Persistent homology of complex networks

@article{Horak2009PersistentHO,
  title={Persistent homology of complex networks},
  author={Danijela Horak and Slobodan Maleti{\'c} and Milan Rajkovi{\'c}},
  journal={Journal of Statistical Mechanics: Theory and Experiment},
  year={2009},
  volume={2009},
  pages={03034}
}
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 parameterized version of a Betti number. Complex networks with distinct degree distributions exhibit distinct persistent topological features. Persistent topological attributes, shown to be related to the robust quality of networks, also reflect the… 
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TLDR
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References

SHOWING 1-10 OF 51 REFERENCES
Computing Persistent Homology
Abstract We show that the persistent homology of a filtered d-dimensional simplicial complex is simply the standard homology of a particular graded module over a polynomial ring. Our analysis…
Spectral and dynamical properties in classes of sparse networks with mesoscopic inhomogeneities.
  • M. Mitrovic, B. Tadić
  • Mathematics
    Physical review. E, Statistical, nonlinear, and soft matter physics
  • 2009
TLDR
Within the exhaustive spectral analysis for both the adjacency matrix and the normalized Laplacian matrix, the spectral properties, which characterize the mesoscopic structure of sparse cyclic graphs and trees are identified.
Statistical mechanics of complex networks
TLDR
A simple model based on these two principles was able to reproduce the power-law degree distribution of real networks, indicating a heterogeneous topology in which the majority of the nodes have a small degree, but there is a significant fraction of highly connected nodes that play an important role in the connectivity of the network.
Complex networks: Structure and dynamics
Topology of random clique complexes
Topological Persistence and Simplification
TLDR
Fast algorithms for computing persistence and experimental evidence for their speed and utility are given for topological simplification within the framework of a filtration, which is the history of a growing complex.
The neighborhood complex of a random graph
Self-similar community structure in a network of human interactions.
TLDR
The results reveal the self-organization of the network into a state where the distribution of community sizes is self-similar, suggesting that a universal mechanism, responsible for emergence of scaling in other self-organized complex systems, as, for instance, river networks, could also be the underlying driving force in the formation and evolution of social networks.
Simplicial complexes of graphs
Let G be a finite graph with vertex set V and edge set E. A graph complex on G is an abstract simplicial complex consisting of subsets of E. In particular, we may interpret such a complex as a fami…
Small worlds: the dynamics of networks between order and randomness
  • Jie Wu
  • Computer Science
    SGMD
  • 2002
Everyone knows the small-world phenomenon: soon after meeting a stranger, we are surprised to discover that we have a mutual friend, or we are connected through a short chain of acquaintances. In his…
...
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