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Spreading processes in Multilayer Networks
Mostafa Salehi, Rajesh Sharma, Moreno Marzolla, Matteo Magnani, Payam Siyari, Danilo Montesi
TL;DR
Spreading processes in multilayer networks remain less consolidated than single-network diffusion despite their relevance to information and pathogen propagation. The paper reviews multilayer models, results, and applications, finding varied layer-dependent behavior while identifying substantial empirical and theoretical open problems.
Problem
Real diffusion often crosses multiple networks, but multilayer spreading remains less understood than spreading on single networks.
Method
The paper reviews models, results, applications, and research directions for spreading processes in multilayer networks.
Results
Multilayer models show that layer coupling, network structure, and individual features influence spreading outcomes, including epidemic phases and spreading speed.
Takeaways & Limitations
The review organizes current multilayer-spreading knowledge and identifies unexplored areas requiring further study.
Takeaways & Limitations
No real datasets combining multilayer diffusion and underlying network structure were available in the reviewed literature; existing studies were simulation-based or analytic.
Abstract
from arXiv · showhide
Several systems can be modeled as sets of interconnected networks or networks with multiple types of connections, here generally called multilayer networks. Spreading processes such as information propagation among users of an online social networks, or the diffusion of pathogens among individuals through their contact network, are fundamental phenomena occurring in these networks. However, while information diffusion in single networks has received considerable attention from various disciplines for over a decade, spreading processes in multilayer networks is still a young research area presenting many challenging research issues. In this paper we review the main models, results and applications of multilayer spreading processes and discuss some promising research directions.
1 INTRODUCTION
The paper reviews spreading processes in multilayer networks, where real diffusion often crosses multiple networks rather than remaining within one. It organizes the topic around models, their results, and applications.
- Online information and epidemics commonly switch among multiple networks, so real spreading phenomena are seldom constrained to a single monoplex network.
- Multilayer networks encompass models involving multiple networks or multiple relationship types, including interconnected and multiplex networks.
- The field extends established network-spreading research while introducing effects from layer structures and their interdependence.
- The review provides a homogeneous overview of current results and identifies unexplored areas in multilayer diffusion.
- Its analysis covers how spreading processes are modeled, what results those models produce, and how results can be exploited in applications.
2 PRELIMINARIES
This section defines multilayer networks, spreading processes, cascades, and diffusion variables, then surveys modeling approaches and the limited availability of empirical multilayer-diffusion data.
- A multilayer network contains multiple monoplex layers, with nodes potentially shared across layers and edges connecting nodes within or between layers.
- Spreading processes describe propagation of pathogens, rumors, behaviors, or news across intra-layer and inter-layer connections.
- A multilayer cascade records timestamped transfers between node-layer pairs, while the diffusion network contains the connections actually traversed.
- Four inter-layer traversal types include same-node and other-node diffusion, alongside other-node intra-layer diffusion.
- Existing multilayer-diffusion research consists of simulation-based or analytic studies because collecting both diffusion data and the underlying multilayer network is difficult.
- Key variables include transmissibility, epidemic thresholds, infection or cascade size, infection rate, epidemic dynamics, cascade velocity, recall, and precision.
3 MODELING SPREADING PROCESSES IN MULTILAYER NETWORKS
The paper reviews how spreading processes in multilayer networks are modeled, emphasizing epidemic-like and decision-based models alongside mathematical analysis techniques. It also surveys generalized models and reported findings about epidemic thresholds and individual features.
- Real multilayer spreading data are difficult to obtain, so modeling is used to understand and analyze diffusion dynamics.
- Existing models are categorized as epidemic-like models, where neighbors determine infection, and decision-based models, where agents adopt behavior based on others.
- SIR and SIS models represent spreading with recovery or reinfection dynamics, beginning from initially infected seed nodes and using infection rate β.
- Multilayer models can assign different transmissibilities to intra-layer and inter-layer connections because infection may diffuse across them at different speeds.
- The generalized epidemic mean-field model supports epidemic-like spreading with more complex states across multiplex network layers than SIS and SIR models.
- Analyses use generating functions, branching processes, percolation theory, and microscopic Markov-chain approximations to study spreading dynamics.
- Branching processes cannot handle transmission probabilities that depend on a destination agent’s infection history, such as immunity in SIR dynamics.
- Microscopic Markov-chain approximation analyzes node-level epidemic behavior, and one multiplex SIS study reduces independent layer dynamics to a single network governed by an effective contagion matrix.
4 SPREADING DYNAMICS ON MULTILAYER NETWORKS
Spreading dynamics in multilayer networks depend on underlying network properties, and reducing multiplex layers to a monoplex can discard information and produce wrong conclusions.
- The speed and pattern of spreading are influenced by properties of the underlying multilayer network.
- Aggregating multiplex layers into a weighted monoplex can support epidemic-threshold and infection-size analysis for SIR and SIS models.
- Disregarding the inherent multiplex structure can lead to information loss and wrong conclusions.
4.1 Interconnected Networks
In interconnected networks, spreading is shaped by inter-layer coupling strength, topology, density, and degree correlations. These properties can alter epidemic phases, thresholds, and infection sizes.
- Inter-layer coupling affects diffusion through spectral properties of the combinatorial supra-Laplacian, whose behavior depends strongly on interaction strength between layers.
- Strong coupling produces either an epidemic state or a disease-free state, whereas weak coupling can create a mixed phase in which only one layer is epidemic.
- In the weakly coupled mixed phase, increasing inter-layer links affects the layer with more intra-layer links while leaving the other layer’s epidemic unchanged.
- The interconnection-topology measure Ω quantifies coupling in two-layer networks, with larger Ω indicating stronger coupling.
- Inter-layer link density is the ratio d = m/(nA × nB) of existing to possible inter-layer links, with 1 representing complete interconnection.
- Simulation results indicate that degree-based inter-layer connection patterns have less effect on infection size than interconnection density.
- For a two-layer SIS network, the epidemic threshold is 1/λ1(M + αN), where α controls infection between layers and N represents inter-layer links.
- Inter-layer patterns can be characterized through correlations between nodes’ intra-layer and inter-layer degrees.
4.2 Multiplex Networks
In multiplex networks, spreading depends on both cross-layer similarity and within-layer structure. Studies show that clique size, layer correlations, and network size can produce different epidemic outcomes.
- Information diffusion in a social-physical multiplex network depends on how links are distributed both between layers and within the same layer.
- Larger physical-network cliques make information spread faster in the analyzed social-physical multiplex model.
- Increasing average clique size from 1 to 2 raises the percentage of individuals receiving the message from 14% to 80%.
- A larger online social network may not produce an outbreak in a social-physical multiplex network.
- Degree-degree correlation and average neighbor similarity are used to measure similarity between multiplex layers.
- Strong positive degree-degree correlation across layers can produce a low epidemic threshold and a relatively smaller infection size.
4.2.3 Layer-switching cost
Layer-switching cost captures the extra difficulty of moving a spreading process between layers. Studies find that this overhead changes epidemic thresholds, with effects depending on layer density and infection-rate differences.
- Layer-switching cost: Layer-switching cost is modeled through differences between intra- and inter-layer transmissibilities in SIR dynamics.The cost reflects the additional effort or reduced effective infection rate associated with changing communication channels.
- Layer-switching cost: When layers have the same average degree, larger differences between intra- and inter-layer infection rates increase the epidemic threshold.Higher crossing overhead makes spreading to other layers more difficult.
- Layer-switching cost: Greater disparity in average degree lowers the epidemic threshold for a fixed rate difference when one layer becomes denser.Denser layers facilitate spreading, although a threshold in degree disparity can alter this relationship.
- Layer-switching cost: In multiplex SIS dynamics, the layer with the largest eigenvalue controls the epidemic threshold of the entire network.This result uses a contact-contagion formulation with distinct intra- and inter-layer infection rates.
4.2.4 Diffusion velocity
Diffusion velocity depends on how layers are coupled and on the topology of inter-layer connections. Although coupling can accelerate spreading, obstructed or inefficient cross-layer paths can slow it.
- Diffusion velocity: Coupling two multiplex layers can speed up spreading by making additional links available across communication channels.The reported result concerns information diffusion and is supported by studies of multiplex spreading dynamics.
- Diffusion velocity: Blocking an inter-layer link that connects shortest paths distributed across layers slows cascade spreading in multiplex networks.This simulation-based result highlights the role of strategically positioned inter-layer links.
- Diffusion velocity: Spreading speed can decrease when link types or paths are topologically inefficient, motivating further study of multilayer diffusion velocity.Related random-walk results also connect visitation time to topology, inter-layer-link strength, and walk type.
- Diffusion velocity: Table 4 organizes studied spreading properties alongside network type, inter-layer connection structure, spreading models, measures, and theoretical approaches.The table uses abbreviations for network configurations, diffusion measures, and methods such as generating functions and mean-field theory.
4.2.5 Partially overlapped multiplex networks
Partial node overlap changes epidemic thresholds and diffusion outcomes by coupling otherwise distinct layers. Increasing overlap connects the behavior of the layers, while overlap can enlarge the affected population.
- Partially overlapped multiplex networks: Overlap among layers adds robustness to multilayer networks, while node behavior may differ across layers.Partially overlapped multiplexes contain only a fraction of nodes in every layer.
- Partially overlapped multiplex networks: In a two-layer social-physical network, epidemic diffusion can occur in the combined network even when neither individual network percolates.The study models information spreading through physical communication and social-network links.
- Partially overlapped multiplex networks: The fraction of nodes receiving information is significantly larger in the overlapped network than when the networks are disjoint.This result concerns SIR dynamics on the two-layer social-physical network.
- Partially overlapped multiplex networks: The epidemic threshold depends on both each layer’s topology and the fraction q of nodes shared between layers.As q approaches zero, diffusion is concentrated mainly in layer A; as q approaches one, it approaches the fully overlapped multiplex case.
- Partially overlapped multiplex networks: Shared nodes allow the layer with lower propagating capability to affect the epidemic threshold of the other layer.This implication follows from the reported threshold expression for the two-layer system.
4.2.6 Interacting spreading processes
Interacting spreading processes can compete, cooperate, or modify one another across multiplex layers. Their outcomes depend on initial conditions, network structure, degree correlations, layer overlap, and process dynamics.
- Interacting spreading processes: Multiple diseases or other spreading processes can occur concurrently on the same network and generate interacting cascades.Different processes may produce different cascade dynamics while affecting one another.
- Interacting spreading processes: When processes can become extinct, one process may dominate the other even when both diffusion rates exceed the epidemic threshold.Domination time also depends on the number of infected nodes at the beginning of the process.
- Interacting spreading processes: Game-theoretical models study competing rumors and firms allocating resources to maximize product adoption in social networks.The rumor results indicate that starting first is not always advantageous.
- Interacting spreading processes: Cross-immunity is more effective when high-degree nodes in different layers are connected, whereas positive degree correlation improves immunization efficiency and overlap facilitates disease invasion.These findings come from analytic and dynamical studies of interacting SIR processes on multiplex networks.
- Interacting spreading processes: Mean-field analysis derives epidemic thresholds for concurrent processes, with both thresholds depending on network structure and each process’s dynamics.The framework extends to SIR, SIS, and SEIR epidemic models.
- Interacting spreading processes: Computer-science models address misinformation limitation and interacting memes using extensions of independent-cascade and SIS-type models.These approaches analyze multiple cascades over multiplex networks.
4.2.7 Diffusion of Innovations
Multilayer innovation diffusion depends on interaction types and network coupling, while limited node resources can reverse how coupling affects epidemic spreading.
- Multiplex innovation studies examine global cascade conditions and cascade size using an extension of Watts’ threshold model.
- Without resource constraints, positively correlated coupling produces a lower epidemic threshold than negatively correlated coupling.
- Under resource constraints, spreading is less efficient with positively correlated coupling than with negatively correlated networks.The constrained SIR model limits the number of neighbors each node can infect at every step.
5 APPLICATIONS
Multilayer spreading research supports forward prediction to steer diffusion and backward prediction to infer or control it, across influence, disease, networking, and malware applications.
- Applications include cascade analysis, influence maximization, and selecting sensor locations to detect spreading quickly.
- Forward prediction steers a network toward a desired state, whereas backward prediction estimates how information diffuses or supports detection and immunization.
- 5.1 Influence Maximization: Influence maximization seeds strategic nodes to spread messages quickly to many nodes, but monoplex k-shell rankings can lose effectiveness in interconnected networks.
- 5.1 Influence Maximization: A message-survival metric can select among equivalent messages the one likely to reach a higher fraction of nodes in a multilayer graph.
- 5.2 Immunization Strategies: Awareness and prevention layers can reduce epidemic waves or raise infection thresholds, although prevention may sometimes help disease persist.
- 5.2 Immunization Strategies: SAIS-based studies identify spectral centrality as determining an information-propagation overlay, with small-subgroup monitoring contributing to disease prevention.
- 5.3 Epidemic Routing in Delay-Tolerant Networking: In delay-tolerant networks, finding routes under latency and energy constraints is a forward-prediction application of multilayer networks.
- 5.4 Malware Propagation in the Internet: Malware can exploit multiple physical communication layers and application interactions, making multilayer analysis useful for predicting infections and placing countermeasures.
6 CONCLUSION AND OPEN PROBLEMS
The field remains unsettled, with open problems in empirical validation, visualization, and the joint evolution of network structure and spreading.
- Information diffusion in multilayer networks remains active and not yet consolidated, requiring new ideas and algorithms alongside extensions of monoplex understanding.
- Real multilayer diffusion datasets are difficult to collect, and existing studies are based on simulation or analytic methods rather than real datasets.
- Effective visualization remains unclear because multilayer diffusion combines time-dependent dynamics with multiple network layers.
- Changing an underlying network to prevent infection does not always reduce spreading in adaptive-network settings.