Euler characteristic heuristic for approximating the distribution of the largest eigenvalue of an orthogonally invariant random matrix

2008 ◽  
Vol 138 (11) ◽  
pp. 3357-3378 ◽  
Author(s):  
Satoshi Kuriki ◽  
Akimichi Takemura
Keyword(s):  
Euler Characteristic ◽  
Random Matrix ◽  
Largest Eigenvalue ◽  
The Largest Eigenvalue

scholarly journals Analysis of the Global Banking Network by Random Matrix Theory

2021 ◽  
Vol 8 ◽  
Author(s):  
Ali Namaki ◽  
Reza Raei ◽  
Jamshid Ardalankia ◽  
Leila Hedayatifar ◽  
Ali Hosseiny ◽  
...  
Keyword(s):  
Random Matrix Theory ◽  
Random Matrix ◽  
Matrix Theory ◽  
Final Analysis ◽  
The United States ◽  
Mean Value ◽  
Largest Eigenvalue ◽  
Global Banking ◽  
The Largest Eigenvalue

Since the financial crisis of 2008, the network analysis of financial systems has attracted a lot of attention. In this paper, we analyze the global banking network via the method of Random Matrix Theory. By applying that method on a cross border lending network, it is shown that while the connectivity between different parts of the network has risen and the profile of transactions has diversified, the role of hubs remains important in the weighted perspective. The largest eigenvalue of the transaction matrix as the leading mode of the system shows sharp growth since 2002. As well, it is observed that its growth has diminished since 2008. This indicates that the crisis of 2008 has left a long-lasting footprint on the financial system. Analyzing the mean value of the participation ratio reveals the fact that the role of countries in forming small modes, has increased since 2002. In our final analysis, we provide snapshots of the hubs in the network over time. We observe that the share of countries in total transactions is not equal to their share in shaping the eigenvector of the largest eigenvalue. In 2018 for example, while the United Kingdom leads the share of transactions, it is the United States that has the largest value in the leading eigenvector. The proposed technique in the paper can be useful for analyzing different types of interaction networks between countries.

scholarly journals Spectral Gap of the Largest Eigenvalue of the Normalized Graph Laplacian

2021 ◽  
Author(s):  
Jürgen Jost ◽  
Raffaella Mulas ◽  
Florentin Münch
Keyword(s):  
Lower Bound ◽  
Bipartite Graph ◽  
Spectral Gap ◽  
Complete Bipartite Graph ◽  
Graph Laplacian ◽  
New Method ◽  
Single Edge ◽  
Largest Eigenvalue ◽  
Complement Graph ◽  
The Largest Eigenvalue

AbstractWe offer a new method for proving that the maxima eigenvalue of the normalized graph Laplacian of a graph with n vertices is at least $$\frac{n+1}{n-1}$$ n + 1 n - 1 provided the graph is not complete and that equality is attained if and only if the complement graph is a single edge or a complete bipartite graph with both parts of size $$\frac{n-1}{2}$$ n - 1 2 . With the same method, we also prove a new lower bound to the largest eigenvalue in terms of the minimum vertex degree, provided this is at most $$\frac{n-1}{2}$$ n - 1 2 .

scholarly journals A note on a walk-based inequality for the index of a signed graph

2021 ◽  
Vol 9 (1) ◽  
pp. 19-21
Author(s):  
Zoran Stanić
Keyword(s):  
Adjacency Matrix ◽  
Signed Graph ◽  
Arbitrary Length ◽  
Largest Eigenvalue ◽  
The Largest Eigenvalue

Abstract We derive an inequality that includes the largest eigenvalue of the adjacency matrix and walks of an arbitrary length of a signed graph. We also consider certain particular cases.

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