Abstract
Yang, Wang, and Motter [Phys. Rev. Lett. 109, 258701 (2012)] analyzed a model for network observability transitions in which a sensor placed on a node makes the node and the adjacent nodes observable. The size of the connected components comprising the observable nodes is a major concern of the model. We analyze this model in random heterogeneous networks with degree correlation. With numerical simulations and analytical arguments based on generating functions, we find that negative degree correlation makes networks more observable. This result holds true both when the sensors are placed on nodes one by one in a random order and when hubs preferentially receive the sensors. Finally, we numerically optimize networks with a fixed degree sequence with respect to the size of the largest observable component. Optimized networks have negative degree correlation induced by the resulting hub-repulsive structure; the largest hubs are rarely connected to each other, in contrast to the rich-club phenomenon of networks.
Highlights
In power-grid networks, the state of a node, i.e., the complex voltage, can be determined by so-called phase measurement units (PMUs; we call them sensors in the following)
We study the model of network observability transitions proposed in Ref. [3] in the case of random and degree-based placement of sensors in uncorrelated and correlated networks
We examined observability transitions in correlated networks
Summary
In power-grid networks, the state of a node, i.e., the complex voltage, can be determined by so-called phase measurement units (PMUs; we call them sensors in the following). The results were theoretically derived in the case of random placement of sensors and uncorrelated heterogeneous networks They proposed a heuristic algorithm to determine the order in which the nodes receive the sensors for efficient observation of networks with a community structure. In many network models with degree correlation, negative degree correlation makes the critical node or link occupation probability above which a giant component emerges large, which makes the network less robust [4,6,7,9] Motivated by these previous studies, in the present paper we examine the effect of degree correlation on observability transitions. Similar to the percolation problem, we focus on the size of the LOC, defined as the largest connected component composed of directly or indirectly observable nodes
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