Iterative structural coarse-graining for contagion dynamics in complex networks

TL;DR

The paper presents Iterative Structural Coarse-Graining (ISCG), a scalable framework for reducing large, complex networks while preserving contagion dynamics under the SIR model. By aggregating dense subgraphs into weighted super-nodes via $k$-clique coarse-grained networks (CGNs) and enforcing two fidelity conditions, ISCG maintains both macroscopic outbreak sizes and microscopic node-level infection probabilities across scales. Theoretical results establish exact preservation under $\beta \ge \hat{\beta}_k$ and provide practical mappings for inter- and intra-clique transmission, with extensive experiments on real networks showing strong fidelity and substantial reduction. Beyond reduction, ISCG enables effective solutions to influence maximization, edge immunization, and sentinel surveillance, outperforming traditional adaptive centrality approaches and offering flexible approximate reductions via $k$-plexes. The framework thus delivers a robust, multi-scale tool for analyzing contagion and related dynamical processes in large-scale networks.

Abstract

Contagion dynamics in complex networks drive critical phenomena such as epidemic spread and information diffusion,but their analysis remains computationally prohibitive in large-scale, high-complexity systems. Here, we introduce the Iterative Structural Coarse-Graining (ISCG) framework, a scalable methodology that reduces network complexity while preserving key contagion dynamics with high fidelity. Importantly, we derive theoretical conditions ensuring the precise preservation of both macroscopic outbreak sizes and microscopic node-level infection probabilities during network reduction. Under these conditions, extensive experiments on diverse empirical networks demonstrate that ISCG achieves significant complexity reduction without sacrificing prediction accuracy. Beyond simplification, ISCG reveals multiscale structural patterns that govern contagion processes, enabling practical solutions to longstanding challenges in contagion dynamics. Specifically, ISCG outperforms traditional adaptive centrality-based approaches in identifying influential spreaders, immunizing critical edges, and optimizing sentinel placement for early outbreak detection, offering superior accuracy and computational efficiency. By bridging computational efficiency with dynamical fidelity, ISCG provides a transformative framework for analyzing large-scale contagion processes, with broad applications for epidemiology, information dissemination, and network resilience.

Figures (15)

• Figure 1: Illustration of the iterative structural coarse-graining (ISCG) framework for preserving contagion dynamics.a Schematic of the coarse-graining method. Maximal cliques in the network (e.g., orange circle) are merged into super-nodes. Each super-node inherits all links from the merged clique, and its weight is updated as the sum of the constituent nodes. For multiple links between a super-node and another node, a single link is retained with a weight updated equal to the sum of the original links. Initially, all nodes and edges are assigned a weight of 1. This process produces a weighted network at each iteration step ($t=1$, second column), and the procedure continues until the network is reduced to a single node. b Construction of $k$-clique coarse-graining networks (CGNs). The ISCG method is applied iteratively to reduce the network until all cliques in the network are smaller than size $k$, resulting in a $k$-clique CGN ($G_{R=k}$). The size of each node represents its weight, while the thickness of each link corresponds to the weight of the edge. b1 Original network (e.g., a co-authorship network). b2 5-clique CGN. b3 4-clique CGN. b4 3-clique CGN. c Perserving the final outbreak size ($s$) in SIR dynamics. The final outbreak size ($s$) in the original network is inferred from the contagion dynamics on the $k$-clique CGN. c1 For a given contagion configuration ($\mathcal{M}$) on $G_{R=4}$, all nodes represented by a super-node are assumed to be infected if the super-node itself is infected in $\mathcal{M}$. This maps to the corresponding configuration $\mathcal{H}\{M\}$ in the original network, where $s$ is estimated by summing the number of nodes in the recovered ($R$) state. c2 Two conditions are required to preserve $s$ in the original network: (1) $\beta \geq \hat{\beta_k}$ where $\hat{\beta_k}$ is the minimum transmission probability required for a seed node to infect all nodes within a $k$-clique ($G_{\Delta_k}$). (2) $\beta_{w_{ij}(k)} = 1-(1-\beta)^{e^w_{ij}(k)}$, where $\beta_{w_{ij}(k)}$ is the effective transmission probability for a weighted edge with weight $e^w_{ij}(k)$ in $G_{R=k}$.
• Figure 1: Illustration of infection probabilities in coarse-grained networks (CGNs) and $k$-clique structures. a Probability mapping in $4$-clique CGNs. a1 Example of a 4-clique CGN showing node $i$ infecting node $j$ through an edge with weight 2. a2 Corresponding contagion configuration in the original network, highlighting the infection of all nodes within node $j$’s clique. a3 Case 1: Node $i$ infects one node in $j$’s clique via one of the two links, and the infected node spreads the contagion to all remaining nodes in the $4$-clique. a4 Case 2: Node $i$ infects two nodes simultaneously, with the infected nodes subsequently spreading the contagion to the rest of the $4$-clique. b Infection dynamics in $k$-clique structures (for $k =2$ to $5$): b1-b4 Scenarios illustrating how a single seed node infects all other nodes in a $k$-clique. The left side of each brackets shows the initial seed infecting all nodes, while the right side lists possible configurations enabling this event. Solid arrows within the clique denote successful transmissions, and dashed arrows represent failed attempts.
• Figure 2: Minimum transmission probability required to preserve the contagion behavior in $k$-clique CGNs.a Schematic representation of a single seed node infecting all nodes within a $k$-clique ($k$ = 2 to 10). Susceptible nodes are shown in orange, while the initially infected nodes are highlighted in blue. b Probability $\Lambda^k_{1\rightarrow(k-1)}$ of a single seed node fully infecting a $k$-clique as a function of the transmission probability $\beta$. Circle markers indicate numerical simulation results, while solid lines represent theoretical predictions obtained from Eqs .\ref{['eq:eqs3']} and \ref{['eq:eqs4']}. The vertical lines denote the minimum transmission probability $\hat{\beta}_k$, where $\Lambda^k_{1\rightarrow(k-1)}=1$. c Relationship between $\hat{\beta}_k$ and $k$-clique size for different recovery probabilities $\mu$. Larger cliques require smaller $\hat{\beta}_k$ for full contagion. d Influence of multiple initial seed nodes on $\hat{\beta}_k$ for varying $k$-clique sizes with $\mu=1$. The difference in $\hat{\beta}_k$ between single-seed and multi-seed scenarios diminishes as clique size increases.
• Figure 2: Validation of the ISCG framework in preserving SIR dynamics.a A toy network $G$, consisting of two isolated nodes ($A$ and $F$) and a 4-clique substructure ($B$, $C$, $D$, and $E$). b The corresponding 4-clique CGN $G_{R=4}$ where the 4-clique is reduced to a super-node ($H$) through the ISCG process. c The probability of node $A$ infecting node $F$ in the original network (G) and the 4-clique CGN ($G_{R=4}$) is plotted as a function of the transmission probability $\beta$. The figure also shows the probability $\Lambda^{4}_{1\rightarrow3}(\beta)$, representing the likelihood of a single seed infecting all other nodes in the 4-clique. Simulation results (markers) are compared with theoretical predictions (dashed lines), confirming strong agreement. Vertical dashed lines indicate key thresholds: transmission probability $\hat{\beta_4}$ where $\Lambda^{4}_{1\rightarrow3} =1$, and the point where $P(A \rightarrow F|G) = P(A \rightarrow F|G_{R=4})$, demonstrating the accuracy of the ISCG framework in preserving contagion dynamics.
• Figure 3: Network reduction performance of $k$-clique CGNs on four real-world networks.a Distribution of clique size in the original networks, following a power-law relationship $P(k)\sim k^{-\alpha}$, where $\alpha$ is fitted using maximum likelihood estimation. b Size of the largest clique as a function of iteration steps during the reduction process. The curves reveal that merging smaller cliques often generates new $k$-cliques, extending the reduction process as $k$ decreases. c Distribution of the size of newly formed cliques during the coarse-graining process (excluding those present in the original network). The size of these newly formed cliques also follow a power-law distribution, consistent with the original network. d Proportion of nodes remaining in $k$-clique CGNs relative to the original network. The proportion decreases sharply as $k$ decreases, particularly for smaller $k$. e Proportion of links remaining in $k$-clique CGNs relative to the original network. The number of links decreases exponentially, resulting in substantial network simplification.