I think I can see some issues with your specific example that are easily corrected, but you have also struck upon some more general phenomena that I think merits some discussion as well.
Your specific example
I think the problem with the specific example you've given is that you're being a little too loose with
- the meanings of your nodes/variables
- the language around cause and effect
Meanings of nodes/variables
As an example of (1), I'll focus on the town of residence <--> sex relationship.
In this scenario, "sex" is ambiguous because it can be applied at the level of an individual, but from the context you've given, I think you really mean to aggregate it at the level of the "town of residence".
It might be better to call it something like "sex ratio" to make that more clear.
More importantly, "town of residence" implies some special coupling between a town and its residents.
But there are aspects of a town that are independent of its residents (e.g. location, geography, weather, resources, etc).
Why not be more general and have that variable simply represent the concept of "town (identity)"?
Now it is plain to see that we cannot have sex ratio --> town, as there is no way the sex of a town's residents can influence its very identity; towns do not spontaneously change from one to another if their residents' sex ratio changes.
However the culture of, jobs available in, and policies instituted by a town can certainly affect the sex distribution of its inhabitants through birth, death, and immigration, so you can have town --> sex ratio.
To give a concrete example, at the time of writing, New York has a higher ratio of female:male (1.096) than Los Angeles (1.019).
If you move from one city to the other you will observe the change in sex ratio.
However, you cannot change New York into Los Angeles no matter how hard you manipulate the sex ratio or other attributes of the cities.
You may argue "well what if changing the sex ratio influences the culture and policies of the town" - that is a perfectly valid hypothesis; but the culture and policies of the town do not describe the town in its entirety, they are simply some subset of its attributes. So you can have relations like town --> sex ratio --> {culture, policies}.
Language of cause and effect
That previous fix alone should solve your problem, but I'll address point (2) as well about the language around cause and effect.
For example, when you state
sex may be a reason to change residence (e.g. local policies discriminate women and they move to another town)
as evidence for the relation sex --> town of residence, I think this is actually better re-framed as the town creating discriminatory culture/policies, causing women to leave, i.e. town --> discriminatory culture/policies --> sex.
Even though someone may feel that their sex is the "reason" they need to move (and I feel so sorry for anyone who may think this way!), an individual's sex can't be the cause of their need to move, because it cannot be changed (at least not very easily).
The thing that changed and caused this individual to want to move is the creation of discriminatory culture and policy!
You may recall that we just said sex ratio --> {culture, policy} a couple paragraphs ago, but now we're saying {culture, policy} --> sex ratio is possible too... no, there is not a conflict here but you'll have to read the section below on "feedback loops" to see the resolution.
Some more general phenomena
Tightly coupled and missing variables
It may be the case that you have two or more variables that appear very tightly coupled in some way, but are actually related through a missing variable, like a common cause or influence.
For example, consider several light bulbs connected together in series (e.g. like Christmas lights).
As you may be aware, when wired this way, when one of the bulbs burns out, the whole thing stops working.
When one bulb is on, they all are on; when one is off, they all are off; everything is highly correlated.
If someone were to observe that the lights are off, they would not be able to tell which bulb is the one that caused the problem.
Did bulb $b$ burning out cause the other ones to stop working, or was it because of bulb $b^\prime$?
The direction of causality is very unclear here and the naive causal graph where nodes represent the on/off state of each bulb would be fully connected and bidirectional (and therefore not a DAG).
However, if we could inspect and test each individual bulb, then we could in principle figure out which one was broken, and integrate that into our model of the on/off states of the lights.
In this scenario you need more data, depicted by additional nodes in the DAG besides the on/off status of each bulb representing their working/broken state.
Some common causes are a little simpler.
For example, if all the bulbs in our string of lights are off, it could be that the power to the entire thing is off!
This is usually a much simpler hypothesis, but we still need additional data, depicted by a node in the DAG representing power to the lights.
It's also possible, but less common, that two variables may share a common influence, or descendant node in the DAG that is implicitly being conditioned on.
For example, a company's revenue might have two factors: sales volume and per unit price (pretend that these factors are totally independent), which are causally related via the "collider model" volume --> revenue <-- price.
If you implicitly condition on companies with a specific level of revenue (for example by using data from just the Fortune 500), you'll likely find a negative relationship between volume and price (think about volume * price = revenue while holding revenue constant), which could lead you to conflicting causal conclusions like
- high sales volume causes low prices because if it is so easy to produce more product, the barrier to entry is low and competition in the market is high, suppressing prices
- high prices cause low sales volume because there is less demand at higher prices
In the collider model, both of these are wrong (though the second is actually true in real life) because there is no causal path between the two nodes.
Feedback loops
Sometimes you may have a scenario where there is some kind of feedback loop where A influences B, then B influences A again (or some even longer cycle like A -> B -> ... -> Z -> A), and this time it's not just an apparent cycle, you know for sure that it's real.
This happens all the time in physics, economics, chemistry, etc:
- the sun pulls on the earth via gravity, then the earth pulls on the sun, and so on back and forth forever, causing them to orbit each other
- prices go up, which causes workers to demand higher wages; higher wages mean higher costs for firms, which causes firms to increase prices to maintain profit margins (i.e. inflation)
- in a chemical reaction, reagents A + B interact to form C, but C will also spontaneously dissociate back into A + B, causing the relative number of each species to evolve until equilibrium is reached
The thing that all these examples have in common is the flow of time.
When A influences B, it happens at a specific moment in time, and then when B influences A, that happens at a different, later, moment in time.
The duration between these moments could be arbitrarily small in principle so for all practical purposes they could appear to be occurring simultaneously.
Or if this feedback is happening slowly enough we could approximate this process with larger time steps (e.g. months or years might be reasonable in the inflation example).
But the key point is that we can unravel an apparent relation like A <--> B into something like {A(t), B(t)} --> {A(t+1), B(t+1)}.
Now you have sequential data that can be modeled by time series, differential equations, etc.
Summary/TL;DR
In conclusion, when you encounter a scenario like this where it is difficult to reason about the direction of causality, here's what I would advise you consider to get yourself unstuck:
- are you being too loose with your definitions?
- can you rename a node in order to make its meaning more precise? (e.g.
sex -> sex ratio)
- can you refine/generalize a concept that might be too specific? (e.g.
town of residence -> town)
- can you split a more general concept into smaller pieces or attributes?(e.g.
town may have attributes {culture, policies, jobs, demographics, ...})
- are you considering the appropriate level of aggregation (e.g.
sex of an individual vs sex ratio of an entire town)
- are you sure you're thinking about the direction of causality correctly?
- how possible/practical is it to modify/manipulate a certain variable? (e.g. the sex of an individual vs. the sex ratio of a town)
- if the state of A changes, but the state of B remains the same, any change in the state of C was potentially caused by the change in A, and A might not be influenced by B (i.e.
A --> C <-- B might be a decent hypothesis)
- think about "what would happen if I changed X?" and "how could X be changed?"
- could your model have omitted variables that could explain things more fully?
- could your variables have common causes you are not taking into account? (e.g. all light bulbs can be turned on/off by power, or missing variables for working/broken state)
- could your variables have common influences that you are unknowingly/implicitly conditioning on? (e.g.
volume --> revenue <-- price induces acausal relationship between volume/price when conditioning on revenue)
- can you decompose the relationship at one moment into a sequence of events over time?
