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Limiting behaviour of Branching Processes and Online Social Networks

Khushboo Agarwal

TL;DR

The thesis develops a broad framework for total-current population-dependent branching processes (BPs) and analyzes their limiting behavior using stochastic-approximation (SA) techniques tied to autonomous ODEs. A novel hovering-around behavior near saddle points emerges as a potential limiting regime, extending SA-Lyapunov theory beyond traditional attractor convergence. The work introduces three BP variants relevant to online social networks: BP with attack (competition/ownership transfer), BP with unnatural deaths, and saturated total-population dependent BP (STP-BP) capturing re-forwarding saturation. These BP insights are translated into OSN applications, including robust fake-post detection via crowd signals and a participation mean-field game to incentivize truthful tagging, as well as a viral content analysis in competitive markets. Together, the study advances BP theory, SA methodology, and practical OSN design, offering deterministic ODE approximations, finite-time trajector tracking, and new equilibrium concepts for crowd-based information propagation.

Abstract

The literature considers multi-type Markov branching processes (BPs), where the offspring distribution depends only on the living (current) population. We analyse the total-current population-dependent BPs where the offspring distribution can also depend on the total (dead and living) population. Such a generalization is inspired by the need to accurately model content propagation over online social networks (OSNs). The key question investigated is the time-asymptotic proportion of the populations, which translates to the proportional visibility of the posts on the OSN. We provide the answer using a stochastic approximation (SA) technique, which has not been used in the existing BP literature. The analysis is derived using a non-trivial autonomous measurable ODE. Interestingly, we prove the possibility of a new limiting behaviour for the stochastic trajectory, named as hovering around. Such a result is not just new to the theory of BPs but also to the SA based literature. Later, we explore three new variants of BPs: (i) any living individual of a population can attack and acquire the living individuals of the other population, in addition to producing its offspring; (ii) the individuals can die due to abnormal circumstances, and not just at the completion of their lifetimes; (iii) the expected number of offspring decreases as the total-population increases, leading to the saturation of the total-population. Such variants aid in analysing unexplored aspects of content propagation over OSNs: (i) competition in advertisement posts for similar products; (ii) controlling fake-post propagation, while not affecting the sharing of real-post; (iii) impact of re-forwarding the posts. We also designed and analysed a participation (mean-field) game where the OSN lures the users with a reward-based scheme to provide their opinion about the actuality of the post (fake or real).

Limiting behaviour of Branching Processes and Online Social Networks

TL;DR

The thesis develops a broad framework for total-current population-dependent branching processes (BPs) and analyzes their limiting behavior using stochastic-approximation (SA) techniques tied to autonomous ODEs. A novel hovering-around behavior near saddle points emerges as a potential limiting regime, extending SA-Lyapunov theory beyond traditional attractor convergence. The work introduces three BP variants relevant to online social networks: BP with attack (competition/ownership transfer), BP with unnatural deaths, and saturated total-population dependent BP (STP-BP) capturing re-forwarding saturation. These BP insights are translated into OSN applications, including robust fake-post detection via crowd signals and a participation mean-field game to incentivize truthful tagging, as well as a viral content analysis in competitive markets. Together, the study advances BP theory, SA methodology, and practical OSN design, offering deterministic ODE approximations, finite-time trajector tracking, and new equilibrium concepts for crowd-based information propagation.

Abstract

The literature considers multi-type Markov branching processes (BPs), where the offspring distribution depends only on the living (current) population. We analyse the total-current population-dependent BPs where the offspring distribution can also depend on the total (dead and living) population. Such a generalization is inspired by the need to accurately model content propagation over online social networks (OSNs). The key question investigated is the time-asymptotic proportion of the populations, which translates to the proportional visibility of the posts on the OSN. We provide the answer using a stochastic approximation (SA) technique, which has not been used in the existing BP literature. The analysis is derived using a non-trivial autonomous measurable ODE. Interestingly, we prove the possibility of a new limiting behaviour for the stochastic trajectory, named as hovering around. Such a result is not just new to the theory of BPs but also to the SA based literature. Later, we explore three new variants of BPs: (i) any living individual of a population can attack and acquire the living individuals of the other population, in addition to producing its offspring; (ii) the individuals can die due to abnormal circumstances, and not just at the completion of their lifetimes; (iii) the expected number of offspring decreases as the total-population increases, leading to the saturation of the total-population. Such variants aid in analysing unexplored aspects of content propagation over OSNs: (i) competition in advertisement posts for similar products; (ii) controlling fake-post propagation, while not affecting the sharing of real-post; (iii) impact of re-forwarding the posts. We also designed and analysed a participation (mean-field) game where the OSN lures the users with a reward-based scheme to provide their opinion about the actuality of the post (fake or real).
Paper Structure (86 sections, 38 theorems, 285 equations, 27 figures, 3 tables, 2 algorithms)

This paper contains 86 sections, 38 theorems, 285 equations, 27 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Contributions
  3. Towards BP-literature
  4. Towards SA-literature
  5. Towards OSNs
  6. Thesis outline
  7. Preliminaries and New results
  8. Branching processes (BPs)
  9. Classification of BPs
  10. Current population-dependent BP
  11. New Variants of BPs
  12. Ordinary differential equations (ODEs)
  13. Existing ODEs
  14. Our result
  15. Asymptotic behaviour of ODE \ref{['eqn_our_ODE']}
  16. ...and 71 more sections

Key Result

Theorem 2.1

Let $f:[a, b] \times \mathbb{R} \to \mathbb{R}$ be a continuous function and let $\alpha, \beta: [a, b] \times \mathbb{R} \to \mathbb{R}$ be two continuous and differentiable functions such that $\alpha < \beta$ and $\dot{\alpha} \leq f(t, \alpha(t)), \dot{\beta} \geq f(t, \beta(t))$. Then, every s

Figures (27)

  • Figure 1: Attractor set
  • Figure 2: Saddle set
  • Figure 3: Repeller set
  • Figure 4: Asymptotic limits for one-dimensional ODE
  • Figure 5: Saddle or repeller point of \ref{['eqn_one_dim_ODE']} leads to saddle point of \ref{['eqn_our_ODE']}; here $g_1(x^*) > 0$
  • ...and 22 more figures

Theorems & Definitions (72)

  • Theorem 2.1: Comparison Result
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Theorem 2.9
  • Definition 2.10
  • ...and 62 more