Interdisciplinary and physics challenges of Network Theory

Network theory has unveiled the underlying structure of complex systems such as the Internet or the biological networks in the cell. It has identified universal properties of complex networks, and the interplay between their structure and dynamics. After almost twenty years of the field, new challenges lie ahead. These challenges concern the multilayer structure of most of the networks, the formulation of a network geometry and topology, and the development of a quantum theory of networks. Making progress on these aspects of network theory can open new venues to address interdisciplinary and physics challenges including progress on brain dynamics, new insights into quantum technologies, and quantum gravity.Comment: (7 pages, 4 figures

Distinct dynamical behavior in Erd\H{o}s-R\'enyi networks, regular random networks, ring lattices, and all-to-all neuronal networks

Neuronal network dynamics depends on network structure. In this paper we study how network topology underpins the emergence of different dynamical behaviors in neuronal networks. In particular, we consider neuronal network dynamics on Erd\H{o}s-R\'enyi (ER) networks, regular random (RR) networks, ring lattices, and all-to-all networks. We solve analytically a neuronal network model with stochastic binary-state neurons in all the network topologies, except ring lattices. Given that apart from network structure, all four models are equivalent, this allows us to understand the role of network structure in neuronal network dynamics. Whilst ER and RR networks are characterized by similar phase diagrams, we find strikingly different phase diagrams in the all-to-all network. Neuronal network dynamics is not only different within certain parameter ranges, but it also undergoes different bifurcations (with a richer repertoire of bifurcations in ER and RR compared to all-to-all networks). This suggests that local heterogeneity in the ratio between excitation and inhibition plays a crucial role on emergent dynamics. Furthermore, we also observe one subtle discrepancy between ER and RR networks, namely ER networks undergo a neuronal activity jump at lower noise levels compared to RR networks, presumably due to the degree heterogeneity in ER networks that is absent in RR networks. Finally, a comparison between network oscillations in RR networks and ring lattices shows the importance of small-world properties in sustaining stable network oscillations.Comment: 9 pages, 4 figure

Fermionic Networks: Modeling Adaptive Complex Networks with Fermionic Gases

We study the structure of Fermionic networks, i.e., a model of networks based on the behavior of fermionic gases, and we analyze dynamical processes over them. In this model, particle dynamics have been mapped to the domain of networks, hence a parameter representing the temperature controls the evolution of the system. In doing so, it is possible to generate adaptive networks, i.e., networks whose structure varies over time. As shown in previous works, networks generated by quantum statistics can undergo critical phenomena as phase transitions and, moreover, they can be considered as thermodynamic systems. In this study, we analyze Fermionic networks and opinion dynamics processes over them, framing this network model as a computational model useful to represent complex and adaptive systems. Results highlight that a strong relation holds between the gas temperature and the structure of the achieved networks. Notably, both the degree distribution and the assortativity vary as the temperature varies, hence we can state that fermionic networks behave as adaptive networks. On the other hand, it is worth to highlight that we did not find relation between outcomes of opinion dynamics processes and the gas temperature. Therefore, although the latter plays a fundamental role in gas dynamics, on the network domain its importance is related only to structural properties of fermionic networks.Comment: 19 pages, 5 figure

Granular Response to Impact: Topology of the Force Networks

Impact of an intruder on granular matter leads to formation of mesoscopic force networks seen particularly clearly in the recent experiments carried out with photoelastic particles, e.g., Clark et al., Phys. Rev. Lett., 114 144502 (2015). These force networks are characterized by complex structure and evolve on fast time scales. While it is known that total photoelastic activity in the granular system is correlated with the acceleration of the intruder, it is not known how the structure of the force network evolves during impact, and if there is a dominant features in the networks that can be used to describe intruder's dynamics. Here, we use topological tools, in particular persistent homology, to describe these features. Persistent homology allows quantification of both structure and time evolution of the resulting force networks. We find that there is a clear correlation of the intruder's dynamics and some of the topological measures implemented. This finding allows us to discuss which properties of the force networks are most important when attempting to describe intruder's dynamics. Regarding temporal evolution of the networks, we are able to define the upper bound on the relevant time scale on which the networks evolve

Correlations between structure and dynamics in complex networks

Previous efforts in complex networks research focused mainly on the topological features of such networks, but now also encompass the dynamics. In this Letter we discuss the relationship between structure and dynamics, with an emphasis on identifying whether a topological hub, i.e. a node with high degree or strength, is also a dynamical hub, i.e. a node with high activity. We employ random walk dynamics and establish the necessary conditions for a network to be topologically and dynamically fully correlated, with topological hubs that are also highly active. Zipf's law is then shown to be a reflection of the match between structure and dynamics in a fully correlated network, as well as a consequence of the rich-get-richer evolution inherent in scale-free networks. We also examine a number of real networks for correlations between topology and dynamics and find that many of them are not fully correlated.Comment: 16 pages, 7 figures, 1 tabl