Well, in the model I mentioned in answer to your question 1 (random walkers at stationarity on an undirected, connected network), then, if the number of walkers is high enough that fluctuations are negligible, the inflow to any node is equal to its outflow. Therefore, also for a subset of S nodes, the net flow will be zero. The flow either in or out of a node in this setting is proportional to its degree. So for the subset S, its net inflow (equal to its net outflow) will be the total number of walkers on the network (sum of transactions) multiplied by the number of links that connect S to the rest of the network, divided by the total number of links in the network.

Of course, for a more realistic model things might be different. May I ask what it is that you are trying to model?

Sam

Dear Sam,

I have explained a bit what I want to do as an added comment to your answer for my question 1. If I repeat it briefly, I am working on monetary economics and want to know what determines the demand for money. Or more precisely, I want to know if the concept “demand for money” can have any operational meaning.

Let a node be an individual, or a legal person like a stock company and others. It makes many transactions, say buy materials, sell its products, pay salaries and return principals of its borrowing. For each transaction, money flow takes place. The first trouble with my inquiry is we know very little what determines the volume of each transaction. It will be reasonable to assume the volumes obey a power-law distribution, but I have no idea about dynamical properties. But this is only a starting point. What I want to know is how money is related the smooth going of various transactions.

For example, take a single person. Out-flow of its money cannot exceed in-flow eternally. The cumulative net out-flow can only exceed its money balance which it had at the initial point of a period. If by a needs of transaction, out-flow may exceeds the in-flow plus money balance, it has to appeal to a bank to lend it certain amount of money. But this credit gaining is not automatic. So if we assume this credit offering takes time for example a week, then the most simple observation will be that if money balance distributed among agents will determine the maximal transaction volume for the economy in a week.

This kind of arguments leads to the (rather notorious) quantity theory of money. This theory is attacked by many monetary economists but still very little is known how the money balance distribution and the possible transaction volumes are related. I wonder if any knowledge, or any suggestive hints are obtained in the fields of network theory, be it a random network theory or scale free network theory or any other related fields.

If Sam or any other person has any information about these questions, I will be very lucky and glad.

Yoshinori

I think neoclassical theory maintains that the amount of money banks can create is ultimately controlled by the government via capital reserve ratios or other regulation determining the "money multiplier". However, the reality seems to be that banks basically lend as much as they can find (moderately solvent) borrowers to take on (see lines of credit); they can always find cash later if they need it. This would explain why M0/M4 has been steadily falling in all developed countries for decades, and why combined debt is quite a bit more than the total money supply. (What a ridiculous situation: we owe more money than exists!)

Anyway, that rant was just to say that your idea sounds good to me even if it is not popular with monetary economists.

Good luck,

Sam