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Behaviors of susceptible-infected epidemics on scale-free networks with identical infectivity
Tao Zhou, Jian-Guo Liu, Wen-Jie Bai, Guanrong Chen, Bing-Hong Wang
TL;DR
The paper asks how SI epidemics behave when degree does not determine infectivity, a distinction relevant to spreading processes with constrained forwarding capacity. It studies equal-infectivity SI dynamics on scale-free networks and compares spreading strategies using local information. The authors report that small-degree-preferring strategies spread more efficiently than large-degree-preferring strategies.
Problem
Earlier SI studies assumed infectivity equaled degree, leaving unclear whether rapid propagation arose from degree heterogeneity, infectivity heterogeneity, or both.
Source: §I · p. 2Method
The model gives every infected node the same fixed number of contacts per time step and applies the SI process to scale-free networks.
Source: §II · p. 2Results
Small-degree-preferring strategies are more efficient than large-degree-preferring strategies, and the optimal β for shortest saturation time is −0.3.
Source: §VI · pp. 5–6Takeaways & Limitations
The authors conclude that redundant contacts matter more than shorter paths in the studied fast-spreading strategy.
Source: §VI · p. 6
Abstract
from arXiv · showhide
In this article, we proposed a susceptible-infected model with identical infectivity, in which, at every time step, each node can only contact a constant number of neighbors. We implemented this model on scale-free networks, and found that the infected population grows in an exponential form with the time scale proportional to the spreading rate. Further more, by numerical simulation, we demonstrated that the targeted immunization of the present model is much less efficient than that of the standard susceptible-infected model. Finally, we investigated a fast spreading strategy when only local information is available. Different from the extensively studied path finding strategy, the strategy preferring small-degree nodes is more efficient than that preferring large-degree nodes. Our results indicate the existence of an essential relationship between network traffic and network epidemic on scale-free networks.
I. INTRODUCTION
The section motivates an SI model for processes without recovery and where infectivity may not track degree. It questions the assumption that hubs are necessarily superspreaders.
I. INTRODUCTION
- SI dynamics are relevant when recovery and death can be ignored, including early epidemic outbreaks and information broadcasting.In broadcasting, received nodes can forward information and generally do not lose it.Source: §I · p. 1
- Prior SI work on scale-free networks found approximately exponential infection growth, but assumed infectivity equals node degree.That assumption leaves unclear whether rapid propagation reflects the degree distribution, the infectivity distribution, or both.Source: §I · pp. 1–2
- Some spreading processes have limited, similar forwarding capacity across nodes, making degree-based infectivity assumptions unsuitable.Examples include wireless broadcasting, sexual contact, invitation systems, and network marketing.Source: §I · p. 2
- The standard hub ≡ superspreader assumption can fail when hubs cannot contact all neighbors or have no greater capacity to spread.The section distinguishes topological degree from an individual's or node's actual spreading capacity.Source: §I · p. 2
II. MODEL
The model assigns all infected nodes the same infectivity: each makes a fixed number of contacts per time step. Infection probability depends on the spreading rate and contacts with infected nodes.
II. MODEL
- The model uses a constant population N partitioned into susceptible S(t) and infected I(t) individuals.The infected density is i(t) = I(t)/N.Source: §II · p. 2, Figure 1 · p. 2
- The spreading rate λ is the per-time-step rate at which a susceptible acquires infection from an infected neighbor.The susceptible individual's infection probability is specified using θ(x,t−1), the number of contacts with infected individuals at the preceding step.Source: §II · p. 2
- Each infected node makes a fixed A contacts per time step, with repeated contacts to one neighbor allowed.Contacts between two infected nodes are counted, although they do not affect epidemic dynamics; the process starts from one randomly selected infected node.Source: §II · p. 2
III. SPREADING VELOCITY
The present model's spreading velocity is much less than standard SI, while its early exponential timescale is proportional to the spreading rate and nearly independent of the degree exponent.
III. SPREADING VELOCITY
- The analytical comparison matches the present model's average infectivity to standard SI by setting A = ⟨k⟩.The derivation uses a mean-field approximation to obtain the infected-density evolution.Source: §III · pp. 2–3
- For different spreading rates, early infected-density curves collapse when time is rescaled as λt.The single-log curves are approximately straight at early times, indicating exponential growth.Source: Figure 2 · p. 3, §III · p. 3
- The present model's spreading velocity is much less than standard SI, while the early exponential timescale c is proportional to λ and nearly independent of γ.The reported simulations also show that λ, rather than degree-distribution heterogeneity, governs epidemic behavior.Source: §III · pp. 3–4
- Simulations vary γ from 2.0 to 4.0 on scale-free configuration networks, alongside tests of spreading rates from 0.0001 to 0.01.The reported simulations use networks with average degree ⟨k⟩ = 6.Source: Figure 3 · p. 3, §III · p. 3
IV. TARGETED IMMUNIZATION
The section compares targeted vaccination of high-degree nodes in the standard and identical-infectivity SI models. The strategy markedly slows standard SI spreading but is much less effective in the present model.
IV. TARGETED IMMUNIZATION
- Random vaccination is inefficient on scale-free networks, whereas vaccinating random neighbors of randomly selected nodes can favor high-degree nodes when degree data are unavailable.When node degrees are known, the cited approach selects the highest-degree nodes for vaccination.Source: §IV · pp. 4–5
- Vaccinating even f = 0.001 of the population sharply reduces the standard SI model’s spreading velocity, while targeted immunization has a much weaker effect in the present model.In the present model, the exponential-growth slope changes only slightly as hub nodes are vaccinated.Source: §IV · p. 5
- Targeted immunization selects the fraction f of highest-degree nodes for vaccination; these Nf nodes cannot be infected, although their contacts are still counted.The comparison is with the standard SI model, where infectivity is tied to degree, and the present model, where it is identical across nodes.Source: §IV · p. 5
V. FAST SPREADING STRATEGY
The proposed fast-spreading strategy uses only local information and biases contacts toward neighbors according to degree. Its performance reflects competing shorter-path and redundant-contact effects, with the unbiased strategy outperforming extreme degree preferences.
V. FAST SPREADING STRATEGY
- Each infected node contacts a neighbor with probability proportional to k_y^β, using only local network information.The degree k_y belongs to the neighbor being contacted.Source: §V · p. 5
- The strategy balances a shorter path effect (SPE), associated with preferring high-degree nodes, against a redundant contact effect (RCE), since contacts to already infected nodes do not advance spreading.Large β can increase redundant contacts and slow the spread despite the shorter-path possibility.Source: §V · p. 5
- Strategies with β = 1 or 2 and β = −1 or −2 spread less efficiently than the unbiased strategy with β = 0.For β > 0, redundant contacts dominate and larger β slows spreading; for β < 0, the unbiased strategy leads early, but is later surpassed by the alternatives.Source: §V · p. 5
VI. CONCLUSION AND DISCUSSION
The conclusion argues that identical infectivity changes how scale-free network epidemics behave and makes targeted immunization less effective than in standard SI. It also reports that, with local information, preferring small-degree nodes is more efficient for fast spreading.
VI. CONCLUSION AND DISCUSSION
- In the identical-infectivity model, early infected-population growth is exponential and depends on spreading rate λ rather than the power-law exponent γ; targeted immunization is much less efficient than in standard SI.The authors attribute epidemic behavior to infectivity heterogeneity rather than degree heterogeneity.Source: §VI · p. 5
- The local-information strategy preferring small-degree nodes is more efficient than strategies preferring large-degree nodes.The authors interpret this as evidence that redundant contacts matter more than shorter paths.Source: §VI · pp. 5–6
- Under the saturation-time criterion—the first time infected density exceeds 0.9—the optimal β is −0.3.The criterion is used to estimate epidemic efficiency.Source: §VI · p. 6
- Traffic studies also report larger throughput for routing strategies favoring small-degree nodes, associated with avoiding possible congestion at high-degree nodes.The authors suggest there may be common features between network traffic and epidemic spreading on scale-free networks.Source: §VI · p. 6