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Competing activation mechanisms in epidemics on networks
Claudio Castellano, Romualdo Pastor-Satorras
TL;DR
The paper examines whether hubs or dense cores sustain epidemic activity across different spreading processes and network topologies. It compares activity on networks, maximum k-cores, and hub-centered stars, alongside threshold estimates. It finds that transient outbreaks are core-led, while steady SIS activity can be hub- or core-led depending on heterogeneity, with correlations able to mix mechanisms.
Problem
Which network vertices sustain epidemic activity remains an open question, with hubs and dense cores proposed as competing explanations.
Source: Effects of correlations · p. 7, Discussion · p. 9Method
The study compares activity and threshold estimates for whole networks, maximum k-cores, and hub-centered stars, including SIS simulations on correlated networks.
Source: §2 · p. 23, §5 · p. 26, §6 · p. 28Results
Transient SIR outbreaks are core-triggered, while uncorrelated SIS activation is hub-led above γ = 5/2 and core-led below; correlations can mix the mechanisms.
Source: Discussion · p. 9, §2 · pp. 22–23Takeaways & Limitations
The vertices sustaining epidemic activity vary with the spreading process and the network's topological features.
Source: Discussion · p. 9For larger γ, simulations use larger systems to distinguish the competing threshold scales.
Source: §1 · p. 22
Abstract
from arXiv · showhide
In contrast to previous common wisdom that epidemic activity in heterogeneous networks is dominated by the hubs with the largest number of connections, recent research has pointed out the role that the innermost, dense core of the network plays in sustaining epidemic processes. Here we show that the mechanism responsible of spreading depends on the nature of the process. Epidemics with a transient state are boosted by the innermost core. Contrarily, epidemics allowing a steady state present a dual scenario, where either the hub independently sustains activity and propagates it to the rest of the system, or, alternatively, the innermost network core collectively turns into the active state, maintaining it globally. In uncorrelated networks the former mechanism dominates if the degree distribution decays with an exponent larger than 5/2, and the latter otherwise. Topological correlations, rife in real networks, may perturb this picture, mixing the role of both mechanisms.
Results
The activation mechanism for SIS depends on degree heterogeneity: on uncorrelated networks, hubs dominate above γ = 5/2, while the maximum k-core is activated collectively below it. SIR spreading is instead controlled by the maximum k-core, while degree correlations can mix the mechanisms in real networks.
Activation mechanisms for SIS in uncorrelated networks
- For uncorrelated networks, SIS activity is hub-triggered for γ > 5/2 and collectively core-triggered for γ < 5/2.The isolated hub-centered star and maximum k-core show distinct activation behavior; threshold estimates support the same crossover.Source: Activation mechanisms for SIS in… · pp. 5–6
The SIR model
- In SIR, the maximum k-core controls spreading and sets the HMF threshold, while the hub-centered star has a threshold constant independent of qmax.The whole-network transition closely tracks the maximum k-core transition.Source: The SIR model · p. 7
Effects of correlations
- Correlations can alter the SIS pattern: Movies and PGP follow the core-led expectation, but strong disassortativity in AS maps mixes hub and core mechanisms.In AS maps, activity can occur even when both the hub and maximum k-core appear subcritical.Source: Effects of correlations · p. 8
Discussion
The activation mechanism varies with epidemic dynamics and network structure: transient SIR outbreaks are triggered by the innermost core, while steady-state processes can be sustained by either the hub or the core. Degree correlations can shift the balance between these mechanisms.
Discussion
- For processes without a steady state, such as SIR, the innermost core is the main trigger of infection and sets the epidemic threshold.Source: Discussion · p. 9
- For endemic processes, either the hub or the innermost core can sustain epidemic activity, depending on network heterogeneity.Source: Discussion · p. 9
- Strong degree correlations can shift the balance between hub- and core-based activation mechanisms.Source: Discussion · p. 9
- The identified super-spreaders may inform immunization strategies tailored to a contact network and merit investigation in other spreading and dynamical processes.Source: Discussion · p. 9
Methods
The methods define the SIS and SIR dynamics and the network core, derive a star-graph SIR benchmark, and describe three real network datasets used in the analysis.
The SIS and SIR epidemic models
- SIS permits repeated infection and an endemic state above λc, whereas SIR ends in an absorbing state and separates finite-fraction outbreaks from infinitesimal ones at λc.Source: The SIS and SIR epidemic models · p. 10
- The k-core decomposition repeatedly removes vertices below successive degree thresholds; the maximum k-core is the set removed at the next iteration.Vertices in a k-core of index k have degree at least k.Source: The k-core decomposition · pp. 10–11
SIR model on a star-graph
- For SIR on a star of size qmax + 1, the expected outbreak size averages outcomes from infections starting at the hub or a leaf.The large-qmax approximation is used to obtain the average final density in equation (3).Source: SIR model on a star-graph · p. 11
Quantitative features of the real networks considered
- The real-network datasets are an Internet autonomous-system map, the PGP user network, and an actor-collaboration network.Source: Quantitative features of the real… · p. 12
- The study summarizes topological features of the real-network datasets in Table 1.Source: Quantitative features of the real… · p. 12
Tables
The table caption identifies topological features of the real networks, while the figure captions describe comparisons of activity across whole networks, maximum k-cores, and hubs.
Tables
- Table 1 summarizes real-network topological features, including network size N, average degree, maximum degree, and maximum k-core index kS.Source: Table 1 · p. 17, Tables · p. 17
Figure Captions
- In uncorrelated networks, SIS activity at γ = 2.1 tracks the maximum k-core, whereas at γ = 2.75 it tracks an active hub and subcritical core.The caption compares whole-network, maximum-k-core, and isolated-hub activity across spreading rates.Source: Figure Captions · p. 19
- For SIR on uncorrelated networks, the transition to a finite infected fraction tracks the maximum k-core, while hub activity follows the star-graph prediction.Source: Figure Captions · p. 20
- In the Movies and PGP networks the maximum k-core dominates SIS activation, while in the correlated AS network the hub dominates.Source: Figure Captions · p. 21
1 Details of SIS numerical simulations.
The SIS simulations use infected-node updates on UCM networks, start with all nodes infected, and average repeated dynamics on individual network instances.
1 Details of SIS numerical simulations.
- SIS simulations choose an infected node, remove it, infect each healthy neighbor with probability λ, and advance time by the inverse number of occupied nodes.The substrate networks are generated with the uncorrelated configuration model.Source: §1 · p. 22
- All nodes start infected; experiments starting from a single infected vertex are left for future work.Source: §1 · p. 22
- Results average at least 10 dynamical realizations on one network, not across network samples, to avoid obscuring geometric fluctuations.Source: §1 · p. 22
- Larger γ networks are larger to better separate the hub and heterogeneous-mean-field threshold scales.The stated scales are 1/√qmax and ⟨q⟩/⟨q2⟩.Source: §1 · p. 22
2 Results for additional values of γ.
In uncorrelated scale-free networks, the mechanism sustaining SIS activity depends on whether the degree exponent γ is above or below 5/2. Simulations show global activation tracks the hub for larger γ and the maximum k-core for smaller γ.
2 Results for additional values of γ.
- For γ < 5/2, global SIS activation is associated with an active maximum k-core while the hub remains subcritical.At γ = 2.3, the maximum k-core is responsible for overall network activity.Source: §2 · pp. 22–23, Figure 5 · p. 23
- For γ > 5/2, global activation is associated with an active hub while the maximum k-core remains subcritical.This hub-versus-core pattern is also confirmed for additional γ values.Source: §2 · pp. 22–23, Figure 5 · p. 23
- The simulations compare infection density on the whole network, the maximum k-core, and the largest hub as an isolated star.Networks use the UCM algorithm and include γ = 2.9, 2.6, and 2.3.Source: §2 · p. 23, Figure 5 · p. 23
3 Degree distribution of the maximum k-core.
The maximum k-core’s degree properties scale proportionally with network size in the studied uncorrelated scale-free networks. This supports using heterogeneous mean-field theory to estimate the scaling of its epidemic threshold.
3 Degree distribution of the maximum k-core.
- The topological analysis focuses on the maximum k-core’s average degree and maximum degree.Source: §3 · p. 24
- The maximum k-core’s computed degree properties scale proportionally with network size, indicating that its degree distribution remains narrow.The analysis plots these properties against qmax ~ N^1/2 for networks with γ = 2.5.Source: §3 · p. 24
- The observed scaling supports the assumption that heterogeneous mean-field theory correctly determines the maximum k-core’s epidemic-threshold scaling.Source: §3 · p. 24
4 Degree distributions and correlations in real net-
The three real networks have slowly decaying high-degree distributions with exponents close to 2, but their degree correlations differ. Movies and PGP are mildly assortative, whereas AS is strongly disassortative and its correlations persist after degree-preserving randomization.
4 Degree distributions and correlations in real networks.
- The Movies, PGP, and AS degree distributions decay slowly at large q, with exponents close to γ = 2.Source: works · p. 25
- Movies and PGP show mild assortativity, while AS is strongly disassortative, connecting high-degree vertices typically to low-degree ones.Source: works · p. 25, Figure 8 · p. 26
- Degree-preserving randomization removes the weak assortativity in Movies and PGP but leaves AS’s strong correlations apparent.Source: works · pp. 25–26, Figure 8 · p. 26
5 Values of the thresholds
Threshold estimates agree with the predicted activation mechanism for uncorrelated networks and for Movies and PGP, but AS correlations mix hub and core roles. A later AS map likewise shows global activity without either the hub or maximum k-core apparently active.
5 Values of the thresholds
- For UCM networks, measured SIS thresholds closely follow the paper’s predictions; Movies and PGP are closer to the HMF prediction associated with the maximum k-core.Source: §5 · p. 26, Table 2 · p. 27
- In AS, the inverse largest eigenvalue is close to 1/√qmax and well above the HMF prediction, consistent with correlations mixing hub and maximum-core roles.Source: §5 · pp. 26–27, Table 2 · p. 27
- In the later AS2 map, the global network is active at spreading rates where neither the hub nor the maximum k-core is apparently active.The authors report this result in other AS maps as well and conclude that strong correlations mix activation mechanisms.Source: §5 · pp. 27–28
6 SIS dynamics on Weber-Porto networks.
Weber–Porto SIS tests show that degree correlations can alter which activation mechanism operates: disassortative examples favor the hub, while assortative examples can favor the maximum k-core.
6 SIS dynamics on Weber-Porto networks.
- The authors simulate SIS dynamics on Weber–Porto networks, comparing activity on the whole network, maximum k-core, and an isolated largest-hub star.They vary degree exponent γ and nearest-neighbor degree exponent α; positive α indicates assortative correlations and negative α disassortative correlations.Source: §6 · pp. 28, 31, Table 3 · p. 28
- With disassortative correlations, 1/ΛN coincides with 1/√qmax for γ=2.1, α=−0.5 and is in very good agreement for γ=3.5, α=−0.5.For γ=2.1, the largest hub drives the global transition; for γ=3.5, the network has no k-core structure.Source: §6 · p. 30
- For γ=3.5 and α=2, 1/ΛN is much smaller than 1/√qmax, and the maximum k-core triggers the transition.For γ=2.1 and α=0.5, 1/ΛN is smaller than the HMF prediction, which is itself smaller than 1/√qmax; assortative correlations strengthen the k-core role for γ<5/2.Source: §6 · p. 30