Abstract: The spontaneous ability of spatially extended systems to self-organize in space and time is proverbial and has been raised to paradigm in modern science. Collective behaviors are widespread in nature and mirror, at the macroscopic level, the microscopic interactions at play among elementary constituents. Convection instabilities in fluid dynamics, rhythms production and insect swarms are representative examples that emblematize the remarkable capacity of physical and biological systems to yield coherent dynamics. Instabilities triggered by random fluctuations are often patterns precursors. The imposed perturbation shakes e.g. an homogeneous equilibrium, seeding a resonant amplification mechanism that eventually materializes in magnificent patchy motifs, characterized by a vast gallery of shapes and geometries. Exploring possible routes to pattern formation, and unraveling novel avenues to symmetry breaking instability, is hence a challenge of both fundamental and applied importance. In the so-called modulational instability deviations from a periodic waveform are reinforced by nonlinearity, leading to spectral-sidebands and the breakup of the waveform into a train of pulses. The phenomenon was first conceptualized for periodic surface gravity waves on deep water by Benjamin and Feir, and for this reason it is customarily referred to as the Benjamin-Feir instability. The Benjamin-Feir instability has been later on discussed in the context of the Complex Ginzburg-Landau equation, a quintessential model for non linear physics. Here we revisit the Benjamin-Feir instability in the framework of the Complex Ginzburg-Landau equation, modified with the inclusion of a drift term. This latter is rigorously derived from a stochastic description of the microscopic coupling between adjacent oscillators. Generalized Benjamin-Feir instabilities occur, stimulated by the drift, outside the region of parameters for which the classical Benjamin-Feir instability is manifested. This observation, grounded on a detailed mathematical theory, contributes to considerably enrich the landscape of known instabilities, along a direction of investigation that can be experimentally substantiated.

Abstract: The N-Intertwined Mean Field Approximation (NIMFA) exponentially reduces the number of differential equations that need to be calculated in the Markovian susceptible-infected-susceptible (SIS) model in networks [1]. The non-zero steady state of NIMFA has been proved globally asymptotically stable when the number of initially infected nodes is larger than 0 and the effective spreading rate is above the NIMFA epidemic threshold [2,3]. Moreover, an accuracy criterion in the steady state has been derived in [4]. However, the accuracy of NIMFA is also determined by initial conditions of the epidemic process, which escaped the attention so far. We find that the virus die-out probability at an arbitrary time, which is determined both by the initial conditions and the network topology, impacts the accuracy of NIMFA. New results will be presented to show how the virus die-out probability influences the accuracy of time-dependent NIMFA prevalence in networks and a novel correction function for NIMFA has been found. Furthermore, the virus die-out probability is equivalent to the gambler’s ruin problem [5, page 231] in complete graphs and can also be solved numerically as a birth and death process in complete graphs. References: [1] P. Van Mieghem, J. Omic, and R. Kooij, IEEE/ACM Trans. on Networking, vol. 17, no. 1, pp. 1–14, Feb. 2009. [2] A. Khanafer, T. Başar, and B. Gharesifard, in 2014 American Control Conference, 2014, pp. 3579–3584. [3] S. Bonaccorsi, S. Ottaviano, D. Mugnolo, and F. Pellegrini, SIAM J. Appl. Math., vol. 75, no. 6, pp. 2421–2443, Jan. 2015. [4] P. Van Mieghem and R. van de Bovenkamp, Phys. Rev. E, vol. 91, no. 3, p. 032812, Mar. 2015. [5] P. Van Mieghem, Performance analysis of complex networks and systems. Cambridge: Cambridge University Press, 2014.

Abstract: Many methods have been developed to estimate the inter-dependence (called also coupling, information flow or Granger causality) between interacting variables of a dynamical system. Typically, this problem regards complex systems observed from multivariate time series, such as financial markets, climatic phenomena, brain activity and earthquakes. The methods tested so far are found to estimate the true underlying complex network with varying success. Here, a difficult but realistic setting of non-observed important variables (or subsystems) of the complex network is considered. In particular, the unobserved variables are assumed to be hidden sources of the complex network. The performance of different connectivity (Granger causality) measures on settings of unobserved source is evaluated with simulations on linear and nonlinear stochastic processes and dynamical systems of many variables. The results show that though some connectivity measures (of those tested) identify correctly the true complex network from the multivariate time series when no source is involved, all connectivity measures fail to estimate the true connections when a driving, but unobserved, hub is present in the true complex network. An example from finance is added to illustrate the problem.

Abstract: We will sketch an algorithmic take, i.e. message-passing algorithms, on networks and its relevance to some questions and insights in complex systems. Recently, message-passing algorithms have been shown to be an efficient, scalable approach to solve hard computational problems ranging from detecting community structures in networks to simulating probabilisitic epidemic dynamics on networks. The objective of the talk is two-fold. On on hand, we will discuss how the non-backtracking nature of message-passing avoids an “echo-chamber effects” of signal flow and thus makes a good tool to consider for problems in networks. On the other hand, we will also argue why insights gained from algorithms are equally important when exploring questions at the boundaries of scientific studies, such as networks and complex systems.

Abstract: Hypergraphs generalise graphs and networks by allowing edges to contain many vertices. An oriented hypergraph edge is a ‘simplex’. Simplices generalize oriented network edges: is a 1-dimensional edge, is a 2-dimensional triangle, is 3-dimensional tetrahedron, and … with (p+1) vertices being a p-dimensional polyhedron. Multiplex networks allow many relations to hold between sets of vertices. The ‘explicit relation’ notation can represent this: != . This generalizes to ‘hypersimplices’ with explicit relations on many vertices, != . Multiplex hypernetworks are formed from hypersimplices defined by many n-ary relations. Hypernetworks are foundational in a new theory of multilevel systems. The network family is integrated as follows (sorry if diagram doesn’t work): __________________orientation _____________________________many relations hypergraphs --------------> simplicial complexes ---------------> multiplex hypernetworks ^ . . . . . . . . . . . . . .__________. . . . . . . . ^ . . . . . . . . . . . . . __________. . . . . . . . . .^ | > 2 vertices. . . __________ . . . . . . . | > 2 vertices . . . . __________ . . . . . . . . . | > 2 vertices |. . . . . . . . . . . . . .__________ . . . . . . . . |. . . . . . . . . . . . . . .__________. . . . . . . . . | graphs -------------------------> networks -----------------------> multiplex networks __________________orientation _____________________________many relations It will be briefly explained how these structures contribute to understanding complexity.