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Hi Ian,

What one can say from a graph-theoretic point of view is that what
you're asking depends on how densely and closely (parsimoniously) egos
are knitted to each other.

In other words, this is a question of how tidily multiple egos are
extracted from the whole network.

To give a couple of examples let me denote by SE the set of egos and
by FF the set of friend's friends of  all egos such that any actor in
FF is not an ego. (|SE| denoted the the total number of egos and |FF|
the total number of friend's friends: notice that, in general, FF is a
multi-set.)

A trivial case is when every actor in the network is an ego, in which
case |FF| is empty.

Further, there is the following interesting definition in graph
theory: A set S of actors is called "dominating set of actors" if any
actor is either in S or adjacent to an actor in S. Furthermore, a set
S of actors is called a "minimum dominating set" (MDS) if the
cardinality of S is minimum among the cardinalities of any other
dominating set. In addition, the cardinality of a minimum dominating
set is called "domination number" of the network.

These notions are quite useful because they allow us to answer the
following question: How should egos be selected from a network in such
a way that (1) every actor that is not selected as an ego may be
adjacent to an ego (i.e., it may be an alter of some ego) and (2) the
number of so selected egos becomes minimal?

Apparently, the answer to this question (from the point of view of
graph domination theory) is to select egos in such a way that SE
becomes a MDS, i.e., so that |SE| might become equal to the domination
number of the network. Let me call "set of dominating egos" (SDE) this
selection of egos.

Obviously an upper bound of |SDE| is n/2, where n is the total number
of actors in the network. However, it is not hard to show that a lower
bound of |SDE| is (diam + 1)/3, where diam is the network diameter.

--Moses

> ***** To join INSNA, visit http://www.insna.org *****
> Hello!
>
> Is anyone aware of any empirical social network studies that evaluate an
> ego's accuracy in reporting (or being aware of) his friends' friends (2nd
> order connections)?  Even better would be extending this to 3rd, 4th, 5th,
> etc order.
>
> I'm familiar with the literature on social network search, such as
> Granovetter's famous six degrees experiment and Duncan Watts' more recent
> email version, but this is not really what I'm looking for.
>
> I want to know how aware is an individual of network connections that exist
> beyond their immediate ties.
>
> The closest literature I have found is David Krackhardt's cognitive social
> structure work, however, the networks are small networks and I would prefer
> to see data where a larger diameter is possible.
>
> I appreciate any leads you might have.
>
> Kind Regards,
>
> Ian
>
> Ian McCulloh, Ph.D.
> Johns Hopkins University
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