Abstract
We introduce the concept of network susceptibilities quantifying the response of the collective dynamics of a network to small parameter changes. We distinguish two types of susceptibilities: vertex susceptibilities and edge susceptibilities, measuring the responses due to changes in the properties of units and their interactions, respectively. We derive explicit forms of network susceptibilities for oscillator networks close to steady states and offer example applications for Kuramoto-type phase-oscillator models, power grid models, and generic flow models. Focusing on the role of the network topology implies that these ideas can be easily generalized to other types of networks, in particular those characterizing flow, transport, or spreading phenomena. The concept of network susceptibilities is broadly applicable and may straightforwardly be transferred to all settings where networks responses of the collective dynamics to topological changes are essential.
Highlights
INTRODUCTIONSusceptibility constitutes a key concept in physics, from statistical mechanics to condensed matter theory and experiments
We introduce the concept of network susceptibilities quantifying the response of the collective dynamics of a network to small parameter changes
Susceptibility constitutes a key concept in physics, from statistical mechanics to condensed matter theory and experiments
Summary
Susceptibility constitutes a key concept in physics, from statistical mechanics to condensed matter theory and experiments. While ideal solids are organized in the form of perfectly periodic crystals with, e.g., nearest-neighbor interactions, many natural and engineered complex systems are organized in networks with a rich variety of their underlying interaction topologies [1,2] The susceptibility of such a networked system, i.e., its response to changes in their parameters, is essentially determined by their topology. As a key class of network dynamics, we analyze the susceptibilities of oscillator networks describing the dynamics of various natural and manmade systems We define both vertex susceptibilities and edge susceptibilities to qualitatively and quantitatively distinguish the responses to changes of single-unit and single-interaction properties, respectively.
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