Long lived topological features are distinguished from short lived ones (considered as topological noise) in simplicial complexes constructed from complex networks. A new topological invariant, persistent homology, is determined and presented as a parametrized version of a Betti number. Complex networks with distinct degree distributions exhibit distinct persistent topological features. Persistent topological attributes, shown to be related to robust quality of networks, also reflect defficiency in certain connectivity properites of networks. Random networks, networks with exponential conectivity distribution and scale-free networks were considered for homological persistency analysis.

A general method for constructing simplicial complex from observed time series of dynamical systems based on the delay coordinate reconstruction procedure is presented. The obtained simplicia 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 computation of homology of the invariant set of some well known dynamical systems which display chaotic behavior. Persistent homology of simplicial complex and its relationship with the embedding dimensions are examined by studying the

In online communications, patterns of conduct of individual actors and use of emotions in the process can lead to a complex social graph exhibiting multilayered structure and mesoscopic communities. Using simplicial complexes representation of graphs, we investigate in-depth topology of online social network which is based on MySpace dialogs. The network exhibits original community structure. In addition, we simulate emotion spreading in this network that enables to identify two emotion-propagating layers. The analysis resulting in three structure vectors quantifies the graph's architecture at different topology levels. Notably, structures emerging through shared links, triangles and tetrahedral faces, frequently occur and range from tree-like to maximal 5-cliques and their respective complexes. On the other hand, the structures which spread only negative or only positive emotion messages appear to have much simpler topology consisting of links and triangles. Furthermore, we introduce the node's structure vector which represents the number of simplices at each topology level in which the node resides. The total number of such simplices determines what we define as the node's topological dimension. The presented results suggest that the node's topological dimension provides a suitable measure of the social capital which measures the agent's ability to act as a broker in compact communities, the so called Simmelian brokerage. We also generalize the results to a wider class of computer-generated networks. Investigating components of the node's vector over network layers reveals that same nodes develop different socio-emotional relations and that the influential nodes build social capital by combining their connections in different layers.

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Abstract. Topological, algebraic and combinatorial properties of simplicial complexes which are constructed from networks (graphs) are examined from the statistical point of view. We show that basic statistical features of scale free networks are preserved by topological invariants of simplicial complexes and similarly statistical properties pertaining to topological invariants of other types of networks are preserved as well. Implications and advantages of such an approach to various research areas involving network concepts are discussed.