Critical and collective effects in graphs and networks - 2019

Since the scientific revolution, interpretable closed-form mathematical models have been instrumental for advancing our understanding of the world. Think, for example, of Newton’s law of gravitation, and how it has enabled us to predict astronomical phenomena with great accuracy and to gain deep insights about seemingly unrelated physical phenomena. With the data revolution, we may now be in a position to uncover new mathematical models for many systems, from physics and chemistry to the social sciences. However, to deal with increasing amounts of data, we need approaches that are able to extract these models automatically, without supervision, and with guarantees of asymptotically finding the correct model. In this talk I will review standard machine learning approaches and discuss their limitations in terms of getting interpretable models. Then, I will present a Bayesian "machine scientist" that deals rigorously with model plausibilities and  also explores systematically the space of models, using the analogy between Bayesian inference, information theory, and statistical mechanics. The machine scientist is able to obtain closed-form mathematical models from data, and to make out-of-sample predictions that are more accurate than those of standard machine learning approaches.

Symmetry breaking--the phenomenon in which the symmetry of a system is not inherited by its stable states--underlies pattern formation, superconductivity, and numerous other effects. In this talk, I will report on the recently established possibility of converse symmetry breaking, an emergent network phenomenon in which the stable states are symmetric only when the system itself is not. In particular, I will present an experimental demonstration of this phenomenon as well as concrete applications to network optimization and control. The presentation will also discuss how converse symmetry breaking challenges the fundamental and widely held assumption that identical agents are necessarily more likely to exhibit similar behavior. I will show that it can, in fact, give rise to beneficial effects of heterogeneity in numerous complex systems in which interacting entities are required to exhibit coordinated behavior. Through this presentation, I hope to convey that our research in network science is now not only benefiting from statistical and nonlinear physics, but also fostering foundational discoveries in these areas. 

Systems where time evolution follows a multiplicative process are ubiquitous in physics. We study a toy model for such systems where each time step is given by multiplication with an independent random NxN matrix with complex Gaussian elements, the complex Ginibre ensemble. This model allows to explicitly compute the Lyapunov exponents and local correlations amongst them, when the number of factors M becomes large. While the smallest eigenvalues always remain deterministic, which is also the case for many chaotic quantum systems, we identify a critical double scaling limit N~M for the rest of the spectrum. It interpolates between the known deterministic behavior of the Lyapunov exponents for M>>N (or N fixed) and universal random matrix statistics for M<<N (or M fixed), characterizing chaotic behavior. After unfolding this agrees with Dyson's Brownian Motion starting from equidistant positions in the bulk and at the soft edge of the spectrum. This universality statement is further corroborated by numerical experiments, multiplying different kinds of random matrices. It leads us to conjecture a much wider applicability in complex systems, that display a transition from deterministic to chaotic behavior.

In this talk I will present recent advances on percolation theory including progress on the large deviation of percolation and percolation on hyperbolic manifolds. The common theme that we will address is the investigation of when discontinuous percolation transitions can be observed.

The theory of large deviation of percolation [1-3] aims at characterizing the uncertain response of finite networks to perturbations. Therefore, this theory addresses important challenges posed by the study of the robustness of networks far from the thermodynamic limit such as ecosystems, power-grids and infrastructures. In this context we will show that when aggravating configuration of the initial damage are considered the percolation becomes discontinuous.

In is known that for the Farey graph, constituting a special type of hyperbolic manifold formed by simplicial complexes, percolation can be discontinuous. Here we investigate how robust this result is with respect to the extension to higher dimensional percolation, i.e. topological percolation to hyperbolic manifolds formed by simplicial complexes of dimension d=3 [4].

[1] Bianconi, G., 2018. Rare events and discontinuous percolation transitions. Physical Review E, 97(2), p.022314.

[2] Bianconi, G., 2019. Large deviation theory of percolation on multiplex networks. Journal of Statistical Mechanics: Theory and Experiment, 2019(2), p.023405.

[3] Coghi, F., Radicchi, F. and Bianconi, G., 2018. Controlling the uncertain response of real multiplex networks to random damage. Physical Review E, 98(6), p.062317.

[4] Bianconi, G. and Ziff, R.M., 2018. Topological percolation on hyperbolic simplicial complexes. Physical Review E, 98(5), p.052308.

Face-to-face interaction networks portray social interactions in closed settings such as schools, hospitals, offices, university campuses, etc. A typical representation consists of a series of network snapshots. The agents (nodes) in each snapshot are individuals and an edge between any two agents represents a direct face-to-face interaction. Analyses of such networks have uncovered universal properties, such as the heavy-tailed distributions of the interaction duration and time between consecutive interactions. In the first part of this talk, we will show that these universal properties along with the formation of connected components that appear recurrently in such systems, find a natural explanation in the assumption that the agents of the temporal network reside in a hidden similarity space. Distances between the agents in this space act as similarity forces directing their motion towards other agents in the physical space and determining the duration of their interactions. The modeling approach we consider bears similarities to N-body simulations and Langevin dynamics, while the resulting probability that two agents interact as a function of their similarity distance is reminiscent of a Fermi-Dirac distribution. Based on this observation, in the second part of the talk we will establish a connection to the Newtonian/Hyperbolic latent-space model of traditional complex networks. Our findings are also applicable to human proximity networks where interactions are not exactly face-to-face, and could potentially be applicable to other types of time-varying networks.