Persistence of the Jordan center in Random Growing Trees

The Jordan center of a graph is defined as a vertex whose maximum distance to other nodes in the graph is minimal, and it finds applications in facility location and source detection problems. We study properties of the Jordan Center in the case of random growing trees. In particular, we consider a regular tree graph on which an infection starts from a root node and then spreads along the edges of the graph according to various random spread models. For the Independent Cascade (IC) model and the discrete Susceptible Infected (SI) model, both of which are discrete time models, we show that as the infected subgraph grows with time, the Jordan center persists on a single vertex after a finite number of timesteps. Finally, we also study the continuous time version of the SI model and bound the maximum distance between the Jordan center and the root node at any time.Comment: 28 pages, 14 figure

Moments of general time dependent branching processes with applications

In this paper, we give sufficient conditions for a Crump-Mode-Jagers process to be bounded in $L_k$ for a given $k>1$. This result is then applied to a recent random graph process motivated by pairwise collaborations and driven by time-dependent branching dynamics. We show that the maximal degree has the same rate of increase as the degree process of a fixed vertex.Comment: 12 page

Attribute network models, stochastic approximation, and network sampling and ranking algorithms

We analyze dynamic random network models where younger vertices connect to older ones with probabilities proportional to their degrees as well as a propensity kernel governed by their attribute types. Using stochastic approximation techniques we show that, in the large network limit, such networks converge in the local weak sense to randomly stopped multitype branching processes whose explicit description allows for the derivation of asymptotics for a wide class of network functionals. These asymptotics imply that while degree distribution tail exponents depend on the attribute type (already derived by Jordan (2013)), Page-rank centrality scores have the \emph{same} tail exponent across attributes. Moreover, the mean behavior of the limiting Page-rank score distribution can be explicitly described and shown to depend on the attribute type. The limit results also give explicit formulae for the performance of various network sampling mechanisms. One surprising consequence is the efficacy of Page-rank and walk based network sampling schemes for directed networks in the setting of rare minorities. The results also allow one to evaluate the impact of various proposed mechanisms to increase degree centrality of minority attributes in the network, and to quantify the bias in inferring about the network from an observed sample. Further, we formalize the notion of resolvability of such models where, owing to propagation of chaos type phenomenon in the evolution dynamics for such models, one can set up a correspondence to models driven by continuous time branching process dynamics.Comment: 48 page

Evolving Inhomogeneous Random Structures

We introduce general models of evolving, inhomogeneous random structures, where in each of the models either one or several nodes arrive at a time, and are equipped with random, independent weights. In the two evolving tree models we study, an existing vertex is chosen at each time-step with probability proportional to its fitness function, which is a function of its weight, and possibly the weights of its neighbours, and the newly arriving node(s) connect to it. The third models, with parameter $d$ consist of evolving sequences of $(d-1)$-dimensional simplicial complexes. At each time-step a $(d-1)$-simplex is sampled with probability proportional to a function of the weights of the vertices the $(d-1)$-simplex contains. In both variants, Model~\textbf{A} and Model~\textbf{B}, for each subset $S$ of size $(d-2)$, we add the simplex consisting of $S$ and the single new-coming vertex. Additionally, in Model~\textbf{B}, the selected simplex is removed from the simplicial complex. In each of the models we study the limiting proportion of vertices in the structure with a given degree, showing that, in general, this limit exists in probability, and behaves like a type of \emph{generalised geometric distribution}. In the evolving tree models, we actually study a more general quantity: the empirical measures associated with the number of vertices with a given degree and weight. With regards to this quantity, when normalised by the size of the network, we also show that the limit exists and belongs to a certain universal class. Depending on various assumptions, we prove that for any measurable set, the measure of that set converges either almost surely or in probability to its measure under this deterministic limit. In the evolving tree models, we also study another quantity: the empirical measure corresponding to the proportion of edges in the structure with endpoint having a given weight. We show that, when normalised by the number of edges in the tree, under certain assumptions, this quantity also converges to a deterministic limiting measure, in the sense that for any measurable set, the measure of that set converges either almost surely. However, when the trees take certain forms, which we call the GPAF-tree, or the PANI-tree, we show that interesting, non-trivial behaviour can emerge when these assumptions fail. In particular, with regards to the GPAF-tree, we show that this model can exhibit condensation where a positive proportion of edges accumulate around vertices with weight that maximises the reinforcement of their fitness, or, more drastically, have a degenerate limiting degree distribution where the entire proportion of edges accumulate around these vertices. We also show that the condensation phenomenon extends to the more general PANI-tree model. As we will show, the latter two models have limiting distribution of degrees that behaves like an `averaged' power law, which may be of interest when considering them as toy models for the evolution of complex networks