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# Controllability, Identification, and Randomness in by Marzieh Nabi-Abdolyousefi

By Marzieh Nabi-Abdolyousefi

This interdisciplinary thesis contains the layout and research of coordination algorithms on networks, id of dynamic networks and estimation on networks with random geometries with implications for networks that help the operation of dynamic structures, e.g., formations of robot autos, allotted estimation through sensor networks. the consequences have ramifications for fault detection and isolation of large-scale networked structures and optimization versions and algorithms for subsequent new release plane energy structures. the writer unearths novel functions of the method in power structures, resembling residential and commercial shrewdpermanent strength administration systems.

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Where d(G) is the diameter of the graph and κ(G) = (1/n) i=2 Although the spectra of the Laplacian provides important insights, as noted above, into the structural properties of the network, we now proceed to explore the possibility of complete identification of the underlying network using its graph spectra complemented with a sieve method. 2 Graph Characterization via Graph Sieve In this section, we provide an overview of the graph sieve procedure—that in conjunction with the identified Laplacian spectra—leads to a more confined search for the network in the black box.

Define the rightmost adjacency set A R (i) containing the di largest index nodes different from i. t. {i, j} ∈ / E } as the forbidden neighbors of node i. Note that X (i) originally might contain some nodes according to the forbidden set X (d). Create the set A R (1) and X (1) for node 1: connect node 1 to n (this never breaks graphicality). Set X (1) = {n}. Define the new sequence d = {d1 − |X (1)|, d2 , . . , dn }. Let k = n − 1. 1. Connect another edge of 1 to k. 3. 2. 1. 3. If the test passes, keep (save) the connection, add the node k to the forbidden set X (1) and update the degree sequence d = {d1 − |X (1)|, d2 , .

1. Let the number of partitions of rm into n − r˜ integers between 1 and n − 1 be denoted by Pn−˜r (rm ). Then Pn−˜r (rm ) = Pn−˜r −1 (rm − 1) + (n − r˜ )Pn−˜r (rm − 1). 3). The partitioning of rm into m components can be generated in increasing lexicographic order by starting with d1 = d2 = . . = dm−1 = 1, dm = rm − m + 1 and continuing as follows. To obtain the next partition from the first one, scan the elements from right to left, stopping at the right most di such that dm − di ≥ 2. Replace d j by di + 1 for j = i, i + 1, .