In December 2008, Luis Bettencourt and David Kaiser reported their findings[1] from studies of research collaboration networks, which included their discovery that, as coauthorship networks in a particular field reach the point of forming a single giant component of interconnected authors that dwarfs all other coauthor groups in that field, the growth near that point depends in a universal way on the average number of coauthors per author.  In particular, the fraction of coauthor links that belong to the giant component appears to be proportional to ( - k_c)^0.35, where k_c, which marks the critical point, depends on the research field.[2]  The remarkable fact is that the exponent, 0.35, fits the data for networks in several quite distinct fields.  This value apparently isn’t common to networks in general, though.  I had wondered what features of a network do determine the exponent’s value. 

Many physical systems exhibit critical-point transitions like the formation of a giant component in networks—e.g., iron magnets lose their magnetism at a certain critical temperature, and the sharp difference between the densities of water and water vapor disappears above a critical pressure.  Yet systems with very different interaction mechanisms can, near critical points, be described by the same mathematical functions.  These functions have a particular kind of underlying similarity:  the functions remain the same when the size of the system is shrunk (or, equivalently, when the minimum significant distance in the functions is enlarged), so that the change in size is equivalent to a change in the scale of the system’s critical variables.  The fact that very different physical systems are described by the same functions near their critical points indicates that the systems have certain features in common, which are unaltered by the shrinkage.  These features include the number of the systems’ dimensions, the number of components in the variable that exhibits the critical-point behavior, and whether the systems’ components can interact over long distances that don’t tend toward zero as the system is shrunk.[3] 

These features, however, seem to be the same in all networks, and so wouldn’t explain why some networks don’t exhibit the behavior common to the coauthor networks investigated by Bettencourt and Kaiser.  Since, potentially, any two vertices in a network might be linked (e.g., any two authors might become coauthors), the space they occupy has up to one less dimension than the number of vertices, which means that all networks with infinitely many vertices (the “thermodynamic limit” usually analyzed in critical-point studies) are essentially infinite-dimensional; the fraction of a network belonging to a giant component is a scalar variable (one component); and since any two vertices of a network are either linked or not linked, all vertices are in one sense nearest neighbors, so there are no long-range interactions (though the probability of two vertices’ being linked may well depend on the lengths of paths between them through other vertices).  Shrinking an infinite-dimensional network by combining sets of vertices into new single vertices doesn’t change the network’s dimensionality, or make the giant component’s size a multiple variable, or add any long-range interactions. 

There is, however, another feature of physical systems that may be unaffected by the shrinkage:  the symmetry of the function that describes interactions between the system’s components.[4]  For different networks, these functions can be very different.  In some networks, the probability that two vertices are linked may depend only on whether both are linked to the same other vertex, or on how many such other vertices they both link to in common.  But Bettencourt and Kaiser raise the point that the set of concepts an author can work with— an internal vertex state—has an important influence on whether that author can work with another author to produce joint reports.[5]  The probability that two investigators become coauthors should be higher, the more relevant concepts they can both work with.  If we represent distinct concepts by orthogonal axes in a multidimensional concept space, so that similar concepts or sets of concepts are represented by nonorthogonal axes, we can represent the internal state of an author as an axis in the concept space.  The probability of linkage between two authors would thus depend on the angles between their axes in concept space. 

If the probability of any set of coauthorship linkages among an entire author network were completely symmetrical with respect to directions in the concept space, then the probability’s dependence on authors’ concept sets would be irrelevant to how the network coalesces into a giant component—linkage probabilities that don’t depend on concept sets at all are just as symmetrical under rotations of the concept space.  But if the space of concept sets is, for any reason, not isotropic (e.g., if some sets of concepts are likelier to exist in the same mind than other sets), then arbitrary rotations of all the authors’ concept axes will generally vary the coauthorship probabilities.  The coauthorship-probability function would thus be less symmetrical, which might mean that the giant component’s size has a different dependence on the average number of coauthors per author. 

If a network’s giant-component growth near its critical point depends on the symmetry of its linkage probability function in this way, we should be able to reproduce the critical-point behavior of real coauthorship networks by constructing an artificial network whose linkage probability has the same kind of symmetry, and produce other critical-point behaviors from linkage probabilities of different symmetry.  Put differently, if linkage probabilities of only the appropriate limited symmetry reproduced the critical-point growth of a coauthorship network’s giant component, this would strongly suggest that concept space, for whatever reasons, has the same kind of limited symmetry when it comes to research collaboration.