Reaction-Diffusion Systems and beyond on Complex Networks

Many patterns that we see in nature can be explained by the reaction-diffusion mechanism (genial idea of Alan Turing!). It involves two (or more) species existing on a spatial support, like black and white pigmentation cells on the skin of zebras. They move around by following the diffusion rule (which is how fluids move) and in each point where they meet they react (like a chemical reaction, hence the name). Both processes alone would bring to a homogeneous occupation of space by the two quantities, but when the two mechanisms coexist…boom! Wonderful patterns may emerge! This can happen in a continuous space but in a discrete one too, like the nodes of a complex network.

My (and collaborators) contributions:

• In networks the possibility that patterns emerge depends on topology and on the specific reaction. In a regime with patterns we can hence play with the network structure (with the Laplacian operator, for those who are familiar with) to make patterns disappear.
• We discovered that, under some conditions, patterns can emerge on networks even when the species do not follow diffusion but a generalized non-local interaction scheme.
• If, instead, we replace diffusion by random walk we obtain a totally different setting, which can be useful to learn something about the network structure (functional centralities, reveal symmetries and degree correlations).

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Topological Control for networks, ecosystems and games

We talk about topological control when we have a dynamical system on a network and we modify the network structure to obtain some specific result.

My (and collaborators) contributions:

• Suppose we have a reaction-diffusion system on a network, in a regime where heterogeneous spatio-temporal patterns appear. We can play with the network structure (with the Laplacian operator, for those who are familiar with) to make patterns disappear.
• Analogously, we can find different network topologies that correspond to the same (or similar) spatio-temporal patterns.
• A similar spectral analysis can be used to modify a network in order to make stable an ecosystem of interacting (cooperative and/or competitive) species.
• Such spectral control, however, implies a global modification of the species interaction matrix. A local modification can be represented by the addition of a single node (an added species) to control and make stable a desired equilibrium. It is tested on a mice microbiota dataset.
• A totally different mechanism of control can be applied to a network which is explored by randomly moving agents, seeking for something that they can find on some specific node (an information if the network represents the web, or an item to buy if they are moving in a city). We developed a method (based on a mathematical game) to find an optimal position for the seeked item which is strategical from the point of view of the supplier so as to attract the maximum number of customers randomly passing by.

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Temporal Graph Generation

Interestingly, the small substructures that compose the complex topology of a network are sometimes recurrent, emerging as essential ingredients for the specific network at hand. Those are generally represented by subnetworks composed by a small number of nodes characterized by a specific structure of connections. The substructure which are recognized as the most significative take the name of motifs. The fact that a specific meso-scale structure is repeated many times in a specific network becomes a characterizing feature for this last and can represent a way to couple similar networks and separate different ones.

My (and collaborators) contributions:

• We have developed a method which allows to identify the most significant motifs in different social contexts characterized by temporal networks of interactions (egocentric temporal motifs). The method is very efficient and allows to distinguish among different social contexts. [Open code & video tutorial]
• Moreover, the possibility to identify the significant meso-scale structures in a temporal network allows to generate synthetic temporal networks that mimic the real ones. That is useful when only partial data are available or larger data are needed for simulations or analyses, and allows to avoid privacy issues in case of sensitive personal data. [Open code]

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Higher-Order Networks

The complexity of many biological, social and technological systems stems from the richness of the interactions among their units. Over the past decades, a variety of complex systems has been successfully described as networks whose interacting pairs of nodes are connected by links. Yet, from human communications to chemical reactions and ecological systems, interactions can often occur in groups of three or more nodes and cannot be described simply in terms of dyads. Until recently little attention has been devoted to the higher-order architecture of real complex systems. However, a mounting body of evidence is showing that taking the higher-order structure of these systems into account can enhance our modeling capacities and help us understand and predict their dynamical behavior.

Higher-order networks take into account group interactions among nodes and can be described by two different models: hypergraphs and simplicial complexes.

My (and collaborators) contributions:

• A review of structural and dynamical applications.
• A redefinition of random walk process on hypergraphs.
• A generalization of synchronization process to higher-order networks.
• An analysis of social interaction data by the higher-order perspective.
• Temporal higher-order networks and a contagion model which takes into account different order connections.

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Applications to Society: Digital Contact Tracing and Econo-physics

Many mathematical frameworks can be applied to reproduce real-world scenarios. This allows us to analyze real behaviors in a mathematical or numerical model and to explore the effects of interventions and parameters settings. In other words, we can play and see the consequences of different actions in a safe environment, without making any harm to the real population!
We can for instance simulate an economic setting, with firms, banks, and possible bankruptcy, and observe how it evolves in time. We did it in a paper published on Physical Review Letters, where we also observed economic crises and proved that they can spontaneously emerge from the natural evolution of the initial setting.
Or we can simulate a Covid-19 epidemic on a generic mathematical population and see how many people are infected in time. Moreover, we can insert a digital contact tracing app in this generic population to see if we are able to correctly trace contagious contacts. You can read this in a paper published on Nature Communications, where we quantify the effect of digital contact tracing, with the help of datasets of real physical contacts in various social settings. The results are however general and not bounded to specific environments.
Still Covid-related, we can also monitor social contexts (like summerschools) by assessing the risk occurring through contacts between individuals.

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Urban Science (A Sim City Model)

The spontaneous emergence of spatial patterns, which arise from microscopic rules and break the intrinsic symmetry of space, is ubiquitous in nature, from the skin pigmentation of some animals to the regular shape of snowflakes, or the embryonic cellular structure. Curious patterns can also be found in urban areas, constituted by the different specific use of each piece of land. Each city is characterized by its own particular pattern; the land use pattern is strictly related to the mobility, the social structure of the city and, more in general, to its livability. Clearly, the organization of land use affects mobility and travel demand of people and has an impact on how the transport lines (be they highways, metro lines, or long connections in general) are built, and viceversa. We are working on a simple framework for simulating urban land use patterns starting from microscopic rules which determine the local benefit and cost of assigning a particular typology to each building, i.e. Residential, Commercial and Business, Industrial and Green or empty areas. The theoretical city that we obtain with the proposed model follows a temporal evolution before reaching a stable configuration, where the four typologies coexist, shaping peculiar aggregation patterns.

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Dissemination

The first mission of University, in Italy, is teaching, the second one is doing research, and the third one is the outreach, or all the activities that connect University and Society for the valorization of knowledge and for the general well-being.

I consider this activity of paramount importance to bring non-scientists closer to the scientific world and to create awareness about the importance of fundamental scientific research for everyday life. For this reason I’m trying to do my best! So far (all references are in italian, sorry to all non-italian speaking readers!):

• I have been entertainer at the laboratory-show ScienzEstate in 2016 and in 2018. ScienzEstate is a big outreach event organized by the University of Florence, where many research groups expose their activity by organizing presentations and games involving people of every age.
• I have been occasional speaker for the scientific broadcast RadioMoka (radio hosts: Franco Bagnoli, Giovanna Pacini). Radio Moka is an appendix of CaffeScienza, which organizes weekly meetings of science talks for everyone in Florence.
• I have been interviewed by Marco Serra for the FBK broadcast “Senti chi ricerca” where I illustrated my research path.
• I have been interviewed by Giuseppe d’Agostino for the Rai Radio1 broadcast “Terre di confine”.
• The article that I published on Nature Communications about digital contact tracing in the Covid-19 pandemic has received much attention from the media. I have been contacted by Nature Italy, a digital magazine of scientific journalism focused on summarizing and making accessible to everyone major current scientific discoveries, with bilingual issues in English and Italian. An article has been published explaining the main results of our research and highlighting the importance of digital contact tracing before the advent of Covid vaccines. Other media have shown interest in this work and I have been interviewed by three radio programs on scientific dissemination in Italy: "Senti che scienza!” at Radio Dolomiti, Radio Music Trento and Radio Moka.
• I have been entertainer at the big outreach event called La notte dei ricercatori in 2021 in Trento. I and my colleagues exposed our research about big data and computational social science by showing interactive tools and making people play with network generation and privacy issues (when to trust and when not to trust).

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