Index
Equilibrium Statistical Mechanics
Bipartite Spin Systems
are formed by linking spins of two weighted lattices, often modeled as fully connected graphs. Depending on whether the spins interact in a ferromagnetic or disordered manner, these systems exhibit unique behaviors, particularly in their free energy, which leads to a degenerate self-consistency equation. These models can be extended beyond fully connected interactions by incorporating elements such as Bernoulli-distributed bond dilution. Additionally, it has been demonstrated that analogous neural networks can be mapped onto bipartite models, significantly broadening the potential applications of these systems.
References A. Barra, G. Genovese, F. Guerra
J. Phys. A: Math. Theor. 44 245002 A. Barra, G. Del Ferraro, D. Tantari
EPJB (2013) 86:332 A. Barra, E. Agliari
J. Stat. Mech. (2011) P02027
are formed by linking spins of two weighted lattices, often modeled as fully connected graphs. Depending on whether the spins interact in a ferromagnetic or disordered manner, these systems exhibit unique behaviors, particularly in their free energy, which leads to a degenerate self-consistency equation. These models can be extended beyond fully connected interactions by incorporating elements such as Bernoulli-distributed bond dilution. Additionally, it has been demonstrated that analogous neural networks can be mapped onto bipartite models, significantly broadening the potential applications of these systems.
References A. Barra, G. Genovese, F. Guerra
J. Phys. A: Math. Theor. 44 245002 A. Barra, G. Del Ferraro, D. Tantari
EPJB (2013) 86:332 A. Barra, E. Agliari
J. Stat. Mech. (2011) P02027
Non-equilibrium Statistical Mechanics
Current Large Deviations
is an important topic in Interacting Particle Systems (IPS). There are models characterised by particles hopping between the sites of a discrete lattice according to transition probabilities which can depend on various parameters. For instance, if the probability depends solely on the number of particles at the departure site, the IPS is called the Zero Range Process. If the probability decreases with the number of particles at the arrival site and increases with the number of particles at the departure site, it is referred to as the Misanthrope Process. While many properties of IPS have been well-documented in the stationary regime (long observation times), it is also crucial to understand their behavior out of the stationary regime. Specifically, analyzing the current as a physical observable allows for the study of these behaviors from both microscopic and macroscopic perspectives. Theoretical results in this domain can be validated using numerical simulation techniques derived from large deviation theory, such as the Cloning Algorithm.
References P. Chleboun, S. Grosskinsky
JSP (2014) 154:432-465 S. Grosskinsky, G. M. Schutz, H. Spohn
JSP (2003) 113:389-410 M. R. Evans, B. Waclaw
J. Phys. A: Math. Theor. 47 095001
is an important topic in Interacting Particle Systems (IPS). There are models characterised by particles hopping between the sites of a discrete lattice according to transition probabilities which can depend on various parameters. For instance, if the probability depends solely on the number of particles at the departure site, the IPS is called the Zero Range Process. If the probability decreases with the number of particles at the arrival site and increases with the number of particles at the departure site, it is referred to as the Misanthrope Process. While many properties of IPS have been well-documented in the stationary regime (long observation times), it is also crucial to understand their behavior out of the stationary regime. Specifically, analyzing the current as a physical observable allows for the study of these behaviors from both microscopic and macroscopic perspectives. Theoretical results in this domain can be validated using numerical simulation techniques derived from large deviation theory, such as the Cloning Algorithm.
References P. Chleboun, S. Grosskinsky
JSP (2014) 154:432-465 S. Grosskinsky, G. M. Schutz, H. Spohn
JSP (2003) 113:389-410 M. R. Evans, B. Waclaw
J. Phys. A: Math. Theor. 47 095001
Network Optimisation
Clustering over Signed Graphs
involves associating each edge with a dichotomous variable, either +1 or -1, resulting in a signed adjacency matrix. By allowing the nodes to also take values of +1 or -1, one can explore the node configuration that maximizes the quadratic form associated with the graph. This optimization can be approached using various techniques from algebra, such as spectral analysis, or physics, like the Ginzburg-Landau functional. A significant variant of this problem arises when some node values are predetermined. This modification disrupts standard techniques and necessitates the development of novel optimization strategies. See more details here.
References M. Cucuringu
Journal of Complex Networks, 3 (3):469-506 M. Cucuringu, A. Singer, D. Cowburn
Information and Inference: A Journal of the IMA, 1 (1), pp. 2167 M. Cucuringu, Y. Lipman , A. Singer
ACM Transactions on Sensor Networks, 8 (3), pp. 1-42
involves associating each edge with a dichotomous variable, either +1 or -1, resulting in a signed adjacency matrix. By allowing the nodes to also take values of +1 or -1, one can explore the node configuration that maximizes the quadratic form associated with the graph. This optimization can be approached using various techniques from algebra, such as spectral analysis, or physics, like the Ginzburg-Landau functional. A significant variant of this problem arises when some node values are predetermined. This modification disrupts standard techniques and necessitates the development of novel optimization strategies. See more details here.
References M. Cucuringu
Journal of Complex Networks, 3 (3):469-506 M. Cucuringu, A. Singer, D. Cowburn
Information and Inference: A Journal of the IMA, 1 (1), pp. 2167 M. Cucuringu, Y. Lipman , A. Singer
ACM Transactions on Sensor Networks, 8 (3), pp. 1-42
Quantitative Social Sciences
Immigration Phenomena and Interaction with Native Citizens
Statistical Mechanics (SM) offers a powerful framework for describing scenarios involving numerous interacting agents. By dividing individuals into two groups—immigrants and native citizens—and allowing them to interact, we can model social dynamics using a bipartite Curie-Weiss model, akin to systems with two sets of spins. This mathematical approach can also incorporate various sociological indicators. Recently, SM techniques have been successfully applied to study marriages between immigrants and native citizens in Spain, providing insights into integration trends. The next step is to extend these techniques to other European countries, even without focusing on immigration. This approach could potentially reveal cultural fractures among native citizens within a country.
References A. Barra, P. Contucci
Europhysics Letters 89, 68001-68007 A. Barra, P. Contucci,. R. Sandell, C. Vernia
Nature Scientific Reports 3, 4174 E. Agliari, A. Barra, P. Contucci,. R. Sandell, C. Vernia
New Journal of Physics 16, 103034
Statistical Mechanics (SM) offers a powerful framework for describing scenarios involving numerous interacting agents. By dividing individuals into two groups—immigrants and native citizens—and allowing them to interact, we can model social dynamics using a bipartite Curie-Weiss model, akin to systems with two sets of spins. This mathematical approach can also incorporate various sociological indicators. Recently, SM techniques have been successfully applied to study marriages between immigrants and native citizens in Spain, providing insights into integration trends. The next step is to extend these techniques to other European countries, even without focusing on immigration. This approach could potentially reveal cultural fractures among native citizens within a country.
References A. Barra, P. Contucci
Europhysics Letters 89, 68001-68007 A. Barra, P. Contucci,. R. Sandell, C. Vernia
Nature Scientific Reports 3, 4174 E. Agliari, A. Barra, P. Contucci,. R. Sandell, C. Vernia
New Journal of Physics 16, 103034
Blockchain Cybersecurity Modelling
Stochastic Approaches to Consensus
Blockchain and general-purpose distributed ledgers represent foundational technologies that drive significant innovation within the infrastructures and socio-economic systems. These peer-to-peer (P2P) technologies enable the secure dissemination of information across networks without the need for trusted intermediaries or central authorities to enforce consensus. A key area of research involves developing minimalistic stochastic models to understand the dynamics of blockchain-based consensus. By applying random-walk theory, researchers model block propagation delays across different network topologies and classify blockchain systems based on two emergent properties.
References C.J. Tesson, P. Tasca, F. Iannelli
arXiv preprint arXiv:2106.06465, 2021 P.-Y. Piriou, J.-F. Dumas
14th European Dependable Computing Conference (EDCC), 2018 A. Dembo, S. Kannan, E.N. Tas, D. Tse, P. Viswanath, X. Wang, O. Zeitouni
Proceedings of the 2020 ACM SIGSAC Conference on Computer and Communications Security
Blockchain and general-purpose distributed ledgers represent foundational technologies that drive significant innovation within the infrastructures and socio-economic systems. These peer-to-peer (P2P) technologies enable the secure dissemination of information across networks without the need for trusted intermediaries or central authorities to enforce consensus. A key area of research involves developing minimalistic stochastic models to understand the dynamics of blockchain-based consensus. By applying random-walk theory, researchers model block propagation delays across different network topologies and classify blockchain systems based on two emergent properties.
References C.J. Tesson, P. Tasca, F. Iannelli
arXiv preprint arXiv:2106.06465, 2021 P.-Y. Piriou, J.-F. Dumas
14th European Dependable Computing Conference (EDCC), 2018 A. Dembo, S. Kannan, E.N. Tas, D. Tse, P. Viswanath, X. Wang, O. Zeitouni
Proceedings of the 2020 ACM SIGSAC Conference on Computer and Communications Security
Physics-inspired ML
Mean Curvature Flow for Clustering
involves the continuous deformation of a set's boundary, where higher curvature regions deform faster than flatter ones. Level set methods efficiently solve this deformation process, accommodating topological changes. For clustering, this concept is applied to partition a weighted undirected graph with both positive and negative edge weights. The goal is to create clusters with predominantly positive internal edges and predominantly negative edges between clusters. Using a graph-based diffuse interface model with the Ginzburg–Landau functional, adapted from the Merriman–Bence–Osher (MBO) scheme, the method minimizes positive inter-cluster edge weights while maximizing negative ones. This scalable approach can incorporate labeled data for semi-supervised learning. Tests on synthetic and real-world datasets, including financial correlation matrices, have shown promising results compared to state-of-the-art methods.
References A. Bertozzi, A. Flenner
Multiscale Modeling & Simulation, 2012 - SIAM E. Merkurjev, T. Kostic,. A. Bertozzi
Journal on Imaging Sciences, 2013 - SIAM Y. van Gennip, N. Guillen, B. Osting, A. Bertozzi
Milan Journal of Mathematics, 2014
involves the continuous deformation of a set's boundary, where higher curvature regions deform faster than flatter ones. Level set methods efficiently solve this deformation process, accommodating topological changes. For clustering, this concept is applied to partition a weighted undirected graph with both positive and negative edge weights. The goal is to create clusters with predominantly positive internal edges and predominantly negative edges between clusters. Using a graph-based diffuse interface model with the Ginzburg–Landau functional, adapted from the Merriman–Bence–Osher (MBO) scheme, the method minimizes positive inter-cluster edge weights while maximizing negative ones. This scalable approach can incorporate labeled data for semi-supervised learning. Tests on synthetic and real-world datasets, including financial correlation matrices, have shown promising results compared to state-of-the-art methods.
References A. Bertozzi, A. Flenner
Multiscale Modeling & Simulation, 2012 - SIAM E. Merkurjev, T. Kostic,. A. Bertozzi
Journal on Imaging Sciences, 2013 - SIAM Y. van Gennip, N. Guillen, B. Osting, A. Bertozzi
Milan Journal of Mathematics, 2014
Rare Events Simulations
Catastrophe Events Generation
Given the increase in catastrophic events related to climate change, pandemics, and cyber attacks, it is crucial to develop statistical models capable of generating such rare events to quantify their economic impact. For power plants, digital twins can be created using a set of partial differential equations (PDEs) based on initial conditions that ensure stationary states of the main observables. Rare-event studies are then conducted on simulated power systems where grid-scale batteries provide regulation and emergency frequency control ancillary services. Utilizing a model of random power disturbances at each bus, the skipping sampler, a Markov Chain Monte Carlo algorithm for rare-event sampling, is employed to build conditional distributions of power disturbances leading to two types of instability: frequency excursions outside the normal operating band and load shedding. The study examines potential saturation in benefits and competition between the two services as the battery's maximum power output increases.
References J. Moriarty, J. Vogrinc, A. Zocca
Statistics and Computing, 2021 P. Mancarella, J. Moriarty, A. Philpott, A. Veraart, S. Zachary, B. Zwart
Philosophical Transactions of the Royal Society A, 2021 J. Bierkens, P. Fearnhead, G. Roberts
The Annals of Statistics, 2019
Given the increase in catastrophic events related to climate change, pandemics, and cyber attacks, it is crucial to develop statistical models capable of generating such rare events to quantify their economic impact. For power plants, digital twins can be created using a set of partial differential equations (PDEs) based on initial conditions that ensure stationary states of the main observables. Rare-event studies are then conducted on simulated power systems where grid-scale batteries provide regulation and emergency frequency control ancillary services. Utilizing a model of random power disturbances at each bus, the skipping sampler, a Markov Chain Monte Carlo algorithm for rare-event sampling, is employed to build conditional distributions of power disturbances leading to two types of instability: frequency excursions outside the normal operating band and load shedding. The study examines potential saturation in benefits and competition between the two services as the battery's maximum power output increases.
References J. Moriarty, J. Vogrinc, A. Zocca
Statistics and Computing, 2021 P. Mancarella, J. Moriarty, A. Philpott, A. Veraart, S. Zachary, B. Zwart
Philosophical Transactions of the Royal Society A, 2021 J. Bierkens, P. Fearnhead, G. Roberts
The Annals of Statistics, 2019