# Hierarchical invasion of cooperation in complex networks

## TL;DR

This paper investigates how cooperative behavior spreads in complex networks, revealing that invasion follows a hierarchical pattern from high to low degree nodes and emphasizing the importance of initial conditions for successful cooperation spread.

## Contribution

It introduces a hierarchical invasion model for cooperation in complex networks, highlighting the role of node degree and initial conditions in the spread of cooperation.

## Key findings

- Invasion proceeds hierarchically from high to low degree nodes.
- Successful invasion requires a balanced initial number of cooperators.
- Optimal initial nodes are high but not too high degree, at intermediate values.

## Abstract

The emergence and survival of cooperation is one of the hardest problems still open in science. Several factors such as the existence of punishment, fluctuations in finite systems, repeated interactions and the formation of prestige may all contribute to explain the counter-intuitive prevalence of cooperation in natural and social systems. The characteristics of the interaction networks have been also signaled as an element favoring the persistence of cooperators. Here we consider the invasion dynamics of cooperative behaviors in complex topologies. The invasion of a heterogeneous network fully occupied by defectors is performed starting from nodes with a given number of connections (degree) $k_0$. The system is then evolved within a Prisoner's Dilemma game and the outcome is analyzed as a function of $k_0$ and the degree $k$ of the nodes adopting cooperation. Carried out using both numerical and analytical approaches, our results show that the invasion proceeds following preferentially a hierarchical order in the nodes from those with higher degree to those with lower degree. However, the invasion of cooperation will succeed only when the initial cooperators are numerous enough to form a cluster from which cooperation can spread. This implies that the initial condition must be a suitable equilibrium between high degree and high numerosity, which usually takes place, when possible, at intermediate values of $k_0$. These findings have potential applications, as they suggest that, in order to promote cooperative behavior on complex networks, one should infect with cooperators \emph{high but not too high} degree nodes.