Linear processes on complex networks

2020 ◽  
Vol 8 (4) ◽  
Author(s):  
Ivan Jokić ◽  
Piet Van Mieghem
Keyword(s):  
Graph Theory ◽  
Complex Networks ◽  
Large Scale ◽  
Linear Time ◽  
State Space Model ◽  
Internal Dynamics ◽  
Dynamic Interactions ◽  
Linear Processes ◽  
Explicit Analytical Solution ◽  
Time Invariant

Abstract This article studies the dynamics of complex networks with a time-invariant underlying topology, composed of nodes with linear internal dynamics and linear dynamic interactions between them. While graph theory defines the underlying topology of a network, a linear time-invariant state-space model analytically describes the internal dynamics of each node in the network. By combining linear systems theory and graph theory, we provide an explicit analytical solution for the network dynamics in discrete-time, continuous-time and the Laplace domain. The proposed theoretical framework is scalable and allows hierarchical structuring of complex networks with linear processes while preserving the information about network, which makes the approach reversible and applicable to large-scale networks.

Complete moment convergence for the dependent linear processes with application to the state observers of linear-time-invariant systems

2020 ◽  
Vol 65 (4) ◽  
pp. 725-745
Author(s):  
Chao Lu ◽  
Chao Lu ◽  
Xuejun J Wang ◽  
Xuejun J Wang ◽  
Yi Wu ◽  
...  
Keyword(s):  
Linear Time ◽  
The State ◽  
Complete Moment Convergence ◽  
Linear Processes ◽  
Linear Time Invariant ◽  
Moment Convergence ◽  
State Observers ◽  
Linear Time Invariant Systems ◽  
Time Invariant

Пусть $X_t=\sum_{j=-\infty}^{\infty}A_j\varepsilon_{t-j}$ - зависимый линейный процесс, где $\{\varepsilon_n, n\in \mathbf{Z}\}$ - последовательность $m$-обобщенных отрицательно зависимых ($m$-END) случайных величин с нулевым средним, которая стохастически доминируется случайной величиной $\varepsilon$, и пусть $\{A_n, n\in \mathbf{Z}\}$ - другая последовательность случайных величин с нулевым средним, обладающая свойством $m$-END. При подходящих условиях установлена полная моментная сходимость для зависимых линейных процессов. В частности, приведены достаточные условия полной моментной сходимости. В качестве приложения исследуется сходимость наблюдателей состояния для линейных стационарных систем.

scholarly journals Explaining Savings for Visuomotor Adaptation: Linear Time-Invariant State-Space Models Are Not Sufficient

2008 ◽  
Vol 100 (5) ◽  
pp. 2537-2548 ◽  
Author(s):  
Eric Zarahn ◽  
Gregory D. Weston ◽  
Johnny Liang ◽  
Pietro Mazzoni ◽  
John W. Krakauer
Keyword(s):  
State Space ◽  
Linear Time ◽  
State Space Model ◽  
Visuomotor Adaptation ◽  
Invariant State ◽  
Visuomotor Rotation ◽  
Body Memory ◽  
Linear Time Invariant ◽  
And Coping ◽  
Time Invariant

Adaptation of the motor system to sensorimotor perturbations is a type of learning relevant for tool use and coping with an ever-changing body. Memory for motor adaptation can take the form of savings: an increase in the apparent rate constant of readaptation compared with that of initial adaptation. The assessment of savings is simplified if the sensory errors a subject experiences at the beginning of initial adaptation and the beginning of readaptation are the same. This can be accomplished by introducing either 1) a sufficiently small number of counterperturbation trials (counterperturbation paradigm [ CP]) or 2) a sufficiently large number of zero-perturbation trials (washout paradigm [ WO]) between initial adaptation and readaptation. A two-rate, linear time-invariant state-space model (SSMLTI,2) was recently shown to theoretically produce savings for CP. However, we reasoned from superposition that this model would be unable to explain savings for WO. Using the same task (planar reaching) and type of perturbation (visuomotor rotation), we found comparable savings for both CP and WO paradigms. Although SSMLTI,2 explained some degree of savings for CP it failed completely for WO. We conclude that for visuomotor rotation, savings in general is not simply a consequence of LTI dynamics. Instead savings for visuomotor rotation involves metalearning, which we show can be modeled as changes in system parameters across the phases of an adaptation experiment.

scholarly journals A Method To Improve The Time Of Computing For Detecting Community Structure In Social Network Graph

2019 ◽  
Vol 8 (6) ◽  
pp. 933-937
Keyword(s):  
Community Structure ◽  
Social Network ◽  
Complex Networks ◽  
Large Scale ◽  
Linear Time ◽  
A Priori ◽  
Label Propagation ◽  
Data Sets ◽  
Network Graph ◽  
Priori Information

Identifying communities has always been a fundamental task in analysis of complex networks. Currently used algorithms that identify the community structures in large-scale real-world networks require a priori information such as the number and sizes of communities or are computationally expensive. Amongst them, the label propagation algorithm (LPA) brings great scaslability together with high accuracy but which is not accurate enough because of its randomness. In this paper, we study the equivalence properties of nodes on social network graphs according to the labeling criteria to shorten social network graphs and develop label propagation algorithms on shortened graphs to discover effective social networking communities without requiring optimization of the objective function as well as advanced information about communities. Test results on sample data sets show that the proposed algorithm execution time is significantly reduced compared to the published algorithms. The proposed algorithm takes an almost linear time and improves the overall quality of the identified community in complex networks with a clear community structure.

Order Reduction and Control of Large-Scale Linear Time Periodic Dynamical Systems

1999 ◽  
Author(s):  
Venkatesh Deshmukh ◽  
S. C. Sinha
Keyword(s):  
Large Scale ◽  
Linear Time ◽  
Parametric Instability ◽  
Order Reduction ◽  
System Stability ◽  
Order System ◽  
Control Laws ◽  
Reduced Order ◽  
Time Invariant ◽  
Time Periodic

Abstract This paper provides methodology for designing reduced order controllers for large-scale, linear systems represented by differential equations having time periodic coefficients. The linear time periodic system is first converted into a form in which the system stability matrix is time invariant. This is achieved by the application of Liapunov-Floquet transformation. Then a system called an auxiliary system is constructed which is a completely time invariant. Order reduction algorithms are applied to this system to obtain a reduced order system. The control laws are calculated for the reduced order system by minimizing the least square error between the auxiliary and the transformed system. These control laws when transformed back to time varying domain provide the desired control action. The schemes formulated are illustrated by designing full state feedback and output feedback controllers for a five mass inverted pendulum exhibiting parametric instability.

Minimum Robust Sensor Placement for Large Scale Linear Time-Invariant Systems: A Structured Systems Approach*

2013 ◽  
Vol 46 (27) ◽  
pp. 417-424 ◽  
Author(s):  
Xiaofei Liu ◽  
Yilin Mo ◽  
Sergio Pequito ◽  
Bruno Sinopoli ◽  
Soummya Kar ◽  
...  
Keyword(s):  
Large Scale ◽  
Systems Approach ◽  
Linear Time ◽  
Sensor Placement ◽  
Linear Time Invariant ◽  
Structured Systems ◽  
Linear Time Invariant Systems ◽  
Time Invariant
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