Expository Research Papers - Ipsen, Ilse (SIAM review, 2011)
The two papers in this issue deal with bifurcations and social network analysis. Bifurcation, in its original sense, means division into two branches. In the context of dynamical systems, a bifurcation occurs when a small change in a parameter causes a sudden qualitative change in the solution. Mike Jeffrey and S. J. Hogan study dynamical systems where a state vector $x$ is defined by a system of ordinary differential equations $$\frac{dx}{dt}= f(x,t;μ)$$ and $μ$ represents the parameter. Here it is the function $f$ that is responsible for the bifurcations, because $f$ is only piecewise smooth. Piecewise smoothness occurs in applications ranging from engineering and medicine to biology and ecology, often in situations where impact or friction is modeled. Furthermore the authors assume that the discontinuity in $f$ is confined to a smooth manifold of codimension one, the so-called switching manifold, where the solution $x$ remains continuous but may fail to be unique. As a consequence, trajectories of $x$ either cross through the switching manifold or slide along it. If the latter give rise to a bifurcation, then it is, aptly named, a sliding bifurcation. This is a succinct and well-written paper and it makes a number of contributions: first, identifying those points on the switching manifold where switching bifurcations can occur, namely at certain singularities (folds, cusps, saddles, and bowls); second, deriving a complete classification of all one-parameter sliding bifurcations; and third, discovering new sliding bifurcations, which all turn out to be catastrophic. In the second paper, "Comparing Community Structure to Characteristics in Online Collegiate Social Networks," Amanda Traud, Eric Kelsic, Peter Mucha, and Mason Porter examine how relationships on a social networking site are correlated with demographic traits. In particular, they inspect the complete Facebook pages of five U.S. universities from a single snapshot in September 2005 and ask questions like: If people are friends on Facebook, are they likely to live in the same dormitory, major in the same subject, or have gone to the same high school? Computationally, Facebook users are represented as nodes of a graph, friendships as edges, and social communities as clusters of nodes. Identifying clusters, i.e., community detection, is done with an "unsupervised" algorithm that amounts to optimizing a modularity quality function, which depends only on the edge structure of the graph. This algorithmically computed community must now be compared to communities associated with demographic traits. The authors investigate different similarity measures, based on pair counting, for performing the comparisons. Along the way they face a number of issues: How to display and compare the communities visually? How to compute the correlations accurately and efficiently? What to do about missing data? This is an exciting paper, drawing on tools from graph partitioning, optimization, data clustering, contingency tables, and statistical psychology. It might also be the only paper in a SIAM journal that explains why the social structure at Caltech is different from that of the University of North Carolina.