In the last years, several authors studied a class of continuous-time semi-Markov processes obtained by time-changing Markov processes by hitting times of independent subordinators. Such processes are governed by integrodifferential convolution equations of generalized fractional type. The aim of this paper is to develop a discrete-time counterpart of such a theory and to show relationships and differences with respect to the continuous time case. We present a class of discrete-time semi-Markov chains which can be constructed as time-changed Markov chains and we obtain the related governing convolution type equations. Such processes converge weakly to those in continuous time under suitable scaling limits.

Let Φ be a uniformly distributed random k-SAT formula with n variables and m clauses. Non-rigorous statistical mechanics ideas have inspired a message passing algorithm called Belief propagation guided decimation for finding satisfying assignments of Φ. This algorithm can be viewed as an attempt at implementing a certain thought experiment that we call the decimation process. In this paper we identify a variety of phase transitions in the decimation process and link these phase transitions to the performance of the algorithm.

We give a common description of Simon, Barabási-Albert, II-PA and Price growth models, by introducing suitable random graph processes with preferential attachment mechanisms. Through the II-PA model, we prove the conditions for which the asymptotic degree distribution of the Barabási-Albert model coincides with the asymptotic in-degree distribution of the Simon model. Furthermore, we show that when the number of vertices in the Simon model (with parameter α) goes to infinity, a portion of them behave as a Yule model with parameters (λ, β) = (1 − α, 1), and through this relation we explain why asymptotic properties of a random vertex in Simon model, coincide with the asymptotic properties of a random genus in Yule model. As a by-product of our analysis, we prove the explicit expression of the in-degree distribution for the II-PA model, given without proof in [16]. References to traditional and recent applications of the these models are also discussed.MSC2010 : 05C80, 90B15.Furthermore, it should be noticed that Barabási-Albert model exhibits an asymptotic degree distribution that equals the Yule-Simon distribution in correspondence of a specific choice of the parameters and still presents power-law characteristics for more general choices of the parameters. The same happens also for other preferential attachment models.Yule, Simon and Barabási-Albert models share the preferential attachment paradigm that seems to play an important role in the explanation of the scale-freeness of real networks. However, the mathematical tools classically used in their analysis are different. This makes difficult to understand in which sense models producing very similar asymptotic distributions are actually related one another. Although often remarked and heuristically justified, no rigorous proofs exist clarifying conditions for such result. Different researchers from different disciplines, for example theoretical physicists and economists asked themselves about the relations between Simon, Barabási-Albert, Yule and also some other models closely related to these first three (sometimes confused in the literature under one of the previous names). Partial studies in this direction exist but there is still a lack of clarifying rigorous results that would avoid errors and would facilitate the extension of the models.The existing results refer to specific models and conditions but there is not a unitary approach to the problem. For instance, in [4], the authors compared the distribution of the number of occurrences of a different word in Simon model, when time goes to infinity, with the degree distribution in the Barabási-Albert model, when the number of vertices goes to infinity. In [21], an explanation relating the asymptotic distribution of the number of species in a random genus in Yule model and that of the number of different words in Simon model appears. More recently, following a heuristic argument, Simkin and Roychowdhury [20] gave a justification of the relation between Yule and Simon models.The aim of this paper is to study...

Complex networks in different areas exhibit degree distributions with heavy upper tail. A preferential attachment mechanism in a growth process produces a graph with this feature. We herein investigate a variant of the simple preferential attachment model, whose modifications are interesting for two main reasons: to analyze more realistic models and to study the robustness of the scale free behavior of the degree distribution.We introduce and study a model which takes into account two different attachment rules: a preferential attachment mechanism (with probability 1 − p) that stresses the rich get richer system, and a uniform choice (with probability p) for the most recent nodes. The latter highlights a trend to select one of the last added nodes when no information is available. The recent nodes can be either a given fixed number or a proportion (αn) of the total number of existing nodes. In the first case, we prove that this model exhibits an asymptotically power-law degree distribution. The same result is then illustrated through simulations in the second case. When the window of recent nodes has constant size, we herein prove that the presence of the uniform rule delays the starting time from which the asymptotic regime starts to hold.The mean number of nodes of degree k and the asymptotic degree distribution are also determined analytically. Finally, a sensitivity analysis on the parameters of the model is performed.Other models for different real world networks request the use of different growth paradigms and do not present the scale free property. For example, some networks exhibit the small-world phenomenon, in which sub-networks have connections between almost any two nodes within them. Furthermore, most pairs of nodes are connected by at least one short path. On the other hand, one of the most studied models, the Erdös-Rényi random graph, does not exhibit either the power-law behavior for the degree distribution of its nodes, nor the small-world phenomenon [4,12,13,14,15]. Mathematically, the growth of networks can be modeled through random graph processes, i.e. a family (G t ) t∈N of random graphs (defined on a common probability space), where t is interpreted as time. Different features of the model are then described as properties of the corresponding random graph process. In particular, the interest often focuses on the degree, deg(v, t), of a vertex v at time t, that is on the total number of incoming and (or) outgoing edges to and (or) from v, respectively. In this framework, new nodes of the Barabási-Albert model link with higher probabilities with nodes of higher degree. An important feature of preferential attachment models is an asymptotic power-law degree distribution: the fraction P (k) of vertices in the network with degree k, goes as P (k) ∼ k −γ , with γ > 0, for large values of k. Real world modeling instances motivated the proposal of generalizations of the Barabási-Albert model, see e.g. [3,6,7,10,16,17,18,20,22]. A common characteristic of many of these models is the presence of the...

We determine the asymptotic size of the largest component in the 2-type binomial random graph G(n, P ) near criticality using a refined branching process approach. In G(n, P ) every vertex has one of two types, the vector n describes the number of vertices of each type, and any edge {u, v} is present independently with a probability that is given by an entry of the probability matrix P according to the types of u and v.We prove that in the weakly supercritical regime, i.e. if the 'distance' to the critical point of the phase transition is given by an ε = ε(n) → 0, with probability 1 − o(1), the largest component in G(n, P ) contains asymptotically 2ε n 1 vertices and all other components are of size o(ε n 1 ).