论文标题
多路复用网络上的超扩散
Hyper-diffusion on multiplex networks
论文作者
论文摘要
多路复用网络描述了其交互可能具有不同性质的系统，并且对于了解简单图的框架以外的网络的复杂性至关重要。最近，有人指出，将注意力限制在成对相互作用上也是一个限制，因为绝大多数复杂系统都包括强烈影响其动态的高阶相互作用。在这里，我们提出了多路复用网络上的超扩散，这是一个动力学过程，其中，由于存在链接重叠时存在高阶相互作用，因此每个单层的扩散与其他层中的扩散相结合。我们表明，在双链网络上的高扩散可以由超拉普拉斯式描述，其中由多链接$（1,1）$连接的每组四个复制节点之间的四体相互作用的强度可以通过参数$δ__{11}}} \ ge 0 $调用。当$δ_{11} = 0 $时，超拉丝拉斯人还原为标准的下拉普拉斯人，在两层捕获成对的相互作用。通过结合光谱图理论的工具，应用拓扑和网络科学，当$δ_{11}> 0 $时，我们提供了对双工网络上超扩散的一般理解，包括在Fiedler上的理论界限，超拉普拉奇人的最大特征和最大的特征范围，以及$Δ__} $Δ_{11}} $Δ__{11}} $δ_{11} \ gg1 $。尽管多路复用网络上的过度扩散并不意味着层之间直接``质量转移''（即，每一层中的复制节点的平均状态是动态的保守数量），但我们发现，两个层的动力学是在稳态相互作用的稳定状态时会伴随着同步的fieder nofe n ofer-forder nofe n ofer-forder nofe n of for-ny-fore n of for-ny-fied n of for-glied n of for-glape and fied and fied and fied and fied and fied and。本地化在双工网络的单层中。
Multiplex networks describe systems whose interactions can be of different nature, and are fundamental to understand complexity of networks beyond the framework of simple graphs. Recently it has been pointed out that restricting the attention to pairwise interactions is also a limitation, as the vast majority of complex systems include higher-order interactions that strongly affect their dynamics. Here, we propose hyper-diffusion on multiplex networks, a dynamical process in which diffusion on each single layer is coupled with the diffusion in other layers thanks to the presence of higher-order interactions occurring when there exists link overlap. We show that hyper-diffusion on a duplex network can be described by the Hyper-Laplacian in which the strength of four-body interactions among every set of four replica nodes connected by a multilink $(1,1)$ can be tuned by a parameter $δ_{11}\ge 0$. The Hyper-Laplacian reduces to the standard lower Laplacian, capturing pairwise interactions at the two layers, when $δ_{11}=0$. By combining tools of spectral graph theory, applied topology and network science we provide a general understanding of hyper-diffusion on duplex networks when $δ_{11}>0$, including theoretical bounds on the Fiedler and the largest eigenvalue of Hyper-Laplacians and the asymptotic expansion of their spectrum for $δ_{11}\ll1$ and $δ_{11}\gg1$. Although hyper-diffusion on multiplex networks does not imply a direct ``transfer of mass" among the layers (i.e. the average state of replica nodes in each layer is a conserved quantity of the dynamics), we find that the dynamics of the two layers is coupled as the relaxation to the steady state becomes synchronous when higher-order interactions are taken into account and the Fiedler eigenvalue of the Hyper-Laplacian is not localized in a single layer of the duplex network.
