Local weak limits for collapsed branching processes with random out-degrees

TL;DR

The paper establishes local weak limits for CBP, a dynamic directed graph model formed by collapsing random-sized families from a Markovian birth process, showing that the local neighborhood of a typical vertex converges to a marked CTBP stopped at an independent exponential time with rate $\lambda$. A precise strong coupling is constructed between CBP in-components and the limiting CTBP, which implies that finite collections of uniformly chosen vertices converge to i.i.d. copies of the limit. This framework unifies and extends local limit descriptions for directed preferential and uniform attachment, and yields tractable descriptions of the joint in-degree and PageRank distributions, including tail bounds and phase transitions controlled by the attachment function via functionals $G_k$. The results provide a robust analytic tool for understanding the local geometry and centrality measures in evolving networks and open avenues for refined asymptotics and joint limits of in- and out-components. Overall, the work deepens the connection between dynamic random graphs and continuous-time branching processes, with potential applications to PageRank analysis and tail behavior in complex networks.

Abstract

We obtain local weak limits in probability for Collapsed Branching Processes (CBP), which are directed random networks obtained by collapsing random-sized families of individuals in a general continuous-time branching process. The local weak limit of a given CBP, as the network grows, is shown to be a related continuous-time branching process stopped at an independent exponential time. The proof involves the construction of an explicit coupling of the in-components of vertices with the limiting object. We also show that the in-components of a finite collection of uniformly chosen vertices locally weakly converge (in probability) to i.i.d. copies of the above limit, reminiscent of propagation of chaos in interacting particle systems. We obtain as special cases novel descriptions of the local weak limits of directed preferential and uniform attachment models. We also outline some applications of our results for analyzing the limiting in-degree and PageRank distributions. In particular, upper and lower bounds on the tail of the in-degree distribution are obtained and a phase transition is detected in terms of the growth rate of the attachment function governing reproduction rates in the branching process.

Figures (3)

• Figure 1: Collapsed branching process. On the left the tree $\mathcal{T}(\sigma_{S_6})$, on the right the corresponding graph $G(V_6, E_6)$.
• Figure 2: The lifted tree $\mathcal{T}(\sigma_{S_n})$ and the in-component of vertex $i$, $\mathcal{G}_i^{(n)}$. Nodes in each family $V(j)$, $j \geq 1$ are depicted as if they were all born at the same time and are labeled according to the vertices of $G(V_n, E_n)$ they give rise to. In this figure, $V(1) = \{1,2\}$, $V(i) = \{S_i\}$ and $V(n) = \{S_n\}$. This depiction of $\mathcal{G}_i^{(n)}$ shows a successful coupling with its local limit.
• Figure 3: Coupling of $\mathcal{G}_i^{(n)}$ and $\mathcal{T}_i^c(t_{n,i})$. On the left we have a depiction of $\mathcal{G}_i^{(n)}$, and on the right, its coupled tree $\mathcal{T}_i^c(t_{n,i})$. Vertices $i, \kappa_i(2)$, and $\kappa_i(5)$ are in $J_i$, while vertex $\kappa_i(3)$ is in $J_i^*$. Vertex $\kappa_i(4)$ is not in $J_i \cup J_i^*$ since it was skipped in step 3(b). The miscoupling caused by vertex $\kappa_i(3)$ generated one 'dummy node' in $\mathcal{T}_i^c(t_{n,i})$ with an independently generated mark, the one depicted as an offspring of node $2$. Vertex $\kappa_i(5)$ did not cause a miscoupling since it did not create a self-loop nor did it attach to more than one vertex in $J_i$.

Theorems & Definitions (32)

• Remark 1
• definition 1
• Theorem 1
• Corollary 1
• Corollary 2
• Proposition 1
• Proposition 2
• Remark 2
• Proposition 3
• Remark 3