Abstract: Random matrix theory (RMT) is a tool to study complex quantum systems. In this talk I will present the basic idea of RMT, and how it can be used to study open quantum systems and present some of our results concerning decoherence. In particular, we shall show how purity decay can be predicted with this tool, show that generic quantum open systems display non-markovian behaviour and how this behaviour is smeared out when the coupling between the system of interest and the environment is weak. The transition from non-Markovian to Markovian dynamics for generic environments , Phys. Rev. A 93, 012113 (2016) Random density matrices versus random evolution of open systems, J. Phys. A: Math. Theor. 48 425005 (2015) A random matrix theory of decoherence, New J. Phys. 10, 115016 (2008)

Abstract: We discuss the average relaxation properties of the internal memory in models of pure memristive networks. We consider the simplest linear model of memristor unit and introduce a dynamical equation for the evolution of internal memory in terms of projection operators. We find that for the case of passive components the dynamics is described by an orthogonal projection operator and for the case of active elements by a non-orthogonal projector. We analyze the average properties of internal memory parameters for random projection operators, and find that this is well described by a slow relaxation evolution if no active components are present. We provide a simple explanation for the emerging slow relaxation as a superposition of exponential relaxations with broad time scales range.

Abstract: Many online collaboration networks struggle to gain user activity and become self-sustaining due to the ramp-up problem or dwindling activity within the system. Prominent examples include online encyclopedias such as (Semantic) MediaWikis, Question and Answering portals such as StackOverflow, online ontology editors and repositories, such as WebProtégé or BioPortal, and many others. Only a small fraction of these systems manage to reach self-sustaining activity, a level of activity that prevents the system from reverting to a non-active state. In this paper, we model and analyze activity dynamics in synthetic and empirical collaboration networks. Our approach is based on two opposing and well-studied principles: (i) without incentives, users tend to lose interest to contribute and thus, systems become inactive, and (ii) people are susceptible to actions taken by their peers (social or peer influence). With the activity dynamics model that we introduce in this paper we can represent typical situations of such collaboration networks. For example, activity in a collaborative network, without external impulses or investments, will vanish over time, eventually rendering the system inactive. However, by appropriately manipulating the activity dynamics and/or the underlying collaboration networks, we can jump-start a previously inactive system and advance it towards an active state. To be able to do so, we first describe our model and its underlying mechanisms. We then provide illustrative examples of empirical datasets and characterize the barrier that has to be breached by a system before it can become self-sustaining in terms of critical mass and activity dynamics. Additionally, we expand on this empirical illustration and introduce a new metric p—the Activity Momentum—to assess the activity robustness of collaboration networks. Full paper: http://dl.acm.org/citation.cfm?id=2873060

Abstract: The competition for the attention of users is a central element of the Internet. Crucial issues are the origin and predictability of big hits, the few items that capture a big portion of the total attention. We address these issues analyzing 10 million time series of videos’ views from YouTube. We find that the average gain of views is linearly proportional to the number of views a video already has, in agreement with usual rich-get-richer mechanisms and Gibrat’s law, but this fails to explain the prevalence of big hits. The reason is that the fluctuations around the average views are themselves heavy tailed. Based on these empirical observations, we propose a stochastic differential equation with Lévy noise as a model of the dynamics of videos. We show how this model is substantially better in estimating the probability of an ordinary item becoming a big hit, which is considerably underestimated in the traditional proportional-growth models.

Abstract: At first, we say that a ego-community structure is a probability measure defined on the set of network nodes. Any subset of nodes may engender its own ego-community structure around. Many community detection algorithms can be modified to yield a result of this type, for instance, the personalized pagerank. Next, we present a continuous version of Viard-Latapy-Magnien link streams, that we call "temporal density". Classical kernel density estimation is used to move from discrete link streams to their continuous counterparts. Using matrix perturbation theory we can prove that ego-community structure changes smoothly when the network evolves smoothly. This is very important, for example, for visualization purposes. Combining the temporal density and personalized pagerank methods, we are able to visualize and study the evolution of the ego-community structures of complex networks with a large number of temporal links. We illustrate and validate our approach using "Primary school temporal network data" provided by sociopatterns.org, and we show how the temporal density can be applied to the study of very large datasets, such as a collection of tweets written by European Parliament candidates during European Parliament election in 2014. Main Topic: Foundations of Complex Systems Sub Topic: Social networks

Abstract: Many (experimental) sciences don't work or assume actual infinity. Finitely Supported Mathematics (FSM) is introduced as a mathematics dealing with a more relaxed notion of (in)finiteness. FSM has strong connections with the Fraenkel-Mostowski (FM) permutative model of Zermelo-Fraenkel set theory with atoms. However, FSM can characterize infinite algebraic structures using their finite supports. More exactly, FSM is ZF mathematics rephrased in terms of finitely supported structures by using an infinite set of atoms. In FSM, 'sets' are replaced either by `invariant sets' (sets endowed with some group actions satisfying a finite support requirement) or by `finitely supported sets' (finitely supported elements in the powerset of an invariant set), and developed a theory of `invariant algebraic structures'. We describe FSM by using principles (rather than axioms), and the principles of constructing FSM have historical roots both in the definition of Tarski `logical notions' and in the Erlangen Program of F.Klein for the classification of geometries according to invariants under suitable groups of transformations. There exist other connections between FSM, admissible sets and Gandy machines. The main principle of constructing FSM is that all the structures have to be invariant or finitely supported. As a consequence, we cannot obtain a property in FSM only by involving a ZF result without an appropriate proof reformulated according to the finite support requirement. Moreover, not every ZF result can be directly reformulated in terms of finitely supported objects because, given an invariant set, some of its subsets might be non-finitely supported (an example is given by a simultaneously infinite and coinfinite subset of the invariant set of all atoms). We have specific techniques of reformulating ZF properties of algebraic structures in FSM. More details are presented in the papers published by the authors in the last 2 years.

Abstract: We study the problem of determining all possible asymptotic dynamics of Boolean Networks (BNs) such as Discrete Hopfield Nets, Sequential and Synchronous Dynamical Systems, and (finite) Cellular Automata. Viewing BNs as an abstraction for a broad variety of decentralized cyber-physical, biological, social and socio-technical systems, we discuss similarities and differences between open vs. closed such decentralized systems, in an admittedly simplified but rigorous mathematical setting. We revisit the problem of enumerating all possible dynamical evolutions of a large-scale decentralized complex system abstracted as a BN. We show that, in general, the problem of enumerating possible dynamics is provably computationally hard for both "open" and "closed" variants of BNs, even when all of the following restrictions simultaneously hold: i) the local behaviors (that is, node update rules) are very simple, monotone Boolean-valued functions; ii) the network topology is sparse; and iii) either there is no external environment impact on the system modeled as a BN (this case captures "closed systems"), or the model of the environment and how it influences individual nodes in the BN, is of a rather simple, deterministic nature. Our results should be viewed as lower bounds on the complexity of possible behaviors of "the real" large-scale cyber-physical, biological, social and other decentralized systems, with some far-reaching implications insofar as (un)predictability of possible dynamics of such systems.

Abstract: Networks are the skeleton that support dynamical processes in complex systems. Links in many real-world networks activate and deactivate in correspondence to the sporadic interactions between the elements of the system. Activation patterns may be irregular or bursty and play an important role on the dynamics of processes taking place in the network. Most of recent results point towards a delay of these processes due to the interplay between topology and link activation. Social networks and information or disease spreading processes are paradigmatic examples of this situation. Besides the burstiness, several correlations may appear in the process of link activation: Memory effects imply temporal correlations and the existence of communities in the network may mediate the activation patterns of internal an external links. Here, we study how these different types of correlations influence dynamical systems on the networks. As paradigmatic examples, we consider the SI spreading and the voter model on networks. As noted in the literature, the relation between topology and activation leads to a delay in the dynamics. However, we find that memory effects can notably accelerate the models' arrival at the absorbing states. A theoretical explanation about how this phenomenon occurs is provided. Furthermore, we show that when both types of correlations are present, the final dynamics crucially depends on the mix. The characteristic times of the dynamics suffers a divergence for some particular correlation combinations. Some mixes between topology and memory notably speed up the dynamics, while others strongly slow it down. Mixed correlations, topological and memory effects, are commonly present in any real system, so understanding their non-trivial competition is of great importance. In this sense, the SI and voter models are simple benchmark dynamics, but we expect our results to be generalizable to more elaborated dynamical processes. The complete work is available in https://arxiv.org/abs/1604.04155

Abstract: We introduce a new type of evolution equation for 1-dimensional complex systems where the order of differentiation is itself one of the variables. We show that such ultra-fast growing systems with evolution determined by the variable order of differentiation can be mapped into a fractional differential equation and further into a Volterra integral equation. We elaborate on the existence and stability of the evolution solutions for various initial conditions and we present several case studies. The core of this approach is related to the observational connection between the evolution of the degree of complexity, and the rate of accelerated change on one hand, and the degree of time non-locality (history dependent) of the model equation, on the other hand. Since the latter quantity is connected to the number of neighbors or steps taken into account in discretized models, it results the need for a new type of equation whose order of differentiation changes in a dynamical way. We present applications of this approach in: nonlinear evolution equations for long-term memory systems, [1], fast growing computer/internet systems (e.g. Kryder’s or Nielsen's laws), [2], and accelerating change systems like populations (e.g. Reed’s and Carlson’s laws, Ribeiro model). We also present some novel applications developed from this model on cell growing, phase transitions, and avalanches. References [1] Spectral decomposition of nonlinear systems with memory A. Svenkeson, B. Glaz, S. Stanton, and B. J. West, Phys. Rev. E 93 (2016) 022211. [2] A. Ludu, Boundaries of a Complex World (Springer-Verlag, Heidelberg 2016).

Abstract: In academia, nepotism has been blamed for poor graduate career support, gender inequality, and emigration of the intelligentsia. To support this idea Allesina reported an unnatural scarcity of distinct surnames among tenured faculties in Italy while Ferlazzo and Sdoia repeated the same analysis in the UK, finding a more objective expression of social capital. Albeit with very careful consideration of surnames’ distributions across regions and time, surname clustering can be used to reflect family ties or kinship, and interpreted in relation to measures of social capital (including corruption, income inequality, scientific output). Here, we examine co-authorship patterns in the health science literature over five decades, by country using over 21 million papers indexed by the MEDLINE®/PubMed® database. Our analysis shows that kinship in the health literature has increased over the past fifty years with substantial differences between nations. I.e. Italy and Poland exhibited a dramatic increase in kinship starting from very low values and crossing the overall trend in the early eighties. We also observed low kinship among countries with low perceived corruption, and an association with income inequality. Investigating the co-authorship network from top publishing countries, we found that authors who are part of a kin tend to have a larger degree and occupy central positions in network. We could interpret this as increased information flows and allied activities such as grant applications, emanating from influential individuals who are more commonly kin co-authors. Our results also highlight that the local structure of collaborations of a kin co-author is usually very centralized, while authors who are not part of a kin tend to create ‘democratic’ structures. Finally, the analysis of mixing patterns strongly supports the idea that important kin authors form robust collaborations among their peers while do not collaborate with scientists who are not part of a kin.