The topology of the directed clique complex as a network invariant

Masulli, Paolo; Villa, Alessandro E. P.

doi:10.1186/s40064-016-2022-y

Cited by 33 publications

(30 citation statements)

References 36 publications

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“…While making the necessary choice, it is imperative to remark that there are three essentially different types of directed triangles which can arise in complex networks ( Fig. 1(b)), and of these three possible choices, we opt for the first configuration to be the directed simplicial complex following the recent work by Masulli & Villa [48] and Courtney & Bianconi [25]. Note that the first configuration shown in Fig.…”

Section: Augmented Forman-ricci Curvature For Directed Networkmentioning

confidence: 99%

Discrete Ricci curvatures for directed networks

Saucan

Sreejith

Vivek-Ananth

et al. 2019

Chaos, Solitons & Fractals

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A goal in network science is the geometrical characterization of complex networks. In this direction, we have recently introduced Forman's discretization of Ricci curvature to the realm of undirected networks. Investigation of this edge-centric network measure, Forman-Ricci curvature, in diverse model and real-world undirected networks revealed that the curvature measure captures several aspects of the organization of undirected complex networks. However, many important real-world networks are inherently directed in nature, and the definition of the Forman-Ricci curvature for undirected networks is unsuitable for the analysis of such directed networks. Hence, we here extend the Forman-Ricci curvature for undirected networks to the case of directed networks. The simple mathematical formula for the Forman-Ricci curvature of a directed edge elegantly incorporates vertex weights, edge weights and edge direction. Furthermore we have compared the Forman-Ricci curvature with the adaptation to directed networks of another discrete notion of Ricci curvature, namely, the well established Ollivier-Ricci curvature. However, the two above-mentioned curvature measures do not account for higher-order correlations between vertices. To this end, we adjusted Forman's original definition of Ricci curvature to account for directed simplicial complexes and also explored the potential of this new, augmented type of Forman-Ricci curvature, in directed complex networks. * asamal@imsc.res.in arXiv:1809.07698v1 [math.MG] 15 Sep 2018Curvature represents a central concept in geometry, and it certainly is the focal point of differential and Riemannian geometry, which formally quantifies our intuition of the deviation of an object from being flat. In the classical context, where the notion of curvature originated, several types of curvature have been considered [26]. Among the different notions of curvature, it appears that the concept of Ricci curvature is the most useful, after a proper discretization process, for the analysis of graphs or networks [14-18, 20-22, 24]. To this end, let us note that, in differential geometry, Ricci curvature first appears in the context of the so called Jacobi equation, that represents a linearization of the equation for geodesics [26] and, as such, governs both the growth of volumes and the dispersion rate of geodesics [26,27]. Furthermore, Ricci curvature is also an essential ingredient in the so called Bochner-Weitzenböck formula, that connects between the Laplacian on a manifold and its geometry, as expressed by its curvature. By its very definition through the above mentioned equation of geodesics, Ricci curvature depends on a direction, which then for graphs or networks translates into the fact that it should be a measure associated to an edge rather than to a vertex. Among the several curvature measures [13][14][15][16][17][18][19][20][21][22][28][29][30][31][32] that have been proposed for geometrical characterization of complex networks, two different discretizations of the Ricci curvature, Ollivier-...

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Section: Augmented Forman-ricci Curvature For Directed Networkmentioning

confidence: 99%

Discrete Ricci curvatures for directed networks

Saucan

Sreejith

Vivek-Ananth

et al. 2019

Chaos, Solitons & Fractals

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“…For instance, persistent homology [14] has been employed across fields, such as contagion maps [68] and materials science [69]. In neuroscience, it has also yielded quite impactful results [31,33,37,57,[70][71][72]. In this sense, the Blue Brain Project recently provided persuasive support based both on empirical data and theoretical insights for the hypothesis that the brain network comprises topological structures in up to eleven dimensions [51].…”

Section: Discussionmentioning

confidence: 99%

Topological phase transitions in functional brain networks

Santos

Raposo

Coutinho‐Filho

et al. 2018

Preprint

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Functional brain networks are often constructed by quantifying correlations among brain regions.Their topological structure includes nodes, edges, triangles and even higher-dimensional objects.Topological data analysis (TDA) is the emerging framework to process datasets under this perspective. In parallel, topology has proven essential for understanding fundamental questions in physics. Here we report the discovery of topological phase transitions in functional brain networks by merging concepts from TDA, topology, geometry, physics, and network theory. We show that topological phase transitions occur when the Euler entropy has a singularity, which remarkably coincides with the emergence of multidimensional topological holes in the brain network. Our results suggest that a major alteration in the pattern of brain correlations can modify the signature of such transitions, and may point to suboptimal brain functioning. Due to the universal character of phase transitions and noise robustness of TDA, our findings open perspectives towards establishing reliable topological and geometrical biomarkers of individual and group differences in functional brain network organization. 1 2 I. INTRODUCTIONTopology aims to describe global properties of a system that are preserved under continuous deformations and are independent of specific coordinates, while differential geometry is usually associated with the system's local features [1]. As they relate to the fundamental understanding of how the world around us is intrinsically structured, topology and differential geometry have had a great impact on physics [2, 3], materials science [4], biology [5], and complex systems [6], to name a few. Importantly, topology has provided [7-11] strong arguments towards associating phase transitions with major topological changes in the configuration space of some model systems in theoretical physics. More recently, topology has also started to play a relevant role in describing global properties of real world, data-driven systems [12]. This emergent research field is called topological data analysis (TDA) [13,14].In parallel to these conceptual and theoretical advances, the topology of complex networks and their dynamics have become an important field in their own right [15,16]. The diversity of such networks ranges from the internet to climate dynamics, genomic, brain and social networks [15,16]. Many of these networks are based upon intrinsic correlations or similarities relations among their constituent parts. For instance, functional brain networks are often constructed by quantifying correlations between time series of activity recorded from different brain regions in an atlas spanning the entire brain [16].Here we report the discovery of topological phase transitions in functional brain networks.We merge concepts of TDA, topology, geometry, physics, and high-dimensional network theory to describe the topological evolution of complex networks, such as the brain networks, as function of their intrinsic correlation level. We consider tha...

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“…A k-simplex represents a group of k + 1 countries where pairwise interactions exist between any two members, and the overall flow structure can be characterized by the unidirectional flow from a unique source (that sends to every country) to a unique sink (that receives from every country). Masulli and Villa [27] used the same construction to investigate how resulting topological invariants, the Euler characteristic and network degree, can be used to assess specific functional and dynamical properties of directed networks. This definition can be modified to fit an appropriate alternate model.…”

Section: Directed Clique Complexesmentioning

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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The topology of the directed clique complex as a network invariant

Cited by 33 publications

References 36 publications

Discrete Ricci curvatures for directed networks

Discrete Ricci curvatures for directed networks

Topological phase transitions in functional brain networks

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

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