Persistent Path Homology of Directed Networks

Chowdhury, Samir; Mémoli, Facundo

doi:10.1137/1.9781611975031.75

Cited by 48 publications

(61 citation statements)

References 20 publications

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“…3(e) is in fact the sum of two 2-paths, and hence regarded as a two-dimensional simplex generator in a path complex. This however, as Chowdhury and Mémoli [19] point out, shows that path homology is able to differentiate this particular motiff from other types of cycles. Another construction due to [31] is via ordered tuple complexes where simplices are ordered tuples (v 0 , v 1 , .…”

Section: Directed Clique Complexesmentioning

confidence: 86%

“…Several works have employed this approach in studying the topology of undirected networks [14][15][16][17]. An extension of this approach to directed networks using path complexes has been explored by Chowdhury and Mémoli [18,19], and via neighborhood complex by Horak et al [20]. In [17], Petri et al introduced a filtration that allowed them to compute the persistent homology of weighted undirected networks, and remarked that their methods are amenable to extension to directed networks following the directed clique construction of Palla et al [21].…”

Section: Introductionmentioning

confidence: 99%

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Tracing patterns and shapes in remittance and migration networks via persistent homology

Ignacio

Darcy

2019

EPJ Data Sci.

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Pattern detection in network models provides insights to both global structure and local node interactions. In particular, studying patterns embedded within remittance and migration flow networks can be useful in understanding economic and sociologic trends and phenomena and their implications both in regional and global settings. We illustrate how topo-algebraic methods can be used to detect both local and global patterns that highlight simultaneous interactions among multiple nodes, giving a more holistic perspective on the network fabric and a higher order description of the overall flow structure of directed networks. Using the 2015 Asian net migration and remittance networks, we build and study the associated directed clique complexes whose topological features correspond to specific flow patterns in the networks. We generate diagrams recording the presence, persistence, and perpetuity of patterns and show how these diagrams can be used to make inferences about the characteristics of migrant movement patterns and remittance flows.

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Section: Directed Clique Complexesmentioning

confidence: 86%

Section: Introductionmentioning

confidence: 99%

Tracing patterns and shapes in remittance and migration networks via persistent homology

Ignacio

Darcy

2019

EPJ Data Sci.

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“…We remark that a persistent homology framework for the directed flag complex has been proposed by [Tur16], but the computational aspects of this construction have not been addressed in the current literature. Another approach for computing persistence diagrams from asymmetric networks, which bypasses the construction of any simplicial complex and operates directly at the chain level is given in [CM17]. Some other interesting questions relate to cycle networks: for example, we would like to obtain a characterization of the Rips persistence diagrams of cycle networks for any dimension k ě 1.…”

Section: Discussionmentioning

confidence: 99%

A functorial Dowker theorem and persistent homology of asymmetric networks

Chowdhury

Mémoli

2018

J Appl. and Comput. Topology

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We study two methods for computing network features with topological underpinnings: the Rips and Dowker persistent homology diagrams. Our formulations work for general networks, which may be asymmetric and may have any real number as an edge weight. We study the sensitivity of Dowker persistence diagrams to asymmetry via numerous theoretical examples, including a family of highly asymmetric cycle networks that have interesting connections to the existing literature. In particular, we characterize the Dowker persistence diagrams arising from asymmetric cycle networks. We investigate the stability properties of both the Dowker and Rips persistence diagrams, and use these observations to run a classification task on a dataset comprising simulated hippocampal networks. Our theoretical and experimental results suggest that Dowker persistence diagrams are particularly suitable for studying asymmetric networks. As a stepping stone for our constructions, we prove a functorial generalization of a theorem of Dowker, after whom our constructions are named. (S. Chowdhury) DEPARTMENT OF MATHEMATICS, THE OHIO STATE UNIVERSITY. PHONE: (614) 292-6805. (F. Mémoli)

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“…They study online social networks (OSN). They define the distance between two nodes as the number of hopes on the shortest path between these nodes and create 0-,1-and 2-dimensional persistence VR 0-3 dim Betti numbers PPI, brain and simulated weighted networks [58] VR 0-2 dim PD Economy networks [59] VR 0-1 dim PD Finance networks [23] CL 0-11 dim PD Random, email and scale-free networks [27] FMG 0 dim PD Road networks [70] VSF 0 dim zigzag PD Dynamic biological networks [30] PPH 1 dim PD Cycle networks [68] POW 0-1 dim PD Dynamic communication networks [60] VFB 0-1 dim PD Attributed social networks [69] POW 0-1 dim PD Online social networks [55] VFB 0-1 dim PD Social, medical and biological networks [20,51,52] VR 1 dim PB Social, infrastructural and biological networks barcodes using this distance in the power filtration (POW). They analyze the original and anonymized OSNs using the barcodes.…”

Section: Multiple Graphs Analysismentioning

confidence: 99%

Persistence homology of networks: methods and applications

2019

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Information networks are becoming increasingly popular to capture complex relationships across various disciplines, such as social networks, citation networks, and biological networks. The primary challenge in this domain is measuring similarity or distance between networks based on topology. However, classical graph-theoretic measures are usually local and mainly based on differences between either node or edge measurements or correlations without considering the topology of networks such as the connected components or holes. In recent years, mathematical tools and deep learning based methods have become popular to extract the topological features of networks. Persistent homology (PH) is a mathematical tool in computational topology that measures the topological features of data that persist across multiple scales with applications ranging from biological networks to social networks.In this paper, we provide a conceptual review of key advancements in this area of using PH on complex network science. We give a brief mathematical background on PH, review different methods (i.e. filtrations) to define PH on networks and highlight different algorithms and applications where PH is used in solving network mining problems. In doing so, we develop a unified framework to describe these recent approaches and emphasize major conceptual distinctions. We conclude with directions for future work. We focus our review on recent approaches that get significant attention in the mathematics and data mining communities working on network data. We believe our summary of the analysis of PH on networks will provide important insights to researchers in applied network science.

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Persistent Path Homology of Directed Networks

Cited by 48 publications

References 20 publications

Tracing patterns and shapes in remittance and migration networks via persistent homology

Tracing patterns and shapes in remittance and migration networks via persistent homology

A functorial Dowker theorem and persistent homology of asymmetric networks

Persistence homology of networks: methods and applications

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