Andrea Collevecchio

On a Preferential Attachment and Generalized Pólya's Urn Model

We study a general preferential attachment and Polyas 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.

On the transience of processes defined on Galton–Watson trees

We introduce a simple technique for proving the transience of certain processes defined on the random tree

On the Push&Pull Protocol for Rumor Spreading

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On the Push&Pull Protocol for Rumour Spreading: [Extended Abstract]

generated by a supercritical branching process. We prove the transience for once-reinforced random walks on

The Worm Process for the Ising Model is Rapidly Mixing

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Longest paths in random Apollonian networks and largest r-ary subtrees of random d-ary recursive trees

, that is, a generalization of a result of Durrett, Kesten and Limic [Probab. Theory Related Fields 122 (2002) 567–592]. Moreover, we give a new proof for the transience of a family of biased random walks defined on

The Probability of Nontrivial Common Knowledge

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On the Coupling Time of the Heat-Bath Process for the Fortuin–Kasteleyn Random–Cluster Model

. Other proofs of this fact can be found in [Ann. Probab. 16 (1988) 1229–1241] and [Ann. Probab. 18 (1990) 931–958] as part of more general results. A similar technique is applied to a vertex-reinforced jump process. A by-product of our result is that this process is transient on the 3-ary tree. Davis and Volkov [Probab. Theory Related Fields 128 (2004) 42–62] proved that a vertex-reinforced jump process defined on the b-ary tree is transient if b≥4 and recurrent if b=1. The case b=2 is still open.

On the push & pull protocol for rumour spreading

The asynchronous push&pull protocol, a randomized distributed algorithm for spreading a rumour in a graph G, is defined as follows. Independent exponential clocks of rate 1 are associated with the vertices of G, one to each vertex. Initially, one vertex of G knows the rumour. Whenever the clock of a vertex x rings, it calls a random neighbour y: if x knows the rumour and y does not, then x tells y the rumour (a push operation), and if x does not know the rumour and y knows it, y tells x the rumour (a pull operation). The average spread time of G is the expected time it takes for all vertices to know the rumour, and the guaranteed spread time of G is the smallest time t such that with probability at least 1 − 1∕n, after time t all vertices know the rumour. The synchronous variant of this protocol, in which each clock rings precisely at times 1, 2, …, has been studied extensively.

BOUNDS ON THE SPEED AND ON REGENERATION TIMES FOR CERTAIN PROCESSES ON REGULAR TREES

The asynchronous push&pull protocol, a randomized distributed algorithm for spreading a rumour in a graph G, is defined as follows. Independent exponential clocks of rate 1 are associated with the vertices of G, one to each vertex. Initially, one vertex of G knows the rumour. Whenever the clock of a vertex x rings, it calls a random neighbour y: if x knows the rumour and y does not, then x tells y the rumour (a push operation), and if x does not know the rumour and y knows it, y tells x the rumour (a pull operation). The average spread time of G is the expected time it takes for all vertices to know the rumour, and the guaranteed spread time of G is the smallest time t such that with probability at least 1 - 1/n, after time t all vertices know the rumour. The synchronous variant of this protocol, in which each clock rings precisely at times 1,2,..., has been studied extensively. We prove the following results for any n-vertex graph: in either version, the average spread time is at most linear even if only the pull operation is used, and the guaranteed spread time is within a logarithmic factor of the average spread time, so it is O(n log n). In the asynchronous version, both the average and guaranteed spread times are Omega(log n). We give examples of graphs illustrating that these bounds are best possible up to constant factors. We also prove the first theoretical relationships between the guaranteed spread times in the two versions. Firstly, in all graphs the guaranteed spread time in the asynchronous version is within an O(log n) factor of that in the synchronous version, and this is tight. Next, we find examples of graphs whose asynchronous spread times are logarithmic, but the synchronous versions are polynomially large. Finally, we show for any graph that the ratio of the synchronous spread time to the asynchronous spread time is O(n2/3).