Laplacian Spectra and Synchronization Processes on Complex Networks

Chen, Juan; Lu, Jun-an; Zhan, Choujun; Chen, Guanrong

doi:10.1007/978-1-4614-0754-6_4

Cited by 22 publications

(14 citation statements)

References 47 publications

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“…determined by the a linear combination of eigenmodes of the graph Laplacian, since all eigenmodes destabilise simultaneously. It is known that the graph Laplacian can be used to predict phase-locked patterns (Chen, Lu, Zhan, & Chen, 2012) and has indeed been used to predict empirical FC from SC (Abdelnour, Dayan, Devinsky, Thesen, & Raj, 2018). Following from this, the eigenmodes of the Jacobian in Equation 5 can be used as simple, easily computable proxy for the FC matrix when the system is poised at a local instability.…”

Section: It Has Been Shown Inmentioning

confidence: 99%

The role of node dynamics in shaping emergent functional connectivity patterns in the brain

Forrester

Crofts

Sotiropoulos

et al. 2020

Network Neuroscience

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The contribution of structural connectivity to functional brain states remains poorly understood. We present a mathematical and computational study suited to assess the structure–function issue, treating a system of Jansen–Rit neural mass nodes with heterogeneous structural connections estimated from diffusion MRI data provided by the Human Connectome Project. Via direct simulations we determine the similarity of functional (inferred from correlated activity between nodes) and structural connectivity matrices under variation of the parameters controlling single-node dynamics, highlighting a nontrivial structure–function relationship in regimes that support limit cycle oscillations. To determine their relationship, we firstly calculate network instabilities giving rise to oscillations, and the so-called ‘false bifurcations’ (for which a significant qualitative change in the orbit is observed, without a change of stability) occurring beyond this onset. We highlight that functional connectivity (FC) is inherited robustly from structure when node dynamics are poised near a Hopf bifurcation, whilst near false bifurcations, and structure only weakly influences FC. Secondly, we develop a weakly coupled oscillator description to analyse oscillatory phase-locked states and, furthermore, show how the modular structure of FC matrices can be predicted via linear stability analysis. This study thereby emphasises the substantial role that local dynamics can have in shaping large-scale functional brain states.

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Section: It Has Been Shown Inmentioning

confidence: 99%

The role of node dynamics in shaping emergent functional connectivity patterns in the brain

Forrester

Crofts

Sotiropoulos

et al. 2020

Network Neuroscience

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“…Dynamic phenomena are ubiquitous in nature and play a key role within various contexts in sociology [3], and technology [4]. To date, the problem of how the structural properties of a network influences the convergence and stability of its synchronized states has been extensively investigated and discussed, both numerically and theoretically [5,6,7,8,9], with special attention given to networks of coupled oscillators [10,11,12,13].…”

Section: Introductionmentioning

confidence: 99%

Algebraic connectivity of interdependent networks

Martı́n-Hernández

Wang

Mieghem

et al. 2014

Physica A: Statistical Mechanics and its Applications

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The algebraic connectivity µ N −1 , i.e. the second smallest eigenvalue of the Laplacian matrix, plays a crucial role in dynamic phenomena such as diffusion processes, synchronization stability, and network robustness. In this work we study the algebraic connectivity in the general context of interdependent networks, or network-of-networks (NoN). The present work shows, both analytically and numerically, how the algebraic connectivity of NoNs experiences a transition. The transition is characterized by a saturation of the algebraic connectivity upon the addition of sufficient coupling links (between the two individual networks of a NoN). In practical terms, this shows that NoN topologies require only a fraction of coupling links in order to achieve optimal diffusivity. Furthermore, we observe a footprint of the transition on the properties of Fiedler's spectral bisection.

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“…Communication efficiency was studied via analyzing changes to the global topological distance, expressed as the average shortest path, and the diameter of the network (the longest of the shortest paths), while undergoing edge removal. Network synchronizability was studied by computing spectral graph metrics of the graph Laplacian (Chung, 1997 ; Boccaletti et al, 2006 ; Arenas et al, 2008 ; Chen et al, 2012 ). Specifically, the second smallest eigenvalue, λ 2 , which is known as algebraic connectivity (Fiedler, 1973 ), and is an important measure of graph robustness, i.e., its magnitude indicates connectedness.…”

Section: Introductionmentioning

confidence: 99%

“…In addition, the largest absolute eigenvalue (λ N ), called the spectral radius was also computed. Synchronizability, as the stability of a synchronized state, is maximal in unweighted, fully random networks with uniform degree distribution, and depends strongly on λ 2 (Nishikawa et al, 2003 ; Atay et al, 2006 ; Almendral and Díaz-Guilera, 2007 ; Chen et al, 2012 ). The higher the maximal eigenvalue, the easier the network reaches the synchronization regime.…”

Section: Introductionmentioning

confidence: 99%

Network Path Convergence Shapes Low-Level Processing in the Visual Cortex

Varga

Soós

Jákli

et al. 2021

Front. Syst. Neurosci.

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Hierarchical counterstream via feedforward and feedback interactions is a major organizing principle of the cerebral cortex. The counterstream, as a topological feature of the network of cortical areas, is captured by the convergence and divergence of paths through directed links. So defined, the convergence degree (CD) reveals the reciprocal nature of forward and backward connections, and also hierarchically relevant integrative properties of areas through their inward and outward connections. We asked if topology shapes large-scale cortical functioning by studying the role of CD in network resilience and Granger causal coupling in a model of hierarchical network dynamics. Our results indicate that topological synchronizability is highly vulnerable to attacking edges based on CD, while global network efficiency depends mostly on edge betweenness, a measure of the connectedness of a link. Furthermore, similar to anatomical hierarchy determined by the laminar distribution of connections, CD highly correlated with causal coupling in feedforward gamma, and feedback alpha-beta band synchronizations in a well-studied subnetwork, including low-level visual cortical areas. In contrast, causal coupling did not correlate with edge betweenness. Considering the entire network, the CD-based hierarchy correlated well with both the anatomical and functional hierarchy for low-level areas that are far apart in the hierarchy. Conversely, in a large part of the anatomical network where hierarchical distances are small between the areas, the correlations were not significant. These findings suggest that CD-based and functional hierarchies are interrelated in low-level processing in the visual cortex. Our results are consistent with the idea that the interplay of multiple hierarchical features forms the basis of flexible functional cortical interactions.

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Laplacian Spectra and Synchronization Processes on Complex Networks

Cited by 22 publications

References 47 publications

The role of node dynamics in shaping emergent functional connectivity patterns in the brain

The role of node dynamics in shaping emergent functional connectivity patterns in the brain

Algebraic connectivity of interdependent networks

Network Path Convergence Shapes Low-Level Processing in the Visual Cortex

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