Geometric and Combinatorial Aspects of NonEquilibrium Statistical Mechanics

Non-Equilibrium Statistical Mechanics is a broad subject. Grossly speaking, it deals with systems which have not yet relaxed to an equilibrium state, or else with systems which are in a steady non-equilibrium state, or with more general situations. They are characterized by external forcing and internal fluxes, resulting in a net production of entropy which quantifies dissipation and the extent by which, by the Second Law of Thermodynamics, time-reversal invariance is broken. In this thesis we discuss some of the mathematical structures involved with generic discrete-state-space non-equilibrium systems, that we depict with networks in all analogous to electrical networks. We define suitable observables and derive their linear regime relationships, we discuss a duality between external and internal observables that reverses the role of the system and of the environment, we show that network observables serve as constraints for a derivation of the minimum entropy production principle. We dwell on deep combinatorial aspects regarding linear response determinants, which are related to spanning tree polynomials in graph theory, and we give a geometrical interpretation of observables in terms of Wilson loops of a connection and gauge degrees of freedom. We specialize the formalism to continuous-time Markov chains, we give a physical interpretation for observables in terms of locally detailed balanced rates, we prove many variants of the fluctuation theorem, and show that a well-known expression for the entropy production due to Schnakenberg descends from considerations of gauge invariance, where the gauge symmetry is related to the freedom in the choice of a prior probability distribution. As an additional topic of geometrical flavor related to continuous-time Markov chains, we discuss the Fisher-Rao geometry of nonequilibrium decay modes, showing that the Fisher matrix contains information about many aspects of non-equilibrium behavior, including non-equilibrium phase transitions and superposition of modes. We establish a sort of statistical equivalence principle and discuss the behavior of the Fisher matrix under time-reversal. To conclude, we propose that geometry and combinatorics might greatly increase our understanding of nonequilibrium phenomena.

Graph algorithms for bioinformatics

Biological data are inherently interconnected: protein sequences are connected to their annotations, the annotations are structured into ontologies, and so on. While protein-protein interactions are already represented by graphs, in this work I am presenting how a graph structure can be used to enrich the annotation of protein sequences thanks to algorithms that analyze the graph topology. We also describe a novel solution to restrict the data generation needed for building such a graph, thanks to constraints on the data and dynamic programming. The proposed algorithm ideally improves the generation time by a factor of 5. The graph representation is then exploited to build a comprehensive database, thanks to the rising technology of graph databases. While graph databases are widely used for other kind of data, from Twitter tweets to recommendation systems, their application to bioinformatics is new. A graph database is proposed, with a structure that can be easily expanded and queried.

Unsupervised reinforcement learning via state entropy maximization

Reinforcement Learning (RL) provides a powerful framework to address sequential decision-making problems in which the transition dynamics is unknown or too complex to be represented. The RL approach is based on speculating what is the best decision to make given sample estimates obtained from previous interactions, a recipe that led to several breakthroughs in various domains, ranging from game playing to robotics. Despite their success, current RL methods hardly generalize from one task to another, and achieving the kind of generalization obtained through unsupervised pre-training in non-sequential problems seems unthinkable. Unsupervised RL has recently emerged as a way to improve generalization of RL methods. Just as its non-sequential counterpart, the unsupervised RL framework comprises two phases: An unsupervised pre-training phase, in which the agent interacts with the environment without external feedback, and a supervised fine-tuning phase, in which the agent aims to efficiently solve a task in the same environment by exploiting the knowledge acquired during pre-training. In this thesis, we study unsupervised RL via state entropy maximization, in which the agent makes use of the unsupervised interactions to pre-train a policy that maximizes the entropy of its induced state distribution. First, we provide a theoretical characterization of the learning problem by considering a convex RL formulation that subsumes state entropy maximization. Our analysis shows that maximizing the state entropy in finite trials is inherently harder than RL. Then, we study the state entropy maximization problem from an optimization perspective. Especially, we show that the primal formulation of the corresponding optimization problem can be (approximately) addressed through tractable linear programs. Finally, we provide the first practical methodologies for state entropy maximization in complex domains, both when the pre-training takes place in a single environment as well as multiple environments.

Network assimilation methodology and sectorization of water distribution systems by graph mining and optimization algorithm

Water Distribution Networks (WDNs) play a vital importance rule in communities, ensuring well-being band supporting economic growth and productivity. The need for greater investment requires design choices will impact on the efficiency of management in the coming decades. This thesis proposes an algorithmic approach to address two related problems:(i) identify the fundamental asset of large WDNs in terms of main infrastructure;(ii) sectorize large WDNs into isolated sectors in order to respect the minimum service to be guaranteed to users. Two methodologies have been developed to meet these objectives and subsequently they were integrated to guarantee an overall process which allows to optimize the sectorized configuration of WDN taking into account the needs to integrated in a global vision the two problems (i) and (ii). With regards to the problem (i), the methodology developed introduces the concept of primary network to give an answer with a dual approach, of connecting main nodes of WDN in terms of hydraulic infrastructures (reservoirs, tanks, pumps stations) and identifying hypothetical paths with the minimal energy losses. This primary network thus identified can be used as an initial basis to design the sectors. The sectorization problem (ii) has been faced using optimization techniques by the development of a new dedicated Tabu Search algorithm able to deal with real case studies of WDNs. For this reason, three new large WDNs models have been developed in order to test the capabilities of the algorithm on different and complex real cases. The developed methodology also allows to automatically identify the deficient parts of the primary network and dynamically includes new edges in order to support a sectorized configuration of the WDN. The application of the overall algorithm to the new real case studies and to others from literature has given applicable solutions even in specific complex situations.

Learning representations for graph-structured socio-technical systems

The recent widespread use of social media platforms and web services has led to a vast amount of behavioral data that can be used to model socio-technical systems. A significant part of this data can be represented as graphs or networks, which have become the prevalent mathematical framework for studying the structure and the dynamics of complex interacting systems. However, analyzing and understanding these data presents new challenges due to their increasing complexity and diversity. For instance, the characterization of real-world networks includes the need of accounting for their temporal dimension, together with incorporating higher-order interactions beyond the traditional pairwise formalism. The ongoing growth of AI has led to the integration of traditional graph mining techniques with representation learning and low-dimensional embeddings of networks to address current challenges. These methods capture the underlying similarities and geometry of graph-shaped data, generating latent representations that enable the resolution of various tasks, such as link prediction, node classification, and graph clustering. As these techniques gain popularity, there is even a growing concern about their responsible use. In particular, there has been an increased emphasis on addressing the limitations of interpretability in graph representation learning. This thesis contributes to the advancement of knowledge in the field of graph representation learning and has potential applications in a wide range of complex systems domains. We initially focus on forecasting problems related to face-to-face contact networks with time-varying graph embeddings. Then, we study hyperedge prediction and reconstruction with simplicial complex embeddings. Finally, we analyze the problem of interpreting latent dimensions in node embeddings for graphs. The proposed models are extensively evaluated in multiple experimental settings and the results demonstrate their effectiveness and reliability, achieving state-of-the-art performances and providing valuable insights into the properties of the learned representations.

Supporting requirement elicitation and ontology testing in knowledge graph engineering

Knowledge graphs and ontologies are closely related concepts in the field of knowledge representation. In recent years, knowledge graphs have gained increasing popularity and are serving as essential components in many knowledge engineering projects that view them as crucial to their success. The conceptual foundation of the knowledge graph is provided by ontologies. Ontology modeling is an iterative engineering process that consists of steps such as the elicitation and formalization of requirements, the development, testing, refactoring, and release of the ontology. The testing of the ontology is a crucial and occasionally overlooked step of the process due to the lack of integrated tools to support it. As a result of this gap in the state-of-the-art, the testing of the ontology is completed manually, which requires a considerable amount of time and effort from the ontology engineers. The lack of tool support is noticed in the requirement elicitation process as well. In this aspect, the rise in the adoption and accessibility of knowledge graphs allows for the development and use of automated tools to assist with the elicitation of requirements from such a complementary source of data. Therefore, this doctoral research is focused on developing methods and tools that support the requirement elicitation and testing steps of an ontology engineering process. To support the testing of the ontology, we have developed XDTesting, a web application that is integrated with the GitHub platform that serves as an ontology testing manager. Concurrently, to support the elicitation and documentation of competency questions, we have defined and implemented RevOnt, a method to extract competency questions from knowledge graphs. Both methods are evaluated through their implementation and the results are promising.