Networks and Cycles: A Persistent Homology Approach to Complex Networks

Petri, Giovanni; Scolamiero, Martina; Donato, Irene; Vaccarino, Francesco

doi:10.1007/978-3-319-00395-5_15

Cited by 27 publications

(24 citation statements)

References 14 publications

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“…In the present paper, we study bifurcations in WFP and ANC dynamics by examining data that are generated by several contagions on a given noisy geometric network. Our methodology is grounded in the field of computational topology [ 47 , 48 ], and we note that there has been rapidly intensifying interest (see, e.g., [ 49 – 52 ]) in using tools from computational topology to study structural features in networks and for machine learning [ 53 ]. In taking this perspective, we introduce a map from the network nodes to points in a metric space based on contagion dynamics.…”

mentioning

confidence: 99%

Topological data analysis of contagion maps for examining spreading processes on networks

et al. 2015

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Social and biological contagions are influenced by the spatial embeddedness of networks. Historically, many epidemics spread as a wave across part of the Earth’s surface; however, in modern contagions long-range edges—for example, due to airline transportation or communication media—allow clusters of a contagion to appear in distant locations. Here we study the spread of contagions on networks through a methodology grounded in topological data analysis and nonlinear dimension reduction. We construct “contagion maps” that use multiple contagions on a network to map the nodes as a point cloud. By analyzing the topology, geometry, and dimensionality of manifold structure in such point clouds, we reveal insights to aid in the modeling, forecast, and control of spreading processes. Our approach highlights contagion maps also as a viable tool for inferring low-dimensional structure in networks.

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confidence: 99%

Topological data analysis of contagion maps for examining spreading processes on networks

et al. 2015

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“…In [20,51,52], persistent homology is used to detect particular non-local structural as US air passenger networks, C. Elegans's neuronal network [53], the online messages network [54], gene network, network of mentions and re-tweet between Twitter users, school face-to-face contact network, co-authorship networks. While the gene network and airport network are in class 1, co-authorship networks and twitter network are in class 2.…”

Section: Multiple Graphs Analysismentioning

confidence: 99%

“…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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“…Many adaptations and refinements have been proposed to make persistent homology computationally efficient [32,33] and suited for a wide variety of data sets. In particular, Petri et al [16,17] applied persistent homology on the filtered clique complex for weighted undirected networks. As suggested in their paper, we modify their approach by adopting the directed k-clique construction of Palla et al [21].…”

Section: Filtrations Persistence and Perpetuitymentioning

confidence: 99%

“…3.2. 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].…”

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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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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Networks and Cycles: A Persistent Homology Approach to Complex Networks

Cited by 27 publications

References 14 publications

Topological data analysis of contagion maps for examining spreading processes on networks

Topological data analysis of contagion maps for examining spreading processes on networks

Persistence homology of networks: methods and applications

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

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