Go Viral, or Not: Rate-Optimal Control for Resource-Constrained Branching Processes

TL;DR

This paper develops a control framework for multi-type branching processes under resource constraints, providing optimal growth strategies with applications in viral marketing, political blog analysis, and cancer treatment.

Contribution

It introduces a new class of controlled branching processes with resource constraints and derives optimal growth strategies, including closed-form solutions for specific cases.

Findings

Optimal population growth achieved by fixed proportional species distribution.

Application to US blogosphere shows significant gains over uniform strategies.

Cancer model estimates more conservative treatment intensities under worst-case scenarios.

Abstract

We propose and analyze a new class of controlled multi-type branching processes with a per-step linear resource constraint, motivated by potential applications in viral marketing and cancer treatment. We show that the optimal exponential growth rate of the population can be achieved by maintaining a fixed proportion among the species, for both deterministic and stochastic branching processes. In the special case of a two-type population and with a symmetric reward structure, the optimal proportion is obtained in closed-form. In addition to revealing structural properties of controlled branching processes, our results are intended to provide the practitioners with an easy-to-interpret benchmark for best practices, if not exact policies. As a proof of concept, the methodology is applied to the linkage structure of the 2004 US Presidential Election blogosphere, where the optimal growth rate demonstrates sizable gains over a uniform selection strategy, and to a two-compartment cell-cycle kinetics model for cancer growth, with realistic parameters, where the robust estimate for minimal treatment intensity under a worst-case growth rate is noticeably more conservative compared to that obtained using more optimistic assumptions.