# Description of spreading dynamics by microscopic network models and macroscopic branching processes can differ due to coalescence

## TL;DR

This paper demonstrates that microscopic network models and macroscopic branching processes can differ significantly due to coalescence, and proposes a non-linear estimator to accurately infer model parameters across system sizes.

## Contribution

It analytically shows how coalescence causes discrepancies between microscopic and macroscopic spreading descriptions and introduces a non-linear estimator to correct parameter inference.

## Key findings

- Coalescence causes non-linear effects in spreading dynamics.
- A universal non-linear scaling function describes coalescence impact.
- The proposed estimator accurately infers microscopic parameters from macroscopic data.

## Abstract

Spreading processes are conventionally monitored on a macroscopic level by counting the number of incidences over time. The spreading process can then be modeled either on the microscopic level, assuming an underlying interaction network, or directly on the macroscopic level, assuming that microscopic contributions are negligible. The macroscopic characteristics of both descriptions are commonly assumed to be identical. In this work, we show that these characteristics of microscopic and macroscopic descriptions can be different due to coalescence, i.e., a node being activated at the same time by multiple sources. In particular, we consider a (microscopic) branching network (probabilistic cellular automaton) with annealed connectivity disorder, record the macroscopic activity, and then approximate this activity by a (macroscopic) branching process. In this framework, we analytically calculate the effect of coalescence on the collective dynamics. We show that coalescence leads to a universal non-linear scaling function for the conditional expectation value of successive network activity. This allows us to quantify the difference between the microscopic model parameter and established macroscopic estimates. To overcome this difference, we propose a non-linear estimator that correctly infers the model branching parameter for all system sizes.

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Source: https://tomesphere.com/paper/1905.10402