Abstract
Complex networks are essentially heterogeneous not only in the basic properties of the constituent nodes, such as their degree, but also in the effects that these have on the global dynamical properties of the network. Networks of coupled identical phase oscillators are good examples for analyzing these effects, since an overall synchronized state can be considered a reference state. A small variation of intrinsic node parameters may cause the system to move away from synchronization, and a new phase-locked stationary state can be achieved. We propose a measure of phase dispersion that quantifies the functional response of the system to a given local perturbation. As a particular implementation, we propose a variation of the standard Kuramoto model in which the nodes of a complex network interact with their neighboring nodes, by including a node-dependent frustration parameter. The final stationary phase-locked state now depends on the particular frustration parameter at each node and also on the network topology. We exploit this scenario by introducing individual frustration parameters and measuring what their effect on the whole network is, measured in terms of the phase dispersion, which depends only on the topology of the network and on the choice of the particular node that is perturbed. This enables us to define a characteristic of the node, its functionability, that can be computed analytically in terms of the network topology. Finally, we provide a thorough comparison with other centrality measures.
Highlights
Synchronization has become one of the most paradigmatic examples of emergent properties in complex systems,1,2 since the degree of interaction between the oscillatory units of a discrete system shows that a variety of macroscopic states are available
We introduce a dynamic model based on FKM: the general frustrated Kuramoto model (GFKM), which enables us to characterize each node individually by means of an intrinsic frustration parameter
In order to assess the impact on the whole system of a node being perturbed, we make use of the GFKM described in Eq (6) and the effect measure defined in Eq (7)
Summary
Synchronization has become one of the most paradigmatic examples of emergent properties in complex systems, since the degree of interaction between the oscillatory units of a discrete system shows that a variety of macroscopic states are available. Highly functional nodes can be potentially dangerous in systems where tiny perturbations can produce cascadelike effects that completely disrupt the network dynamics. An example of this is offered by the transfer networks of power grids, which have been widely studied in the field of complex networks, focusing on their structure to assess the damage of failures.. III, we introduce a new oscillatory model based on the previous one: the general frustrated Kuramoto model (GFKM) as well as a measure to quantify the phase dispersion of the system when a perturbation is produced This model is the building block of our main contribution, that is, the definition of a new centrality measure: functionability. Further details concerning the validity of the model, as well as an example of a toy network, can be found in the Appendix
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