Scaling in local to global condensation of wealth on sparse networks

Introduction

Generalized Yard-Sale (gYS) model is a wealth exchange model we introduced in our recent study which exhibits some very fascinating dynamics. We carefully delineate our observations in the paper, while in here we aim to enhance visualization on what is really happening during the simulations.

Model

$N$ individuals start with equal wealth ($w_i=1$). Upon simulation, we update the wealth via following rule: with probability $1-p$, a randomly picked link (say, the end nodes are $i$ and $j$) transfers $\varepsilon \min(w_i,w_j)$ amount of wealth where $\varepsilon \le 1$ is a constant, and with probability $p$, the link transfers $\varepsilon w_i$ amount of wealth if the direction is $i \rightarrow j$. The tranfer direction is randomly chosen with equal probability. We do this update $N$ times during $t \rightarrow t+1$ in simulation time.

We present simulation results on highly sparse and heterogeneous scale-free network with $\gamma=2.5, \langle k \rangle=4$.

Results

Emergence of local & global condensate

Here, we plot individual nodes' wealth $w_i$ and the wealth variance $\sigma^2$ of the total individuals ($N=97, p=10^{-6}, \varepsilon=0.05$).

We note that the local condensate is characterized by almost completely frozen wealth of all nodes (video link below). It is followed by relaxation in which the frozen wealth dissolves and roam around in a random-walk-like manner across significantly longer time-scales. This transfer of wealth is driven by the aforementioned exchange scheme which occurs with probability $p$, and the average time period for each jump is proportional to $1/\varepsilon p$ as we elucidate in the paper.

As a consequence of the dynamic movement, and the coalescence of nearest neighbors' wealth, global condensate arises in which the total wealth is collected by a single monopolistic node.

     Coalescence             Transfer

In the following videos, wealth of an individual is respresented by the size of a node, the simulation time is displayed on top left, and the (time-averaged) amount of transferred wealth is visualized with an edge's brightness.

Initial growth ($10^1 < t < 10^4$)

Video link (Google Drive)

Local condensate ($10^5 < t < 10^6$)

Video link (Google Drive)

Relaxation ($10^7 < t < 10^8$)

Video link (Google Drive)

Global condensate ($10^8 < t < 10^9$)

Video link (Google Drive)