arXiv subjects

Mark Yarrow

Publications and source records attributed to Mark Yarrow.

4 recordsLinked to original sources

arXiv2022

Competing types in preferential attachment graphs with community structure

We extend the two-type preferential attachment model of Antunovi\'c, Mossel and R\'acz, where each new vertex takes its type according to a defined rule based on the types of its neighbours, to incorporate community structure, and investigate whether the proportions of vertices of each type synchronise between communities. The behaviour depends both on the choice of community structure and on the type assignment rule. For essentially all cases where the single community model has more than one possible limit, communities may fail to synchronise for weakly interacting communities. Even when the single community model almost surely converges to a deterministic limit, synchronisation is not guaranteed. However, we give natural conditions on the assignment rule and, for two communities, on the structure, either of which will imply synchronisation to this limit, and each of which is essentially best possible. We also give an example where the proportions of types almost surely do not converge, which is impossible in the single community model.

math.PR↗

arXiv2019

Preferential attachment with location-based choice: Degree distribution in the noncondensation phase

We consider the preferential attachment model with location-based choice introduced by Haslegrave, Jordan and Yarrow as a model in which condensation phenomena can occur [Haslegrave et al. 2020]. In this model every vertex carries an independent and uniformly drawn location. Starting from an initial tree the model evolves in discrete time. At every time step, a new vertex is added to the tree by selecting $r$ candidate vertices from the graph with replacement according to a sampling probability proportional to these vertices' degrees. The new vertex then connects to one of the candidates according to a given probability associated to the ranking of their locations. In this paper, we introduce a function that describes the phase transition when condensation can occur. Considering the noncondensation phase, we use stochastic approximation methods to investigate bounds for the (asymptotic) proportion of vertices inside a given interval of a given maximum degree. We use these bounds to observe a power law for the asymptotic degree distribution described by the aforementioned function. Hence, this function fully characterises the properties we are interested in. The power law exponent takes the critical value one at the phase transition between the condensation - noncondensation phase.

math.PR↗

arXiv2018

Graphs that are cospectral for the distance Laplacian

The distance matrix $\mathcal{D}(G)$ of a graph $G$ is the matrix containing the pairwise distances between vertices, and the distance Laplacian matrix is $\mathcal{D}^L(G)=T(G)-\mathcal{D}(G)$, where $T(G)$ is the diagonal matrix of row sums of $\mathcal{D}(G)$. We establish several general methods for producing $\mathcal{D}^L$-cospectral graphs that can be used to construct infinite families. We provide examples showing that various properties are not preserved by $\mathcal{D}^L$-cospectrality, including examples of $\mathcal{D}^L$-cospectral strongly regular and circulant graphs. We establish that the absolute values of coefficients of the distance Laplacian characteristic polynomial are decreasing, i.e., $|\delta^L_{1}|\geq \dots \geq |\delta^L_{n}|$ where $\delta^L_{k}$ is the coefficient of $x^k$.

math.CO↗

arXiv2018

Condensation in preferential attachment models with location-based choice

We introduce a model of a preferential attachment based random graph which extends the family of models in which condensation phenomena can occur. Each vertex has an associated uniform random variable which we call its location. Our model evolves in discrete time by selecting $r$ vertices from the graph with replacement, with probabilities proportional to their degrees plus a constant $\alpha$. A new vertex joins the network and attaches to one of these vertices according to a given probability associated to the ranking of their locations. We give conditions for the occurrence of condensation, showing the existence of phase transitions in $\alpha$ below which condensation occurs. The condensation in our model differs from that in preferential attachment models with fitness in that the condensation can occur at a random location, that it can be due to a persistent hub, and that there can be more than one point of condensation.

math.PR↗