A new technique allows researchers to create real system 'cartographic' maps at different scales: “Multiscale unfolding of real complex networks by geometric renormalization”
UBICS projects 2018/03/19
A new technique allows researchers to create real system 'cartographic' maps at different scales
There is something in common in the following systems: the Internet, the world airport network, the human proteome, music, Darwin’s On the Origin of Species, the mailing system of a company, the human metabolic network and the brain of Drosophila fly. All of these are complex networks with similar properties. This was used by a team of the Institute of Complex Systems of the University of Barcelona (UBICS) to work on a technique that allows researchers to represent these systems geometrically at different scales as if they were “cartographic” maps.
UBICS researchers M. Ángeles Serrano, Guillermo García-Pérez and Marián Boguñá, who conducted the study published in Nature Physics, applied the renormalization group technique to real systems. “This technique allows us to explore a system at different resolution levels, such as a kind of inverted microscope that allows us to zoom out and widen the scale at which we make the observation”, notes the ICREA research lecturer M. Ángeles Serrano, director of the study.
“Being able to move around a network at several scales is very important in systems in which you have many interacting elements, such as the networks we studied. These systems are multiscale networks, that is, their structure or associated processes result from a mix of structures and processes at different scales”, says Guillermo García-Pérez, first author of the study. “Each scale has specific data, but scales are also interrelated between them”, he says.
Representing reality as complex networks
The UB researchers applied the technique they developed to the above mentioned systems. Although they are different, all of them can be defined in the shape of nodes and connections. We know about some cases, such as the Internet; but in others, for example in music, researchers regarded chords as nodes and connections as the proximity of these chords in modern music songs.
In any case, all these systems can be defined as complex networks because they have a property known as small-world, i.e., the nodes are connected between them in a few steps. “It is because of the small-world property that it had been impossible to split structural scales in real complex networks, and in order to do so, we had to develop geometric maps on each one of them so we could define the distances between nodes”, says the lecturer Marián Boguñá.
Moreover, these networks fulfill two more features: on the one hand, they have a heterogeneous connectivity –i.e., there are elements with a high connectivity and others with low connectivity-, and on the other hand, they display many node groupings in a triangular shape (clustering).
“This is the first time a really geometric renormalization group has been defined in complex networks”, notes M. Ángeles Serrano, who adds “We can now build maps of complex networks in the most cartographical sense of the word, real maps where elements or nodes have positions and distance between them”. “These maps –continues the researcher- are not only attractive visual representations but they are full of meaning and they allow us to find out information on the systems and to navigate through them”. In this sense, “we can increase the system navigability if we take into account the information provided by the renormalization group, which allows us to unfold networks at the different scales that build them up, and which, in addition, turn out to be self-similar, that is, they have the same organization at different scales”, highlights the researcher.
These results can also be applied to make reduced versions of the original networks at smaller scales and which have the same properties. “The possibility of having reduced copies has a great potential; for instance, they can serve as a test bench to assess expensive processes in original networks, such as new Internet routing protocols”, concludes Serrano.
Being able to move around a network at several scales is very important in systems in which you have many interacting elements.
These networks fulfill two more features: on the one hand, they have a heterogeneous connectivity –i.e., there are elements with a high connectivity and others with low connectivity-, and on the other hand, they display many node groupings in a triangular shape (clustering).
G. García-Pérez, M. Boguñá and M. Á. Serrano. “Multiscale unfolding of real complex networks by geometric renormalization”. Nature Physics, March 19, 2018. Doi: 10.1038/s41567-018-0072-5
Symmetries in physical theories denote invariance under some transformation, such as self-similarity under a change of scale. The renormalization group provides a powerful framework to study these symmetries, leading to a better understanding of the universal properties of phase transitions. However, the small-world property of complex networks complicates application of the renormalization group by introducing correlations between coexisting scales. Here, we provide a framework for the investigation of complex networks at different resolutions. The approach is based on geometric representations, which have been shown to sustain network navigability and to reveal the mechanisms that govern network structure and evolution. We define a geometric renormalization group for networks by embedding them into an underlying hidden metric space. We find that real scale-free networks show geometric scaling under this renormalization group transformation. We unfold the networks in a self-similar multilayer shell that distinguishes the coexisting scales and their interactions. This in turn offers a basis for exploring critical phenomena and universality in complex networks. It also affords us immediate practical applications, including high-fidelity smaller-scale replicas of large networks and a multiscale navigation protocol in hyperbolic space, which betters those on single layers.