Modelling and Study of t , Peak and Effective Diameter in Temporal Networks

TL;DR

The paper tackles how to define and quantify diameter in temporal networks, where connections evolve over time, by introducing a formal framework with three time-aware metrics: $\tilde{\mathcal{D}}$ (effective diameter), $*\mathcal{D}$ (peak diameter), and $\tau\mathcal{D}$ (\tau-diameter). It develops a flow-based temporal-path model, defines temporal reachability sets, and derives analytical expressions—such as $|\mathcal{R}_t(i)| \approx N\bigl(1 - e^{- t \langle \hat{k} \rangle / N}\bigr)$ and $\tilde{\mathcal{D}}$, $\tau\mathcal{D}$, $\tau\mathcal{D} \approx \frac{\ln(N/3)}{\ln(1 + \langle \hat{k} \rangle / N)}$—to predict diffusion dynamics. Validation across synthetic distributions and four real-world temporal networks shows accurate predictions and reveals that $\tilde{\mathcal{D}}$ decreases with higher average degree while $\tau\mathcal{D}$ and $*\mathcal{D}$ are more sensitive to node removal, with practical implications for epidemic control and robustness. The work provides a bridge between theory and empirical data for time-dependent connectivity, offering tools to assess resilience and guide interventions in dynamic systems. The findings underscore the importance of temporal structure in diffusion processes and suggest future enhancements to capture burstiness, node/edge attributes, and domain-specific scenarios.

Abstract

Understanding how information, diseases, or influence spread across networks is a fundamental challenge in complex systems. While network diameter has been extensively studied in static networks, its definition and behavior in temporal networks remain underexplored due to their dynamic nature. In this study, we present a formal mathematical framework for analyzing diameter in temporal networks and introduce three time-aware metrics: Effective Diameter , Peak Diameter (*D), and t-Diameter (tD), each capturing distinct temporal aspects of connectivity and diffusion. Our approach combines theoretical analysis with empirical validation using four real-world datasets: high school, hospital, conference, and workplace contact networks. We simulate flow propagation on temporal networks and compare the observed diameters with the proposed theoretical Equations. Across all datasets, our model demonstrates high accuracy, with low RMSE and absolute error values. Furthermore, we observe that the effective diameter decreases with increasing average degree and increases with network size. The results also show that tD and *D are more sensitive to node removal, highlighting their relevance for applications such as epidemic modeling. By bridging formal modeling and empirical data, our framework offers new insights into the temporal dynamics of networked systems and provides tools for assessing robustness, controlling information spread, and optimizing interventions in time-sensitive environments.

Figures (9)

• Figure 1: A nine-node network with changing connections over time. (a) Static view, (b) Temporal connections at each time step.
• Figure 2: Flow propagation from node $v_6$ in the network of Figure \ref{['connectiontimefig']}, illustrating the reachable set at each time step.
• Figure 3: Comparison of effective diameter ($\backsim \mathcal{D}$) between theoretical predictions and simulations for different distributions: (a) normal, (b) Pareto, and (c) Poisson. All simulations are conducted with $N=500$ nodes, and the plots depict the diameter across various values of the average degree $\langle \hat{k} \rangle$.
• Figure 4: Comparison between the network effective diameter ($\backsim \mathcal{D}$) in simulations and the theoretical equation for different link distributions. The plots correspond to (a) normal, (b) Pareto, and (c) Poisson distributions. In these networks, $\langle \hat{k} \rangle = 5$, and the plots illustrate the diameter for varying values of $N$.
• Figure 5: Effect of network parameters on effective diameter ($\sim \mathcal{D}$). (a) The impact of network size $N$ on the effective diameter for different average degrees ($\langle \hat{k} \rangle \in \{10, 15, 20, 25\}$). (b) The effect of average degree on the effective diameter for networks of sizes $N \in \{1000, 5000, 10000\}$.

Theorems & Definitions (14)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Definition 5
• Definition 6
• Theorem 2.1
• proof
• Lemma 2.2
• proof