If there is no relation between to entities then the relation distance will be infinite and there will be no interaction.  However, the converse is not true.  A finite relation distance does not imply an interaction.  This is very similar to the situation with correlation, in that correlation does not imply a causation relationship.  The behavior of two entities may be correlated or have a finite relation distance and not be casually connected.  The correlation or the relation distance may be due to an interaction with a third party.  An example may be useful.

Consider 4 persons interacting; A, B, C, and D.  The specific interaction does not matter, however the relation distance will apply only for the specific type of interaction.  Possible examples are eating a meal together, having sex, playing chess, etc. 

Case #1

• A and B interact 1 out of every 100 hours.
• C and D interact 1 out of every 100 hours.
• No other interactions between these 4 persons. 

Case #2

• A and B interact 1 out of every 100 hours.
• B and C interact 1 out of every 100 hours.
• C and D interact 1 out of every 100 hours.
• No other interactions between these 4 persons. 

We can calculate the value of J, and the correlation between each pair combination of persons as follows.  I have also calculated, which can be thought of as the correlation distance just a  is defined as the IDA relation distance or.

Relation Distance Example

In Case #1 where our 4 persons are divided into 2 couples and there is no interactions between the couples then J and =0 and the distances are infinite except for the A-B and C-D interactions.  When in Case #2 we introduce an interaction between B and C, we have J > 0 for all pairings and is finite. 

This example shows that both J and  can non-zero even when there is no direct interaction.  One should also note that changing a relation distance causes other relation distances to change such as the A-B relation distance going from 12.37729 to 16.44923 when the B-C relation distance is changed.  This is similar to the physical dynamics situation that when the distance between 2 objects is changed by moving one of the objects all of the distances between that object and other objects change.  That is one cannot change just a single physical or single relation distance.  In the case of physical dynamics we see this intuitively because of the 3 dimensional structure of the space that physical objects must occupy.  The situation in relation space is a bit more complex and not so intuitively clear.

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