# Characterizing the dynamical importance of network nodes and links.

@article{Restrepo2006CharacterizingTD, title={Characterizing the dynamical importance of network nodes and links.}, author={J. Restrepo and E. Ott and B. Hunt}, journal={Physical review letters}, year={2006}, volume={97 9}, pages={ 094102 } }

The largest eigenvalue of the adjacency matrix of networks is a key quantity determining several important dynamical processes on complex networks. Based on this fact, we present a quantitative, objective characterization of the dynamical importance of network nodes and links in terms of their effect on the largest eigenvalue. We show how our characterization of the dynamical importance of nodes can be affected by degree-degree correlations and network community structure. We discuss how our… Expand

#### 196 Citations

Dynamical and spectral properties of complex networks

- Physics
- 2007

Dynamical properties of complex networks are related to the spectral properties of the Laplacian matrix that describes the pattern of connectivity of the network. In particular we compute the… Expand

Network Evolution Based on Centrality

- 2010

We study the evolution of networks when the creation and decay of links are based on the position of nodes in the network measured by their centrality. We show that the same network dynamics arises… Expand

Node importance for dynamical process on networks: a multiscale characterization.

- Mathematics, Medicine
- Chaos
- 2011

A multiscale node-importance measure that can characterize the importance of the nodes at varying topological scale is proposed by introducing a kernel function whose bandwidth dictates the ranges of interaction, and meanwhile, by taking into account the interactions from all the paths a node is involved. Expand

Network evolution based on centrality.

- Mathematics, Physics
- Physical review. E, Statistical, nonlinear, and soft matter physics
- 2011

A discontinuous transition in the network density between hierarchical and homogeneous networks is found, depending on the rate of link decay, and this evolution mechanism leads to double power-law degree distributions, with interrelated exponents. Expand

Centrality and Network Analysis: A Perturbative Approach to Dynamical Importance

- Mathematics
- 2011

The purpose of this paper is to investigate methods for analyzing networks from an algebraic perspective. The main focus will be on the dominant eigenpair of the adjacency matrix representing a… Expand

Approximating spectral impact of structural perturbations in large networks.

- Mathematics, Physics
- Physical review. E, Statistical, nonlinear, and soft matter physics
- 2010

A theory for estimating the change of the largest eigenvalue of the adjacency matrix or the extreme eigenvalues of the graph Laplacian when small but arbitrary set of links are added or removed from the network is developed. Expand

Bounding network spectra for network design

- Physics
- 2007

The identification of the limiting factors in the dynamical behaviour of complex systems is an important interdisciplinary problem which often can be traced to the spectral properties of an… Expand

Interconnecting Networks: The Role of Connector Links

- Computer Science
- 2016

This chapter examines the effects that the particular way in which networks get connected exerts on each of the individual networks and describes how choosing the adequate connector links between networks may promote or hinder different structural and dynamical properties of a particular network. Expand

Eigenvector Localization in Real Networks and Its Implications for Epidemic Spreading

- Computer Science, Physics
- ArXiv
- 2018

The spectral properties of the adjacency matrix, in particular its largest eigenvalue and the associated principal eigenvector, dominate many structural and dynamical properties of complex networks.… Expand

Analysis of relative influence of nodes in directed networks.

- Mathematics, Medicine
- Physical review. E, Statistical, nonlinear, and soft matter physics
- 2009

The global properties of networks such as hierarchy and position of shortcuts rather than local properties of the nodes, such as the degree, are shown to be the chief determinants of the influence of nodes in many cases. Expand

#### References

SHOWING 1-10 OF 54 REFERENCES

The Structure and Function of Complex Networks

- Physics, Computer Science
- SIAM Rev.
- 2003

Developments in this field are reviewed, including such concepts as the small-world effect, degree distributions, clustering, network correlations, random graph models, models of network growth and preferential attachment, and dynamical processes taking place on networks. Expand

Complex networks: Structure and dynamics

- Physics
- 2006

Coupled biological and chemical systems, neural networks, social interacting species, the Internet and the World Wide Web, are only a few examples of systems composed by a large number of highly… Expand

Resilience to damage of graphs with degree correlations.

- Mathematics, Medicine
- Physical review. E, Statistical, nonlinear, and soft matter physics
- 2003

This work investigates the problems of site and bond percolation on graphs with degree correlations and their connection with spreading phenomena and obtains some general expressions that allow the computation of the transition thresholds or their bounds. Expand

Emergence of scaling in random networks

- Computer Science, Physics
- Science
- 1999

A model based on these two ingredients reproduces the observed stationary scale-free distributions, which indicates that the development of large networks is governed by robust self-organizing phenomena that go beyond the particulars of the individual systems. Expand

Statistical mechanics of complex networks

- Computer Science, Physics
- ArXiv
- 2001

A simple model based on these two principles was able to reproduce the power-law degree distribution of real networks, indicating a heterogeneous topology in which the majority of the nodes have a small degree, but there is a significant fraction of highly connected nodes that play an important role in the connectivity of the network. Expand

Synchronization in large directed networks of coupled phase oscillators.

- Physics, Medicine
- Chaos
- 2006

This work studies the emergence of collective synchronization in large directed networks of heterogeneous oscillators by generalizing the classical Kuramoto model of globally coupled phase oscillators to more realistic networks and considers the case of networks with mixed positive-negative coupling strengths. Expand

Generalized percolation in random directed networks.

- Mathematics, Physics
- Physical review. E, Statistical, nonlinear, and soft matter physics
- 2005

We develop a general theory for percolation in directed random networks with arbitrary two-point correlations and bidirectional edges--that is, edges pointing in both directions simultaneously. These… Expand

Synchronization in networks with random interactions: theory and applications.

- Mathematics, Medicine
- Chaos
- 2006

Some recent results on synchronization in randomly coupled networks and applications to neuroscience, in particular, networks of Hodgkin-Huxley neurons, are included. Expand

Scale-free topology of e-mail networks.

- Physics, Medicine
- Physical review. E, Statistical, nonlinear, and soft matter physics
- 2002

The resulting network exhibits a scale-free link distribution and pronounced small-world behavior, as observed in other social networks, implying that the spreading of e-mail viruses is greatly facilitated in real e- mail networks compared to random architectures. Expand

Onset of synchronization in large networks of coupled oscillators.

- Mathematics, Medicine
- Physical review. E, Statistical, nonlinear, and soft matter physics
- 2005

The theory of the transition from incoherence to coherence in large networks of coupled phase oscillators is studied and it is found that the theory describes the transition well in situations in which the mean-field approximation fails. Expand