Physica A 404 (2014) 92–105 Contents lists available at ScienceDirect Physica A journal homepage: www.elsevier.com/locate/physa Algebraic connectivity of interdependent networks J. Martín-Hernández a,∗ , H. Wang a , P. Van Mieghem a , G. D’Agostino b a Faculty of Electrical Engineering, Mathematics and Computer Science, P.O Box 5031, 2600 GA Delft, The Netherlands b ENEA Centro Ricerche Casaccia, via Anguillarese 301, I-00123 Roma (RM), Italy highlights • We prove that the spectra of interdependent graphs experience a phase transition. • The transition is characterized by the saturation of diffusion processes. • An expression for the transition is found as a function of a coupling constant. • The transition point depends on the network types and the link addition strategy. article info Article history: Received 7 August 2013 Received in revised form 12 November 2013 Available online 1 March 2014 Keywords: Network of networks Synchronization Laplacian Spectral properties System of systems abstract The algebraic connectivity μ N−1 , i.e. the second smallest eigenvalue of the Laplacian matrix, plays a crucial role in dynamic phenomena such as diffusion processes, synchronization stability, and network robustness. In this work we study the algebraic connectivity in the general context of interdependent networks, or network-of-networks (NoN). The present work shows, both analytically and numerically, how the algebraic connectivity of NoNs experiences a transition. The transition is characterized by a saturation of the algebraic connectivity upon the addition of sufficient coupling links (between the two individual networks of a NoN). In practical terms, this shows that NoN topologies require only a fraction of coupling links in order to achieve optimal diffusivity. Furthermore, we observe a footprint of the transition on the properties of Fiedler’s spectral bisection. © 2014 Elsevier B.V. All rights reserved. 1. Introduction In the last decades, there has been a significant advance in understanding the structure and functioning of complex networks [1,2]. Statistical models of networks are now widely used to describe a broad range of complex systems, from networks of human contacts to interactions amongst proteins. In particular, emerging phenomena of a population of dynamically interacting units has always fascinated scientists. Dynamic phenomena are ubiquitous in nature and play a key role within various contexts in sociology [3], and technology [4]. To date, the problem of how the structural properties of a network influences the convergence and stability of its synchronized states has been extensively investigated and discussed, both numerically and theoretically [5–9], with special attention given to networks of coupled oscillators [10–13]. In the present work, we focus on the second smallest eigenvalue μ N−1 of a graph’s Laplacian matrix, also called algebraic connectivity. This metric plays an important role on, among others, synchronization of coupled oscillators, network robustness, consensus problems, belief propagation, graph partitioning, and distributed filtering in sensor networks [14–18]. For example, the time it takes to synchronize Kuramoto oscillators upon any network scales with the inverse of μ N−1 [19–22]. In other words, larger values of μ N−1 enable synchronization in both discrete and continuous-time systems, even in the presence of transmission delays [23,24]. As a second application, graphs with ‘‘small’’ algebraic connectivity ∗ Corresponding author. Tel.: +31 152782132. E-mail addresses: j.martinhernandez@tudelft.nl, Javier.Martin.Hernandez@gmail.com (J. Martín-Hernández), h.wang@tudelft.nl (H. Wang), p.f.a.vanmieghem@tudelft.nl (P. Van Mieghem), gregorio.dagostino@enea.it (G. D’Agostino). http://dx.doi.org/10.1016/j.physa.2014.02.043 0378-4371/© 2014 Elsevier B.V. All rights reserved.