Julian Pfeifle

Overlapping Community Search for social networks

Finding decompositions of a graph into a family of clusters is crucial to understanding its underlying structure. While most existing approaches focus on partitioning the nodes, real-world datasets suggest the presence of overlapping communities. We present OCA, a novel algorithm to detect overlapped communities in large data graphs. It outperforms previous proposals in terms of execution time, and efficiently handles large graphs containing more than 108 nodes and edges.

Gale duality bounds for roots of polynomials with nonnegative coefficients

We bound the location of roots of polynomials that have nonnegative coefficients with respect to a fixed but arbitrary basis of the vector space of polynomials of degree at most d. For this, we interpret the basis polynomials as vector fields in the real plane, and at each point in the plane analyze the combinatorics of the Gale dual vector configuration. This approach permits us to incorporate arbitrary linear equations and inequalities among the coefficients in a unified manner to obtain more precise bounds on the location of roots. We apply our technique to bound the location of roots of Ehrhart and chromatic polynomials. Finally, we give an explanation for the clustering seen in plots of roots of random polynomials.

Polytopality and Cartesian products of graphs

We study the question of polytopality of graphs: when is a given graph the graph of a polytope? We first review the known necessary conditions for a graph to be polytopal, and we present three families of graphs which satisfy all these conditions, but which nonetheless are not graphs of polytopes.Our main contribution concerns the polytopality of Cartesian products of non-polytopal graphs. On the one hand, we show that products of simple polytopes are the only simple polytopes whose graph is a product. On the other hand, we provide a general method to construct (non-simple) polytopal products whose factors are not polytopal.

Webs of stars or how to triangulate free sums of point configurations

Abstract The triangulations of point configurations which decompose as a free sum are classified in terms of the triangulations of the summands. The methods employ two new partially ordered sets associated with any triangulation of a point set with one marked point, the web of stars and the stabbing poset. Triangulations of smooth Fano polytopes are discussed as a case study.