletion', 'sec:poset']}). Right column: Some combinations of these causal arrows are remarkably parallelizable, such as the model we call Distributed Affine Preferential Attachment (DAPA). In addition to a flexible power law, it naturally contains a phase transition between several well-studied sparse growth rates: from constant average degree, to logarithmic, to polynomial (\ref{['sec:ourppamodel']}).
  • Figure 2: Types of causal arrows between dyads that share a node.The black edges represent the dyads of the growing network and colors represent different types of causal arrows between them.
  • Figure 3: Types of causal arrows between dyads that do not share a node. Horizontal dashed lines indicate the relative ordering of the nodes in the parent dyad ${( )}$ with respect to the nodes in the child dyad $(ij)$.
  • Figure 4: Sparsity and power-laws in the DAPA model.
  • Figure 5: Causal graphs for each of the types of causal arrows between dyads that share a node. Causal meta-DAGs between dyads of a growing network with $5$ nodes that are compatible with network models having the following types of causal arrows:Hub (top-left); Path (top-right); Old (bottom-left); and New (bottom-right).
  • ...and 3 more figures

Theorems & Definitions (4)

  • Theorem 1: Deletion-invariant causal meta-DAGs
  • Theorem 2: Deletion and marginalization-invariant causal meta-DAGs
  • Theorem 3: Phase transitions in the average degree of the DAPA model
  • Theorem 4: Power-law degree distributions of the DAPA model