### Abstract

Rotation dynamics of eigenvectors of modular network adjacency matrices under random perturbations are presented. In the presence of q communities, the number of eigenvectors corresponding to the q largest eigenvalues form a "community" eigenspace and rotate together, but separately from that of the "bulk" eigenspace spanned by all the other eigenvectors. Using this property, the number of modules or clusters in a network can be estimated in an algorithm-independent way. A general argument and derivation for the theoretical detectability limit for sparse modular networks with q communities is presented, beyond which modularity persists in the system but cannot be detected. It is shown that for detecting the clusters or modules using the adjacency matrix, there is a "band" in which it is hard to detect the clusters even before the theoretical detectability limit is reached, and for which the theoretically predicted detectability limit forms the sufficient upper bound. Analytic estimations of these bounds are presented and empirically demonstrated.

Original language | English |
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Article number | 062312 |

Journal | Physical Review E - Statistical, Nonlinear, and Soft Matter Physics |

Volume | 93 |

Issue number | 6 |

DOIs | |

Publication status | Published - 20 Jun 2016 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Condensed Matter Physics
- Statistical and Nonlinear Physics
- Statistics and Probability

### Cite this

*Physical Review E - Statistical, Nonlinear, and Soft Matter Physics*,

*93*(6), [062312]. https://doi.org/10.1103/PhysRevE.93.062312

**Eigenvector dynamics under perturbation of modular networks.** / Sarkar, Somwrita; Chawla, Sanjay; Robinson, P. A.; Fortunato, Santo.

Research output: Contribution to journal › Article

*Physical Review E - Statistical, Nonlinear, and Soft Matter Physics*, vol. 93, no. 6, 062312. https://doi.org/10.1103/PhysRevE.93.062312

}

TY - JOUR

T1 - Eigenvector dynamics under perturbation of modular networks

AU - Sarkar, Somwrita

AU - Chawla, Sanjay

AU - Robinson, P. A.

AU - Fortunato, Santo

PY - 2016/6/20

Y1 - 2016/6/20

N2 - Rotation dynamics of eigenvectors of modular network adjacency matrices under random perturbations are presented. In the presence of q communities, the number of eigenvectors corresponding to the q largest eigenvalues form a "community" eigenspace and rotate together, but separately from that of the "bulk" eigenspace spanned by all the other eigenvectors. Using this property, the number of modules or clusters in a network can be estimated in an algorithm-independent way. A general argument and derivation for the theoretical detectability limit for sparse modular networks with q communities is presented, beyond which modularity persists in the system but cannot be detected. It is shown that for detecting the clusters or modules using the adjacency matrix, there is a "band" in which it is hard to detect the clusters even before the theoretical detectability limit is reached, and for which the theoretically predicted detectability limit forms the sufficient upper bound. Analytic estimations of these bounds are presented and empirically demonstrated.

AB - Rotation dynamics of eigenvectors of modular network adjacency matrices under random perturbations are presented. In the presence of q communities, the number of eigenvectors corresponding to the q largest eigenvalues form a "community" eigenspace and rotate together, but separately from that of the "bulk" eigenspace spanned by all the other eigenvectors. Using this property, the number of modules or clusters in a network can be estimated in an algorithm-independent way. A general argument and derivation for the theoretical detectability limit for sparse modular networks with q communities is presented, beyond which modularity persists in the system but cannot be detected. It is shown that for detecting the clusters or modules using the adjacency matrix, there is a "band" in which it is hard to detect the clusters even before the theoretical detectability limit is reached, and for which the theoretically predicted detectability limit forms the sufficient upper bound. Analytic estimations of these bounds are presented and empirically demonstrated.

UR - http://www.scopus.com/inward/record.url?scp=84975318812&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84975318812&partnerID=8YFLogxK

U2 - 10.1103/PhysRevE.93.062312

DO - 10.1103/PhysRevE.93.062312

M3 - Article

AN - SCOPUS:84975318812

VL - 93

JO - Physical Review E

JF - Physical Review E

SN - 2470-0045

IS - 6

M1 - 062312

ER -