Extended Persistent Homology Distinguishes Simple and Complex Contagions with High Accuracy

TL;DR

The paper introduces extended persistent homology (EPH) as a topological feature to distinguish simple from complex contagions on networks and to predict their parameters. By applying a reverse-infection-time filtration and computing EPH-based summaries on network contagion data, the authors show high accuracy in classifying contagion type and robust regression performance for parameters $q$ (simple contagion) and $\theta$ (complex contagion) across real-world networks and under partial observation and noise. The approach highlights the role of cycle lifetimes in contagion dynamics and demonstrates that EPH-derived features outperform baseline topology-informed metrics. The work suggests wide applicability to seeding, vaccination, quarantine planning, and network inference, providing a principled, geometry-aware lens on contagion processes. The combination of topology-driven features with SI-based contagion models offers a scalable, interpretable toolkit for understanding and controlling diffusion phenomena on complex networks.

Abstract

The social contagion literature makes a distinction between simple (independent cascade or bond percolation processes that pass infections through edges) and complex contagions (bootstrap percolation or threshold processes that require local reinforcement to spread). However, distinguishing simple and complex contagions using observational data poses a significant challenge in practice. Estimating population-level activation functions from observed contagion dynamics is hindered by confounding factors that influence adoptions (other than neighborhood interactions), as well as heterogeneity in individual behaviors and modeling variations that make it difficult to design appropriate null models for inferring contagion types. Here, we show that a new tool from topological data analysis (TDA), called extended persistent homology (EPH), when applied to contagion processes over networks, can effectively detect simple and complex contagion processes, as well as predict their parameters. We train classification and regression models using EPH-based topological summaries computed on simulated simple and complex contagion dynamics on three real-world network datasets and obtain high predictive performance over a wide range of contagion parameters and under a variety of informational constraints, including uncertainty in model parameters, noise, and partial observability of contagion dynamics. EPH captures the role of cycles of varying lengths in the observed contagion dynamics and offers a useful metric to classify contagion models and predict their parameters. Analyzing geometrical features of network contagion using TDA tools such as EPH can find applications in other network problems such as seeding, vaccination, and quarantine optimization, as well as network inference and reconstruction problems.

Figures (27)

• Figure 1: Panel \ref{['fig:panel1']} shows how to determine whether a tie is long or short based on the length of the cycle it creates. A 3-cycle indicates a short tie, and longer cycles signify longer ties (assuming that there are no edges crossing the cycle). Panel \ref{['fig:panel3']}: To redefine cycle length for network contagions, consider a simple contagion in a cycle with node A initially infected, where the infection spreads with certainty ($\theta = \infty$, $q = 1$). In this example, the time required for all nodes to become infected equals half the cycle's length (path length between A and B). However, to better reflect contagion dynamics, we propose using the total infection time as an extended definition of the cycle length, rather than the traditional cycle length that counts the number of nodes. This extension allows for a more accurate representation of the contagion process in relation to the cycle lengths. In more complex scenarios, such as multiple initially infected nodes, this definition must be adjusted again to account for the varied dynamics of contagion. Panel \ref{['fig:panel4']}: For example, if nodes A, C and D are all initially infected, then the infection spreads not just from node A, but from the collective influence of nodes A, C, and D together. Panel \ref{['fig:panel5']}: To accurately model the spread dynamics from the collective influence of nodes A, C, and D, we can treat the initial infected nodes A, C, and D as a single entity. This grouping divides the original cycle into two other independent cycles (labeled cycle 1 and cycle 2 in Figure \ref{['fig:panel5']}), where the contagion spreads separately. Using the reverse of the infection steps as filtration in EPH results in a scenario that is completely similar to the one shown (considering initially infected nodes as a single block). This demonstrates why we use EPH with the negative infection order as filtration.
• Figure 1: Short ties indicate 3-cycle
• Figure 2: On the left, violin plots showing the distribution of Extended Persistent Homology (EPH) values under simple (red) and complex (blue) contagion models across varying values of threshold $\theta$ and simple contagion probability $q$. Each distribution is derived from 200 simulations on the Email dataset, with the contagion process terminated once $85\%$ of the nodes were infected. For complex contagion ($\theta < \infty$), increasing $\theta$ leads to higher EPH values, reflecting greater temporal dispersion in infection times due to more stringent activation requirements. A similar trend is observed in simple contagion ($\theta = \infty$) as $q$ decreases. On the right, EPH distributions for the same parameter settings, with simulations halted after three infection steps. Compared to simple contagion, complex contagion exhibits greater sensitivity in both the shape and variance of the EPH distribution under parameter changes, highlighting its increased topological and geometric variability.
• Figure 2: All nodes will be infected after 4 steps.
• Figure 3: Panel 3A: Classification accuracy (with $95\%$ confidence intervals) as a function of the threshold $\theta$ for distinguishing complex contagion types using EPH features. Models were trained and tested on data generated under the same $\theta$ values. Inset: Confusion matrix for a binary classifier differentiating between simple and complex contagion based on EPH. Each contagion type was simulated 800 times. For simple contagion, $q$ was randomly selected from $\{0.02, 0.03, 0.04, 0.05\}$. For complex contagion, $q$ was fixed at $0.02$, and $\theta$ was varied over $\{2, 3, \dots, 7\}$. Panel 3B: Polynomial regression (using second-order polynomials) of threshold $\theta$ on EPH for complex contagion. Inset: Prediction accuracy measured as the percentage of samples for which the regressor predicts $\theta$ exactly, within $\pm 1$, and with absolute error greater than one (no cases were identified in the latter category). Panel 3C: Polynomial regression (using third-order polynomials) of transmission probability $q$ on EPH under simple contagion. Inset: Residual plot showing deviation of predicted values from observed $q$.