Optimal Filter Design for Consensus on Random Directed Graphs

Optimal design of consensus acceleration graph filters relates closely to the eigenvalues of the consensus iteration matrix. This task is complicated by random networks with uncertain iteration matrix eigenvalues. Filter design methods based on the spectral asymptotics of consensus iteration matrices for large-scale, random undirected networks have been previously developed both for constant and for time-varying network topologies. This work builds upon these results by extending analysis to large-scale, constant, random directed networks. The proposed approach uses theorems by Girko that analytically produce deterministic approximations of the empirical spectral distribution for suitable non-Hermitian random matrices. The approximate empirical spectral distribution defines filtering regions in the proposed filter optimization problem, which must be modified to accommodate complex-valued eigenvalues. Presented numerical simulations demonstrate good results. Additionally, limitations of the proposed method are discussed.

[1]  Soummya Kar,et al.  Consensus + innovations distributed inference over networks: cooperation and sensing in networked systems , 2013, IEEE Signal Processing Magazine.

[2]  Laura Cottatellucci,et al.  Spectral properties of random matrices for stochastic block model , 2015, 2015 13th International Symposium on Modeling and Optimization in Mobile, Ad Hoc, and Wireless Networks (WiOpt).

[3]  Stephen P. Boyd,et al.  Fast linear iterations for distributed averaging , 2003, 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475).

[4]  Pascal Frossard,et al.  Polynomial Filtering for Fast Convergence in Distributed Consensus , 2008, IEEE Transactions on Signal Processing.

[5]  Geert Leus,et al.  Distributed Autoregressive Moving Average Graph Filters , 2015, IEEE Signal Processing Letters.

[6]  J. Moura,et al.  Topology for Global Average Consensus , 2006, 2006 Fortieth Asilomar Conference on Signals, Systems and Computers.

[7]  Geert Leus,et al.  Filtering Random Graph Processes Over Random Time-Varying Graphs , 2017, IEEE Transactions on Signal Processing.

[8]  Reza Olfati-Saber,et al.  Consensus and Cooperation in Networked Multi-Agent Systems , 2007, Proceedings of the IEEE.

[9]  José M. F. Moura,et al.  Spectral statistics of lattice graph structured, non-uniform percolations , 2017, 2017 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).

[10]  José M. F. Moura,et al.  Optimal Filter Design for Signal Processing on Random Graphs: Accelerated Consensus , 2018, IEEE Transactions on Signal Processing.

[11]  V. Girko,et al.  Theory of stochastic canonical equations , 2001 .

[12]  Audra E. Kosh,et al.  Linear Algebra and its Applications , 1992 .

[13]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.

[14]  V. Marčenko,et al.  DISTRIBUTION OF EIGENVALUES FOR SOME SETS OF RANDOM MATRICES , 1967 .

[15]  Pascal Frossard,et al.  Chebyshev polynomial approximation for distributed signal processing , 2011, 2011 International Conference on Distributed Computing in Sensor Systems and Workshops (DCOSS).

[16]  Stephen P. Boyd,et al.  A scheme for robust distributed sensor fusion based on average consensus , 2005, IPSN 2005. Fourth International Symposium on Information Processing in Sensor Networks, 2005..

[17]  Reza Olfati-Saber,et al.  Flocking for multi-agent dynamic systems: algorithms and theory , 2006, IEEE Transactions on Automatic Control.

[18]  Alejandro Ribeiro,et al.  Weak law of large numbers for stationary graph processes , 2017, 2017 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).

[19]  Alain Sarlette,et al.  Accelerating Consensus by Spectral Clustering and Polynomial Filters , 2017, IEEE Transactions on Control of Network Systems.

[20]  Hyunjoong Kim,et al.  Functional Analysis I , 2017 .

[21]  José M. F. Moura,et al.  Consensus state gram matrix estimation for stochastic switching networks from spectral distribution moments , 2017, 2017 51st Asilomar Conference on Signals, Systems, and Computers.

[22]  Soummya Kar,et al.  Finite-time distributed consensus through graph filters , 2014, 2014 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).

[23]  Santiago Segarra,et al.  Optimal Graph-Filter Design and Applications to Distributed Linear Network Operators , 2017, IEEE Transactions on Signal Processing.

[24]  R. Couillet,et al.  Random Matrix Methods for Wireless Communications: Estimation , 2011 .

[25]  Carlos Sagüés,et al.  Chebyshev Polynomials in Distributed Consensus Applications , 2011, IEEE Transactions on Signal Processing.

[26]  George Cybenko,et al.  Dynamic Load Balancing for Distributed Memory Multiprocessors , 1989, J. Parallel Distributed Comput..

[27]  E. Wigner.  On the Distribution of the Roots of Certain Symmetric Matrices , 1958 .

[28]  José M. F. Moura,et al.  Graph signal processing: Filter design and spectral statistics , 2017, 2017 IEEE 7th International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP).