Period abroad at NaXys, Namur Institute for Complex Systems, University of Namur, Sara Nicoletti – June-August 2019

I spent three months of my PhD in Belgium at the University of Namur. I collaborated with Professor Timoteo Carletti, member of the scientific board of NaXys, research institute for Complex Systems (www.naxys.be). Professor Carletti is my coauthor in the two first papers that I published during my PhD and, in the latest three months, we worked to another paper that I’m going to publish. My research project is devoted, in general, to complex networks an their applications, in particular to reaction-diffusion models where we have two contributions, one by the reaction-diffusion equations and the other one by the topology of the networks, characterized by the Laplacian matrix. In my first two papers I studied some specific topologies of networks in order to investigate the well-kown effect of non-normality in the considered models. In a different approach we are now trying to create a network with specific diffusive properties, in particular a Laplacian matrix with a desired spectrum. The procedure is based on the eigenvalue decomposition: by fixing a set of eigenvectors we are able to choose any desired complex spectrum for the Laplacian matrix (the real spectrum case is discussed in [1]). The networks that we obtain have some interesting properties, for example are fully connected and some links have negative weights (see Figure 1). Because of the nature of the connections we perform a sparsification procedure that permits to preserve the structure of the adjacency matrix and to keep the Laplacian eigenvalues almost fixed, that is in a region that surrounds the original ones. The idea of this method is simply to cut links with small weights and it allows to obtain a connected network with an half of zeros links. In order to take advantage of the presence of negative links in these networks we are thinking to some application to models where signed Laplacian are often used, like for example consensus models with antagonistic interactions or, more interestingly, microgrids [2, 3].

Bibliography

[1] A. Forrow, F. G. Woodhouse, J. Dunkel, Functional Control of Network Dynamics Using Designed Laplacian SpectraPhys. Rev. X 8, 041043 (2018).

[2] Song Y., Hill D. J. and Liu T.,Small-Disturbance Angle Stability Analysis of Microgrids: A Graph Theory Viewpoint 2015 IEEE Conference on Control Applications (CCA), Part of 2015 IEEE Multi-Conference on Systems and Control, September 21-23, 2015. Sydney, Australia

[3] Chen W., Wang D., Liu J., Başar T., Johansson K. H., Qiu L., On Semidefiniteness of Signed Laplacians with Application to Microgrids IFAC-PapersOnLine 49-22 (2016) 097–102