Persistent homology of complex networks

Horak, Danijela; Maletić, Slobodan; Rajković, Milan

doi:10.1088/1742-5468/2009/03/p03034

Cited by 197 publications

(179 citation statements)

References 19 publications

Supporting

Mentioning

177

Contrasting

Unclassified

Order By: Relevance

“…Persistent Homology [38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53][54][55] is a means of topological data analysis. Now let us use an example to show how the topological data analysis methods can overcome the limitations of geometrical methods.…”

Section: Persistent Homologymentioning

confidence: 99%

Topological Data Analysis for the Characterization of Atomic Scale Morphology from Atom Probe Tomography Images

2018

View full text Add to dashboard Cite

Atom probe tomography (APT) represents a revolutionary characterization tool for materials that combine atomic imaging with a time-of-flight (TOF) mass spectrometer to provide direct space three-dimensional, atomic scale resolution images of materials with the chemical identities of hundreds of millions of atoms. It involves the controlled removal of atoms from a specimen's surface by field evaporation and then sequentially analyzing them with a position sensitive detector and TOF mass spectrometer. A paradox in APT is that while on the one hand, it provides an unprecedented level of imaging resolution in three dimensions, it is very difficult to obtain an accurate perspective of morphology or shape outlined by atoms of similar chemistry and microstructure. The origins of this problem are numerous, including incomplete detection of atoms and the complexity of the evaporation fields of atoms at or near interfaces. Hence, unlike scattering techniques such as electron microscopy, interfaces appear diffused, not sharp. This, in turn, makes it challenging to visualize and quantitatively interpret the microstructure at the "meso" scale, where one is interested in the shape and form of the interfaces and their associated chemical gradients. It is here that the application of informatics at the nanoscale and statistical learning methods plays a critical role in both defining the level of uncertainty and helping to make quantitative, statistically objective interpretations where heuristics often dominate. In this chapter, we show how the tools of Topological Data Analysis provide a new and powerful tool in the field of nanoinformatics for materials characterization.

show abstract

Section: Persistent Homologymentioning

confidence: 99%

Topological Data Analysis for the Characterization of Atomic Scale Morphology from Atom Probe Tomography Images

2018

View full text Add to dashboard Cite

show abstract

“…deformation of the excursion sets by tabulating the occurrence of critical values [14]. This framework is general enough for dealing with a wide variety of noisy multivariate data including brain images [17,18], networks [19,20] and gene expression [21].…”

Section: It Is Known Thatmentioning

confidence: 99%

Sparse topological data recovery in medical images

Chung

Lee

Kim

et al. 2011

2011 IEEE International Symposium on Biomedical Imaging: From Nano to Macro

View full text Add to dashboard Cite

For medical image analysis, the test statistic of the measurements is usually constructed at every voxels in space and thresholded to determine the regions of significant signals. This thresholding produces a small patch of regions around the critical values of the test statistic. It is known that the probability of the critical values bigger than a specific threshold can be computed as the expectation of the Euler characteristic of the patch. Motivated by this topological connection, we present a new computational framework of modeling various functional measurements as topological objects.The level set associated with functional measurements can be approximated using a simplicial complex consisting of nodes and links. The existence of links basically determine the underlying topological structure of the signal. The strength of links can be modeled using an underdetermined linear model. By incorporating sparsity into the model, the links can be sparsely obtained making interpretation and visualization of the simplicial complex easier. The main contribution of this paper is showing the relationship between sparse topological structures to the sparse regression framework. We apply this novel framework in constructing a structural brain network model.

show abstract

“…The most important tool of TDA is the persistent homology method [12,13], which is proven as useful in many real-world applications. The abundance of applications covers a broad range of phenomena in biological and medical science, like breast cancer research [14], brain science [15][16][17][18][19][20][21], biomolecules [22][23][24], evolution [25] and bacteria [26], followed by the applications in sensor networks [27,28], signal analysis [29], image processing [30], musical data [31], text mining [32], phase space reconstruction of dynamical systems [33,34], as well as complex networks related to either dynamics taking place on networks [35] or structural properties [36,37].…”

Section: Introductionmentioning

confidence: 99%

Multilevel Integration Entropies: The Case of Reconstruction of Structural Quasi-Stability in Building Complex Datasets

Maletić

Zhao

2017

Entropy

Self Cite

View full text Add to dashboard Cite

Abstract:The emergence of complex datasets permeates versatile research disciplines leading to the necessity to develop methods for tackling complexity through finding the patterns inherent in datasets. The challenge lies in transforming the extracted patterns into pragmatic knowledge. In this paper, new information entropy measures for the characterization of the multidimensional structure extracted from complex datasets are proposed, complementing the conventionally-applied algebraic topology methods. Derived from topological relationships embedded in datasets, multilevel entropy measures are used to track transitions in building the high dimensional structure of datasets captured by the stratified partition of a simplicial complex. The proposed entropies are found suitable for defining and operationalizing the intuitive notions of structural relationships in a cumulative experience of a taxi driver's cognitive map formed by origins and destinations. The comparison of multilevel integration entropies calculated after each new added ride to the data structure indicates slowing the pace of change over time in the origin-destination structure. The repetitiveness in taxi driver rides, and the stability of origin-destination structure, exhibits the relative invariance of rides in space and time. These results shed light on taxi driver's ride habits, as well as on the commuting of persons whom he/she drove.

show abstract

Persistent homology of complex networks

Cited by 197 publications

References 19 publications

Topological Data Analysis for the Characterization of Atomic Scale Morphology from Atom Probe Tomography Images

Topological Data Analysis for the Characterization of Atomic Scale Morphology from Atom Probe Tomography Images

Sparse topological data recovery in medical images

Multilevel Integration Entropies: The Case of Reconstruction of Structural Quasi-Stability in Building Complex Datasets

Contact Info

Product

Resources

About