Modeling financial transactions via random walks on temporal networks

TL;DR

This framework analytically derives heavy-tailed distributions for the stationary balances and transaction sizes of financial transactions by enforcing fund conservation by reproducing observed correlations between inflows and outflows.

Abstract

We model financial transactions as random walks on activity-driven temporal networks. By enforcing fund conservation, our framework analytically derives heavy-tailed distributions for the stationary balances and transaction sizes. Crucially, the latter is driven by variance in the spending propensity of individuals. Calibrated with empirical data from a closed, digital currency community, the model also reproduces observed correlations between inflows and outflows. Our findings provide a path for understanding emergent properties of the circulation of money.

Figures (4)

• Figure 1: Probability distributions $P(m)$ (a) and $P(w)$ (b) for different values of the precision parameter $\xi$. Yellow dashed line in panel (a) shows a power-law scaling $P(m) \sim m^{-2}$. Vertical dotted lines correspond to $m = K_0/s_\mathrm{max}$ (a) and $w = K_0$ (b). Gray line in panel (a) corresponds to the analytical solution of $P(m)$ obtained in the limit of $\xi \to \infty$ and small $s$, see Eq. (18) of the SM. Solid lines correspond to numerical integrations of Eq. (29) (a) and Eq. (43) (b) of the SM. We used $N = 10^4$, $M = 10^7$, $\sigma = 0$, $s_\mathrm{max} = 1.0$, and perfect correlation $b = a$.
• Figure 2: Probability distribution of time-average spending propensities $h(s)$ (a) and spending fractions $P(q)$ (b), observed in the data (bars) and obtained from simulations of the model (points), for different values of the precision parameter $\xi$ and a shape parameter $k = 0.75$.
• Figure 3: Probability distributions of balances $P(m)$ (blue) and transaction sizes $P(w)$ (orange), observed in the data (dashed lines) and obtained from numerical simulations of the model with $\xi = 1$ and $k = 0.75$ (points).
• Figure 4: Joint probability distribution of in-degree ($k_\mathrm{in})$ and out-degree ($k_\mathrm{out})$ of individuals, observed in the dataset (a) and the model with $\xi = 1$ and $k = 0.75$ (b). Joint probability distribution of total money received ($m_\mathrm{in})$ and total money spent ($m_\mathrm{out}$) by individuals, observed in the dataset (c) and the model with $\xi = 1$ and $k = 0.75$ (d). Each point in the plots represents a single user. The color represents the density value of users estimated through a kernel density estimation using Gaussian kernels.