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John, as you've been requesting a formal-mathematical representation
of your problem, what I'd like to propose is the formalism of the
theory of hypergraphs and intersection graphs.
You might know that a hypergraph H is a couple (X, E), where X is a
(finite) set of vertices and E is a family of nonempty subsets of X
such that their union is X. It might sound a little confusing but in
the terminology of thypergraph theory the subsets belonging to E are
called "edges" (and I will keep the quotation marks in order to
distinguish them from what we commonly understand as edges in a
Now, what you have in your problem is a sequence of hypergraphs Hj =
(X, Ej), all of them defined over the same set of vertices but being
connected through different families of "edges."
In this way, the union of all "edges" in hypergraphs Hj forms an
intersection graph, which is an undirected graph with vertices all the
"edges" of the hypergraphs such that two "edges" are connected as far
as these "edges" have a nonempty intersection.
Claude Berge's "Graphs and Hypergraphs" and McKee & McMorris' "Topics
in Intersection Graph Theory" are two standard references in the
literature of hypergraphs and intersection graphs (respectively).
You might find also helpful the corresponding Wikipedia articles:
On Wed, Feb 11, 2015 at 7:03 AM, John McCreery <[log in to unmask]> wrote:
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> <Continuing my meditations on the role of teams in network chemistry>
> In their article "Structural Redundancy and Multiplicity in Corporate in
> Corporate Networks" Connections 30(2) 2010, Roy Barnes and Tracy Burkett
> begin with the following statement.
> "At the second meeting of the Politics and Interlocking Directorates
> Research Community, Burris (2006, p. 2) called for a "Sociology of Elites"
> in which corporate interlocks would be 'only one thread in a much denser
> fabric of social ties among corporations and corporate elites." Moreover,
> Burris cautioned those interested in pursuing a research agenda involving
> interlocking directorates "not to reify or isolate director interlocks from
> other social networks within which they are embedded.'"
> A bit later, they state that,
> "the men and women of the corporate elite are simultaneously directors of
> corporations and trustees or directors of museums, research universities,
> and members of social clubs."
> These statements are certainly true and, in my case, they speak to personal
> experience. While a member of the American Chamber of Commerce in Japan and
> working for Hakuhodo, Japan's 2nd largest advertising agency, I was asked if
> it would be useful for the account executives I worked with to join the
> Chamber. Yes, I would say, but you can't just join; you have to take part.
> You also need to realize that the Chamber's members have many non-business
> ties. They belong to the same churches and clubs; they play on the same
> teams (football, rugby, tennis, etc.); their children attend the same
> schools. You will have to participate and do more than exchange business
> cards if you want to develop new business with them.
> That said, these kinds of relationships are not the kind of relationships
> involved in forming teams. Joining a club and sharing other ties with other
> members is not being recruited or assigned to play a specific role required
> by a project. When an American football team needs a quarterback, its scouts
> are not searching for fullbacks or defensive tackles (unless, of course, the
> team also needs to fill these positions). Producing a TV commercial requires
> a TV cameraman [cinematographer] and not a still photographer. And only in
> rare instances will the cameraman also be the producer or film director. In
> these instances, the 2-mode relationships of team members to projects and
> the resulting "working together" 1-mode relationships among team members do
> not arise informally. They emerge as the result of a need for specialists to
> get a job done. It is that need that triggers a search when a new team is
> formed to see (1) who has the necessary skills, (2) is currently available,
> and (3) especially desirable to have on board.
> It is, of course, true that the individuals assembled to form a team may
> also be linked by other types of relationships. In Japan, it is not uncommon
> for team members to have graduated from the same schools. They frequently
> work for the same agencies or belong to the same professional associations.
> By working together they may become friends. When they have worked together
> on successful projects, like the ads that made it into the data set with
> which I am working, they are highly likely to spread the word about each
> other's talent. Alternatively, teams may become sites at which bitter feuds
> erupt. But these are not the relationships that lead directly to team
> membership, which always turn on a project's need for specific skills and
> the need to assemble a team whose members possess those skills. Which brings
> me to the heart of my claim--what credits data provide, is information on the
> distribution of skills and of the people who provide those skills, within
> specific industries. Network analysis tools can be used to examine how
> networks are created as teams are formed and team members move between
> projects. One specific set of questions is how specific skills and the
> relationships they channel contribute to shaping the networks that
> constitute not an industry.
> This is the background to the technical issues on which I am seeking advice.
> John McCreery
> The Word Works, Ltd., Yokohama, JAPAN
> Tel. +81-45-314-9324
> [log in to unmask]
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