Characteristics of Small Social Networks

Two dozen networks are analyzed using three parameters that attempt to capture important properties of social networks: leadership L, member bonding B, and diversity of expertise D. The first two of these parameters have antecedents, the third is new. A key part of the analysis is to examine networks at multiple scales by dissecting the entire network into its n subgraphs of a given radius of two edge steps about each of the n nodes. This scale-based analysis reveals constraints on what we have dubbed “cognitive” networks, as contrasted with biological or physical networks. Specifically, “cognitive” networks appear to maximize bonding and diversity over a range of leadership dominance. Asymptotic relations between the bonding and diversity measures are also found when small, nearly complete subgraphs are aggregated to form larger networks. This aggregation probably underlies changes in a regularity among the LBD parameters; this regularity is a U-shaped function of networks size, n, which is minimal for networks around 80 or so nodes. 1.0 Overview One of the ways science advances is to discover non-trivial relations among measurements of an entity, which then give insights into the nature of that entity. Classic examples include: (a) the PV/T relation between pressure, P, temperature, T, and the volume, V, of a gas, (b) the Strahler number, Ni/(Ni+1) which characterizes the branching structure of rivers or trees, or (c) supply-demand relations. In the study of social networks, these kinds of discoveries of parametric relations are rare (but see Barabasi & Albert 2002, Watts & Strogatz 1998, Newman, 2003). Using a multi-scale analysis, we report three new properties that are characteristic of one important class of social networks. Although our specific measurement parameters might be questioned, they were chosen to have relevance to the building of “cognitive” social networks, as contrasted with networks such as the spread of infectious diseases or telecommunications infrastructure. Two of the three parameters chosen are closely allied with previous measures. These choices, however, are not the key issue. Rather, can these parameters lead to the discovery of new properties characteristic of a class of social networks? Will these discoveries of new relationships lead to insights about the structure of social networks?

[1]  M E J Newman,et al.  Community structure in social and biological networks , 2001, Proceedings of the National Academy of Sciences of the United States of America.

[2]  R. Rothenberg,et al.  Risk network structure in the early epidemic phase of HIV transmission in Colorado Springs , 2002, Sexually transmitted infections.

[3]  W. Zachary,et al.  An Information Flow Model for Conflict and Fission in Small Groups , 1977, Journal of Anthropological Research.

[4]  Jure Leskovec,et al.  Patterns of Influence in a Recommendation Network , 2006, PAKDD.

[5]  Carolyn J. Anderson,et al.  A p* primer: logit models for social networks , 1999, Soc. Networks.

[6]  S. Brenner,et al.  The structure of the nervous system of the nematode Caenorhabditis elegans. , 1986, Philosophical transactions of the Royal Society of London. Series B, Biological sciences.

[7]  Robin J. Wilson,et al.  An Atlas of Graphs , 1999 .

[8]  Duncan J. Watts,et al.  Collective dynamics of ‘small-world’ networks , 1998, Nature.

[9]  Albert,et al.  Emergence of scaling in random networks , 1999, Science.

[10]  Mark E. J. Newman.  A measure of betweenness centrality based on random walks , 2005, Soc. Networks.

[11]  Owen Macindoe,et al.  Investigating the fine grained structure of networks , 2010 .

[12]  Albert-László Barabási,et al.  Linked: The New Science of Networks , 2002 .

[13]  Mark S. Granovetter.  The Strength of Weak Ties , 1973, American Journal of Sociology.

[14]  Whitman Richards,et al.  Graph Comparison Using Fine Structure Analysis , 2010, 2010 IEEE Second International Conference on Social Computing.

[15]  Maximino Aldana-Gonzalez,et al.  Linked: The New Science of Networks , 2003 .

[16]  Le Song,et al.  Estimating time-varying networks , 2008, ISMB 2008.

[17]  Frank Harary,et al.  Graph Theory , 2016 .

[18]  Richard H. Schelp,et al.  A Remark on the Number of Complete and Empty Subgraphs , 1998, Comb. Probab. Comput..

[19]  M. Newman,et al.  Finding community structure in networks using the eigenvectors of matrices. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[20]  Christophe Prieur,et al.  Structure of Neighborhoods in a Large Social Network , 2009, 2009 International Conference on Computational Science and Engineering.

[21]  Mark E. J. Newman,et al.  The Structure and Function of Complex Networks , 2003, SIAM Rev..

[22]  T. Vicsek,et al.  Uncovering the overlapping community structure of complex networks in nature and society , 2005, Nature.

[23]  R. Pastor-Satorras,et al.  Structure of cycles and local ordering in complex networks , 2004 .

[24]  S. N. Dorogovtsev,et al.  Evolution of networks , 2001, cond-mat/0106144.

[25]  Donald E. Knuth,et al.  The Stanford GraphBase - a platform for combinatorial computing , 1993 .

[26]  Duncan J Watts,et al.  A simple model of global cascades on random networks , 2002, Proceedings of the National Academy of Sciences of the United States of America.

[27]  D. Lusseau,et al.  The bottlenose dolphin community of Doubtful Sound features a large proportion of long-lasting associations , 2003, Behavioral Ecology and Sociobiology.

[28]  Nicholas C. Wormald,et al.  Representing Small Group Evolution , 2009, 2009 International Conference on Computational Science and Engineering.

[29]  S. Page.  Prologue to The Difference: How the Power of Diversity Creates Better Groups, Firms, Schools, and Societies , 2007 .

[30]  L. Freeman.  Centrality in social networks conceptual clarification , 1978 .