The basic concept of branching process

[albertshx]

Dear all,
I have a question on the basic concept of branching process.
Say, in such a process, different objects of a generation don't
interfere with each other.
Consider an email forwarding analogy where one node forwards an email
to some other users. And those users will continue forwarding
the email to others...thus forming a cascade. However, it's
possible that different nodes may forward the email to one same node.
In this sense, is the process a branching process?

Thank you a lot.

 

[Stephen Tashi]

In this sense, is the process a branching process?

If you take a situation in the real world and ask a question like "Is this a branching process?" or "Is this a Markov process?", you aren't really asking a question that has an objective answer.

The mathematical definitions of stochastic processes say that these processes have certain parts such as "states", "individuals", "descendants" etc. Until you say what things in the real world correspond these parts, then it is impossible to evaluate whether the situation in the real world satisfies the required mathematical assumptions of the process.

You didn't specify such definitions. When you say "in this sense", the phrase is ambiguous. If we ask a general question like "Is this, in some sense" a branching process?" or "Is this, in some sense, a Markov process?" then we allow the answerer complete freedom as to how he defines parts of the process such as "individuals" and "descendants". For example, in the case of email forwarding, you might be thinking of an "individual" as an email message forwarded from person A to person B. However, there is no law of mathematics that says that we must define an "individual" in this manner, so perhaps clever and devious people looking at the same situation might come up with different and more complicated definitions of "individual".

If we take "individual" to mean an email message forwarded from person A to person B then we still have the problem of defining a "descendant". For example, we could assume a simple model where no two emails arrive at the same time and person A always does his forwarding of messages in response to the first copy of a message he recieves. Then we could define the copy he forwards is a "descendant" of the first copy he got and not of the subsequent copies.

In my opinion, in the way the average person would define things, the example of email forwarding is not a branching process. However, I wouldn't put limits on human ingenuity by saying that no way exists to look at it as a branching process.