Data-driven macroscopic dynamics of complex networks using Topological Data Analysis and the Equation-Free Method

TL;DR

This work addresses how to analyze macroscopic dynamics of agent-based networks without explicit macroscopic equations by fusing Topological Data Analysis with the Equation-Free Method. The authors construct a low-dimensional topological observable, the minimal filtration radius $r_{\min;t}$ at which $\beta_1$ appears, and show that a small-landmark representation can reproduce the macroscopic dynamics captured by the full active-density $d_t$. They develop lifting operators based on geometric TDA information and (optionally) simulated annealing, and they use a coarse time-stepper to perform bifurcation and stability analysis, revealing a saddle-node transition near $\epsilon_c\approx0.24$. This framework provides a computationally efficient route to diagnose phase transitions and stability in large-scale agent-based networks, with potential applicability to other social, biological, and epidemiological systems.

Abstract

In this work, we present a computational framework for exploring and analyzing the macroscopic dynamics of complex agent-based network models by integrating Topological Data Analysis with the Equation-Free Method. To demonstrate the effectiveness of our method, we apply it to Erdős--Rényi-type random networks. Central to our approach is a Topological Data Analysis-based filtration process driven by the density of activated network nodes (agents), from which we extract a coarse-grained macroscopic topological observable. This observable is defined via persistent Betti numbers, thus requiring significantly reduced data dimensionality while retaining essential topological features. Subsequently, within the Equation-Free Method framework, we show firstly that a \textit{lifting procedure} can be achieved using topological properties and secondly, a data-driven evolution law that governs the dynamics of this macroscopic variable. Finally, we perform a numerical bifurcation and stability analysis to investigate the global behavior and qualitative transitions of the emergent macroscopic dynamics.

Figures (12)

• Figure 1: Pipeline of the proposed method to analyse the network dynamics, combining topological data analysis with Equation-Free method Gear03Kev09Spil11Marsch14. (A) The microscopic network state can be transformed into a low-dimensional topological representation. To achieve this, we project the network state into the circle (where each agent in the network is represented now as a node in the unit circle), keeping the connectivity structure. Then, we select a small number of landmark points from the set of nodes in the circle. Using the filtration process Adams21topa15, we compute the minimum radius such that Betti1=1. Since our system is stochastic (at each time step), we repeat the same procedure and we average to extract the macroscopic topological value $R$. This procedure is described in section \ref{['sec:topology']}(B) Since we have a low dimensional representation of the network dynamics, i.e., the topological variable $R$, our next objective is to establish (numerically) an evolutionary equation: $R_{t+1}= F_T(R_t;\epsilon)$, using the fine atomistic rules of the agent-based model. This is implemented under the Equation-Free method Gear03Kev09 and presented in section \ref{['sec:results']}. At this point, one can wrap around the coarse time stepper $F_T$ numerical solvers, e.g., Newton or Jacobian-free methods, to compute equilibria or periodic solutions (even unstable ones).
• Figure 2: Time-evolution plots of the activity density for the Erdös-Rényi type network with $K=10^4$ agents, connectivity probability $p=0.001$, initial activity densities $d_0=0.1, 0.3, 0.7, 0.9$, and activation probabilities $\epsilon = 0.18, 0.2, 0.23, 0.25$. For small values of $\epsilon$, two stable states occur (a)--(c), while for $\epsilon$ greater than a critical value the network has only one stable state (d). The high-density solution diminishes (illustrated here for $\epsilon = 0.25$), leading the system, regardless of initial conditions, to converge toward low-density solutions.
• Figure 3: Using Betti numbers to express the dynamics of agent-based network for $\epsilon=0.18, 0.2, 0.23$ and $\epsilon=0.25$. We plot the minimum radius $r_{\min;t}$ where we obtain $Betti_1=1$. For each case of $\epsilon$ and for each time step, we use the state of the network to compute the $r_{\min;t}$. Purple and yellow lines represent solutions that are initialized on a low activation state, while red and blue lines are initialized on the high activation state of the network (a)$\epsilon=0.18$. The system shows bistability, and depending on the initial conditions, the dynamics converge to the low activation state (the case where $r_{\min;t}$ fluctuates around 0.02) or to the high activation state, where the $r_{\min;t}$ fluctuates around 0.002. (b)$\epsilon=0.2$. Similar to (a), the $r_{\min;t}$ in the low activation state fluctuates around 0.017) (c) Simulations for $\epsilon=0.23$. The $r_{\min;t}$ in the low activation state fluctuates around 0.013) (d)$\epsilon=0.25$. Now the system, independently of the initial conditions, converges to the low-activation state. The inset shows initialization from high density, which shows a transition to a low-activation state.
• Figure 4: The transient dynamics of the network are observed when simulating the model with $\epsilon=0.239$, initiated from a density of $d_0=0.8$. (a) Illustrates the phase transition in the macroscopic density $d_t$ over time. (b) Notably, a similar transition occurs in the minimal radius $r_{\min;t}$ within the same time horizon.
• Figure 5: The evolution of macroscopic variables $D_t$ and $R_t$ is shown for different initial activation densities and activation probabilities. For $\epsilon = 0.2 < \epsilon_c$, the system settles into two stable states: a low-activity state with $D = 0.18$ and $R = 0.017$, and a high-activity state with $D = 0.78$ and $R = 0.002$ (see (a) and (b)). In contrast, for $\epsilon = 0.25 > \epsilon_c$, the system evolves toward a single steady state regardless of its initial activity density, characterized by $D = 0.25$ and $R = 0.011$ (see (c) and (d)).