Master of Science / Department of Electrical and Computer Engineering / Faryad Darabi Sahneh / A significant number of processes we observe in nature can be described as a spreading process; any agent which is compelled to survive by replicating through a population, examples include viruses, opinions, and information. Accordingly, a significant amount of thought power has been spent creating tools to aid in understanding spreading processes: How do they evolve? When do they thrive? What can we do to control them? Often times these questions are asked with respect to processes in isolation, when agents are free to spread to the maximum extent possible given topological and characteristic constraints. Naturally, we may be interested in considering the dynamics of multiple processes spreading through the same population, examples of which there are no shortage; we frequently characterize nature itself by the interaction and competition present at all scales of life. Recently the number of investigations into interacting processes, particularly in the context of complex networks, has increased. The roles of interaction among processes are varied from mutually beneficial to hostel, but the goals of these investigations has been to understand the role of topology in the ability of multiple processes to co-survive. A consistent feature of all present works -- within the current authors knowledge -- is that conclusions of coexistence are based on marginal descriptions population dynamics. It is the main contribution of this work to explore the hypothesis that purely marginal population descriptions are insufficient indicators of co-survival between interacting processes. Specifically, evaluating coexistence based on non-zero marginal populations is an over-simplistic definition. We randomly generate network topologies via a community based algorithm, the parameters of which allow for trivially controlling possibility of coexistence. Both marginal and conditional probabilities of each process surviving is measured by stochastic simulations. We find that positive marginal probabilities for both processes existing long term does not necessarily imply coexistence, and that marginal and conditional measurements only agree when layers are strongly anti-correlated (sufficiently distinct). In addition to the present thesis, this work is being prepared for a journal article publication. The second portion of this thesis presents numerical simulations for the Adaptive Contact - Susceptible Alert Infected Susceptible model. The dynamics of interaction between an awareness process and an infectious process are computed over a multilayer network. The rate at which nodes "switch" their immediate neighbors (contacts) when exposed to the infection is varied and numerical solutions to the epidemic threshold are computed according to mean-field approximation. We find two unexpected cases where certain parameter configurations allow the epidemic threshold to either increase above or decrease below the theoretical limits of the layers when considered individually. These computations were performed as part of a separate journal article that has been accepted for publication.

Doctor of Philosophy / Department of Electrical and Computer Engineering / Caterina Scoglio / Society increasingly depends on networks for almost every aspect of daily life. Over the past decade, network science has flourished tremendously in understanding, designing, and utilizing networks. Particularly, network science has shed light on the role of the underlying network topology on the dynamic behavior of complex systems, including cascading failure in power-grids, financial contagions in trade market, synchronization, spread of social opinion and trends, product adoption and market penetration, infectious disease pandemics, outbreaks of computer worms, and gene mutations in biological networks. In the last decade, most studies on complex networks have been confined to a single, often homogeneous network. An extremely challenging aspect of studying these complex systems is that the underlying networks are often heterogeneous, composite, and interdependent with other networks. This challenging aspect has very recently introduced a new class of networks in network science, which we refer to as multilayer and interconnected networks. Multilayer networks are an abstract representation of interconnection among nodes representing individuals or agents, where the interconnection has a multiple nature. For example, while a disease can propagate among individuals through a physical contact network, information can propagate among the same individuals through an online information-dissemination network. Another example is viral information dissemination among users of online social networks; one might disseminate information received from a Facebook contact to his or her followers on Twitter. Interconnected networks are abstract representations where two or more simple networks, possibly with different dynamics over them, are interconnected to each other. For example, in zoonotic diseases, a virus can move from the network of animals, with some transmission dynamics, to a human network, with possibly very different dynamics. As communication systems are evolving more and more toward integration with computing, sensing, and control systems, the theory of multilayer and interconnected networks seems to be crucial to successful communication systems development in cyber-physical infrastructures. Among the most relevant dynamics over networks is epidemic spreading. Epidemic spreading dynamics over simple networks exhibit a clear example where interaction between non-complex dynamics at node level and the topology leads to a complex emergent behavior. A substantial line of research during the past decade has been devoted to capturing the role of the network on spreading dynamics, and mathematical tools such as spectral graph theory have been greatly useful for this goal. For example, when the network is a simple graph, the dominant eigenvalue and eigenvector of the adjacency matrix have been proven to be key elements determining spreading dynamics features, including epidemic threshold, centrality of nodes, localization of spreading sites, and behavior of the epidemic model close to the threshold. More generally, for many other dynamics over a single network, dependency of dynamics on spectral properties of the adjacency matrix, Laplacian matrix, or some other graph-related matrix, is well-studied and rigorously established, and practical applications have been successfully derived. In contrast, limited established results exist for dynamics on multilayer and interconnected networks. Yet, an understanding of spreading processes over these networks is very important to several realistic phenomena in modern integrated and composite systems, including cascading failure in power grids, financial contagions in trade market, synchronization, spread of social opinion and trends, product adoption and market penetration, infectious disease pandemics, and outbreak in computer worms. This dissertation focuses on spreading processes on multilayer and interconnected networks, organized in three parts. The first part develops a general framework for modeling epidemic spreading in interconnected and multilayer networks. The second part solves two fundamental problems: introducing the concept of an epidemic threshold curve in interconnected networks, and coexistence phenomena in competitive spreading over multilayer networks. The third part of this dissertation develops an epidemic model incorporating human behavior, where multi-layer network formulation enables modeling and analysis of important features of human social networks, such as an information-dissemination network, as well as contact adaptation. Finally, I conclude with some open research directions in the topic of spreading processes over multilayer and interconnected networks, based on the resulting developments of this dissertation.

Master of Science / Department of Electrical and Computer Engineering / Caterina M. Scoglio / The generalized epidemic modeling framework simulator (GEMFsim) is a tool designed by Dr. Faryad Sahneh, former PhD student in the NetSE group. GEMFsim simulates stochastic spreading process over complex networks. It was first introduced in Dr. Sahneh’s doctoral dissertation "Spreading processes over multilayer and interconnected networks" and implemented in Matlab. As limited by Matlab language, this implementation typically solves only small networks; the slow simulation speed is unable to generate enough results in reasonable time for large networks. As a generalized tool, this framework must be equipped to handle large networks and contain sufficient support to provide adequate performance. The C language, a low-level language that effectively maps a program to machine in- structions with efficient execution, was selected for this study. Following implementation of GEMFsim in C, I packed it into Python and R libraries, allowing users to enjoy the flexibility of these interpreted languages without sacrificing performance. GEMFsim limitations are not limited to language, however. In the original algorithm (Gillespie’s Direct Method), the performance (simulation speed) is inversely proportional to network size, resulting in unacceptable speed for very large networks. Therefore, this study applied the Next Reaction Method, making the performance irrelevant of network size. As long as the network fits into memory, the speed is proportional to the average node degree of the network, which is not very large for most real-world networks. This study also applied parallel computing in order to advantageously utilize multiple cores for repeated simulations. Although single simulation can not be paralleled as a Markov process, multiple simulations with identical network structures were run simultaneously, sharing one network description in memory.

The complex network theory has been extensively used to understand various natural and artificial phenomena made of interconnected parts. This representation enables the study of dynamical processes running on complex systems, such as epidemics and rumor spreading. The evolution of these dynamical processes is influenced by the organization of the network. The size of some real world networks makes it prohibitive to analyse the whole network computationally. Thus it is necessary to represent it by a set of topological measures or to reduce its size by means of sampling. In addition, most networks are samples of a larger networks whose structure may not be captured and thus, need to be inferred from samples. In this work, we study both problems: the influence of the structure of the network in spreading processes and the effects of sampling in the structure of the network. Our results suggest that it is possible to predict the final fraction of infected individuals and the final fraction of individuals that came across a rumor by modeling them with a beta regression model and using topological measures as regressors. The most influential measure in both cases is the average search information, that quantifies the ease or difficulty to navigate through a network. We have also shown that the structure of a sampled network differs from the original network and that the type of change depends on the sampling method. Finally, we apply four sampling methods to study the behaviour of the epidemic threshold of a network when sampled with different sampling rates and found out that the breadth-first search sampling is most appropriate method to estimate the epidemic threshold among the studied ones. / Vários fenômenos naturais e artificiais compostos de partes interconectadas vem sendo estudados pela teoria de redes complexas. Tal representação permite o estudo de processos dinâmicos que ocorrem em redes complexas, tais como propagação de epidemias e rumores. A evolução destes processos é influenciada pela organização das conexões da rede. O tamanho das redes do mundo real torna a análise da rede inteira computacionalmente proibitiva. Portanto, torna-se necessário representá-la com medidas topológicas ou amostrá-la para reduzir seu tamanho. Além disso, muitas redes são amostras de redes maiores cuja estrutura é difícil de ser capturada e deve ser inferida de amostras. Neste trabalho, ambos os problemas são estudados: a influência da estrutura da rede em processos de propagação e os efeitos da amostragem na estrutura da rede. Os resultados obtidos sugerem que é possível predizer o tamanho da epidemia ou do rumor com base em um modelo de regressão beta com dispersão variável, usando medidas topológicas como regressores. A medida mais influente em ambas as dinâmicas é a informação de busca média, que quantifica a facilidade com que se navega em uma rede. Também é mostrado que a estrutura de uma rede amostrada difere da original e que o tipo de mudança depende do método de amostragem utilizado. Por fim, quatro métodos de amostragem foram aplicados para estudar o comportamento do limiar epidêmico de uma rede quando amostrada com diferentes taxas de amostragem. Os resultados sugerem que a amostragem por busca em largura é a mais adequada para estimar o limiar epidêmico entre os métodos comparados.

The complex network theory has been extensively used to understand various natural and artificial phenomena made of interconnected parts. This representation enables the study of dynamical processes running on complex systems, such as epidemics and rumor spreading. The evolution of these dynamical processes is influenced by the organization of the network. The size of some real world networks makes it prohibitive to analyse the whole network computationally. Thus it is necessary to represent it by a set of topological measures or to reduce its size by means of sampling. In addition, most networks are samples of a larger networks whose structure may not be captured and thus, need to be inferred from samples. In this work, we study both problems: the influence of the structure of the network in spreading processes and the effects of sampling in the structure of the network. Our results suggest that it is possible to predict the final fraction of infected individuals and the final fraction of individuals that came across a rumor by modeling them with a beta regression model and using topological measures as regressors. The most influential measure in both cases is the average search information, that quantifies the ease or difficulty to navigate through a network. We have also shown that the structure of a sampled network differs from the original network and that the type of change depends on the sampling method. Finally, we apply four sampling methods to study the behaviour of the epidemic threshold of a network when sampled with different sampling rates and found out that the breadth-first search sampling is most appropriate method to estimate the epidemic threshold among the studied ones. / Vários fenômenos naturais e artificiais compostos de partes interconectadas vem sendo estudados pela teoria de redes complexas. Tal representação permite o estudo de processos dinâmicos que ocorrem em redes complexas, tais como propagação de epidemias e rumores. A evolução destes processos é influenciada pela organização das conexões da rede. O tamanho das redes do mundo real torna a análise da rede inteira computacionalmente proibitiva. Portanto, torna-se necessário representá-la com medidas topológicas ou amostrá-la para reduzir seu tamanho. Além disso, muitas redes são amostras de redes maiores cuja estrutura é difícil de ser capturada e deve ser inferida de amostras. Neste trabalho, ambos os problemas são estudados: a influência da estrutura da rede em processos de propagação e os efeitos da amostragem na estrutura da rede. Os resultados obtidos sugerem que é possível predizer o tamanho da epidemia ou do rumor com base em um modelo de regressão beta com dispersão variável, usando medidas topológicas como regressores. A medida mais influente em ambas as dinâmicas é a informação de busca média, que quantifica a facilidade com que se navega em uma rede. Também é mostrado que a estrutura de uma rede amostrada difere da original e que o tipo de mudança depende do método de amostragem utilizado. Por fim, quatro métodos de amostragem foram aplicados para estudar o comportamento do limiar epidêmico de uma rede quando amostrada com diferentes taxas de amostragem. Os resultados sugerem que a amostragem por busca em largura é a mais adequada para estimar o limiar epidêmico entre os métodos comparados.

La science moderne des graphes est apparue ces dernières années comme un domaine d'intérêt et a apporté des progrès significatifs à notre connaissance des réseaux. Jusqu'à récemment, les algorithmes d'exploration de données existants étaient destinés à des données structurées / relationnelles, alors que de nombreux ensembles de données nécessitent une représentation graphique, comme les réseaux sociaux, les réseaux générés par des données textuelles, les structures protéiques 3D ou encore les composés chimiques. Il est donc crucial de pouvoir extraire des informations pertinantes à partir de ce type de données et, pour ce faire, les méthodes d'extraction et d'analyse des graphiques ont été prouvées essentielles.L'objectif de cette thèse est d'étudier les problèmes dans le domaine de la fouille de graphes axés en particulier sur la conception de nouveaux algorithmes et d'outils liés à la diffusion d'informations et plus spécifiquement sur la façon de localiser des entités influentes dans des réseaux réels. Cette tâche est cruciale dans de nombreuses applications telles que la diffusion de l'information, les contrôles épidémiologiques et le marketing viral.Dans la première partie de la thèse, nous avons étudié les processus de diffusion dans les réseaux sociaux ciblant la recherche de caractéristiques topologiques classant les entités du réseau en fonction de leurs capacités influentes. Nous nous sommes spécifiquement concentrés sur la décomposition K-truss qui est une extension de la décomposition k-core. On a montré que les noeuds qui appartiennent au sous-graphe induit par le maximal K-truss présenteront de meilleurs proprietés de propagation par rapport aux critères de référence. De tels épandeurs ont la capacité non seulement d'influencer une plus grande partie du réseau au cours des premières étapes d'un processus d'étalement, mais aussi de contaminer une plus grande partie des noeuds.Dans la deuxième partie de la thèse, nous nous sommes concentrés sur l'identification d'un groupe de noeuds qui, en agissant ensemble, maximisent le nombre attendu de nœuds influencés à la fin du processus de propagation, formellement appelé Influence Maximization (IM). Le problème IM étant NP-hard, il existe des algorithmes efficaces garantissant l’approximation de ses solutions. Comme ces garanties proposent une approximation gloutonne qui est coûteuse en termes de temps de calcul, nous avons proposé l'algorithme MATI qui réussit à localiser le groupe d'utilisateurs qui maximise l'influence, tout en étant évolutif. L'algorithme profite des chemins possibles créés dans le voisinage de chaque nœud et précalcule l'influence potentielle de chaque nœud permettant ainsi de produire des résultats concurrentiels, comparés à ceux des algorithmes classiques.Finallement, nous étudions le point de vue de la confidentialité quant au partage de ces bons indicateurs d’influence dans un réseau social. Nous nous sommes concentrés sur la conception d'un algorithme efficace, correct, sécurisé et de protection de la vie privée, qui résout le problème du calcul de la métrique k-core qui mesure l'influence de chaque noeud du réseau. Nous avons spécifiquement adopté une approche de décentralisation dans laquelle le réseau social est considéré comme un système Peer-to-peer (P2P). L'algorithme est construit de telle sorte qu'il ne devrait pas être possible pour un nœud de reconstituer partiellement ou entièrement le graphe en utilisant les informations obtiennues lors de son exécution. Notre contribution est un algorithme incrémental qui résout efficacement le problème de maintenance de core en P2P tout en limitant le nombre de messages échangés et les calculs. Nous fournissons également une étude de sécurité et de confidentialité de la solution concernant la désanonymisation des réseaux, nous montrons ainsi la rélation avec les strategies d’attaque précédemment definies tout en discutant les contres-mesures adaptés. / Modern science of graphs has emerged the last few years as a field of interest and has been bringing significant advances to our knowledge about networks. Until recently the existing data mining algorithms were destined for structured/relational data while many datasets exist that require graph representation such as social networks, networks generated by textual data, 3D protein structures and chemical compounds. It has become therefore of crucial importance to be able to extract meaningful information from that kind of data and towards this end graph mining and analysis methods have been proven essential. The goal of this thesis is to study problems in the area of graph mining focusing especially on designing new algorithms and tools related to information spreading and specifically on how to locate influential entities in real-world networks. This task is crucial in many applications such as information diffusion, epidemic control and viral marketing. In the first part of the thesis, we have studied spreading processes in social networks focusing on finding topological characteristics that rank entities in the network based on their influential capabilities. We have specifically focused on the K-truss decomposition which is an extension of the core decomposition of the graph. Extensive experimental analysis showed that the nodes that belong to the maximal K-truss subgraph show a better spreading behavior when compared to baseline criteria. Such spreaders can influence a greater part of the network during the first steps of a spreading process but also the total fraction of the influenced nodes at the end of the epidemic is greater. We have also observed that node members of such dense subgraphs are those achieving the optimal spreading in the network.In the second part of the thesis, we focused on identifying a group of nodes that by acting all together maximize the expected number of influenced nodes at the end of the spreading process, formally called Influence Maximization (IM). The IM problem is actually NP-hard though there exist approximation guarantees for efficient algorithms that can solve the problem while obtaining a solution within the 63% of optimal classes of models. As those guarantees propose a greedy approximation which is computationally expensive especially for large graphs, we proposed the MATI algorithm which succeeds in locating the group of users that maximize the influence while also being scalable. The algorithm takes advantage the possible paths created in each node’s neighborhood to precalculate each node’s potential influence and produces competitive results in quality compared to those of baseline algorithms such as the Greedy, LDAG and SimPath. In the last part of the thesis, we study the privacy point of view of sharing such metrics that are good influential indicators in a social network. We have focused on designing an algorithm that addresses the problem of computing through an efficient, correct, secure, and privacy-preserving algorithm the k-core metric which measures the influence of each node of the network. We have specifically adopted a decentralization approach where the social network is considered as a Peer-to-peer (P2P) system. The algorithm is built based on the constraint that it should not be possible for a node to reconstruct partially or entirely the graph using the information they obtain during its execution. While a distributed algorithm that computes the nodes’ coreness is already proposed, dynamic networks are not taken into account. Our main contribution is an incremental algorithm that efficiently solves the core maintenance problem in P2P while limiting the number of messages exchanged and computations. We provide a security and privacy analysis of the solution regarding network de-anonimization and show how it relates to previously defined attacks models and discuss countermeasures.

Kollektive biologische Systeme sammeln Informationen und leiten diese intern weiter, um Umweltveränderungen zu detektieren und auf sie zu reagieren. In Tiergruppen können die probabilistischen Entscheidungen von Individuen durch diese kollektive Informationsverarbeitung verbessert werden. Die dem sozialen Austausch zu Grunde liegenden Sinneswahrnehmungen finden jedoch in gängigen Modellen kollektiven Verhaltens kaum Beachtung. Hier untersuche ich, wie der individuelle Zugang zu sozialen Informationen durch visuelle Wahrnehmung und räumliche Gruppenstruktur geformt wird. Zuerst untersuche ich Fluchtwellen in Fischschwärmen in zwei als unterschiedlich riskant wahrgenommenen Kontexten mithilfe empirisch ermittelter visueller Interaktionsnetzwerke. Die beobachtete strukturelle Änderung der Gruppen zwischen den Kontexten erweist sich als essenziell, um die Änderung der Fluchtwellengröße zu erklären und optimiert potenziell die kollektive Informationsverarbeitung im jeweiligen Kontext. Von optimaler Informationsverarbeitung wird in biologischen Systemen oft angenommen, dass sie an Phasenübergängen in deren kollektiver Dynamik stattfindet, sogenannten kritischen Punkten. Die beobachtete strukturelle Änderung ändert den Abstand des Schwarmverhaltens zu einem solchen kritischen Punkt. Jedoch bleiben die Gruppen subkritisch in beiden Kontexten, vermutlich aus Notwendigkeit, tatsächliche Warnungen zu verstärken und falsche zu unterdrücken. Im zweiten Teil vergleiche ich visuelle Netzwerke mit anderen räumlichen Netzwerken bezüglich ihrer Struktur und dem Verhalten von Ausbreitungsprozessen auf ihnen. Einzig visuelle Netzwerke zeigen bei mittleren Gruppendichten Optima in zentralen Netzwerkeigenschaften und behalten realistische Eigenschaften bei hohen und niedrigen Dichten. Abschließend entwickle ich eine analytische Näherung zentraler Netzwerkeigenschaften solcher visuellen Netzwerke. / Collective biological systems gather information and propagate it internally to detect and react to environmental changes. In animal groups the probabilistic decisions of individuals can be improved by this collective information processing. Animals rely on sensory cues for social communication, yet common models of collective behavior neglect this sensory basis of interactions. Here, I investigate how an individual’s access to social information is shaped by visual sensory limitations and spatial group structure. First, escape waves in fish schools are studied under two levels of perceived environmental risk using empirically inferred visual interaction networks. Group-structural change is found to be crucial to explain the observed differences in size of escape waves and potentially optimize collective information processing according to the state of the environment. Optimal information processing in biological systems is often hypothesized to occur at phase transitions in their collective dynamics, so-called critical points. Here, the observed change in group structure modifies the schools’ distance to a critical point. Yet groups stay subcritical in both experimental setups, which may manage a trade-off between sensitivity to true alarms and robustness to false ones. In a second part, visual networks are compared to other spatial networks in structure and behavior of spreading processes on them. Visual networks show a unique dependence on group density with optima in network structural measures at intermediate densities, making them more realistic than other networks at high and low densities. Finally, an analytical approximation of central properties of visual networks is developed. Overall, this thesis identifies group structure as a potential control mechanism of collective information processing, highlights the trade-off associated with criticality in noisy systems and provides a systematic study and analytic approximation of visual sensory networks.

Un grand nombre des problèmes d'optimisation, ainsi que des problèmes inverses, combinatoires ou hors équilibre qui apparaissent en physique statistique des systèmes complexes, peuvent être représentés comme un ensemble des variables en interaction sur un certain réseau. Bien que la recette universelle pour traiter ces problèmes n'existe pas, la compréhension qualitative et quantitative des problèmes complexes sur des graphes a fait des grands progrès au cours de ces dernières années. Un rôle particulier a été joué par des concepts empruntés de la physique des verres de spin et la théorie des champs, qui ont eu beaucoup de succès en ce qui concerne la description des propriétés statistiques des systèmes complexes et le développement d'algorithmes efficaces pour des problèmes concrets.En première partie de cette thèse, nous étudions des problèmes de diffusion sur des réseaux, avec la dynamique hors équilibre. En utilisant la méthode de cavité sur des trajectoires dans le temps, nous montrons comment dériver des équations dynamiques dites "message-passing'' pour une large classe de modèles avec une dynamique unidirectionnelle -- la propriété clef qui permet de résoudre le problème. Ces équations sont asymptotiquement exactes pour des graphes localement en arbre et en général représentent une bonne approximation pour des réseaux réels. Nous illustrons cette approche avec une application des équations dynamiques pour résoudre le problème inverse d'inférence de la source d'épidémie dans le modèle "susceptible-infected-recovered''.Dans la seconde partie du manuscrit, nous considérons un problème d'optimisation d'appariement planaire optimal sur une ligne. En exploitant des techniques de la théorie de champs et des arguments combinatoires, nous caractérisons une transition de phase topologique qui se produit dans un modèle désordonné simple, le modèle de Bernoulli. Visant une application à la physique des structures secondaires de l'ARN, nous discutons la relation entre la transition d'appariement parfait-imparfait et la transition de basse température connue entre les états fondu et vitreux de biopolymère; nous proposons également des modèles généralisés qui suggèrent une correspondance exacte entre la matrice des contacts et la séquence des nucléotides, permettant ainsi de donner un sens à la notion des alphabets effectifs non-entiers. / A large number of optimization, inverse, combinatorial and out-of-equilibrium problems, arising in the statistical physics of complex systems, allow for a convenient representation in terms of disordered interacting variables defined on a certain network. Although a universal recipe for dealing with these problems does not exist, the recent years have seen a serious progress in understanding and quantifying an important number of hard problems on graphs. A particular role has been played by the concepts borrowed from the physics of spin glasses and field theory, that appeared to be extremely successful in the description of the statistical properties of complex systems and in the development of efficient algorithms for concrete problems.In the first part of the thesis, we study the out-of-equilibrium spreading problems on networks. Using dynamic cavity method on time trajectories, we show how to derive dynamic message-passing equations for a large class of models with unidirectional dynamics -- the key property that makes the problem solvable. These equations are asymptotically exact for locally tree-like graphs and generally provide a good approximation for real-world networks. We illustrate the approach by applying the dynamic message-passing equations for susceptible-infected-recovered model to the inverse problem of inference of epidemic origin. In the second part of the manuscript, we address the optimization problem of finding optimal planar matching configurations on a line. Making use of field-theory techniques and combinatorial arguments, we characterize a topological phase transition that occurs in the simple Bernoulli model of disordered matching. As an application to the physics of the RNA secondary structures, we discuss the relation of the perfect-imperfect matching transition to the known molten-glass transition at low temperatures, and suggest generalized models that incorporate a one-to-one correspondence between the contact matrix and the nucleotide sequence, thus giving sense to the notion of effective non-integer alphabets.