COMPLEX NETWORKS AND INFRASTRUCTURAL GRIDS
Institute for Complex Systems, CNR Italy and The London Institute for Mathematical SciencesElectric grids, telecommunication networks, railways, healthcare systems, financial circuits, etc. are infrastructures that are critical for functioning and the welfare of our countries. Most of such infrastructures – for historical reasons – have been developed and designed according to engineering paradigms that are staring to become inadequate to cope with their increasing complexity. Much of this complexity is simply due to increased system size: as statistical physics teaches us, collection of interacting objects exibit emergent phenomena (like phase transitions) that goes beyond to the properties of the single objects and have peculiar characteristics in the infinite size limit. Moreover, the increase of interdependencies among the infrastructures (think as an example of the interdependence among communication networks and electric grids) is adding a further elementcomplexity. Hence, the statistical physics’ approach can enlarge the understanding of the fragilities and vulnerabilities of such critical infrastructures.
In these lectures, we will cover some of the current models of infrastructural grids – both isolated and coupled – hinting out the possible and needed development of the field. In the first lecture, will first start introducing the constitutive equations for gas/oil pipelines and for electric grids Acha . We will then describe some applications of the fiber-bundle model Peires  and of the cavity method Mézard and Parisi  to understanding cascading failures in trasmission power grids Pahwa et al.  and limiting such failures in distribution power grids by introducing self-healing capabilities Quattrociocchi et al. . In the second lecture, we will focus on interating networked infrastructure Rinaldi et al. , D’Agostino and Scala ; we will cover a whole range of models, from the first abstract models of coupled cascading systems Newman et al. , Carreras et al. , Buldyrev et al.  to a recent realistic how energy from renewable sources affect network and markets Mureddu et al. .
For more details see:
 S. Acha. Modelling distributed energy resources in energy service networks. IET Digital Library, 2013.
 F.T. Peires. J. Textile Inst. 17 (1926) 355-368.
 M. Mezard, G. Parisi. The cavity method at zero temperature. J. Stat. Phys. 111 (2003) 1.
 S. Pahwa, C. Scoglio, A. Scala. Abruptness of cascade failures in power grids. Sci. Rep. 4 (2014) 3694.
 W. Quattrociocchi, G. Caldarelli, A. Scala. Self-healing networks: redundancy and structure. PLoS ONE, 9 (2014) 87986.
 S.M. Rinaldi, J.P. Peerenboom, T.K. Kelly. Identifying, understanding, and analyzing critical infrastructure interdependencies. IEEE. Contr. Syst. Mag. 21 (2001) 11-25.
 G. D’Agostino, A. Scala. Networks of networks: the last frontier of complexity, understanding complex systems. Springer International Publishing, 2014.
 D. Newman, et al. In: System sciences, 2005. HICSS ’05. Proceedings of the 38th Annual Hawaii International Conference on (2005), pp. 63c–63c.
 B.A. Carreras, et al. In: Proceedings of the 40th Annual Hawaii International Conference on System Sciences (IEEE Computer Society, Washington, DC, USA, 2007), HICSS ’07, pp. 112.
 S.V. Buldyrev,et al. Catastrophic cascade of failures in interdependent networks. Nature, 464 (2010) 1025.
 M. Mureddu, et al. PLoS ONE, Public Library of Science, 10 (2015) e0135312.