“The solution often turns out more beautiful than the puzzle” - Richard Dawkins
Current research 🧮
I am pursuing a Ph.D. in Applied Mathematics, in Complex Networks in particular. As such, I am obviously interested in anything related to Graph Theory and Linear Algebra, as well as computer simulations involving graphs (e.g. percolation phenomena). However, Complex Networks is an interdisciplinary field, borrowing ideas from Maths, Physics, Biology, Sociology, Computer Science, etc, so it is quite natural for me finding thought-provoking ideas in unexpected places.
However, within the realm of Complex Networks the focus of my thesis is a bit more specific: understanding the properties and applicability of centrality measures, and perhaps coming up with novel ones, solving problems in those already existing. Centrality measures are actually really interesting: they allow us to rank the nodes of a network based on different definitions of importance. This might seem unrelated to the real world, but they are more influential in our day-to-day life than we think: the Google search engine ranks results based on one of these measures, and its success is what made it the huge company it is nowadays. Part of my research revolves around understanding the behavior of this, so called, PageRank centrality measure.
There are some specially intriguing, rather new variants of networks which I have also laid my eyes upon:
- Multilayer networks: these are networks whose interacions fall into different categories, and thus treating those interactions as equal is missing the whole picture. For instance, the connections between a set of individuals are distinct depending of whether individuals are relatives, aquaintances, friends or coworkers.
- Higher order networks: these networks arise from lifting the restriction that interactions be pairwise. Consider, for example, three people writing a paper together. They did not write it “pairwise together”, rather, the three of them should be considered part of the same interaction. This opens a vast world of possibilities, however, the technical difficulties encountered along the way are equally vast.
Other research interests
Complex Systems and Non-linear Dynamics 🌀
Complex Systems are systems where the sum is greater than the parts, in the sense that there is some “global behavior” which can’t be infered from the local dynamics. A classic example of this is flocks of birds: the individual movement of a bird does not directly affect the movement of the flock (in fact, if a bird dies there is no significant change in the flock behavior). This is completely different from, say, a machine: if a gear fails the outcome will be, at best, deteriorated.
Complex Networks are another example of Complex Systems; there are many structural properties arising from the collective structure (topology) of the network, such as certain centralities or connectivity. And we can also discover complexity in dynamical systems built on top of networks, such as collections of oscillators whose interactions follow the graph’s edges.
Besides networks, there are many systems whose dynamics are governed by non-linear differential equations (which I have always found fascinating), giving rise to interesting phenomena such as oscillations and chaos.
(Theoretical) Physics ⚛️
One does not forget its beginnings. I still appreciate and enjoy reading about General Relativity (black holes, inflation, etc) and String Theory (in particular, about the AdS/CFT correspondence). I have mixed feelings about the latter, I highly doubt I’ll ever go back. Regarding the former, I don’t think it is that far apart as of now. In the end, GR is just differential geometry with a physics dress. And, while the discrete nature of graphs seems to be opposite to the infinitesimal nature of differential stuff, weirder connections have been established in both Maths and Physics. In fact I’ve already seen a couple of interesting papers in that direction.
Another area of Physics which I was really fond of was Classical Mechanics, in particular everything related to analytical mechanics. The good news about this is that it does not lie far from the realm of Complex Systems: usually, all the dynamics invoved are inferred from a mechanical system or at least have a mechanical analogue.
Machine Learning 🤖
As a coding enthusiast I am deeply interested in Machine and Deep Learning. In fact, I already have some experience with them, thanks to my short-but-fruitful absence from academia. And given the youth of this fields, there is a lot room for exploration, as well as applicability essentially everywhere. Look no further than parenclitic networks, which are ways to convert supervised datasets into graphs, which can then be studied with our beloved network-theoretic tools.
Another subject which seems to be on the rise these days is that of Graph Neural Networks, a new kind of neural networks whose connections are graph-based rather than uniform. Complex Network theory surely has something to say there.
Others ♾️
Something else I think is really interesting but perhaps falling far from my reach are Cellular Automata, and in particular the Game of Life. I find the idea of deterministic systems leading to unpredictable behavior fascinating. In the case of the Conway’s Game of Life this is the case, using the simplest set of sensible rules one can think of.
Articles 📝
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TitleCo-authorsDescriptionJournal/Preprint
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R. Criado, M. RomanceWe provide an alternative definition of non-undirectedness (i.e. heterogeneity) in hypergraphs, beyond traditional definitions of directed hypergraphs which rely on set-theoretic notions. We discuss different examples of heterogeneous hypergraphs, focusing on their spectral centralities.Journal of Complex Networks, Vol 12, Issue 5, October 2024
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Preprint at arXiv:2403.11825 -
C. Pérez-Corral, M. RomanceWe provide a way to compute spectral centralities of general hypergraphs (in particular, the H-eigenvector centrality). Up until this work there was a major shortcoming, the requirement of the hypergraph to be uniform. We show that suitably adding an extra node does the trick.Preprint at arXiv:2310.20335
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A. Bayani, F. Nazarimehr, S. Jafari, K. Kovalenko, K. Alfaro-Bittner, R. J. Sánchez-García, S. BoccalettiIn this work, the path to synchronization is fully characterized, in the sense that we find a way to predict analytically which clusters can each node belong to, and furthermore we obtain the value of the coupling at which each cluster synchronizes.Nature Communications
15, 4955 (2024)
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Preprint at arXiv:2303.08668 -
R. Criado, G. Vera de Salas, J. YangBased on the concept of the derivative of a hypergraph, which serves as a similarity index akin to Jaccard's one in set theory, we devise a novel way to detect communities in undirected hypergraphs.Chaos, Solitons and Fractals
Vol 177, 114200 (2023) -
P. Khanra, S. Ghosh, D. Aleja, K. Alfaro-Bittner, R. Criado, M. Romance, S. Boccaletti, P. Pal, C. HensThis is somewhat of a "reverse engineering" problem: given a number of nodes, an edge density and a partition of the nodes into clusters such that they are synchronized together, we give a procedure to realize one of such possible networks.Physical Review E
108, 054309 (2023)
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Preprint at arXiv:2302.10548 -
R. Criado, M. RomanceWe obtain some analytical results concerning how controllable the PageRank centrality measure is. For that we develop a novel geometrical approach to centrality rankings, which renders the controllability question almost trivial to answer.Chaos
Vol 33, 083152 (2023)
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Preprint at arXiv:2211.12929 -
D. Aleja, K. Alfaro-Bittner, E. Primo, R. Criado, M. Romance, S. BoccalettiWe attempted to model the effect (in terms of damage) of the spread of a computer virus through a network, in terms of a vectorial compartmental model. The dependence of the final damage on the network's topology is discussed, as well as different viral behaviors.Chaos, Solitons and Fractals
Vol 161, 112310, Aug 2022
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Preprint at arXiv:2205.08345
There are a couple of other papers on the way (1 under review, 2 in preparation).
Talks 🙉
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TitleWhereWhen
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NetPlace SeminarsMar 2024
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Spanish Chapter of the Complex Systems Society 2024Feb 2024
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Mathematical Modelling in Engineering & Human Behaviour 2023Jul 2023
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FisEs (Congreso de Física Estadística) Joven 23May 2023
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BYMAT (Bringing Young MAthematicians Together) 2022Nov 2022
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Mathematical Modelling in Engineering & Human Behaviour 2022Jul 2022