Higher-order clustering patterns in simplicial financial systems.

The interest in induced higher-order relational and multidimensional structures embedded in the financial complex dataset is considered within the applied algebraic topology framework. The aim is to transcend the binary correlations when the interactions of the underlying system are stored in the entries of the cross-correlation matrix. By applying different criteria, we examined aggregations of firms through higher-order clustering of the financial system. The outcome is the extraction of patterns that appear in assemblages of firms due to their multidimensional properties embedded in the cross-correlation matrices. Results are compatible with classifying firms into clusters due to the industry they belong to. Furthermore, the novel and mixed collections of firms are revealed based on the applied mathematical approach. In the broader context, results shed light on the higher-order organization of interactions embedded in the cross-correlation matrix and, as a consequence, extract patterns of collective behavior within a complex system.

Potential grouping of nodes induced by higher-order structures in complex networks.

Complex networks display an organization of elements into nontrivial structures at versatile inherent scales, imposing challenges on a more complete understanding of their behavior. The interest of the research presented here is in the characterization of potential mesoscale structures as building blocks of generalized communities in complex networks, with an integrated property that goes beyond the pairwise collections of nodes. For this purpose, a simplicial complex is obtained from a mathematical graph, and indirectly from time series, producing the so-called clique complex from the complex network. As the higher-order organizational structures are naturally embedded in the hierarchical strata of a simplicial complex, the relationships between aggregation of nodes are stored in the higher-order combinatorial Laplacian. Based on the postulate that aggregation of nodes represents integrated configuration of information, the observability parameter is defined for the characterization of potential configurations, computed from the entries of the combinatorial Laplacian matrix. The framework introduced here is used to characterize nontrivial inherent organizational patterns embedded in two real-world complex networks and three complex networks obtained from heart rate time series recordings of three different subject's meditative states.

Persistent topological features of dynamical systems.

Inspired by an early work of Muldoon et al., Physica D 65, 1-16 (1993), we present a general method for constructing simplicial complex from observed time series of dynamical systems based on the delay coordinate reconstruction procedure. The obtained simplicial complex preserves all pertinent topological features of the reconstructed phase space, and it may be analyzed from topological, combinatorial, and algebraic aspects. In focus of this study is the computation of homology of the invariant set of some well known dynamical systems that display chaotic behavior. Persistent homology of simplicial complex and its relationship with the embedding dimensions are examined by studying the lifetime of topological features and topological noise. The consistency of topological properties for different dynamic regimes and embedding dimensions is examined. The obtained results shed new light on the topological properties of the reconstructed phase space and open up new possibilities for application of advanced topological methods. The method presented here may be used as a generic method for constructing simplicial complex from a scalar time series that has a number of advantages compared to the mapping of the same time series to a complex network.

Scaling of Hamiltonian walks on fractal lattices.

We investigate asymptotical behavior of numbers of long Hamiltonian walks (HWs), i.e., self-avoiding random walks that visit every site of a lattice, on various fractal lattices. By applying an exact recursive technique we obtain scaling forms for open HWs on three-simplex lattice, Sierpinski gasket, and their generalizations: Given-Mandelbrot (GM), modified Sierpinski gasket (MSG), and n -simplex fractal families. For GM, MSG and n -simplex lattices with odd values of n , the number of open HWs Z(N), for the lattice with N>>1 sites, varies as omega(N)}N(gamma). We explicitly calculate the exponent gamma for several members of GM and MSG families, as well as for n-simplices with n=3, 5, and 7. For n-simplex fractals with even n we find different scaling form: Z(N) approximately omega(N)mu(N1/d(f), where d(f) is the fractal dimension of the lattice, which also differs from the formula expected for homogeneous lattices. We discuss possible implications of our results on studies of real compact polymers.