Mike,

Good thought! I love dimensionless numbers. They are central to experiment design.
Let me think more about this and get back to you later -- this reminds me of the modeling of pollutant transport where the diffusion and dispersion coefficient are intended to capture the cumulative effect of Brownian motion and the velocity variation from the mean.

Ming

From: Michael J. Leamy [mailto:mjleamy@mitre.org]
Sent: Friday, January 28, 2005 10:24 AM
To: repast-interest@lists.sourceforge.net
Cc: 'Alfred G. Brandstein'; 'Ming-Pin Wang'
Subject: A convenient dimensionless number

This may be off-topic a bit, but this forum looked like a good one for proposing a dimensionless number that could guide RePast users in appropriately selecting model parameters such as agent density and agent speed.  I apologize if this is too trivial or already well-known.  The motivation for this came from investigating a disease spread model (diffuse-like in nature) in which the initially chosen parameters led to results lacking the expected diffuse behavior.  As background, the model in question has bugs moving randomly about the landscape.  Infection occurs when one sick bug shares a space with another bug.  There are other rules about when a bug becomes immune and such.  One change to the model parameters (increasing the size of the 2D space) yielded the desired diffuse-like behavior.  This inspired a dimensionless number to be defined, which I will call the Brownian number (Br).

For an agent-based model where the location and speed of an agent has physical significance, and where the agent heading is randomly selected at each time step, the Brownian dimensionless serves a useful purpose and can be defined as,

Br = v*delta_t/d

where

v : agent speed [length/time]

delta_t: time step [time]

d: largest model distance [length]

Note that d could be a diagonal metric for a 2D Euclidean space.  Behavior of the model at the limiting cases can easily be surmised.  For Br much less than 1, the agent behavior approaches Brownian – i.e. is local.  For Br greater than 1, the agent behavior is non-physical (agents travel a distance greater than the largest distance in the system).  Somewhere between the two extremes lies a transitional state where the behavior is between local and global.

For diffuse-like simulations (and maybe others) where some type of information is carried through agent communication and where the agents themselves are not expected to travel large distances [such as disease spread over a large geographic area], it is desired that the Br number be far smaller than one.  Ideally, it would be nice if someone identified what value of Br (perhaps 0.2, or 0.01, etc.) is required such that the agent behavior can be safely assumed to be Brownian-like for an entire class of simulation types.  Maybe something like this has been done?

Thanks, Mike.

Dr. Michael J. Leamy

The MITRE Corporation

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