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Provided below is a summary of the responses I received to the following
post:

Original post: (to SOCNET on 5/19/07)

I have computed the geodesic distances for all possible dyadic combinations
of actors in a network. The network is not completely connected and thus for
some dyads, there is no path between them. Theoretically, the geodesic
distance between two such actors is infinity. One approach would be to drop
dyadic observations with an infinite path length from my analysis and treat
them as missing data. However, I am looking for suggestions about assigning
an actual value for distance (or a transformation thereof) for such dyads so
they do not fall out of my analysis. I would appreciate any and all help
from the list. Thanks.


Responses

1) ...from a mathematical point of view, the distance between disconnected
nodes is actually undefined rather than infinite. But inserting infinity
certainly works well enough. The other popular values are N (i.e., one more
than maximum distance possible in a graph with n nodes) and D+1 (where D is
the max diameter of any component).


2) One strategy is to use N+1 (where N = number of nodes in graph); this
will not change the median value much, and allows you to keep the dyad in
your analyses.


3) When I deal with disconnected graphs and distance, I often invert the
distance, because an infinite distance then becomes 0, while a distance of 1
stays as 1 (a distance of 2 is 0.5, 3 is 1/3, etc...).


4)Something to consider: shortest path length may not be the right metric.
If A and B are connected by 37 distinct paths of length 2, is that a closer
association than if A and B are connected via exactly one path of length 2?


5) While analytically, I think you want to be careful about the implications
of dropping infinite distances, I have found it useful for graphing and
similar applications to assign an arbitrary distance of n (the number of
nodes/actors), since the maximum width of a connected graph is n-1. Plotting
infinity, or things like average distance, become incalculable when
infinities are present.


6) In a regression framework, you could create dummy variable for various
distances (e.g., geo distance<3, 3<= dist < 5, etc) and then simply create a
dummy variable for "more than X," where X is the observed maximum. Then, you
could include the pairs in a data set without worrying that you have falsely
assumed a finite distance. The maximal distance is simply a categorical
variable.



Corey Phelps, PhD
Asst. Professor, Management & Organization
University of Washington Business School
Box 353200
Seattle, WA 98195
(206) 543-6579

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