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Teoremi di limite classici e non classici per il modello di Ising su grafi random
No full textIl modello di Ising, introdotto per comprendere le proprietà dei ferromagneti, è uno dei più classici modelli studiati in fisica. Oggi c'è un crescente interesse per il modello di Ising su grafi random, in particolare per le sue applicazioni in diversi campi. In fisica statistica, infatti, questo modello è un paradigma per analizzare fenomeni cooperativi su networks.
Il modello di Ising su grafi random è anche un modello ideale in probabilità per studiare risultati asintotici in presenza di una struttura di dipendenza tra le variabili di Ising e un ulteriore livello di disordine dato dai grafi random. Tenendo conto di questa doppia fonte di disordine in diversi modi, definiamo tre differenti misure.
Per una data assegnazione del grafo la misura Random Quenched coincide con la distribuzione di Boltzmann-Gibbs su tale grafo. Mediando su tutti i possibili grafi random in due modi diversi otteniamo la misura Average Quenched e la misura Annealed.
Lo scopo di questa tesi è quello di derivare risultati asintotici, come la legge dei grandi numeri e il teorema del limite centrale, per la somma delle variabili di spin del modello Ising su grafi random, rispetto alle tre diverse misure.
Nel caso random quenched i nostri risultati si applicano alla classe dei grafi random locally tree-like. I teoremi sono formulati nel regime di unicità fuori della regione critica. Per prima cosa studiamo la legge dei grandi numeri e poi dimostramo il teorema del limite centrale. Come nel caso del modello di Ising su reticolo, la varianza che appare nel teorema del limite centrale è data dalla suscettibilità del modello.
Purtroppo, la strategia generale utilizzata per i grafi random locally tree-like nel caso random quenched non funziona per il CLT nei settings average quanched e annealed. In questi due casi le fluttuazioni dovute alla struttura spaziale diventano rilevanti. Quindi studiamo il modello di Ising su particolari esempi di grafi random.
Nel caso average quenched consideriamo due speciali modelli: il 2-regular configuration model e il configuration model con gradi 1 e 2. In entrambi i casi le nostre prove si basano su calcoli espliciti relativi alla soluzione del classico modello di Ising unidimensionale.
Per quanto riguarda la misura annealed, oltre ai due configuration models precedentemente menzionati, esaminiamo il generalized random graph, dove è presente una transizione di fase. Le nostre dimostrazioni si basano ancora su calcoli espliciti. Nel caso annealed, il modello Ising sul generalized random graph viene ricondotto ad un modello Curie-Weiss inomogeneo. Lavorando nel regime di unicità, calcoliamo prima il limite delle quantità termodinamiche, individuando una temperatura critica annealed, e poi dimostriamo la legge dei grandi numeri e il teorema del limite centrale.
La nostra analisi mostra che le fluttuazioni del grafo random svolgono un ruolo cruciale nei settings average quenched e annelaed. In particolare, la varianza che appare nel teorema del limite centrale è in genere influenzata dalle fluttuazioni del grafo.
Infine analizziamo cosa accade al punto critico. Qui la varianza diverge, quindi il CLT non è più valido ed è necessaria un diverso riscalamento del total spin per ottenere una distribuzione limite non banale.
Investighiamo questo problema per il modello di Ising sul generalized random graph nel setting annealed. Dimostriamo che gli esponenti critici per questo modello corrispondono a quelli di un modello di Ising su grafi random locally tree-like nel caso random quenched. Inoltre, dimostriamo un teorema limite non classico. La dimostrazione mostra che abbiamo bisogno di riscalamenti differenti per la somma degli spin a seconda che la distribuzione dei gradi abbia il momento quarto finito o infinito.The Ising model, introduced to understand ferromagnets, is one of the most classical models studied in physics. Today there is a rising interest in the Ising model on random graphs, mainly due to applications in different fields. Indeed this model is a paradigm model in statistical physics to analyze the effect of cooperative phenomena on networks .
The Ising model on random graphs is also an ideal model in probability to study asymptotic results in the presence of two sources of randomness: a dependence structure between the Ising random variables and an extra level of randomness given by the random graphs. Taking this double randomness into account in various ways three different measures arise.
For a given assignment of the graph the Random Quenched measure coincides with the random Boltzmann-Gibbs distribution on that random graph. Averaging over all possible random graphs in two different ways we obtain the Averaged Quenched and the Annealed measures.
The aim of this thesis is to derive asymptotic results for the sum of the spin variables of the Ising model on random graphs, like the law of large numbers and the central limit theorem, with respect to the three different measures.
In the random quenched setting our results apply to the class of locally tree-like random graphs. The theorems are stated in the uniqueness regime outside of the critical region. We first establish the rate at which the law of large numbers is reached and then we prove a central limit theorem. As in the case of the Ising model on lattice, the variance appearing in the central limit theorem is given by the susceptibility of the model.
Unfortunately, the general strategy used for the local tree-like random graphs in the random quenched setting does not work for the averaged quenched and annealed CLT. In these two settings the fluctuations due to the spatial structure become relevant. Therefore we study the Ising model on particular examples of random graphs.
In the average quenched setting we restrict ourselves to two special models: the 2-regular configuration model and the configuration model with degrees 1 and 2. In both these cases our proofs are based on explicit computations relying on the solution of the classical one-dimensional Ising model.
Regarding the annealed measure, besides the two configuration models aforementioned ,we investigate the generalized random graph, where a phase transition is present. Our proofs are based again on explicit computations. In the annealed approximation, the Ising model on the generalized random graph is reduced to an inhomogeneous Curie-Weiss model. Working in the uniqueness regime, we first compute the limit of thermodynamic quantities, identifying a critical annealed temperature, and then we prove the law of large number and the central limit theorem.
Our analysis shows that fluctuations of the random graph play a crucial role in the averaged quenched and annealed set-up. In particular, the variance of the Gaussian limiting law of the observables satisfying a central limit theorem is in general affected by the graph fluctuations.
Finally we analyze what happens at the critical point. Here the variance diverges, so the CLT breaks down and a different scaling of the total spin is needed to obtain a non-trivial limiting distribution.
We investigate this problem for the Ising model on generalized random graph in the annealed setting. We show that the critical exponents for this model match those of an Ising model on locally tree-like random graphs in the random quenched setting. Further, we prove a non-classical limit theorem. The proof reveals that we need different scalings for the total spin according to whether the degree distribution has a finite fourth moment or not
On the modeling of learning dynamics in large living systems
No full textThis paper deals with the modeling of learning dynamics in a large system of interacting entities. The mathematical approach is based on the kinetic theory on active particles. Their microscopic state is modeled by a scalar variable called activity, which is assumed to be heterogeneously distributed among the particles. Nonlinear interactions lead to collective phenomena of learning. The structure allows the derivation of specic models and of numerical simulations related to real systems
Quenched Central Limit Theorems for the Ising Model on Random Graphs
Full text linkThemain goal of the paper is to prove central limit theorems for the magnetization
rescaled by the square root of N for the Ising model on random graphs with N vertices.Both random quenched
and averaged quenched measures are considered.We work in the uniqueness regime β > βc
or β > 0 and B not equal to 0, where β is the inverse temperature, βc is the critical inverse temperature
and B is the external magnetic field. In the random quenched setting our results apply to
general tree-like random graphs (as introduced by Dembo, Montanari and further studied by
Dommers and the first and third author) and our proof follows that of Ellis in Z^d. For the
averaged quenched setting, we specialize to two particular random graph models, namely
the 2-regular configuration model and the configuration model with degrees 1 and 2. In
these cases our proofs are based on explicit computations relying on the solution of the one
dimensional Ising model
Annealed central limit theorems for the Ising model on random graphs
Full text linkThe aim of this paper is to prove central limit theorems with respect to the annealed measure for the magnetization rescaled
by of Ising models on random graphs. More precisely, we consider the general rank-1 inhomogeneous random graph (or generalized random graph), the 2-regular configuration model and the configuration model with degrees 1 and 2. For the generalized random graph, we first show the existence of a finite annealed inverse critical temperature 0le eta^{mathrm{an}}_c < infty and then prove our results in the uniqueness regime, i.e., the values of inverse temperature and external magnetic field for which either eta<eta^{mathrm{an}}_c and , or eta>0 and . In the case of the configuration model, the central limit theorem holds in the whole region of the parameters and , because phase transitions do not exist for these systems as they are closely related to one-dimensional Ising models. Our proofs are based on explicit computations that are possible since the Ising model on the generalized random graph in the annealed setting is reduced to an inhomogeneous Curie-Weiss model, while the analysis of the configuration model with degrees only taking values 1 and 2 relies on that of the classical one-dimensional Ising model
Ising Critical Behavior of Inhomogeneous Curie-Weiss Models and Annealed Random Graphs
No full textWe study the critical behavior for inhomogeneous versions of the Curie-Weiss model, where the coupling constant Jij(β) for the edge ij on the complete graph is given by Jij(β) = βwiwj/ (∑ k∈[N]wk). We call the product form of these couplings the rank-1 inhomogeneous Curie-Weiss model. This model also arises [with inverse temperature β replaced by sinh (β) ] from the annealed Ising model on the generalized random graph. We assume that the vertex weights (wi)i∈[N] are regular, in the sense that their empirical distribution converges and the second moment converges as well. We identify the critical temperatures and exponents for these models, as well as a non-classical limit theorem for the total spin at the critical point. These depend sensitively on the number of finite moments of the weight distribution. When the fourth moment of the weight distribution converges, then the critical behavior is the same as on the (homogeneous) Curie-Weiss model, so that the inhomogeneity is weak. When the fourth moment of the weights converges to infinity, and the weights satisfy an asymptotic power law with exponent τ with τ∈ (3 , 5) , then the critical exponents depend sensitively on τ. In addition, at criticality, the total spin SN satisfies that SN/ N(τ-2)/(τ-1) converges in law to some limiting random variable whose distribution we explicitly characterize
Going Beyond Counting First Authors in Author Co-citation Analysis
Full text linkThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
Variations on the Author
Full text link“Variations on the Author” discusses two of Eduardo Coutinho’s recent films (Um Dia na Vida, from 2010, and Últimas Conversas, posthumously released in 2015) and their contribution to the general question of documentary authorship. The director’s filmography is characterized by a consistent yet self-effacing form of authorial self-inscription: Coutinho often features as an interviewer that rather than express opinions propels discourses; an interviewer that is good at listening. This mode of self-inscription characterizes him as an author who is not expressive but who is nonetheless markedly present on the screen. In Um Dia na Vida, however, Coutinho is completely absent form the image, while Últimas Conversas, on the contrary, includes a confessional prologue that moves the director from the margins to the center of his films. This article examines the ways in which these works stand out in the filmography of a director who offers new insights into the notion of cinematic authorship
Appropriate Similarity Measures for Author Cocitation Analysis
Full text linkWe provide a number of new insights into the methodological discussion about author cocitation analysis. We first argue that the use of the Pearson correlation for measuring the similarity between authors’ cocitation profiles is not very satisfactory. We then discuss what kind of similarity measures may be used as an alternative to the Pearson correlation. We consider three similarity measures in particular. One is the well-known cosine. The other two similarity measures have not been used before in the bibliometric literature. Finally, we show by means of an example that our findings have a high practical relevance.information science;Pearson correlation;cosine;similarity measure;author cocitation analysis
Dispelling the Myths Behind First-author Citation Counts
Full text linkWe conducted a full-scale evaluative citation analysis study of scholars in the XML research field to explore just how different from each other author rankings resulting from different citation counting methods actually are, and to demonstrate the capability of emerging data and tools on the Web in supporting more realistic citation counting methods. Our results contest some common arguments for the continued
use of first-author citation counts in the evaluation of scholars, such as high correlations between author rankings by first-author citation counts and other citation
counting methods, and high costs of using more realistic citation counting methods that are not well-supported by the ISI databases. It is argued that increasingly available digital full text research papers make it possible for citation analysis studies to go beyond what the ISI databases have directly supported and to employ more
sophisticated methods
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