The graph-theoretical based formulation for the representation of the data-driven structure and the dynamics of complex systems is rapidly imposing as the paramount paradigm [1] across a variety of disciplines, from economics to neuroscience, with biological -omics as a major example. In this framework, the concept of Null Model borrowed from the statistical sciences identifies the elective strategy to obtain a baseline points of modelling comparison [2]. Hereafter, a null model is a graph which matches one specific graph in terms of some structural features, but which is otherwise taken to be generated as an instance of a random network. In this view, the network model introduced by Erdos & Renyi [3], where random edges are generated as independently and identically distributed Bernoulli trials, can be considered the simplest possible null model. In the following years, other null models have been developed in the framework of graph theory, with the detection of the community structure as one of the most important target[4]. In particular, the model described in [5] introduces the concept of a randomized version of the original graph: edges are rewired at random, with each expected vertex degree matching the degree of the vertex in the original graph. Although aimed at building a reference for the community detection, this approach will play a key role in one of the model considered in this thesis. Note that, although being the ï¬ rst problem to be considered, designing null models for the community structures detection is still an open problem [6, 7]. Real world applications of null model in graph theory have also gained popularity in many different scientific areas, with ecology as the ï¬ rst example: see [8] for a comprehensive overview. More recently, interest for network null models arose also in computational biology [9, 10], geosciences [11] and economics [12, 13], just to name a few. In the present work the theoretical design and the practical implementation of a series of algorithms for the construction of null models will be introduced, with applications ranging from functional genomics to game theory for social studies. The four chapters devoted to the presentation of the examples of null model are preceded by an introductory chapter including a quick overview of graph theory, together with all the required notations. The ï¬ rst null model is the topic of the second chapter, where a suite of novel algorithms is shown, aimed at the efficient generation of complex networks under different constraints on the node degrees. Although not the most important example in the thesis, the premiment position dedicated to this topic is due to its strict familiarity with the aforementioned classical null models for random graph construction. Together with the algorithms definition and examples, a thorough theoretical analysis of the proposed solutions is shown, highlighting the improvements with respect to the state-of-the-art and the occurring limitations. Apart from its intrinsic mathematical value, the interest for these algorithms by the community of systems biology lies in the need for benchmark graphs resembling the real biological networks. They are in fact of uttermost importance when testing novel inference methods, and as testbeds for the network reconstruction challenges such as the DREAM series [14, 15, 16]. The following Chapter three includes the most complex application of null models presented in this thesis. The scientific workï¬ eld is again functional genomics, namely the combinatorial approach to the modelling of patterns of mutations in cancer as detected by Next Generation Sequencing exome Data. This problem has a natural mathematical representation in terms of rewiring of bipartite networks and mutual-exclusively mutated modules [17, 18], to which Markov chain updates (switching-steps) are applied through a Switching Algorithm SA. Here we show some crucial improvements to the SA, we analytically derive an approximate lower bound for the number of steps required, we introduce BiRewire, an R package implementing the improved SA and we demonstrate the effectiveness of the novel solution on a breast cancer dataset. A novel threshold-selection method for the construction of co-expression net- works based on the Pearson coefficient is the third and last biological example of null model, and it is outlined in Chapter four. Gene co-expression networks inferred by correlation from high-throughput proï¬ ling such as microarray data represent a simple but effective technique for discovering and interpreting linear gene relationships. In the last years several approach have been proposed to tackle the problem of deciding when the resulting correlation values are statistically significant. This is mostly crucial when the number of samples is small, yielding a non negligible chance that even high correlation values are due to random effects. Here we introduce a novel hard thresholding solution based on the assumption that a coexpression network inferred by randomly generated data is expected to be empty. The theoretical derivation of the new bound by geometrical methods is shown together with two applications in oncogenomics. The last two chapters of the thesis are devoted to the presentation of null models in non-biological contexts. In Chapter 5 a novel dynamic simulation model is introduced mimicking a random market in which sellers and buyers follow different price distributions and matching functions. The random marked is mathematically formulated by a dynamic bipartite graph, and the analytical formula for the evolution along time of the mean price exchange is derived, together with global likelihood function for retrieving the initial parameters under different assumptions. Finally in Chapter 6 we describe how graph tools can be used to model abstraction and strategy (see [19, 20, 21]) for a class of games in particular the TTT solitaire. We show that in this solitaire it is not possible to build an optimal (in the sense of minimum number of moves) strategy dividing the big problems into smaller subproblems. Nevertheless, we ï¬ nd some subproblems and strategies for solving the TTT solitaire with a negligible increment in the number of moves. Although quite simple and far from simulating highly complex real-world situations of decision making, the TTT solitaire is an important tool for starting the exploration of the social analysis of the trajectories of the implementation of winning strategies through different learning procedures [22].

This thesis treats mathematical models for host-parasitoid interactions. It is composed of three parts. In the first part, a class of such models is analyzed theoretically. It focuses on the phenomena of multiple coexistence equilibria of competing parasitoid species. The second part is about a model for determining how a parasitoid release should be timed to optimally control the invasive fruit fly Drosophila suzukii. The third part analyzes an experiment for releasing parasitoids in a greenhouse which is infested by D.suzukii. The models presented are used to discuss how to improve such biological control strategies.

About three quarters of human emerging infectious diseases are caused by zoonotic pathogens, and many of them are spread by vectors such as mosquitoes. Mathematical models nowadays represent very powerful tools to make investigations and predictions for biological dynamical systems, providing helpful insights that can be extremely valuable for several aims. In this thesis, we will focus on a particular mosquito-borne zoonosis, West Nile virus (WNV), a flavivirus of emerging public health relevance in Europe and North America, and its main European vector, Culex pipiens mosquitoes. As the transmission of mosquito-borne diseases is largely driven by the abundance of the vector, to design appropriate control strategies it is crucial to understand the population dynamics of existing vector populations and evaluate how it depends on biotic and environmental factors. This thesis presents some new mathematical models that provide insights on several aspects of mosquito population dynamics by using different statistical and computational approaches, including for instance Linear Models and Markov chain Monte Carlo technique. Specifically, they aim to study the effect of biotic and abiotic factors on Cx. pipiens dynamics by using adult mosquito trapping data, gathered over several years in Northern Italy, to feed theoretical models. Furthermore, the effects of host competition and vector feeding preferences on the dynamics of a vector-borne infection (such as WNV) are investigated through a more theoretical study.

In an optimal visiting problem, we want to control a trajectory that has to pass as close as possible to a collection of target points or regions. We introduce a hybrid control-based approach for the classic problem where the trajectory can switch between a group of discrete states related to the targets of the problem. The model is subsequently adapted to a mean-field game framework, that is when a huge population of agents plays the optimal visiting problem with a controlled dynamics and with costs also depending on the distribution of the population. In particular, we investigate a single continuity equation with possible sinks and sources and the field possibly depending on the mass of the agents. The same problem is also studied on a network framework. More precisely, we study a mean-field game model by proving the existence of a suitable definition of an approximated mean-field equilibrium and then we address the passage to the limit.

In this Thesis, we analyze three variational and geometric problems, that extend classical Euclidean issues of the calculus of variations to more general classes of spaces. The results we outline are based on the articles [Ved21; MV21] and on a forthcoming joint work with Nicolussi Golo and Serra Cassano. In the first place, in Chapter 1 we provide a general introduction to metric measure spaces and some of their properties. In Chapter 2 we extend the classical Talenti’s comparison theorem for elliptic equations to the setting of RCD(K,N) spaces: in addition the the generalization of Talenti’s inequality, we will prove that the result is rigid, in the sense that equality forces the space to have a symmetric structure, and stable. Chapter 3 is devoted to the study of the Bernstein problem for intrinsic graphs in the first Heisenberg group H^1: we will show that under mild assumptions on the regularity any stationary and stable solution to the minimal surface equation needs to be intrinsically affine. Finally, in Chapter 4 we study the dimension and structure of the singular set for p-harmonic maps taking values in a Riemannian manifold.

The first aim of this PhD Thesis is devoted to the study of geodesic distances defined on a subdomain of a Carnot group, which are bounded both from above and from below by fixed multiples of the Carnot–Carathéodory distance. Then one shows that the uniform convergence, on compact sets, of these distances can be equivalently characterized in terms of Gamma-convergence of several kinds of variational problems. Moreover, it investigates the relation between the class of intrinsic distances, their metric derivatives and the sub-Finsler convex metrics defined on the horizontal bundle. The second purpose is to obtain the integral representation of some classes of local functionals, depending on a family of vector fields, that satisfy a weak structure assumption. These functionals are defined on degenerate Sobolev spaces and they are assumed to be not translations-invariant. Then one proves some Gamma-compactness results with respect to both the strong topology of L^p and the strong topology of degenerate Sobolev spaces.

Very few standards exist for tting products to people. Footwear fit is a noteworthy example for consumer consideration when purchasing shoes. As a result, footwear manufacturing industry for achieving commercial success encountered the problem of developing right footwear which is fulfills consumer's requirement better than it's competeries. Mass customization starts with understanding individual customer's requirement and it finishes with fulllment process of satisfying the target customer with near mass production efficiency. Unlike any other consumer product, personalized footwear or the matching of footwear to feet is not easy if delivery of discomfort is predominantly caused by pressure induced by a shoe that has a design unsuitable for that particular shape of foot. Footwear fitter have been using manual measurement for a long time, but the combination of 3D scanning systems with mathematical technique makes possible the development of systems, which can help in the selection of good footwear for a given customer. This thesis, provides new approach for addressing the computerize footwear fit customization in industry problem. The design of new shoes starts with the design of the new shoe last. A shoe last is a wooden or metal model of human foot on which shoes are shaped. Despite the steady increase in accuracy, most available scanning techniques cause some deficiencies in the point cloud and a set of holes in the triangle meshes. Moreover, data resulting from 3D scanning are given in an arbitrary position and orientation in a 3D space. To apply sophisticated modeling operations on these data sets, substantial post-processing is usually required. We described a robust algorithm for filling holes in triangle mesh. First, the advance front mesh technique is used to generate a new triangular mesh to cover the hole. Next, the triangles in initial patch mesh is modified by estimating desirable normals instead of relocating them directly. Finally, the Poisson equation is applied to optimize the new mesh. After obtaining complete 3D model, the result data must be generated and aligned before taking this models for shape analysis such as measuring similarity between foot and shoe last data base for evaluating footwear it. Principle Component Analysis (PCA), aligns a model by considering its center of mass as the coordinate system origin, and its principle axes as the coordinate axes. The purpose of the PCA applied to a 3D model is to make the resulting shape independent to translation and rotation asmuch as possible. In analysis, we applied "weighted" PCA instead of applying the PCA in a classical way (sets of 3D point-clouds) for alignment of 3D models. This approach is based on establishing weights associated to center of gravity of triangles. When all of the models are aligned, an efficient algorithm to cut the model to several sections toward the heel and toe for extracting counters is used. Then the area of each contour is calculated and compared with equal sections in shoe last data base for finding best footwear fit within the shoe last data base.