Source-linked AI summary
Degree mixing in multilayer networks impedes the evolution of cooperation
Zhen Wang, Lin Wang, Matjaz Perc
TL;DR
The paper asks how degree mixing affects cooperation when payoff interactions and strategy updating occur on different network layers. It models two-layer scale-free networks, varies mixing and layer symmetry, and finds that most mixing configurations impede cooperation, with symmetric disassortative mixing providing the stated exception under harsher conditions. The authors use degree-dependent strategy distributions and cluster-size analysis to interpret these outcomes.
Problem
Cooperation research has traditionally focused on isolated networks, despite individuals belonging to multiple networks that can play different evolutionary roles.
Method
The study simulates two-layer scale-free networks in which one layer accumulates payoffs, the other updates strategies, and assortative or disassortative mixing is varied across both layers.
Results
Degree mixing generally impedes cooperation: symmetry-breaking combinations do so, and symmetric assortative mixing also inhibits it, whereas symmetric disassortative mixing can sustain cooperation under harsher conditions.
Takeaways & Limitations
Successful cooperation depends on both hub structure and preserving symmetry between the interaction and updating networks.
Abstract
from arXiv · showhide
Traditionally, the evolution of cooperation has been studied on single, isolated networks. Yet a player, especially in human societies, will typically be a member of many different networks, and those networks will play a different role in the evolutionary process. Multilayer networks are therefore rapidly gaining on popularity as the more apt description of a networked society. With this motivation, we here consider 2-layer scale-free networks with all possible combinations of degree mixing, wherein one network layer is used for the accumulation of payoffs and the other is used for strategy updating. We find that breaking the symmetry through assortative mixing in one layer and/or disassortative mixing in the other layer, as well as preserving the symmetry by means of assortative mixing in both layers, impedes the evolution of cooperation. We use degree-dependent distributions of strategies and cluster-size analysis to explain these results, which highlight the importance of hubs and the preservation of symmetry between multilayer networks for the successful resolution of social dilemmas.
I. INTRODUCTION
The paper extends cooperation research from isolated networks to multilayer societies, distinguishing payoff-accumulation and strategy-updating networks. It investigates how degree mixing and symmetry between these layers affect cooperation.
- Motivation: Multilayer networks better represent societies where individuals belong to multiple networks serving different evolutionary roles.The paper focuses on networks that may be interdependent or multilayered rather than isolated.
- Research scope: The model uses one layer for payoff accumulation and another for strategy updating, allowing the two network structures to differ.This distinction follows earlier work on symmetry breaking between interaction and replacement graphs.
- Research scope: Degree mixing is studied through assortative and disassortative connections in two-layer scale-free networks.Assortative mixing links similar-degree nodes, whereas prior isolated-network work associated disassortativity with hub refuges for cooperators.
- Research question: The study examines whether preserving or breaking symmetry between interaction and updating networks changes the evolution of cooperation.It considers all combinations of degree mixing across the two layers.
- Approach: The paper analyzes social dilemmas using multilayer scale-free networks and explains outcomes through degree-dependent strategy distributions and cooperative cluster sizes.The stated motivation connects the framework to previous multilayer and symmetry-breaking studies.
II. MATHEMATICAL MODEL
The model constructs two scale-free network layers with controlled degree mixing, assigns distinct interaction and updating roles, and simulates strategy evolution across social dilemmas. Cooperation is measured across payoff parameters and Monte Carlo runs.
- Network construction: Scale-free networks begin with average degree < k >= 4 and are corrected to remove spurious degree correlations before controlled mixing is applied.The Xulvi-Brunet-Sokolov algorithm removes correlations and then generates assortative or disassortative networks.
- Network construction: Assortative mixing uses A > 0 to connect similar-degree nodes, whereas disassortative mixing uses A < 0 to separate them.The main results focus on A values in [−0.3, 0.3], matching the range reported for most empirical networks.
- Layer roles: Each player occupies both layers, with AI controlling interaction-network mixing and AU controlling updating-network mixing.Players initially receive cooperation or defection with equal probability.
- Social dilemmas: The games use R = 1 and P = 0, while S and T vary across prisoner’s-dilemma, snowdrift, and stag-hunt parameter regions.The explored ranges are −1 ≤ S ≤ 1 and 0 ≤ T ≤ 2.
- Evolutionary dynamics: Each update compares payoffs earned on the interaction network after a player selects a neighbor on the updating network.The selected player adopts the neighbor’s strategy with Fermi-function probability, using K = 0.1.
- Simulation protocol: Results map the stationary cooperator fraction ρC over an 81×81 T−S grid using networks of 10^4 nodes and the final 10^4 of 10^5 Monte Carlo steps.The baseline corresponds to AI = AU = 0.
A. Symmetry preservation
When both network layers share the same degree-mixing coefficient, assortative mixing inhibits cooperation, while disassortative mixing generally lowers cooperation but can preserve it slightly better under harsh prisoner’s-dilemma conditions.
- Assortative mixing of both layers inhibits cooperation by interconnecting large-degree hubs, which undermines cooperative clusters and promotes defector invasion.
- Disassortative mixing generally lowers cooperation relative to neutral mixing across most of the parameter space.
- Under harsh prisoner’s-dilemma conditions, disassortative mixing makes cooperation slightly more persistent because isolated cooperative hubs resist defector invasion.
B. Symmetry breaking
Across asymmetric degree-mixing arrangements, mixing in either layer or opposite mixing types across layers impairs cooperation relative to the baseline. Stronger mixing worsens outcomes by disrupting the alignment between payoff accumulation, strategy updating, and cooperative hubs.
- Assortative interaction mixing combined with disassortative updating mixing inhibits cooperation, because low-degree defectors can access and invade cooperative hubs.The effect is especially pronounced in the prisoner’s-dilemma and snowdrift regions, and complete cooperator dominance is no longer achievable even at small temptations to defect.
- Disassortative interaction mixing combined with assortative updating mixing also impairs cooperation by disconnecting payoff support from cooperative reinforcement.Interconnected hubs may reinforce cooperation during updating, but disconnected hubs in the interaction layer do not receive appropriately high payoffs.
- Degree mixing confined to the updating layer impairs cooperation regardless of whether the updating network is assortative or disassortative.Here the interaction network remains neutral, while updating coefficients include AU = 0.1, 0.3, −0.1, and −0.3.
- Degree mixing confined to the interaction layer also impairs cooperation, with stronger mixing producing lower evolutionary success across the T −S plane.The updating network remains neutral, while interaction coefficients include AI = 0.1, 0.3, −0.1, and −0.3.
- Mixing reduces cooperators’ ability to occupy hubs, exposing potential followers to defector invasion and lowering overall cooperator density.This mechanism appears for both symmetry-preserving and symmetry-breaking combinations.
C. Analysis of cooperator clusters
Degree mixing disrupts the hub-centered, giant cooperative clusters found under the baseline, causing clusters to disintegrate and become more vulnerable to defector invasion.
- Baseline structure: Under neutral mixing with symmetry breaking, cooperators generally occupy network hubs and form a giant cooperative cluster.This arrangement allows cooperators to benefit strongly from network reciprocity.
- Effects of degree mixing: Degree mixing distorts the baseline organization: cooperative clusters disintegrate and become smaller.The paper links this structural change to the reduced ability of cooperators to retain hubs.
- Effects of degree mixing: Cooperators are no longer able to hold onto hubs, making smaller and more vulnerable cooperative clusters easier for defectors to invade.
- Cluster-size analysis: Figure 9 compares the number and largest size of cooperative clusters across symmetric and asymmetric degree-mixing combinations.Panels a and b show NC, while panels c and d show SC.
- Overall outcome: None of the multilayer mixing combinations improves baseline cooperative support, except symmetry-preserving disassortative mixing under adverse conditions.
IV. DISCUSSION
The study examines all assortative and disassortative mixing combinations across multilayer scale-free networks and finds that symmetry and hub isolation determine whether cooperation is sustained or impaired.
- IV. DISCUSSION: The study considers all three main social-dilemma types and all possible assortative and disassortative mixing combinations.
- IV. DISCUSSION: Only symmetry-preserving disassortative mixing sustains cooperation at harsher conditions than an isolated neutrally mixing scale-free network.The mechanism identified is isolation of hubs caused by disassortative mixing.
- IV. DISCUSSION: Symmetry-preserving assortative mixing impairs cooperation because increased hub interconnectedness favors defector invasion.
- IV. DISCUSSION: When symmetry between interaction and updating networks is broken, every assortative/disassortative combination impairs cooperation regardless of the layer affected.
- IV. DISCUSSION: The conclusion applies to all social dilemmas, with snowdrift and prisoner’s dilemma conditions most affected by degree mixing and symmetry breaking.