Multi-scale Local Network Structure Critically Impacts Epidemic Spread and Interventions

TL;DR

The paper investigates how multi-scale local structure in real interaction networks shapes epidemic spread and responses to interventions. It introduces epidemic Network Community Profiles (epidemic NCP) and Area Above the NCP (AANCP) to quantify local structure across scales and links these metrics to quarantine effectiveness. An extensive set of simulations on 15 real-world networks, plus synthetic models, shows that empirical networks with rich local structure are more controllable under local quarantines than common synthetic rewiring, and that traditional global metrics like the dominant eigenvalue $\lambda_1(\mathbf{A})$ poorly predict containment outcomes. The study also develops two generative models (GeometricCommunities and RandomWalkCommunities) that reproduce observed multi-scale local structure and demonstrates that triangles alone do not fully capture local structure, with hypergraph diffusion providing additional insights. These findings underscore the need for modeling frameworks that incorporate multi-scale local structure and real mobility data to inform epidemic control policies and surveillance.

Abstract

Network epidemic simulation holds the promise of enabling fine-grained understanding of epidemic behavior, beyond that which is possible with coarse-grained compartmental models. Key inputs to these epidemic simulations are the networks themselves. However, empirical measurements and samples of realistic interaction networks typically display properties that are challenging to capture with popular synthetic models of networks. Our empirical results show that epidemic spread behavior is very sensitive to a subtle but ubiquitous form of multi-scale local structure that is not present in common baseline models, including (but not limited to) uniform random graph models (Erdos-Renyi), random configuration models (Chung-Lu), etc. Such structure is not necessary to reproduce very simple network statistics, such as degree distributions or triangle closing probabilities. However, we show that this multi-scale local structure impacts, critically, the behavior of more complex network properties, in particular the effect of interventions such as quarantining; and it enables epidemic spread to be halted in realistic interaction networks, even when it cannot be halted in simple synthetic models. Key insights from our analysis include how epidemics on networks with widespread multi-scale local structure are easier to mitigate, as well as characterizing which nodes are ultimately not likely to be infected. We demonstrate that this structure results from more than just local triangle structure in the network, and we illustrate processes based on homophily or social influence and random walks that suggest how this multi-scale local structure arises.

Figures (40)

• Figure 1: The Network Community Profile (NCP) characterizes local structure in networks by showing the distribution of local conductance bottlenecks at all size-scales. Here, we use it to understand epidemic spreading preferences by plotting conductance bottlenecks derived from epidemic spreading. The resulting epidemic NCP plots in (D) and (E) show a characteristic difference between networks with rich local structure (A and D) compared with a randomized network with the same degree distribution (E). (A) We initiate an SEIR process on a network stating from a given node and show a few steps of the epidemic spreading where orange represents an infected or recovered node. (B) We zoom into a single group of infected nodes at time $t=140$ and assess the conductance of the set (the ratio of cut to volume is 13/295 = 0.04 for this set). (C) This is repeated for 10 or 100 thousand times to get multiple epidemics from multiple start points and evaluated with multiple sets of infected nodes from each epidemic. (D) Each evaluated set yields a single point in the size-vs-conductance space, and the NCP is a 2d histogram of these points. The NCP of the network in (A) shows wide variation in the size-vs-conductance space and indicates more local structure is present than in (E). (E) We compare this to the epidemic NCP obtained by randomizing edges from the network in (A) with a configuration model. In this case, conductance has minimal variability as the size-scale changes, indicating a lack of local structure.
• Figure 2: Networks containing realistic multi-scale local structure are more controllable under local interventions than networks without local structure, due to the widespread prevalence of conductance bottlenecks. (A) Shows a network with local structure that is rewired to two different synthetic variants lacking local structure. For each rewired network sample, 800 simulations - split among differing quarantine capacitites - for an SEIR model were performed. (B) Shows the fraction of total infections as the color (darker is a higher fraction) when the original network is a mobility network based on contacts within Mexico City de2020contact. The vertical axis represents the maximum level of quarantine as a fraction of the total number network size. The epidemic uses a fixed infection and recovery probability (0.02 and 0.05 respectively). As we move from the center to the left, we are increasingly rewiring to a random configuration model (CM) network that preserves node degrees in expectation; as we move from the center to the right we rewrite to a uniform random graph ($G_{n,p}$) that preserves average degree only. The network depicted in (A) is used for illustrative purposes, as the Mexico City network does not admit a simple visual depiction.
• Figure 3: The presence of multi-scale local structure dramatically improves the impact of a quarantine intervention in an epidemic. Rows (A)-(E) show total infections as a function for rewiring (x-axis) and quarantine capacity (y-axis), as in Figure \ref{['fig:uplot-explanation']}, for 15 networks. As we increasingly rewire networks (move away from the center), total infections deviates from those on empirical networks under local interventions. This effect is more pronounced in some networks (Row A) while absent in others (Facebook Interactions, etc). These results were produced using SEIR simulations on 50 node samples for each parameter set (quarantine percent and rewiring). See Appendix \ref{['sec:app-uplots-all']} for details concerning choice of epidemic parameters.
• Figure 4: Using the epidemic threshold, $\frac{\beta}{\gamma}\lambda_1(\boldsymbol{{A}})$, as a measure of epidemic strength would predict that higher values of $\lambda_1$ produce infections that are harder to control. However, the opposite result holds for these empirically measured networks. We display the dominant eigenvalue $\lambda_1(\boldsymbol{{A}})$ of the adjacency matrix ($\boldsymbol{{A}}$) vs the same $\text{CM} \leftarrow \text{Original} \rightarrow \text{GNP}$ rewiring on the horizontal axis (omitted for clarity). As we move from the center to the left (right) we are increasingly rewiring to CM (GNP). The original networks have larger eigenvalues, yet, comparing with Figure \ref{['fig:uplots-all']} shows that almost all of these epidemics require smaller interventions than synthetic variants to impede or halt spreading, contrary to what would be expected from eigenvalues alone.
• Figure 5: Local set structure explored by epidemics in these conductance-vs-size plots for graphs derived from the 15 networks. Mobility networks (row A) have some of the highest diversity in sets explored, whereas networked interactions (rows C and D) have lower diversity in this space. Compare with Figure \ref{['fig:uplots-all']}.