Evolutionary games on multilayer networks: A colloquium

Zhen Wang; Lin Wang; Attila Szolnoki; Matjaz Perc

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Evolutionary games on multilayer networks: A colloquium

Zhen Wang, Lin Wang, Attila Szolnoki, Matjaz Perc

arXiv:1504.04359v1Apr 16, 2015physics.soc-phcs.GTcs.SInlin.AOq-bio.PE

TL;DR

Multilayer networks may better represent interdependent social systems than isolated-network models, but their effects on cooperation require synthesis. This colloquium reviews evolutionary games on multilayer networks and finds that interdependence can promote cooperation beyond isolated-network reciprocity, provided coordination is maintained.

Problem
Understanding how cooperation evolves remains a fundamental challenge, while isolated-network and well-mixed models inadequately represent interdependent social systems.
Method
The colloquium reviews multilayer-network concepts and evolutionary-game studies organized around utility, information, strategy-popularity, and network interdependencies.
Results
Interdependence promotes cooperation through mechanisms including correlated cooperator clusters, information transmission, probabilistic interconnectedness, and coevolution toward optimal coupling.
Takeaways & Limitations
Multilayer interdependence can promote cooperation beyond isolated networks, but its benefits depend on preserving coordination between the interacting layers.

Abstract

from arXiv · show

Networks form the backbone of many complex systems, ranging from the Internet to human societies. Accordingly, not only is the range of our interactions limited and thus best described and modeled by networks, it is also a fact that the networks that are an integral part of such models are often interdependent or even interconnected. Networks of networks or multilayer networks are therefore a more apt description of social systems. This colloquium is devoted to evolutionary games on multilayer networks, and in particular to the evolution of cooperation as one of the main pillars of modern human societies. We first give an overview of the most significant conceptual differences between single-layer and multilayer networks, and we provide basic definitions and a classification of the most commonly used terms. Subsequently, we review fascinating and counterintuitive evolutionary outcomes that emerge due to different types of interdependencies between otherwise independent populations. The focus is on coupling through the utilities of players, through the flow of information, as well as through the popularity of different strategies on different network layers. The colloquium highlights the importance of pattern formation and collective behavior for the promotion of cooperation under adverse conditions, as well as the synergies between network science and evolutionary game theory.

I. INTRODUCTION · II. FROM SINGLE-LAYER TOWARDS MULTILAYER NETWORKS

The introduction frames cooperation as evolutionary game theory’s central challenge and motivates multilayer models because people participate in multiple, interdependent social networks. The next section defines multilayer networks as systems with multiple layers connected by intra- and inter-layer links, extending beyond isolated single-layer assumptions.

I. INTRODUCTION: Cooperation is evolutionary game theory’s fundamental problem because altruistic acts cost performers, benefit others, and underpin transitions to complex societies.The paper describes cooperation as a grand challenge spanning social and natural sciences.
I. INTRODUCTION: Network reciprocity showed that lattice cooperators can survive exploitation by forming compact clusters, even when defectors dominate in well-mixed populations.Cluster formation protects cooperators in social dilemmas.
I. INTRODUCTION: Research is shifting from isolated networks toward interdependent, multiplex, and multilayer networks because small changes in one network can trigger unexpected consequences in another.The introduction presents networks of networks as more realistic for interconnected social interactions.
I. INTRODUCTION: Human strategies may be perceived and rewarded differently across friendships, workplaces, and families, motivating multilayer models of cooperative behavior.Individuals may also choose different strategies across weakly interconnected networks to optimize outcomes.
II. FROM SINGLE-LAYER TOWARDS MULTILAYER NETWORKS: Networks represent real-world entities as nodes connected by links, enabling quantitative descriptions of interactions across scientific disciplines.Examples include neuronal interactions and trade among markets.
II. FROM SINGLE-LAYER TOWARDS MULTILAYER NETWORKS: Single-layer network research assumes connections occur within one isolated infrastructure, which can oversimplify systems where nodes simultaneously belong to multiple networks.The paper states that this issue applies to both natural and social systems.
II. FROM SINGLE-LAYER TOWARDS MULTILAYER NETWORKS: A multilayer network contains M (M ≥2) layers with intra-layer links and inter-layer links connecting nodes across networks, including corresponding nodes that exchange information.The paper uses multilayer as a broad term covering interconnected, interdependent, multiplex, network-of-networks, and multivariate models.

A. Basic concepts and definitions of multilayer networks

Multilayer networks generalize single-layer graphs by combining multiple node sets and both intra-layer and inter-layer connections. Their formalism also accommodates differing layer sizes, partial node counterparts, weighted or directed links, and time-dependent adaptive connections.

Definitions: A single-layer network is represented as G = (V, E), with nodes V and edges E ⊆V ×V; multilayer networks generalize this structure across M layers.The multilayer formulation combines nodes and connections from all network layers.
Definitions: Each layer may contain a different number of nodes, and nodes may have one, several, or no counterparts in other layers.Thus, correspondence between layers is not necessarily complete or one-to-one.
Connections: Multilayer connections comprise intra-layer edges within individual layers and inter-layer edges linking nodes across distinct layers.The connection set is written as EM = {Eα ∪Eαβ; α, β ∈{1, ..., M}, α̸ = β}.
Extensions: Adjacency matrices encode intra-layer and inter-layer links, while multilayer formalisms can additionally represent weighted, directed, and time-dependent adaptive connections.Adaptive connections are modeled with time-dependent adjacency entries, whereas weighted or directed networks introduce additional sets such as WM.

B. The structure of multilayer networks

The section emphasizes that nodes can play different roles in multilayer network structure and dynamics, making node roles crucial to understanding structural properties.

B. The structure of multilayer networks: Node roles must be examined because they influence both multilayer network structure and the dynamical processes occurring on the network.The paper introduces specific quantities for reviewing these structural properties in subsequent subsections.

1. Node degree and related properties

Node degree in multilayer networks extends the single-layer definition by representing a node’s degree across layers, commonly through network aggregation. Because this produces a degree vector, uniform node ranking is difficult, motivating alternatives such as threshold, multidegree, and overlapping degree.

1. Node degree and related properties: Multilayer node degree extends the single-layer count of connected nodes and is commonly represented through network aggregation.For node i, the multilayer degree can be expressed as a vector whose layer-specific components are k_iα.
1. Node degree and related properties: Because node degree is a vector across layers, obtaining a uniform ranking of nodes is difficult.Related formulations include threshold degree, multidegree, and overlapping degree.

2. Clustering coefficient · 3. Degree-degree correlation

The section defines clustering coefficient and degree-degree correlation for single-layer networks, then explains why multilayer networks require broader, layer-aware formulations. It highlights averaging inter- and intra-layer clustering and using cross-layer correlation measures, including Pearson correlation and dependency-based normalization.

2. Clustering coefficient: The local clustering coefficient c_i measures the ratio of existing links among node i’s neighbors to all possible such links, while global C averages c_i across nodes.An alternative definition uses the fraction of closed triples among all possible triads.
2. Clustering coefficient: Multilayer clustering permits a class of definitions because isolated-network formulations have relative freedom, with generic approaches averaging coefficients for inter-layer and intra-layer links.One example uses neighbor sets and subgraph projection networks to define clustering as a function of each network.
3. Degree-degree correlation: Degree correlation measures node mixing: connections between similarly high- or low-degree nodes are assortative, whereas connections between high- and low-degree nodes are disassortative.These patterns correspond to positive and negative correlation coefficient r values, respectively.
3. Degree-degree correlation: Directly extending degree correlation to multilayer networks requires several considerations, and multiple methods have been proposed to determine correlation across network layers.The Pearson correlation coefficient is singled out as attracting the most interest.
3. Degree-degree correlation: The multilayer Pearson formulation uses each layer’s average degree ⟨k^α⟩ and degree standard deviation σ_α to characterize cross-layer degree correlation.The standard deviation is defined from the second moment of node degree and the square of its mean.
3. Degree-degree correlation: Inter degree-degree correlation measures dependence between node degrees across a pair of dependent networks using their degree distributions and joint dependency-link probabilities.The resulting measure is normalized by the maximum value ς_max and can be validated on empirical networks.

C. The classification of multilayer networks

The section uses “multilayer network” as a broad term for systems formed by more than one isolated network. It situates this convention within a diverse field containing overlapping network concepts and terminology.

C. The classification of multilayer networks: “Multilayer network” serves as a general term for networks formed by more than a single, isolated network.The term is used as a proxy for various multilayer configurations.
C. The classification of multilayer networks: The field includes diverse, sometimes overlapping concepts such as multiplex and temporal networks, despite differing names.The paper notes that several network concepts study similar systems under different terminology.
C. The classification of multilayer networks: The general multilayer-network convention is also used in research on topics such as network robustness.The passage notes that multilayer-network robustness has received notable attention.

1. Multiplex networks

Multiplex networks consist of layers containing the same or overlapping nodes, with each layer differing in how those nodes are connected. Examples include scientific collaboration and citation networks, as well as airline routes operated by different carriers.

Multiplex networks: Multiplex networks contain the same set of nodes across layers or share at least some fraction of them.This shared-node structure distinguishes multiplex networks from other multilayer configurations.
Multiplex networks: The layers differ in the way their nodes are connected to each other.Thus, the defining distinction lies in layer-specific connectivity rather than necessarily in node composition.
Multiplex networks: Examples include multiplex representations of scientific collaboration and citation networks, and airport routes organized by different airplane carriers.These examples span social and engineering systems.

2. Interdependent networks · 3. Interconnected networks · D. Algorithms for the generation of multilayer networks

The section distinguishes interdependent networks, linked by cross-layer dependency relations, from interconnected networks, linked by actual physical edges. It then outlines algorithmic approaches for generating multilayer structures through added inter-layer connections, network growth, or parameterized static models.

2. Interdependent networks: Interdependent networks contain multiple networks with little or no node overlap, where node wellbeing across layers is mutually coupled through dependency links.Dependency links represent co-dependence rather than physical connections.
3. Interconnected networks: Interconnected networks likewise contain multiple layers with little or no node overlap, but connect them using actual physical links between nodes.They can therefore be regarded as interconnected communities.
3. Interconnected networks: The multilayer-network classification is summarized through schematic illustrations, empirical observations, and basic properties relevant to evolutionary games.The discussion links network type to the consideration of evolutionary games before developing that topic in detail.
3. Interconnected networks: The paper next directs readers to works presenting algorithms for generating multilayer networks before expanding evolutionary games on multilayer networks in detail.This subsection serves as a transition to the algorithmic literature.
D. Algorithms for the generation of multilayer networks: A straightforward generation strategy first constructs individual layers with traditional algorithms and then inserts inter-layer connections according to requirements such as degree-degree correlation.This approach builds the layers separately before coupling them.
D. Algorithms for the generation of multilayer networks: Growth-based algorithms increase the number of nodes over time and commonly use preferential attachment, including inter-layer connection probabilities proportional to related nodes’ intra-layer degrees.The attachment rule can depend on the intra-layer degree of all related nodes in each layer.
D. Algorithms for the generation of multilayer networks: Static-model approaches require advance knowledge of each layer’s structural properties, then realize the multilayer architecture by adjusting parameters within or across layers.Relevant properties include degree distributions and degree correlations.

III. EVOLUTIONARY GAMES ON MULTILAYER NETWORKS

Evolutionary games on multilayer networks place players across layers, where they interact within and between layers and may adopt strategies from more successful competitors. The key question is how multiplexity and interdependence alter cooperation mechanisms, including through payoffs influenced by players in other layers.

Multilayer game structure: Players occupy nodes on different layers, interact with neighbors within and between layers, and typically adopt strategies from more successful competitors on the same layer.These games extend traditional single-network models by incorporating multilayer interactions.
Research focus: The central research question is whether multiplexity and interdependence preserve, weaken, or strengthen cooperation-supporting mechanisms observed on isolated networks.The colloquium focuses on identifying additional effects generated by interactions among network layers.
Payoff interdependence: Because strategy evolution primarily depends on payoff differences, a player’s payoff may also depend on the state of players in other layers.This cross-layer payoff coupling is introduced as a first way to model interactions between players occupying different layers.

A. Coupling through utilities

Coupling players’ utilities across network layers can promote cooperation by altering payoff feedback and creating beneficial asymmetries or interdependence. The resulting cooperation depends on coupling strength, the distribution of external links, and utility thresholds that determine whether interlayer effects persist.

Utility coupling: Cross-layer utilities combine each player’s own payoff with the external partner’s payoff, with Φ determining the weighting bias.The model specifies Ux = ΦPx + (1 −Φ)Px′ and Ux′ = (1 −Φ)Px′ + ΦPx.
Utility coupling: Stronger utility bias raises aggregate cooperation above isolated-network levels, although symmetry breaking can produce unequal cooperation across layers.The mechanism is attributed to suppressed feedback from individual success, which slows defector invasion relative to cooperative group formation.
Optimal interdependence: An optimal degree of interdependence can maximize cooperation because player inhomogeneity facilitates the formation of homogeneous cooperative groups.The positive effect is reported for both asymmetric and symmetric coupling, indicating that it is not limited to direct support for cooperators through payoff asymmetry.
Heterogeneous interdependence: When only a fraction of players maintains external links, coupling becomes heterogeneous, and coevolution can adjust teaching activity in response to successful or unsuccessful strategy transmission.Successful donors increase teaching activity by Δ, while unsuccessful strategy passes reduce it by the same value.
Threshold-dependent coupling: Utility thresholds can sustain widespread cooperation, but excessively large E prevents distinguished players from percolating and leaves cooperation dependent on weaker single-network reciprocity.The threshold mechanism is illustrated for both the prisoner’s dilemma and public goods games.

B. Alternative ways of coupling

Alternative coupling mechanisms connect otherwise independent populations through strategy popularity, information sharing, social pressure, or shared player identities rather than solely through payoffs. These couplings can promote cooperation by synchronizing strategy evolution and stabilizing cooperative domains, while shared identities define multiplex evolutionary games.

Popularity-based coupling: Strategy adoption can depend on a new strategy’s popularity in another network, in addition to neighbors’ payoffs in the player’s own network.The willingness to adopt a more successful strategy decreases when the old strategy is more frequent in the corresponding player’s neighborhood on the other network.
Popularity-based coupling: Despite being strategy neutral, popularity coupling allows cooperators to survive in parameter regions that yield an all D phase on a single-layer network.The improvement arises from spontaneously synchronized strategy evolution across networks.
Information sharing: Synchronization enhances cross-network strategy correlation, stabilizing compact cooperative domains while simultaneously slowing the evolution of defectors.Information sharing can likewise produce strongly correlated evolution across layers, as illustrated by snapshots of the upper and lower networks.
Playing and learning networks: A payoff-independent two-layer model separates each individual’s playing and learning networks, with social pressure measured by doubt potentially influencing cooperation.The relevant doubt concerns skepticism toward the wisdom of the crowd.
Multiplex networks: When the same players belong to multiple networks simultaneously, the framework is more accurately classified as evolutionary games on multiplex networks.This terminology follows the multilayer-network classification adopted by Gómez-Gardeñes et al.

IV. CONCLUSIONS AND OUTLOOK

Multilayer evolutionary-game models are presented as more apt descriptions of real-life systems than isolated-network or well-mixed models, while theoretical research provides quantitative insights that are difficult to test empirically. The review highlights mechanisms by which network interdependence can promote cooperation and identifies several directions for future research.

Conclusions: Networks of networks are described as more apt models of real-life systems than isolated networks or well-mixed models.The latter approaches remain valuable for proof-of-principle fundamental research.
Conclusions: Theoretical predictions may be difficult to test through human or economic experiments, making mathematical modeling central for obtaining quantitative insights into multilayer networks.The passage argues that this difficulty increases, rather than diminishes, the relevance of theoretical research.
Conclusions: Interdependent network reciprocity can maintain healthy public cooperation under extremely adverse conditions.The review identifies this as a prominent mechanism discovered in evolutionary games on multilayer networks.
Conclusions: Network interdependence can promote cooperation beyond isolated-network limits, provided coordination between interdependent networks is not disturbed.Other mechanisms include non-trivial organization of cooperators across interdependent network layers and probabilistic interconnection.
Outlook: Future work can examine other games on networks of networks, including ultimatum, rock-paper-scissors, naming, and collective-risk social dilemma games.The outlook also emphasizes pattern formation and collective behavior as recurring evolutionary-game topics.