13 research outputs found
Stability of a growth process generated by monomer filling with nearest-neighbour cooperative effects
We study stability of a growth process generated by sequential adsorption of
particles on a one-dimensional lattice torus, that is, the process formed by
the numbers of adsorbed particles at lattice sites, called heights. Here the
stability of process, loosely speaking, means that its components grow at
approximately the same rate. To assess stability quantitatively, we investigate
the stochastic process formed by differences of heights.
The model can be regarded as a variant of a Polya urn scheme with local
geometric interaction
Phase transitions in non-linear urns with interacting types
We investigate reinforced non-linear urns with interacting types, and show that where there are three interacting types there are phenomena which do not occur with two types. In a model with three types where the interactions between the types are symmetric, we show the existence of a double phase transition with three phases: as well as a phase with an almost sure limit where each of the three colours is equally represented and a phase with almost sure convergence to an asymmetric limit, which both occur with two types, there is also an intermediate phase where both symmetric and asymmetric limits are possible. In a model with anti-symmetric interactions between the types, we show the existence of a phase where the proportions of the three colours cycle and do not converge to a limit, alongside a phase where the proportions of the three colours can converge to a limit where each of the three is equally represented
Generating Preferential Attachment Graphs via a P\'olya Urn with Expanding Colors
We introduce a novel preferential attachment model using the draw variables
of a modified P\'olya urn with an expanding number of colors, notably capable
of modeling influential opinions (in terms of vertices of high degree) as the
graph evolves. Similar to the Barab\'asi-Albert model, the generated graph
grows in size by one vertex at each time instance; in contrast however, each
vertex of the graph is uniquely characterized by a color, which is represented
by a ball color in the P\'olya urn. More specifically at each time step, we
draw a ball from the urn and return it to the urn along with a number
(potentially time-varying and non-integer) of reinforcing balls of the same
color; we also add another ball of a new color to the urn. We then construct an
edge between the new vertex (corresponding to the new color) and the existing
vertex whose color ball is drawn. Using color-coded vertices in conjunction
with the time-varying reinforcing parameter allows for vertices added (born)
later in the process to potentially attain a high degree in a way that is not
captured in the Barab\'asi-Albert model. We study the degree count of the
vertices by analyzing the draw vectors of the underlying stochastic process. In
particular, we establish the probability distribution of the random variable
counting the number of draws of a given color which determines the degree of
the vertex corresponding to that color in the graph. We further provide
simulation results presenting a comparison between our model and the
Barab\'asi-Albert network.Comment: 21 pages, 6 figure
On a preferential attachment and generalized P贸lya's urn model
We study a general preferential attachment and P贸lya's urn model. At each step a new vertex is introduced, which can be connected to at most one existing vertex. If it is disconnected, it becomes a pioneer vertex. Given that it is not disconnected, it joins an existing pioneer vertex with probability proportional to a function of the degree of that vertex. This function is allowed to be vertex-dependent, and is called the reinforcement function. We prove that there can be at most three phases in this model, depending on the behavior of the reinforcement function. Consider the set whose elements are the vertices with cardinality tending a.s. to infinity. We prove that this set either is empty, or it has exactly one element, or it contains all the pioneer vertices. Moreover, we describe the phase transition in the case where the reinforcement function is the same for all vertices. Our results are general, and in particular we are not assuming monotonicity of the reinforcement function. Finally, consider the regime where exactly one vertex has a degree diverging to infinity. We give a lower bound for the probability that a given vertex ends up being the leading one, i.e. its degree diverges to infinity. Our proofs rely on a generalization of the Rubin construction given for edge-reinforced random walks, and on a Brownian motion embedding.Preferential attachment; Reinforcement processes; Species sampling sequence; P贸lya's urn process
Phase transitions in non-linear urns with interacting types
We investigate reinforced non-linear urns with interacting types, and show that where there are three interacting types there are phenomena which do not occur with two types. In a model with three types where the interactions between the types are symmetric, we show the existence of a double phase transition with three phases: as well as a phase with an almost sure limit where each of the three colours is equally represented and a phase with almost sure convergence to an asymmetric limit, which both occur with two types, there is also an intermediate phase where both symmetric and asymmetric limits are possible. In a model with anti-symmetric interactions between the types, we show the existence of a phase where the proportions of the three colours cycle and do not converge to a limit, alongside a phase where the proportions of the three colours can converge to a limit where each of the three is equally represented