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Register a new userRecovering the Structural Observability of Composite Networks via Cartesian Product
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Mohammadreza Doostmohammadian
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Observability is a fundamental concept in system inference and estimation. This paper is focused on structural observability analysis of Cartesian product networks. Cartesian product networks emerge in variety of applications including in parallel and distributed systems. We provide a structural approach to extend the structural observability of the constituent networks (referred as the factor networks) to that of the Cartesian product network. The structural approach is based on graph theory and is generic. We introduce certain structures which are tightly related to structural observability of networks, namely parent Strongly-Connected-Component (parent SCC), parent node, and contractions. The results show that for particular type of networks (e.g. the networks containing contractions) the structural observability of the factor network can be recovered via Cartesian product. In other words, if one of the factor networks is structurally rank-deficient, using the other factor network containing a spanning cycle family, then the Cartesian product of the two nwtworks is structurally full-rank. We define certain network structures for structural observability recovery. On the other hand, we derive the number of observer nodes--the node whose state is measured by an output-- in the Cartesian product network based on the number of observer nodes in the factor networks. An example illustrates the graph-theoretic analysis in the paper.
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Let $G=(V,E)$ be a graph and $Gamma $ an Abelian group both of order $n$. A $Gamma$-distance magic labeling of $G$ is a bijection $ell colon Vrightarrow Gamma $ for which there exists $mu in Gamma $ such that $% sum_{xin N(v)}ell (x)=mu $ for all $vin V$, where $N(v)$ is the neighborhood of $v$. Froncek %(cite{ref_CicAus}) showed that the Cartesian product $C_m square C_n$, $m, ngeq3$ is a $mathbb{Z}_{mn}$-distance magic graph if and only if $mn$ is even. It is also known that if $mn$ is even then $C_m square C_n$ has $mathbb{Z}_{alpha}times mathcal{A}$-magic labeling for any $alpha equiv 0 pmod {{rm lcm}(m,n)}$ and any Abelian group $mathcal{A}$ of order $mn/alpha$. %cite{ref_CicAus} However, the full characterization of group distance magic Cartesian product of two cycles is still unknown. In the paper we make progress towards the complete solution this problem by proving some necessary conditions. We further prove that for $n$ even the graph $C_{n}square C_{n}$ has a $Gamma$-distance magic labeling for any Abelian group $Gamma$ of order $n^{2}$. Moreover we show that if $m eq n$, then there does not exist a $(mathbb{Z}_2)^{m+n}$-distance magic labeling of the Cartesian product $C_{2^m} square C_{2^{n}}$. We also give necessary and sufficient condition for $C_{m} square C_{n}$ with $gcd(m,n)=1$ to be $Gamma$-distance magic.
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