Source-linked AI summary
The physics of spreading processes in multilayer networks
Manlio De Domenico, Clara Granell, Mason A. Porter, Alex Arenas
TL;DR
Standard networks can miss important structural complexity in systems with multiple relationship types and interacting subsystems. This survey examines spreading processes on multilayer networks and finds that multilayer structure produces new physical behavior, including enhanced navigability, induced congestion, and new critical properties.
Problem
Traditional networks inadequately represent systems with multiple relationship types and interacting subsystems, motivating a unified mathematical framework for representing their dynamics.
Method
The paper surveys multilayer-network representations and single and coupled spreading dynamics to identify phenomena arising from multilayer structure.
Results
Multilayer spreading and random-walk dynamics exhibit enhanced navigability, induced congestion, and new critical properties.
Takeaways & Limitations
Multilayer structure changes the physical behavior of spreading and random-walk processes.
Takeaways & Limitations
Inferring or imposing inter-layer edges remains an open issue affecting multilayer phenomena.
Abstract
from arXiv · showhide
The study of networks plays a crucial role in investigating the structure, dynamics, and function of a wide variety of complex systems in myriad disciplines. Despite the success of traditional network analysis, standard networks provide a limited representation of complex systems, which often include different types of relationships (i.e., "multiplexity") among their constituent components and/or multiple interacting subsystems. Such structural complexity has a significant effect on both dynamics and function. Throwing away or aggregating available structural information can generate misleading results and be a major obstacle towards attempts to understand complex systems. The recent "multilayer" approach for modeling networked systems explicitly allows the incorporation of multiplexity and other features of realistic systems. On one hand, it allows one to couple different structural relationships by encoding them in a convenient mathematical object. On the other hand, it also allows one to couple different dynamical processes on top of such interconnected structures. The resulting framework plays a crucial role in helping achieve a thorough, accurate understanding of complex systems. The study of multilayer networks has also revealed new physical phenomena that remain hidden when using ordinary graphs, the traditional network representation. Here we survey progress towards attaining a deeper understanding of spreading processes on multilayer networks, and we highlight some of the physical phenomena related to spreading processes that emerge from multilayer structure.
Introduction
Multilayer networks provide a unified framework for representing multidimensional structures and studying their dynamics. This section focuses on spreading processes, where multilayer structure reveals new physical behavior in single or coupled dynamical processes.
- Introduction: A unified mathematical framework for multilayer networks was developed only recently to represent multiple interaction types, subsystems, and time dependence.These structures extend beyond single-layer networks, which contain one entity type and one connection type.
- Introduction: Multilayer structures can significantly change the qualitative behavior of complex systems, including percolation properties and catastrophic cascades of failures.These findings motivate consistent mathematical treatment of multiple connectivity layers.
- Introduction: Coupled spreading processes can make one disease’s onset depend on another’s, producing a critical-point curve and regimes with interdependent or non-interdependent criticality.The curve appears in the phase diagram of parameters governing spreading dynamics.
- Introduction: Multilayer structures can enhance cooperative behavior, providing a novel way for cooperation to survive in structured populations.This is presented as an additional physical phenomenon associated with multilayer organization.
- Introduction: The article surveys spreading processes in two cases: one process running on a multilayer network and different processes coupled across layers.The first case includes continuous or discrete diffusion; the second assigns a dynamical process to each layer.
Structural representation of multilayer networks
Multilayer networks represent multiple connectivity aspects and edge types simultaneously, yielding a richer structure than ordinary networks. Tensor-based representations and layer coupling expose structural, mathematical, and dynamical effects that require careful analysis.
- Multilayer structure: Multilayer networks combine aspects such as interaction types, subsystems, spatial locations, and time, making their structure richer than ordinary networks.Tensors can encode this multidimensional connectivity as multilinear-algebraic objects.
- Edge types: Three edge types distinguish within-layer connections, replica-node connections across layers, and cross-layer connections between distinct entities.These edge types can represent fundamentally different relationships, such as subway links versus transfers between subway and bus stations.
- Mathematical representation: A fourth-order multilayer adjacency tensor encodes relationships between any node i in layer α and node j in layer β, across N nodes and L layers.The same structure can alternatively be represented with adjacency-matrix sets or flattened into supra-adjacency matrices for computation.
- Structural measures: Naively generalizing monolayer measures can produce qualitatively incorrect or nonsensical results, so multilayer structural measures require careful construction.Tensor representations provide greater abstraction and support further mathematical development for complex systems.
- Layer coupling: Inter-layer coupling determines whether layers behave as independent entities or as an effectively single-layer system, with some networks showing a sharp transition between these regimes.Inter-layer weights can encode switching costs in transportation or self-reinforcement in opinion dynamics.
Single and coupled dynamics on multilayer networks
Multilayer networks support single processes shaped by intra- and inter-layer structure, as well as coupled processes that interact across layers. These dynamics reveal phenomena including enhanced diffusion, improved navigability, induced congestion, and dependent or inhibitory spreading transitions.
- Multilayer dynamics comprise single processes on coupled structures and mixed processes that couple separate layer processes through inter-layer connections.
- Single dynamics: Single-process behavior depends on both intra-layer structure and the presence and strength of inter-layer interactions.
- Single dynamics: Diffusion can be faster in a multiplex network than in any layer independently, with weak coupling slowing diffusion and strong coupling approaching the superposition’s mean speed.The transition between these regimes is structural, and multilayer paths can make superposition diffusion faster than diffusion in separate layers.
- Single dynamics: Inter-layer switching enriches random walks, enabling navigability larger than that of an aggregated network and greater resilience to uniformly random failures than individual layers.Multilayer structure can also induce congestion even when every layer would remain decongested independently.
- Coupled dynamical processes: Coupled spreading processes can enhance or inhibit one another, with a metacritical point separating regimes where critical properties are independent or dependent.The metacritical point delineates independence from dependence in the critical properties of the two processes.
Conclusions and perspectives
Multilayer network theory captures multiple relationships, subsystems, and temporal complexity, revealing new physics in spreading and random-walk dynamics. Despite progress, major challenges remain in measuring inter-layer structure, understanding cross-layer dynamical correlations, and developing broader theoretical tools.
- Scope and progress: Multilayer network theory generalizes traditional network theory to represent complicated systems with multiple relationship types, subsystems, and temporal change.The framework seeks to improve understanding of complex systems by incorporating these structural features.
- Emergent physics: Richer spreading and random-walk dynamics reveal enhanced navigability, induced congestion, and new critical properties in multilayer networks.These phenomena also affect coarse-graining networks and evaluating node importance.
- Open challenges: Measuring and interpreting inter-layer edges remains difficult because their weights are less reliable and their roles vary across applications.A related challenge is quantifying dependencies among layers, which are not independent.
- Open challenges: Understanding how network structure propagates dynamical correlations across layers is a major challenge affecting spreading processes and dynamical systems more generally.The manuscript also identifies synchronization and reaction–diffusion systems as important directions for multilayer dynamics.
- Future directions: Latent geometrical spaces offer a promising direction for multilayer network theory, combining geometric and statistical techniques with discrete network analysis.Explicit geometry can support continuum analyses and scrutinize dynamical processes on multilayer networks.