Modeling memory in time-respecting paths on temporal networks

TL;DR

The paper investigates memory in time-respecting paths on temporal networks, introducing a minimal two-parameter framework (MEM and MEM+SBM) that quantifies memory via the horizon $m$ and probability $p$ that a TRP revisits a node. It demonstrates strong, statistically significant memory effects across eight SocioPatterns proximity datasets, with MEM+SBM providing better goodness-of-fit by incorporating community structure. A maximum likelihood approach estimates $p$ and, for MEM+SBM, the community affinities, and comparisons to memoryless null models confirm nontrivial memory that varies with context (e.g., higher in schools and Malawi, lower at conferences). Additionally, a synthetic memory-generating process shows that increased memory slows diffusion on temporal networks, highlighting practical implications for modeling spreading processes and guiding memory-aware data aggregation. Overall, the work provides a tractable, interpretable framework to quantify and simulate memory in TRPs, with implications for diffusion dynamics and temporal-network modeling.

Abstract

Human close-range proximity interactions are the key determinant for spreading processes like knowledge diffusion, norm adoption, and infectious disease transmission. These dynamical processes can be modeled with time-respecting paths on temporal networks. Here, we propose a framework to quantify memory in time-respecting paths and evaluate it on several empirical datasets encoding proximity between humans collected in different settings. Our results show strong memory effects, robust across settings, model parameters, and statistically significant when compared to memoryless null models. We further propose a generative model to create synthetic temporal graphs with memory and use it to show that memory in time-respecting paths decreases the diffusion speed, affecting the dynamics of spreading processes on temporal networks.

Figures (8)

• Figure 1: Memory in time-respecting paths.Top row. A temporal graph with $n = 6$ nodes. We depict a non-backtracking time-respecting path $\mathcal{P}$ on this graph. The walker is located in the white dot marked with the thick line. An arrow points from this node to the node occupied at the following time step. The red dots indicates the node the path comes from. This node cannot be reached by a path starting from the blue node, as it would create a backtrack. The brown dots denote the memory set $\mathcal{M}_a$ at each time step for a memory horizon $m = 4$. Note that the size of the memory set is not constant. Bottom row. The path $\mathcal{P}$ at time $t_7$ with the colors and markers obtained at this time.
• Figure 2: Comparison of the goodness-of-fit between MEM and MEM + SBM models as a function of the memory horizon $m$. Each plot refers to one of the six datasets of Table \ref{['tab:data_table']} with a known node label assignment. These dataset have a known node partition in communities. In Primary, High school 1, 2, 3, each node is assigned to a school class. In the Office and in the Hospital datasets labels are a role attribute. We show the BIC (lower is better) of the MEM model (cyan squares) and of the MEM + SBM model (orange circles) as a function of the memory length $m$ for $t_{\rm res} = 20~s$. While the plots share the same $y$-axis, the results cannot be compared across datasets.
• Figure 3: Maximum likelihood estimates of the memory parameter $p$ and comparison with the null models. Each plot refers to one of the datasets described in Table \ref{['tab:data_table']} and shows the inferred value of the probability $p$ of the time-respecting paths to return to a node, as a function of the memory length $m$. The curve "MEM" (solid line with squares) is obtained from the MEM model of Eq. \ref{['model1']} on the empirical data. The curve "MEM+SBM" (solid line with dots) is obtained from the MEM+SBM model of Eq. \ref{['model2']} on the empirical data. The curve "Null MEM" (dashed line with squares) is obtained from the MEM model (Eq. \ref{['model1']}) on the synthetic data, generated from the Erdős Rényi null model. The curve "Null MEM + SBM" (dashed line with dots) is obtained from the MEM+SBM model Eq. \ref{['model2']} on the synthetic data, generated from the SBM null model. For the Conference and Malawi datasets, we only consider the null model based on Erdős-Rényi random graphs, while for all other datasets, we consider the one generated from stochastic block model graphs. For these datasets, the labels are known attributes that indicate school classes (Primary, High school 1, 2, 3), and role attributes (Office, Hospital). The curves referring to the null models show the mean over $50$ realizations, and the shaded areas represent the standard deviation. For all graphs, we consider the temporal resolution $t_{\rm res} = 20~s$.
• Figure 4: The relation between memory and diffusion speed.Left panel. Relation between the weight $\alpha$ appearing in Eq. \ref{['eq:gen_model']} and the estimated $p$ from the MEM model of Eq. \ref{['model1']} with $m = \hat{m} = 5$. The color- and marker-coded lines refer to three values of the graph density, reported in the legend. The solid line is obtained by averaging over $10$ realizations with fixed parameters, while the shaded line is the $5-95$ confidence interval. Each graph has $n = 250$ nodes, and $T = 300$ snapshots. Right panel. For each of the graphs of the left panel, we run $15$ simulations of a diffusive process of a signal $\bm{u}$ starting from a randomly selected node at $t = 0$. The figure shows the relation between the inferred $p$ (the same as in the left panel) and the entropy of the signal density at the end of the process.
• Figure 5: Maximum likelihood estimates of the memory parameter $p$. Each plot refers to one of the datasets described in Table \ref{['tab:data_table']} and shows the inferred values of the probability $p$ in the MEM model (Eq. \ref{['model1']}), probability of the time-respecting paths to return to a node. Paths are generated as per Section \ref{['sec:methods']} for different memory horizon lengths per path and temporal resolution values $t_{\rm res}$.