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Causality in complex networks

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Sebastian Benthall
Sebastian Benthall

I explore ways to combine complex network science with the Rubin model of conference inference. In broad strokes, I discuss the difference between exogenous shocks and endogenous process, and how granularity in time can be used to tease causality out of a complex system.

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Causality in Complex Networks Sebastian Benthall D-Lab
Why causality? Causal relations are scientifically interesting because when exposed, they are: ● a reliable mechanism ● that supports intervention By understanding causality, we can predict and control. This is the interest behind causal knowledge.
Why this talk? ● A “working group” talk - I don’t claim to be an expert ● I’m working through research problems that are challenging to me ● I’m presenting this work in progress both to inform and solicit feedback ● You are welcome to do the same with your work!
Potential outcomes The Rubin Causal Model (RCM) or potential outcomes framework is ascendant.
Potential outcomes “The causal effect of a treatment on a single individual or unit of observation is the comparison (e.g., difference) between the value of the outcome if the unit is treated and the value of the outcome if the unit is not treated.” (Angrist, Imbens, and Rubin, 1996)
Potential outcomes “The causal effect of a treatment on a single individual or unit of observation is the comparison (e.g., difference) between the value of the outcome if the unit is treated and the value of the outcome if the unit is not treated.” (Angrist, Imbens, and Rubin, 1996)
Potential outcomes Average effect: (10+9+7+12)/4 = 9.5 Controlled Outcome - Yi (0) Treated Outcome - Yi (1) Causal Effect of Treatment Yi (1) - Y(0) Alice 20 30 10 Bob 15 24 9 Cathy 10 17 7 David 22 34 12
Potential outcomes You only ever see some of these. This has been called the Fundamental Problem of Causal Inference Controlled Outcome - Yi (0) Treated Outcome - Yi (1) Causal Effect of Treatment Yi (1) - Y(0) Alice 20 30 10 Bob 15 24 9 Cathy 10 17 7 David 22 34 12
Potential outcomes Stable Unit Treatment Value Assumption "the [potential outcome] observation on one unit should be unaffected by the particular assignment of treatments to the other units"
Potential outcomes Controlled Outcome - Yi (0) Treated Outcome - Yi (1) Causal Effect of Treatment Yi (1) - Y(0) Alice 20 30 10 Bob, if Alice in not treated 15 24 9 Bob, if Alice is treated 18 29 11 Cathy 10 17 7 David 22 34 12
Potential outcomes So for every unit, we have to map out all the variables that can have an effect on the potential outcomes. Spouse treated Unit treated Outcome
Potential outcomes So for every unit, we have to map out all the variables that can have an effect on the potential outcomes. A great tool for this: Pearl’s causal networks.
Causal networks Pearl, Causality, http://bayes.cs.ucla.edu/BOOK-2K/
Note There are differences between Pearl and Rubin’s frameworks but their core concepts are compatible, so says Andrew Gelman, 2009: http://andrewgelman.com/2009/07/07/more_on_pearls/
Causal networks N.B. An interesting thing about causal networks is that the conditional probability distributions can be arbitrarily complex. Also, variables need not just be whole numbers or scalars. They could be a matrix.
Complex networks Now I’m going to talk about networks that are not causal networks. Sometimes in the literature these are called complex networks.
from http://www.spandidos-publications.com/ijo/43/6/1737
from http: //noduslabs. com/radar/type s-networks- random-small- world-scale- free/
Complex networks There are lots of different kinds of networks observed in nature and society. They can differ substantially in their emergent properties.
Complex networks def: emergent property “An emergent property is a property which a collection or complex system has, but which the individual members do not have. A failure to realize that a property is emergent, or supervenient, leads to the fallacy of division.”
Degree Distribution from http://www.alexeikurakin. org/main/lecture4Ext.html
Assortativity from http://iopscience.iop.org/1742- 5468/2008/03/P03008/figures
How to make a graph Different processes for generating graphs have result in graphs with different properties.
How to make a graph ● Erdős–Rényi (ER) model: G(n,p): Create n nodes and create edges with probability p ● Barabási–Albert (BA) model: ○ Begin with a fully connected network of m0 nodes ○ Each new node is connected to m (< m0 ) node probability proportional to degree ki of each node i
How to predict a graph The distribution of degree of an Erdős–Rényi graph is binomial. The distribution of degree of a Barabási– Albert (BA) graph is scale-free/power-law.
Bayes Theorem Recall Bayes Theorem: P(H|D) ∝ P(D|H) P(H)
Bayes Theorem Recall Bayes Theorem: P(H|D) ∝ P(D|H) P(H) If we can show a difference in the likelihood of data under different hypotheses, we can learn something
What process created this graph?
What process created this graph? Histogram of observed node degrees d
What process created this graph? Can in principle compare P(d|ER) and P(d|BA).
What process created this graph? ● We can in principle statistically distinguish hypotheses about generative processes based on emergent properties. ● Is this a causal inference?
Ways to model this Process Graph
Ways to model this All the logic of graph generation is buried in the conditional probability function P(B|A). A Process B Graph
Ways to model this All the logic of graph generation is buried in the conditional probability function P(B|A). “Logical causation” allows no intervention! A Process B Graph
Ways to model this The Barabási–Albert (BA) model suggests this interpretation of graph changing over time. You can model time series data like this. B0 B1 B2 B3 B4
Ways to model this Now we can model a stable transition function within the graph and external causes. B0 B1 B2 B3 B4 C
Ways to model this In other words, endogenous process + exogenous shocks B0 B1 B2 B3 B4 C
Tools in the toolbox If we want to understand the effect of a kind of exogenous shock, and we know the endogenous process, then we can look for natural experiments that expose the treated outcome.
Tools in the toolbox To know the endogenous process, then compute the likelihood of emergent properties. B0 B1 B2 B3 B4 EEE E E
Problems ● The vastness of the hypothesis space of graph generation processes ● Related: how do you choose a prior?
Problems ● Computing the likelihood of a graph’s emergent properties given a particular generation process is ○ tricky math ○ maybe computationally hard
Dissatisfactions ● We have continued to bury much of the mechanism of interest in the conditional probability function. ● Suppose we want to cash this out as a finer- grained mechanism that supports finer interventions ● Can we think at multiple levels of abstraction at once?
Thanks! Contact me at: sb at ischool dot berkeley dot edu

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Causality in complex networks

  • 2. Why causality? Causal relations are scientifically interesting because when exposed, they are: ● a reliable mechanism ● that supports intervention By understanding causality, we can predict and control. This is the interest behind causal knowledge.
  • 3. Why this talk? ● A “working group” talk - I don’t claim to be an expert ● I’m working through research problems that are challenging to me ● I’m presenting this work in progress both to inform and solicit feedback ● You are welcome to do the same with your work!
  • 4. Potential outcomes The Rubin Causal Model (RCM) or potential outcomes framework is ascendant.
  • 5. Potential outcomes “The causal effect of a treatment on a single individual or unit of observation is the comparison (e.g., difference) between the value of the outcome if the unit is treated and the value of the outcome if the unit is not treated.” (Angrist, Imbens, and Rubin, 1996)
  • 6. Potential outcomes “The causal effect of a treatment on a single individual or unit of observation is the comparison (e.g., difference) between the value of the outcome if the unit is treated and the value of the outcome if the unit is not treated.” (Angrist, Imbens, and Rubin, 1996)
  • 7. Potential outcomes Average effect: (10+9+7+12)/4 = 9.5 Controlled Outcome - Yi (0) Treated Outcome - Yi (1) Causal Effect of Treatment Yi (1) - Y(0) Alice 20 30 10 Bob 15 24 9 Cathy 10 17 7 David 22 34 12
  • 8. Potential outcomes You only ever see some of these. This has been called the Fundamental Problem of Causal Inference Controlled Outcome - Yi (0) Treated Outcome - Yi (1) Causal Effect of Treatment Yi (1) - Y(0) Alice 20 30 10 Bob 15 24 9 Cathy 10 17 7 David 22 34 12
  • 9. Potential outcomes Stable Unit Treatment Value Assumption "the [potential outcome] observation on one unit should be unaffected by the particular assignment of treatments to the other units"
  • 10. Potential outcomes Controlled Outcome - Yi (0) Treated Outcome - Yi (1) Causal Effect of Treatment Yi (1) - Y(0) Alice 20 30 10 Bob, if Alice in not treated 15 24 9 Bob, if Alice is treated 18 29 11 Cathy 10 17 7 David 22 34 12
  • 11. Potential outcomes So for every unit, we have to map out all the variables that can have an effect on the potential outcomes. Spouse treated Unit treated Outcome
  • 12. Potential outcomes So for every unit, we have to map out all the variables that can have an effect on the potential outcomes. A great tool for this: Pearl’s causal networks.
  • 13. Causal networks Pearl, Causality, http://bayes.cs.ucla.edu/BOOK-2K/
  • 14. Note There are differences between Pearl and Rubin’s frameworks but their core concepts are compatible, so says Andrew Gelman, 2009: http://andrewgelman.com/2009/07/07/more_on_pearls/
  • 15. Causal networks N.B. An interesting thing about causal networks is that the conditional probability distributions can be arbitrarily complex. Also, variables need not just be whole numbers or scalars. They could be a matrix.
  • 16. Complex networks Now I’m going to talk about networks that are not causal networks. Sometimes in the literature these are called complex networks.
  • 17.
  • 20. Complex networks There are lots of different kinds of networks observed in nature and society. They can differ substantially in their emergent properties.
  • 21. Complex networks def: emergent property “An emergent property is a property which a collection or complex system has, but which the individual members do not have. A failure to realize that a property is emergent, or supervenient, leads to the fallacy of division.”
  • 24. How to make a graph Different processes for generating graphs have result in graphs with different properties.
  • 25. How to make a graph ● Erdős–Rényi (ER) model: G(n,p): Create n nodes and create edges with probability p ● Barabási–Albert (BA) model: ○ Begin with a fully connected network of m0 nodes ○ Each new node is connected to m (< m0 ) node probability proportional to degree ki of each node i
  • 26. How to predict a graph The distribution of degree of an Erdős–Rényi graph is binomial. The distribution of degree of a Barabási– Albert (BA) graph is scale-free/power-law.
  • 27. Bayes Theorem Recall Bayes Theorem: P(H|D) ∝ P(D|H) P(H)
  • 28. Bayes Theorem Recall Bayes Theorem: P(H|D) ∝ P(D|H) P(H) If we can show a difference in the likelihood of data under different hypotheses, we can learn something
  • 29. What process created this graph?
  • 30. What process created this graph? Histogram of observed node degrees d
  • 31. What process created this graph? Can in principle compare P(d|ER) and P(d|BA).
  • 32. What process created this graph? ● We can in principle statistically distinguish hypotheses about generative processes based on emergent properties. ● Is this a causal inference?
  • 33. Ways to model this Process Graph
  • 34. Ways to model this All the logic of graph generation is buried in the conditional probability function P(B|A). A Process B Graph
  • 35. Ways to model this All the logic of graph generation is buried in the conditional probability function P(B|A). “Logical causation” allows no intervention! A Process B Graph
  • 36. Ways to model this The Barabási–Albert (BA) model suggests this interpretation of graph changing over time. You can model time series data like this. B0 B1 B2 B3 B4
  • 37. Ways to model this Now we can model a stable transition function within the graph and external causes. B0 B1 B2 B3 B4 C
  • 38. Ways to model this In other words, endogenous process + exogenous shocks B0 B1 B2 B3 B4 C
  • 39. Tools in the toolbox If we want to understand the effect of a kind of exogenous shock, and we know the endogenous process, then we can look for natural experiments that expose the treated outcome.
  • 40. Tools in the toolbox To know the endogenous process, then compute the likelihood of emergent properties. B0 B1 B2 B3 B4 EEE E E
  • 41. Problems ● The vastness of the hypothesis space of graph generation processes ● Related: how do you choose a prior?
  • 42. Problems ● Computing the likelihood of a graph’s emergent properties given a particular generation process is ○ tricky math ○ maybe computationally hard
  • 43. Dissatisfactions ● We have continued to bury much of the mechanism of interest in the conditional probability function. ● Suppose we want to cash this out as a finer- grained mechanism that supports finer interventions ● Can we think at multiple levels of abstraction at once?
  • 44. Thanks! Contact me at: sb at ischool dot berkeley dot edu