Seminar on Simplicial complexes as models of complex systems by Dr. Slobodan Maletic, Tue 20 May 14:00 @ C209 Seminar Room

Abstract:

Modeling complex systems and in particular complex networks with simplicial complexes proved to be useful for acquiring insight into emerging properties and structural organization which is unattainable with standard methods. We are promoting the topological approach based on simplicial complexes, among other reasons, to overcome the shortcomings of the mainstream methods of the complex networks science, which are constrained to pairwise connections and their aggregations. A simplicial complex representation (intrinsically multidimensional) of complex networks consists in constructing different types of simplicial complexes, either directly from the data or from an existing network. Simplicial communities, defined as the connectivity classes at different resolution levels (simplicial complex dimensions), reflect the ability of this approach to capture higher-order structures, which elude standard approaches. It is shown that there is an abundance of different substructural patterns hidden in complex network whose relationships lead to the overall network structure, and may have a large impact on the function and dynamics of a network. The so far research was based on two types of simplicial complexes obtained from complex networks: the neighborhood complex and the clique complex.

Topological quantities, like structure vectors, Betti numbers and combinatorial Laplacian operator are calculated for diverse models of real-world networks. Properties of spectra of the combinatorial Laplacian operator of simplicial complexes are explored, and the advantages of higher order spectral analysis is discussed and compared with results for the ordinary graph Laplacian. The relationship of properties resulting from combinatorial Laplacian spectra with connectivity properties stored in the Q-vector is analyzed and discussed. The basic statistical features of complex networks are preserved by algebraic topological quantities of simplicial complexes, indicating possible presence of the so far unknown generic mechanisms in the complex networks formation. All results support the necessity of developing a novel research field, called statistical mechanics of simplicial complexes as a unifying theory of the complex systems represented by simplicial complexes.

Momoko Zamami  http: //www.oist.jp