Homotopy Type of the Boolean Complex of a Coxeter System

In any Coxeter group, the set of elements whose principal order ideals are boolean forms a simplicial poset under the Bruhat order. This simplicial poset defines a cell complex, called the boolean complex. In this paper it is shown that, for any Coxeter system of rank n, the boolean complex is homotopy equivalent to a wedge of (n-1)-dimensional spheres. The number of such spheres can be computed recursively from the unlabeled Coxeter graph, and defines a new graph invariant called the boolean number. Specific calculations of the boolean number are given for all finite and affine irreducible Coxeter systems, as well as for systems with graphs that are disconnected, complete, or stars. One implication of these results is that the boolean complex is contractible if and only if a generator of the Coxeter system is in the center of the group. of these results is that the boolean complex is contractible if and only if a generator of the Coxeter system is in the center of the group.Comment: final version, to appear in Advances in Mathematic

Multimodal Network Alignment

A multimodal network encodes relationships between the same set of nodes in multiple settings, and network alignment is a powerful tool for transferring information and insight between a pair of networks. We propose a method for multimodal network alignment that computes a matrix which indicates the alignment, but produces the result as a low-rank factorization directly. We then propose new methods to compute approximate maximum weight matchings of low-rank matrices to produce an alignment. We evaluate our approach by applying it on synthetic networks and use it to de-anonymize a multimodal transportation network.Comment: 14 pages, 6 figures, Siam Data Mining 201

Hexagonal Tilings: Tutte Uniqueness

We develop the necessary machinery in order to prove that hexagonal tilings are uniquely determined by their Tutte polynomial, showing as an example how to apply this technique to the toroidal hexagonal tiling.Comment: 12 figure

Relationship based Entity Recommendation System

With the increase in usage of the internet as a place to search for information, the importance of the level of relevance of the results returned by search engines have increased by many folds in recent years. In this paper, we propose techniques to improve the relevance of results shown by a search engine, by using the kinds of relationships between entities a user is interested in. We propose a technique that uses relationships between entities to recommend related entities from a knowledge base which is a collection of entities and the relationships with which they are connected to other entities. These relationships depict more real world relationships between entities, rather than just simple “is-a” or “has-a” relationships. The system keeps track of relationships on which user is clicking and uses this click count as a preference indicator to recommend future entities. This approach is very useful in modern day semantic web searches for recommending entities of user’s interests