Estimating Nodal Spreading Influence Using Partial Temporal Network

TL;DR

This work addresses predicting nodal spreading influence in temporal networks from partial observations by introducing three walk-based centrality metrics derived from partial temporal neighborhoods: $d_i^{(\phi, m)}$, $\Delta_i^{(\phi, m)}$, and $Z_i^{(\phi, m)}$, which quantify walks, time-respecting paths, and time-prioritized reachability. The authors demonstrate that these metrics, computed from $\mathcal{G}_{i}(\phi,m)$, often outperform classical centrality measures derived from full or partial networks across multiple real-world datasets and infection regimes $\beta$, with performance varying by $\beta$ and parameter choices $(\phi,m,\alpha)$. Results show that when $\beta$ is not small, time-scaled temporal reachability $Z$ best captures spreading potential, while for small $\beta$, simpler walk counts (e.g., $d$) suffice; anomalies occur in dense networks where node influence becomes less distinguishable. The findings provide practical guidance for early identification of influential seeds and intervention targeting using only locally available temporal data, and point to future work on extending to other spreading models, higher-order interactions, and scalable approximations.

Abstract

Temporal networks, whose links are activated or deactivated over time, are used to represent complex systems such as social interactions or collaborations occurring at specific times. Such networks facilitate the spread of information and epidemics. The average number of nodes infected via a spreading process on a network starting from a single seed node over a given period is called the influence of that node. In this paper, we address the question of how to utilize the partially observed temporal network (local and of short duration) around each node, to estimate the ranking of nodes in spreading influence on the full network over a long period. This is essential for target marketing and epidemic/misinformation mitigation where only partial network information is possibly accessible. This would also enable us to understand which network properties of a node, observed locally and shortly after the start of the spreading process, determine its influence. We systematically propose a set of nodal centrality metrics based on partial temporal network information, encoding diverse properties of (time-respecting) walks. It is found that distinct centrality metrics perform the best in estimating nodal influence depending on the infection probability of the spreading process. For a broad range of the infection probability, a node tends to be influential if it can reach many distinct nodes via time-respecting walks and if these nodes can be reached early in time. We find and explain why the proposed metrics generally outperform classic centrality metrics derived from both full and partial temporal networks.

Figures (28)

• Figure 1: (a) A temporal network $G$ with 5 nodes and 6 time steps. The first, second, and third contacts between the same pair of nodes are marked in red, yellow, and blue, respectively. (b) The aggregated network $G^w$ of $G$ along with its link weight. (c) The partial temporal network $\mathcal{G}_{A}(\phi,m)$ where $\phi$ = 0.5 and $m$ = 3, observed around node $A$ and its corresponding weighted aggregated network $\mathcal{G}_{A}^w(\phi,m)$. (d) The list of all 3-hop walks between node $A$ and $D$ in $\mathcal{G}_{A}^w(\phi,m)$. (e) For each walk listed in (d), time stamp of each link is added and the walk is marked as $\times$ if it is not a time-respecting walk and as $\checkmark$ if it is time-respecting.
• Figure 2: Average nodal influence in each real-world network as a function of the infection probability $\beta$.
• Figure 3: The (best) prediction quality $\bar{Q}_k^{max}$ and $\bar{Q}_r^{max}$ of weighted degree mass ($d$), time-scaled temporal degree mass ($\Delta$), and time-scaled temporal reachability ($Z$), respectively, across various combinations of $\phi$ and $\beta$, in network HighScholl11.
• Figure 4: The nodal influence of each node in a randomly selected observation period $[t_0+\tau]$ in the Manufacturing Emails network.
• Figure 5: Average Kendall correlation coefficient $\bar{Q}_k$ between every two proposed centrality metrics in Highschool11 with $m = 2$ and $\alpha = 1$ when $\phi$ = 0.25 and 0.5, respectively.