Consensus Building in Sensor Networks and Long Term Planning for the National Airspace System Page: 19
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Hence, the consensus value is given by x(oo)= [(xl(0) + 2x2(0) + x3(0) +x4(0)) , which
matches the result.
2.3. Dependence of Convergence Time on Network Structure
Convergence time is an important performance measure for consensus building strategies.
In this section, the convergence time is related to the second largest eigenvalue of the system
matrix A. The eigen-analysis provides an approach to directly infer convergence time from
some very simple structural characteristics of a DSN, such as the number and degrees of
sensor nodes and VFCs, etc.
From the last section, the dominant eigenvalue of system matrix A was proved to be
X1 = 1 and its associated right eigen vector vl = lxl. To find other eigenvalues of A,
Courant Fischer theory along with the properities of spectral radius are used. The concept of
spectral radius provides the dependence of convergence time on X2 whereas Courant Fischer
theory concepts are used to relate ,2 directly from its structure through routing matrix H.
2.3.1. Dependence of Convergence Time on A2
Theorem 4: Consider a DSN represented by a tanner graph with routing matrix H.
Supposing that the DSN can reach consensus, the convergence time (i.e., the number of
iterations such that the difference between sensor value and final consensus value is within 6 of
its initial value) is Ioga26 as 6 - 0, where X 2 0 is the second largest eigenvalue associated
with the system matrix A.
Proof- From theorem 2, the values of sensor nodes converge to xf r viwix[O]. There is
only one eigenvalue with value 1, and all the other eigenvalues are real and have magnitude
less than 1.
The case that A2 = 0 is trivial in that the consensus can be reached in one iteration. Lets
focus on the case the A2 0.19
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Akula, Naga Venkata Swathik. Consensus Building in Sensor Networks and Long Term Planning for the National Airspace System, thesis, May 2011; Denton, Texas. (https://digital.library.unt.edu/ark:/67531/metadc67954/m1/25/: accessed April 18, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; .