scholarly journals Node and edge nonlinear eigenvector centrality for hypergraphs

2021 ◽  
Vol 4 (1) ◽  
Author(s):  
Francesco Tudisco ◽  
Desmond J. Higham
Keyword(s):  
Eigenvalue Problems ◽  
Point Of View ◽  
Centrality Measures ◽  
Data Sets ◽  
Power Method ◽  
Eigenvector Centrality ◽  
Nonlinear Eigenvalue Problems ◽  
Pairwise Interactions ◽  
Important Nodes ◽  
Realistic Data

AbstractNetwork scientists have shown that there is great value in studying pairwise interactions between components in a system. From a linear algebra point of view, this involves defining and evaluating functions of the associated adjacency matrix. Recent work indicates that there are further benefits from accounting directly for higher order interactions, notably through a hypergraph representation where an edge may involve multiple nodes. Building on these ideas, we motivate, define and analyze a class of spectral centrality measures for identifying important nodes and hyperedges in hypergraphs, generalizing existing network science concepts. By exploiting the latest developments in nonlinear Perron−Frobenius theory, we show how the resulting constrained nonlinear eigenvalue problems have unique solutions that can be computed efficiently via a nonlinear power method iteration. We illustrate the measures on realistic data sets.

scholarly journals Node and Edge Eigenvector Centrality for Hypergraphs

2021 ◽  
Author(s):  
Francesco Tudisco ◽  
Desmond Higham
Keyword(s):  
Eigenvalue Problems ◽  
Point Of View ◽  
Centrality Measures ◽  
Data Sets ◽  
Power Method ◽  
Eigenvector Centrality ◽  
Nonlinear Eigenvalue Problems ◽  
Pairwise Interactions ◽  
Important Nodes ◽  
Realistic Data

Abstract Network scientists have shown that there is great value in studying pairwise interactions between components in a system. From a linear algebra point of view, this involves defining and evaluating functions of the associated adjacency matrix. Recent work indicates that there are further benefits from accounting directly for higher order interactions, notably through a hypergraph representation where an edge may involve multiple nodes. Building on these ideas, we motivate, define and analyze a class of spectral centrality measures for identifying important nodes and hyperedges in hypergraphs, generalizing existing network science concepts. By exploiting the latest developments in nonlinear Perron-Frobenius theory, we show how the resulting constrained nonlinear eigenvalue problems have unique solutions that can be computed efficiently via a nonlinear power method iteration. We illustrate the measures on realistic data sets.

Measures and metrics

2018 ◽  
Author(s):  
Mark Newman
Keyword(s):  
Network Structure ◽  
Similarity Measures ◽  
Clustering Coefficient ◽  
Centrality Measures ◽  
Degree Centrality ◽  
Eigenvector Centrality ◽  
Structural Balance ◽  
Important Nodes

This chapter describes the measures and metrics that are used to quantify network structure. The chapter starts with a discussion of centrality measures, which are used to identify central or important nodes in networks. Measures discussed include degree centrality, eigenvector centrality, PageRank, closeness, and betweenness. This is followed by a discussion of groupings of nodes like cliques and components, transitivity measures including the clustering coefficient, structural balance in networks, similarity measures, and assortative mixing.

Confidentiality of Statistical Records: A Threat-Monitoring Scheme for On Line Dialogue

1976 ◽  
Vol 15 (01) ◽  
pp. 36-42 ◽  
Author(s):  
J. Schlörer
Keyword(s):  
Statistical Data ◽  
Cost Benefit ◽  
Data Bank ◽  
High Ratio ◽  
Point Of View ◽  
Data Sets ◽  
Monitoring Scheme ◽  
Access Controls ◽  
On Line ◽  
Bona Fide

From a statistical data bank containing only anonymous records, the records sometimes may be identified and then retrieved, as personal records, by on line dialogue. The risk mainly applies to statistical data sets representing populations, or samples with a high ratio n/N. On the other hand, access controls are unsatisfactory as a general means of protection for statistical data banks, which should be open to large user communities. A threat monitoring scheme is proposed, which will largely block the techniques for retrieval of complete records. If combined with additional measures (e.g., slight modifications of output), it may be expected to render, from a cost-benefit point of view, intrusion attempts by dialogue valueless, if not absolutely impossible. The bona fide user has to pay by some loss of information, but considerable flexibility in evaluation is retained. The proposal of controlled classification included in the scheme may also be useful for off line dialogue systems.

Centrality Measures

2018 ◽  
Author(s):  
Ginestra Bianconi
Keyword(s):  
State Of The Art ◽  
Centrality Measures ◽  
Eigenvector Centrality ◽  
Multilayer Networks ◽  
Comprehensive Description ◽  
Multilayer Network ◽  
Suitable Definition ◽  
Katz Centrality ◽  
Pagerank Centrality

Defining the centrality of nodes and layers in multilayer networks is of fundamental importance for a variety of applications from sociology to biology and finance. This chapter presents the state-of-the-art centrality measures able to characterize the centrality of nodes, the influences of layers or the centrality of replica nodes in multilayer and multiplex networks. These centrality measures include modifications of the eigenvector centrality, Katz centrality, PageRank centrality and Communicability to the multilayer network scenario. The chapter provides a comprehensive description of the research of the field and discusses the main advantages and limitations of the different definitions, allowing the readers that wish to apply these techniques to choose the most suitable definition for his or her case study.

scholarly journals The position of the farm holiday in Austrian tourism

2019 ◽  
Vol 4 (1) ◽  
pp. 697-711 ◽  
Author(s):  
Erika Quendler
Keyword(s):  
Empirical Research ◽  
Relative Position ◽  
Secondary Data ◽  
Point Of View ◽  
Local Context ◽  
Supply Side ◽  
Data Sets ◽  
Demand Side ◽  
Tourist Destinations ◽  
Tourist Accommodation

AbstractTourism is vitally important to the Austrian economy. The number of tourist destinations, both farms and other forms of accommodation, in the different regions of Austria is considerably and constantly changing. This paper discusses the position of the ‘farm holiday’ compared to other forms of tourism. Understanding the resilience of farm holidays is especially important but empirical research on this matter remains limited. The term ‘farm holiday’ covers staying overnight on a farm that is actively engaged in agriculture and has a maximum of 10 guest beds. The results reported in this paper are based on an analysis of secondary data from 2000 and 2018 by looking at two types of indicator: (i) accommodation capacity (supply side) and (ii) attractiveness of a destination (demand side). The data sets cover Austria and its NUTS3 regions. The results show the evolution of farm holidays vis-à-vis other forms of tourist accommodation. In the form of a quadrant matrix they also show the relative position of farm holidays regionally. While putting into question the resilience of farm holidays, the data also reveals where farm holidays could act to expand this niche or learn and improve to effect a shift in their respective position relative to the market ‘leaders’. However, there is clearly a need to learn more about farm holidays within the local context. This paper contributes to our knowledge of farm holidays from a regional point of view and tries to elaborate on the need for further research.

MULTIPLE POSITIVE SOLUTIONS OF NONLINEAR EIGENVALUE PROBLEMS ASSOCIATED TO A CLASS OF p-LAPLACIAN LIKE OPERATORS

2003 ◽  
Vol 05 (05) ◽  
pp. 737-759 ◽  
Author(s):  
NOBUYOSHI FUKAGAI ◽  
KIMIAKI NARUKAWA
Keyword(s):  
Positive Solutions ◽  
Elliptic Problem ◽  
Eigenvalue Problems ◽  
Multiple Positive Solutions ◽  
Combined Effects ◽  
Nonlinear Eigenvalue Problems ◽  
Quasilinear Elliptic Problem ◽  
Quasilinear Elliptic ◽  
Nonlinear Eigenvalue

This paper deals with positive solutions of a class of nonlinear eigenvalue problems. For a quasilinear elliptic problem (#) - div (ϕ(|∇u|)∇u) = λf(x,u) in Ω, u = 0 on ∂Ω, we assume asymptotic conditions on ϕ and f such as ϕ(t) ~ tp0-2, f(x,t) ~ tq0-1as t → +0 and ϕ(t) ~ tp1-2, f(x,t) ~ tq1-1as t → ∞. The combined effects of sub-nonlinearity (p0> q0) and super-nonlinearity (p1< q1) with the subcritical term f(x,u) imply the existence of at least two positive solutions of (#) for 0 < λ < Λ.

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