Foundations (F) Session 1
Time and Date: 14:15  15:45 on 19th Sep 2016
Room: M  Effectenbeurszaal
Chair: Sarah de Nigris
123  Driftinduced BenjaminFeir instabilities
[abstract] Abstract: The spontaneous ability of spatially extended systems to selforganize 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 socalled modulational instability deviations from a periodic waveform are reinforced by nonlinearity, leading to spectralsidebands 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 BenjaminFeir instability. The BenjaminFeir instability has been later on discussed in the context of the Complex GinzburgLandau equation, a quintessential model for non linear physics. Here we revisit the BenjaminFeir instability in the framework of the Complex GinzburgLandau 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 BenjaminFeir instabilities occur, stimulated by the drift, outside the region of parameters for which the classical BenjaminFeir 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.

Francesca Di Patti, Duccio Fanelli and Timoteo Carletti 
133  On the Accuracy of Timedependent NIMFA Prevalence of SIS Epidemic Process
[abstract] Abstract: The NIntertwined Mean Field Approximation (NIMFA) exponentially reduces the number of differential equations that need to be calculated in the Markovian susceptibleinfectedsusceptible (SIS) model in networks [1]. The nonzero 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 dieout 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 dieout probability influences the accuracy of timedependent NIMFA prevalence in networks and a novel correction function for NIMFA has been found. Furthermore, the virus dieout 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.

Qiang Liu and Piet Van Mieghem 
36  The effect of hidden source on the estimation of complex networks from time series
[abstract] Abstract: Many methods have been developed to estimate the interdependence (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 nonobserved 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.

Dimitris Kugiumtzis and Christos Koutlis 
447  Messagepassing algorithms in networks and complex system
[abstract] Abstract: We will sketch an algorithmic take, i.e. messagepassing algorithms, on networks and its relevance to some questions and insights in complex systems. Recently, messagepassing 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 twofold. On on hand, we will discuss how the nonbacktracking nature of messagepassing avoids an “echochamber 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.

Munik Shrestha 
555  Hypernetworks: multilevel multidimensional multiplex networks
[abstract] 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 1dimensional edge, is a 2dimensional triangle, is 3dimensional tetrahedron, and … with (p+1) vertices being a pdimensional 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 nary 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.

Jeffrey Johnson 