On Causal Analysis for Heterogeneous Networks Katerina Marazopoulou, - - PowerPoint PPT Presentation

On Causal Analysis for Heterogeneous Networks

Katerina Marazopoulou, David Arbour, David Jensen

August 2017

University of Massachusetts Amherst College of Information and Computer Sciences

KDD Workshop on Causal Discovery

Katerina Marazopoulou On Causal Analysis for Heterogeneous Networks

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source: Visual Complexity

Causal inference in networks: How is the behavior of an individual affected by his/her peers?

Katerina Marazopoulou On Causal Analysis for Heterogeneous Networks

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source: Visual Complexity

How does the presence of multiple relationship types affect causal analysis?

Katerina Marazopoulou On Causal Analysis for Heterogeneous Networks

Outline

• Background: Causal effect estimation on networks
• Causal effect estimation in heterogeneous networks
• Experiments on synthetic data
• Application on real-world dataset

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Katerina Marazopoulou On Causal Analysis for Heterogeneous Networks

Causal Effect Estimation in Networks

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friends

Katerina Marazopoulou On Causal Analysis for Heterogeneous Networks

• Population of n individuals that form an undirected graph
• Binary treatment T and outcome O

Causal Effect Estimation in Networks

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friends

Oi(T = t)

t ∈ {0, 1}n where

• The outcome of a node depends on the global treatment

assignment:

τ(1, 0) = 1 n

n

X

i=1

E[Oi(T = 1) − Oi(T = 0)]

• ATE between global treatment and global control

Katerina Marazopoulou On Causal Analysis for Heterogeneous Networks

Causal effect estimation

1. Treatment assignment 2. Exposure model: When an individual is considered to be treated 3. Analysis: How to estimate the causal quantity of interest

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Estimation procedure for causal inference:

Katerina Marazopoulou On Causal Analysis for Heterogeneous Networks

Gui, Basin, Han. WWW 2015

Estimation procedure for causal inference:

Causal effect estimation

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1. Treatment assignment 2. Exposure model: Fraction neighborhood exposure [Gui et al. 2015] 3. Analysis: Linear regression adjustment [Gui et al. 2015]

Katerina Marazopoulou On Causal Analysis for Heterogeneous Networks

The Gui et al. framework

• Fraction neighborhood exposure model:

The response function depends on a node’s own treatment assignment and the proportion of its treated peers

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g(Ti, λi) = α + βTi + γλi τ(1, 0) = g(Ti = 1, λi = 1) − g(Ti = 0, λi = 0) = β + γ

• ATE:

Katerina Marazopoulou On Causal Analysis for Heterogeneous Networks

Heterogenous Network

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friends coworkers

Katerina Marazopoulou On Causal Analysis for Heterogeneous Networks

Response function

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Heterogeneous networks:

g(Ti, λi) = α + βTi + γλi gf,c(Ti, λi) = α + βTi + γfλf

i + γcλc i

Homogeneous networks:

Katerina Marazopoulou On Causal Analysis for Heterogeneous Networks

Sets of peers

• There are more options than friends and coworkers.
• We can consider any combination of non-overlapping sets of peers

friends and coworkers friends only friends or coworkers but not both

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gf,c(Ti, λi) = α + βTi + γfλf

i + γcλc i

Katerina Marazopoulou On Causal Analysis for Heterogeneous Networks

Peer-sets of interest

• Friends (homogeneous network)
• Coworkers (homogeneous network)
• Friends or coworkers (union as a homogeneous network)
• Disjoint
• Friends-coworkers

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Katerina Marazopoulou On Causal Analysis for Heterogeneous Networks

Sets of peers we consider

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A B C D

Coworkers Friends only Friends and coworkers

A B C D

Coworkers only Friends

A B C D

Coworkers Friends or coworkers

A B C D

Friends

A B C D

friends coworkers Friends Coworkers Friends or coworkers Disjoint Friends-coworkers

Katerina Marazopoulou On Causal Analysis for Heterogeneous Networks

Peer sets of interest: Where are they used?

• Response functions
• ATE estimators
• Outcome generation

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Used for:

Katerina Marazopoulou On Causal Analysis for Heterogeneous Networks

Peer sets of interest: Where are they used?

• Response functions
• ATE estimators
• Outcome generation

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Friends-coworkers Used for:

Friends

A B C D

Coworkers

gf,c(Ti, λi) = α + βTi+γfλf

i +γcλc i

τf,c = β+γf+γc Oi = w0 + w1Ti+wf

2

F[·, i]>Ot DF

i

+wc

2

C[·, i]>Ot DC

i

+ ✏

Katerina Marazopoulou On Causal Analysis for Heterogeneous Networks

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How does ignoring/mis-specifying the type of relationships affect estimation of causal effects?

Katerina Marazopoulou On Causal Analysis for Heterogeneous Networks

Experiments (synthetic data)

Goal: impact on estimation of causal effects

• Generation of graphs

Erdos-Renyi Watts-Strogatz Stochastic block model

• Generation of treatment values
• 1. Independent assignment for every node
• 2. Graph cluster randomization [Ugander et al. 2013]

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Ugander, Karrer, Backstrom, Kleinberg. KDD 2013

Katerina Marazopoulou On Causal Analysis for Heterogeneous Networks

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• Generation of outcome values
• 1. Outcome Interference
• 2. Treatment Interference

Experiments (synthetic data)

Oi,t+1 ∼ w0 + w1Ti + f(Opeers of i,t) + ✏ Oi ∼ w0 + w1Ti + f(Tpeers of i) + ✏

where: ✏ = ✏N(0, 1)

Katerina Marazopoulou On Causal Analysis for Heterogeneous Networks

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Experiment configuration:

• Graph model: Watts-Strogatz
• Treatment assignment: Graph cluster randomization
• Treatment probability: 0.5
• Outcome generation: Treatment interference

Results

Katerina Marazopoulou On Causal Analysis for Heterogeneous Networks

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−6.3 −30.2 0.1 3.8 −12.2 2 −46.2 −11.2 −2.8 −16.7 −3.7 −0.1 2 −4.1 −6.4 −25 3.8 −19.6 −7.7 −19.5 −6.9

Coworkers Disjoint Friends Friends−Coworkers Friends or Coworkers Coworkers Disjoint Friends Friends−Coworkers Friends or Coworkers

Generative model Assumed model

10 20 30 40

Absolute relative bias

Katerina Marazopoulou On Causal Analysis for Heterogeneous Networks

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Experiment configuration:

• Graph model: Watts-Strogatz
• Treatment assignment: Graph cluster randomization
• Treatment probability: 0.5
• Outcome generation: Treatment interference

Results

Katerina Marazopoulou On Causal Analysis for Heterogeneous Networks

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Experiment configuration:

• Graph model: Watts-Strogatz
• Treatment assignment: Graph cluster randomization
• Treatment probability: varying
• Outcome generation: Treatment interference

Results

Katerina Marazopoulou On Causal Analysis for Heterogeneous Networks

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• Generative model:

Coworkers Generative model: Disjoint Generative model: Friends Generative model: Friends−Coworkers Generative model: Friends or Coworkers

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

−30 −20 −10

Treatment probability Relative bias (% over true ate)

Exposure model

• Coworkers

Disjoint Friends Friends−Coworkers Friends or Coworkers

Katerina Marazopoulou On Causal Analysis for Heterogeneous Networks

Model selection

Given a set of alternative models, is it possible to identify the true generating model?

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Procedure:

• Generate synthetic networks and synthetic data (as before).
• Compute BIC for each of the five alternative models.
• Select model with the lowest BIC.

Katerina Marazopoulou On Causal Analysis for Heterogeneous Networks

Model selection

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Erdos−Renyi Stochastic−block−model Watts−Strogatz C0 C1 C2 C3 C0 C1 C2 C3 C0 C1 C2 C3 0.00 0.25 0.50 0.75 1.00

Configuration of coefficients Accuracy of model selection Noise

0.5 1.0 2.0

Katerina Marazopoulou On Causal Analysis for Heterogeneous Networks

Model selection

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Random Random Random

Erdos−Renyi Stochastic−block−model Watts−Strogatz C0 C1 C2 C3 C0 C1 C2 C3 C0 C1 C2 C3 0.00 0.25 0.50 0.75 1.00

Configuration of coefficients Accuracy of model selection Noise

0.5 1.0 2.0

Katerina Marazopoulou On Causal Analysis for Heterogeneous Networks

Real data

• Study on the diffusion of micro financing loans through various social networks
• Survey conducted in 75 villages in southern India
• Village-level survey and follow-up survey on a subsample of individuals for each village
• Individual surveys identify 13 types of social relationships (e.g., friends, relatives,

borrowing money from, going to temple with)

• Individual’s attributes (age, gender, etc)

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Katerina Marazopoulou On Causal Analysis for Heterogeneous Networks

Real heterogeneous network

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Katerina Marazopoulou On Causal Analysis for Heterogeneous Networks

Experimental setup for real data

• Several pairs of social relationships
• Combinations of treatment-outcome variables
• Estimate effect using different response functions

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Katerina Marazopoulou On Causal Analysis for Heterogeneous Networks

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• 0.0

0.2 0.4 0.6 0.8 friends−relatives gender−savings friends−relatives gender−working helping with decisions−relatives gender−working borrowing money−relatives gender−working

Relation1−Relation2 Treatment−Outcome Estimated effect Assumed model

• Rel1

Rel2 Rel1orRel2 Rel1−Rel2 Disjoint

Katerina Marazopoulou On Causal Analysis for Heterogeneous Networks

Summary

• Recent work has extended causal inference frameworks for network data.
• We address the case of heterogeneous networks and causal effect estimation in this

framework.

• Mis-specifying the relational structure of causal dependence can lead to significant bias.
• Model selection for distinguishing among candidate response functions.

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Katerina Marazopoulou On Causal Analysis for Heterogeneous Networks

Directions for future work

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• Formal characterization of bias and variance of ATE estimators for heterogeneous

networks

• Interactions of relational semantics (effect present from multiple relational phenomena)
• Measure of model selection for relational data
• Fully automated methods for choosing appropriate response functions
• Extending A/B testing framework for heterogeneous networks

Katerina Marazopoulou On Causal Analysis for Heterogeneous Networks

Questions?

Thank you!

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