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Layer-switching cost and optimality in information spreading on multiplex networks
Byungjoon Min, Sang-Hwan Gwak, Nanoom Lee, K. -I. Goh
TL;DR
The paper studies how layer-switching costs affect information spreading across multiplex interaction networks, where transmissibility depends on both incoming and outgoing layers. It formulates an analytical framework for this path dependence and finds that spreading outcomes and optimal conditions can respond non-monotonically and abruptly to the cost, depending on network and seed characteristics.
Problem
Layer-switching costs are commonplace in multiplex interactions, but their implications for spreading dynamics had not been explicitly addressed systematically.
Source: Introduction · p. 1Method
The authors formulate a generating-function framework for SIR spreading on locally treelike multiplex networks with layer-wise path-dependent infection rates.
Source: Introduction · p. 2, The Model · p. 2, Analytical framework · p. 3Results
Layer-switching cost can affect epidemic threshold and prevalence non-monotonically, with optimal conditions changing abruptly depending on layer densities and seed infection channel.
Source: Epidemic threshold and prevalence · p. 8, Optimal layer-switching cost for… · p. 8, Summary and Outlook · p. 12Takeaways & Limitations
Explicitly representing multiplex interactions matters because aggregating them into one network could lead to inaccurate conclusions about spreading.
Source: Introduction · p. 2The analytical theory is derived under the assumption of locally treelike random network layers.
Source: Analytical framework · p. 3
Abstract
from arXiv · showhide
We study a model of information spreading on multiplex networks, in which agents interact through multiple interaction channels (layers), say online vs.\ offline communication layers, subject to layer-switching cost for transmissions across different interaction layers. The model is characterized by the layer-wise path-dependent transmissibility over a contact, that is dynamically determined dependently on both incoming and outgoing transmission layers. We formulate an analytical framework to deal with such path-dependent transmissibility and demonstrate the nontrivial interplay between the multiplexity and spreading dynamics, including optimality. It is shown that the epidemic threshold and prevalence respond to the layer-switching cost non-monotonically and that the optimal conditions can change in abrupt non-analytic ways, depending also on the densities of network layers and the type of seed infections. Our results elucidate the essential role of multiplexity that its explicit consideration should be crucial for realistic modeling and prediction of spreading phenomena on multiplex social networks in an era of ever-diversifying social interaction layers.
Introduction
Multiplex social networks combine interaction layers whose interplay can produce collective outcomes beyond single-network dynamics. The paper addresses the previously unstudied role of layer-switching cost in information spreading and develops a path-dependent model and analysis of its effects.
Introduction
- Social media can form an interaction layer alongside existing channels, potentially overcoming traditional spatial and temporal limits.The passage describes this layer as coexisting and cooperating with other interaction layers.Source: Introduction · p. 1
- Layer interactions can produce nontrivial collective outcomes, challenging extensions of single-network frameworks that treat layers as independent.The passage notes that layers may neither function completely autonomously nor dependently.Source: Introduction · p. 1
- Layer-switching cost is a surcharge for transmissions across interaction layers, a commonplace feature not yet systematically addressed in multiplex spreading dynamics.The paper motivates this with platform-dependent information diffusion and the extra effort or delay of switching communication channels.Source: Introduction · pp. 1–2
- Figure 1 illustrates how a link’s transmissibility can vary with the preceding transmission layer.The schematic shows two sample spreading trajectories from a seed infection in layer 1.Source: Figure 1 · p. 2
- The authors formulate a generating-function theory for locally treelike networks to handle transmissibility that depends on both incoming and outgoing transmission layers.They use analytical calculations and numerical simulations to study consequences for spreading and optimality.Source: Introduction · p. 2
The Model
The model extends SIR spreading on multiplex networks by making infection rates depend on both the layer of the incoming transmission and the layer of the outgoing contact. Consequently, transmissibility is path-dependent rather than fixed for each link.
The Model
- The model uses an SIR framework in which infected nodes transmit to susceptible neighbors and recover after time τ.With sharply peaked recovery times, transmissibility is T = 1−e^−λ, where λ = βτ; prevalence is the stationary fraction of recovered nodes.Source: The Model · p. 2
- A type-i transmission is an infection through a layer-i link, and its subsequent infection rate depends on both incoming and outgoing layers.After a type-i infection, same-layer transmission uses λii and transmission through layer j uses λji.Source: The Model · p. 2
- Because transmissibilities depend on the incoming and outgoing transmission types, a link’s transmissibility can change as spreading proceeds.The model commonly considers same-layer rates at least as large as cross-layer rates, but does not require that constraint.Source: The Model · p. 3
Analytical framework
The framework extends SIR and multiplex percolation methods to path-dependent transmissibility in locally treelike networks. It estimates outbreak size through effective incoming transmissibilities, treats outbreak probability by seed layer, and derives the threshold from a stability criterion.
Analytical framework
- The theory is derived for locally treelike random network layers, with the duplex case used for illustration and generalization to more layers described as straightforward.The authors note that the approach may also apply to some non-treelike real networks.Source: Analytical framework · p. 3
Outbreak size
- For outbreak size, a self-consistency method infers effective incoming transmissibilities that yield the same outbreak size as the path-dependent model.It estimates the probability that an infected node was reached through each transmission type, then transforms the model to a path-independent SIR model for calculating outbreak size.Source: Outbreak size · p. 3
Outbreak probability
- Outbreak probability is computed by following path-dependent transmissibilities explicitly, and it can depend on the layer of the seed infection.The type-i seed outbreak probability Pi is obtained from the physical solution of coupled self-consistency equations.Source: Outbreak probability · p. 4
Epidemic threshold
- The epidemic threshold is determined by stability of the trivial fixed point: outbreak requires the Jacobian’s largest eigenvalue Λ to exceed one.The multiplex threshold cannot be larger than the epidemic thresholds of either individual layer.Source: Epidemic threshold · p. 4
Comparison with numerical simulations
- Numerical simulations initialize one seed through a specified layer, apply rates based on incoming and outgoing channels, and estimate outbreak probability and size across runs.The simulation uses a 1% recovered-node threshold to identify runs ending in an outbreak; prevalence is computed as ρi = PiS.Source: Comparison with numerical simulations · p. 4
Duplex ER networks
The duplex ER analysis shows that layer-switching cost interacts with layer-density disparity and infection capacity to shape spreading, sometimes non-monotonically. Optimal cost and interlayer degree correlations further alter prevalence and epidemic thresholds.
Duplex ER networks
- For randomly coupled duplex ER networks, analytical calculations agree well with simulations at N = 104, with near-threshold deviations attributed to finite size.Outbreak probabilities can differ by seed layer, and outbreak probability and size need not be equal above threshold.Source: Duplex ER networks · p. 5
Assessing the effect of layer-switching cost
- The analysis parameterizes layer infection rates and degrees so δλ varies at fixed average infection rate and δz varies at fixed total degree.The layers have equal within-layer rates and equal across-layer rates, with within-layer transmission assumed easier than across-layer transmission.Source: Assessing the effect of… · p. 5, Duplex ER networks · p. 5
- At δz = 0, increasing δλ raises the threshold and lowers prevalence; at δz = 1/2, it can lower the threshold yet suppress prevalence at sufficiently high λ.For δz = 1/2 and small λ, confining spread to the denser layer can instead help avoid trapping in the sparse layer.Source: Assessing the effect of… · p. 5, Figure 2 · p. 6
Infection channel profile
- For δz = 0, increasing δλ suppresses cross-layer transmission while intra-layer channel use remains similar; for δz = 1/2, denser-layer transmission increasingly dominates.At δz = 1/2, total prevalence is more or less insensitive to δλ even as channel composition changes.Source: Figure 2 · p. 6, Infection channel profile · p. 7
Epidemic threshold and prevalence
- For intermediate density disparity (0.2 ≲ δz ≲ 0.7), layer-switching cost has non-monotonic effects and can produce abrupt changes in optimal conditions.At lower or higher disparity, its effect is largely monotonic, though the direction reverses between those regimes.Source: Epidemic threshold and prevalence · p. 8
Optimal layer-switching cost for maximal spreading
- With δz > 0.280, the prevalence-maximizing cost changes with λ and can jump abruptly; below the transition, δλ = 1 is optimal down to threshold.For δz < 0.280, small δλ is advantageous for every λ above threshold.Source: Optimal layer-switching cost for… · p. 8, Figure 4 · p. 8
Effect of interlayer degree correlations
- Interlayer degree correlation lowers the epidemic threshold but, at sufficiently high λ, maximally anticorrelated layers yield greater prevalence than maximally correlated layers.At intermediate prevalence levels, the response to correlation depends on details.Source: Figure 5 · p. 9, Effect of interlayer degree correlations · p. 10
Empirical Twitter network
On the empirical Twitter multiplex, layer-switching cost has a moderate but non-negligible effect on prevalence, with retweets dominating infections. Broad degree distribution and extensive overlap constrain observable threshold and suppression effects.
Empirical Twitter network
- The simulated network has 456,631 accounts, with mean degrees 3.21 for retweets and 1.92 for replies.The layers represent retweet and reply interactions on Twitter; each link records at least one such interaction.Source: Empirical Twitter network · p. 10
- For λ≲1, layer-switching cost changes prevalence by as much as 10–50%, while differences become negligible when λ≳2.5.The broad degree distribution brings the epidemic threshold close to zero, making its response to switching cost unobservable in this dataset.Source: Empirical Twitter network · p. 10
- Infections through the retweet layer predominate because it is denser, while 73.2% of reply links overlap with retweet links.The two-layer network’s giant connected component is dominated by the retweet layer, constraining outbreak suppression for large λ.Source: Empirical Twitter network · p. 10
Infection Rates Dependent on Source Layer
The framework also covers path-dependent infection rates determined primarily by the source layer. In a duplex ER comparison, this parameterization yields higher prevalence and a lower epidemic threshold than the original layer-switching-cost model, and can be more effective for maximizing spread.
Infection Rates Dependent on Source Layer
- The framework applies beyond layer-switching cost to spreading processes with layer-wise path-dependent transmissibilities.One case makes infection rates path-dependent but determined primarily by the source layer.Source: Infection Rates Dependent on Source… · p. 10
- For −1≤c≤1, the source-layer parameterization represents differing credibility: information received in layer 1 for c>0 is more likely to be passed on through either layer.Information received in the other layer may receive less attention and be disregarded.Source: Infection Rates Dependent on Source… · p. 10
- The source-layer-dependent model tends to produce larger prevalence and a lower epidemic threshold than the original layer-switching-cost model.Its parameter setting can also be more effective at maximizing information spreading than the original model’s setting.Source: Infection Rates Dependent on Source… · pp. 10, 12
- Figure 7 maps epidemic threshold and prevalence against δz and δλ for the source-layer-dependent model on duplex ER networks with z0=2.5.Its lower panels show the corresponding maxima and minima over δλ as δz varies.Source: Figure 7 · p. 12
Summary and Outlook
The paper concludes that layer-switching cost and multiplex structure can alter spreading outcomes in nontrivial ways. Its analysis argues that multiplex interactions should be modeled explicitly, while identifying broader settings for future investigation.
Summary and Outlook
- Optimal parameters for maximizing prevalence or minimizing the epidemic threshold can vary non-analytically with layer-switching cost, layer link densities, and seed infection channel.The behavior is reminiscent of abrupt structural transitions in interconnected networks.Source: Summary and Outlook · p. 12
- The authors conclude that multiplex interactions cannot simply be reduced to one aggregated layer when modeling these spreading outcomes.Their preliminary analysis reports qualitatively similar layer-switching-cost effects for the SIS model.Source: Summary and Outlook · p. 12
- The framework is presented as a platform for studying layer-wise path-dependent transmissibility, including path-dependency associated with memory in temporal networks.The impact of memory in temporal networks is identified as a subject of recent study and future inquiry.Source: Summary and Outlook · p. 12