***** To join INSNA, visit http://www.sfu.ca/~insna/ ***** Juan,

Interesting work! Of course, there's lots of stuff on triads in the social network literature, but larger cycles are also important altho maybe not as well understood. For cycles of length 4 in Emmanuel Lazega's lawyers data set, it's worthwhile looking at Lazega and Pattison (1999), Multiplexity, generalized exchange and cooperation in organizations. Social Networks, 21, 67-90.  (The methodological basis for these models examining 4- and higher cycles is in Pattison & Robins in Sociological Methodology 2002.)

Very large cycles that span substantial portions of the network may have a somewhat different effect in terms of global structure compared to more localised 3- and 4-cycles. Your work dovetails rather interestingly with DeCanio, Dibble & Amir-Atefi (2000), Importance of organizational structure for the adoption of innovations, Management Science, 46, 1285-1299.  They set up simulations based on a benefit and a cost function, with the benefit function essentially being optimised for short geodesics, and the cost function minimised for small outdegree. Resulting optimal structures involve mutually intersecting large cycles. This is reminiscent of Duncan Watts' small world model (altho Duncan starts from an exogenous large cycle), whereby a few shortcuts across the cycle reduce path lengths dramatically.  Interesting to consider that two mutually intersecting large cycles may provide shortcuts for each other, intuitively an optimising situation in terms of short path lengths, hence DeCanio et al make a good point.

At the last Sunbelt conference, I argued that for sparse nondirected graphs this global pattern can emerge from locally-specified Markov random graph models (p* models), with a positive 2-star parameter (tendency for multiple network partners) and negative 3-star parameter (cost against having too many partners).  Simulations can produce global structures with large mutually intersecting cycles and associated short paths. A positive triangle parameter as well may result in small world graphs but there are also other possibilities. (It's taken a while but we've just about written this up.)

I like the use of node heterogeneity in your model. Often we try to do a lot with node homogeneity, and sometimes I'm not so sure how far it can take us ...?


Garry Robins

>Dear Carl
>We are people from Economics and the model we developed is a theoretical one
>based on Game Theory framework. More specifically based on
>Bala, V. and S. Goyal (2000). "A noncooperative model of network formation."
>Econometrica 68(5): 1181-1229.
>Our variable is information flow (in abstract sense) and the actors are
>rational agents that maximized the information they receive considering
>costs of connection. It's not a structural (sociological) approach paper but
>a model based on economic assumptions. We looked for Nash equilibria,
>optimal topologies, and the like.
>Full graph it's not optimal because of the redundant links that this
>structure posses and because we made the assumptions that agents cannot deny
>connection when required for others agents. There's some sort of
>hierarchical structure underlying the analysis. It is more an analytical
>exercise than a real-based social network research, but we looked for
>potential real examples where this structure could emerge.
>Thanks for the question and the comments
>Sincerely Yours
>PS: you can download a copy of our draft in
>Juan Manuel Larrosa
>PhD Candidate
>CONICET- Universidad Nacional del Sur
>San Juan y 12 de Octubre
>Planta Baja - Gabinete 5
>(8000) Bahia Blanca
>http://jlarrosa.tripod.com   [updated]
>TE (0291) 459 5101 Ext 2735
>-----Mensaje original-----
>De: Carl Nordlund [mailto:[log in to unmask]]
>Enviado el: Miércoles, 25 de Septiembre de 2002 06:28 p.m.
>Para: Juan Manuel Larrosa; [log in to unmask]
>Asunto: SV: Circular networks examples
>Very interesting that you came to the conclusion that a one-dimensional
>closed-space structure is the most optimal. Some thoughts though:
>What is the structural variable in your model?
>What exactly is optimized in the model?
>Shouldn't a full graph, i.e where all actors are connected to eachother, be
>the most optimal structure (at least from a neoclassical point of view)?
>Carl Nordlund, BA, PhD student
>Human Ecology Division
>-----Ursprungligt meddelande-----
>Från: Social Networks Discussion Forum [mailto:[log in to unmask]]För
>Juan Manuel Larrosa
>Skickat: den 25 september 2002 22:29
>Till: [log in to unmask]
>Ämne: Circular networks examples
>*****  To join INSNA, visit http://www.sfu.ca/~insna/ *****
>I have developed with a colleague of mine an economic model of network
>formation. Results gave us circular network architecture as optimal
>structure. It means that individual are full connected but in row where the
>last one individual must be connected with the first one. A quite simple
>example could be this:
>          A
>           /    \
>        /    \
>       /      \
>        B-------C
>I've read sociological literature where this structure could be assimilated
>to a clique network, but I'm suspicious about other potential implications
>in social networks research. Does anyone remember another social network
>architecture (with real examples) where this circle form could emerge?
>Specific references are welcome...
>Thanks in advance,

Dr Garry Robins,
Department of Psychology,
School of Behavioural Science,
The University of Melbourne,
Victoria 3010,

Tel: 61 3 8344 6354
Fax: 61 3 9347 6618
Web: http://www.psych.unimelb.edu.au/staff/robins.html
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