**Organisers:** Joan Saldaña (UdG)

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Friday Oct 23, 10:30 – 12:40, Zoom room *Pi i Sunyer*

*Pi i Sunyer*

10:30 – 11:10 – **Josep Sardanyés** (Centre de Recerca Matemàtica)

*“Dynamics of cooperation”*

In this talk we will introduce the hypercycle model, originally conceived by Manfred Eigen and Peter Schuster to study the dynamics of prebiotic replicators. Hypercycles are dynamical systems formed by replicators with catalytic activity, thus they have been also employed to investigate cooperation in complex ecosystems at different levels. Following this mathematical model, we will show the dynamics and bifurcations tied to cooperation, from origins of life to models of facilitation in metapopulations and dynamics of semiarid ecosystems. We will focus on the dynamics of plants facilitation and the presence of different tipping points depending on plants dispersion ranges.

11:15 – 11:55 – **Marián Boguñá** (Universitat de Barcelona)

*“Non-Markovian dynamics in SIS epidemic models”*

Albeit epidemic models have evolved into powerful predictive tools for the spread of diseases and opinions, most assume memoryless agents and independent transmission channels. We develop an infection mechanism that is endowed with memory of past exposures and simultaneously incorporates the joint effect of multiple infectious sources. Analytic equations and simulations of the susceptible-infected-susceptible model in unstructured substrates reveal the emergence of an additional phase that separates the usual healthy and endemic ones. This intermediate phase shows fundamentally distinct characteristics, and the system exhibits either excitability or an exotic variant of bistability. Moreover, the transition to endemicity presents hybrid aspects. These features are the product of an intricate balance between two memory modes and indicate that non-Markovian effects significantly alter the properties of spreading processes.

12:00 – 12:40 – **Lucas Lacasa** (Queen Mary University of London)

*“Identifying the hidden multiplex architecture of complex systems (aka a new decomposition theory of non-Markovian dynamics)”*

Elements composing complex systems usually interact in several different ways and as such the interaction architecture is well modelled by a network with multiple layers –a multiplex network–. However only in a few cases can such multi-layered architecture be empirically observed, as one usually only has experimental access to such structure from an aggregated projection. A fundamental challenge is thus to determine whether the hidden underlying architecture of complex systems is better modelled as a single interaction layer or results from the aggregation and interplay of multiple layers. Assuming a prior of intralayer Markovian diffusion, in this talk I will present a method [1] that, using local information provided by a random walker navigating the aggregated network, is able possible to determine in a robust manner whether these dynamics can be more accurately represented by a single layer or they are better explained by a (hidden) multiplex structure. In the latter case, I will also provide a Bayesian method to estimate the most probable number of hidden layers and the model parameters, thereby fully reconstructing its hidden architecture. The whole methodology enables to decipher the underlying multiplex architecture of complex systems by exploiting the non-Markovian signatures on the statistics of a single random walk on the aggregated network. In fact, the mathematical formalism presented here extends above and beyond detection of physical layers in networked complex systems, as it provides a principled solution for the optimal decomposition and projection of complex, non-Markovian dynamics into a Markov switching combination of diffusive modes. I will validate the proposed methodology with numerical simulations of both (i) random walks navigating hidden multiplex networks (thereby reconstructing the true hidden architecture) and (ii) Markovian and non-Markovian continuous stochastic processes (thereby reconstructing an effective multiplex decomposition where each layer accounts for a different diffusive mode). I will also state two existence theorems guaranteeing that an exact reconstruction of the dynamics in terms of these hidden jump-Markov models is always possible for arbitrary finite-order Markovian and fully non-Markovian processes. Finally, using experiments, I will apply the methodology to understand the dynamics of RNA polymerases at the single-molecule level. [1] L. Lacasa, I.P. Mariño, J. Miguez, V. Nicosia, E. Roldan, A. Lisica, S.W. Grill, J. Gómez-Gardeñes, Multiplex decomposition of non-Markovian dynamics and the hidden layer reconstruction problem Physical Review X 8, 031038 (2018)

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Saturday Oct 24, 15:00 – 17:10, Zoom room *Pi i Sunyer*

*Pi i Sunyer*

15:00 – 15:40 – **Tiago Peixoto** (Central European University)

*“Network reconstruction from indirect observations”*

The observed functional behavior of a wide variety of large-scale systems is often the result of a network of pairwise interactions. However, in many cases these interactions are hidden from us, either because they are impossible or very costly to be measured directly, or, in the best case, are measured with some degree of uncertainty. In such situations, we are required to infer the network of interactions from indirect information. In this talk, I present a scalable Bayesian method to perform network reconstruction from indirect data, including noisy measurements and observed network dynamics. This kind of approach allows us to convey in a principled manner the uncertainty present in the measurement, and combined with versatile modelling assumptions can yield good results even when data are scarce. In particular, I describe how the reconstruction approach can be combined with community detection, allowing us to tap into multiple sources of evidence available for the task. I show how this combined approach provides a twofold improvement, by increasing not only the reconstruction accuracy, but also the identification of communities in networks. The latter improvement is possible even in situations where at first we might imagine that reconstruction is superfluous, for example when direct network data are available and measurement errors can be neglected.

15:45 – 16:25 – **Caterina Scoglio** (Kansas State University)

*“A multilayer temporal network model for STD “*

Sexually transmitted diseases (STD) modeling has used contact networks to study the spreading of pathogens. Recent findings have stressed the increasing role of casual partners, often enabled by online dating applications. We study the Susceptible-Infected-Susceptible (SIS) epidemic model –appropriate for STDs– over a two-layer network aimed to account for the effect of casual partners in the spreading of STDs. In this novel model, individuals have a set of steady partnerships (links in layer 1). At certain rates, every individual can switch between active and inactive states and, while active, it establishes casual partnerships with some probability with active neighbors in layer 2 (whose links can be thought as potential casual partnerships). Individuals that are not engaged in casual partnerships are classified as inactive, and the transitions between active and inactive states are independent of their infectious state. We use mean-field equations as well as stochastic simulations to derive the epidemic threshold, which decreases substantially with the addition of the second layer. Interestingly, for a given expected number of casual partnerships, which depends on the probabilities of being active, this threshold turns out to depend on the duration of casual partnerships: the longer they are, the lower the threshold.

16:30 – 17:10 – **Markus Kirkilionis** (University of Warwick)

*“Towards a New Multi-Scale And Multi-Relational Network Theory”*

We introduce the mathematical concept of a hierachical hypergraph, as an ordinary hypergraph plus a discrete linear or partial order based on the natural numbers ℕ. This structural concept is immensely important when dealing with the mathematical abstraction of ‘composition’ in general. Ordinary graphs, as used in network theory, have two major defects: (i) they are based on binary relationships, and (ii) have no multi-scale structure, i.e. all the vertices (nodes) are assumed to be positioned on the same level of system description. But the idea of composition, or equivalently, group membership, is central to most real-world systems. Mathematically, problem (i) can be solved by hypergraphs, as they are able to describe arbitrary n-relations. However, the multi-scale composition problem (ii) needs a further ingredient, and this is an order relation between vertices of the hypergraph. We introduce and investigate the mathematical structure of this idea, the hierachical hypergraph and discuss applications in all areas of science.