Random walks are the simplest way to explore or search a graph and have revealed a very useful tool to investigate and characterize the structural properties of complex networks from the real world. For instance, they have been used to identify the modules of a given network, its most central nodes and paths, or to determine the typical times to reach a target. Although various types of random walks whose motion is biased on node properties, such as the degree, have been proposed, which are still amenable to analytical solution, most if not all of them rely on the assumption of linearity and independence of the walkers. In this work we introduce a class of nonlinear stochastic processes describing a system of interacting random walkers moving over networks with finite node capacities. The transition probabilities that rule the motion of the walkers in our model are modulated by nonlinear functions of the available space at the destination node, with a bias parameter that allows to tune the tendency of the walkers to avoid nodes occupied by other walkers. First, we derive the master equation governing the dynamics of the system, and we determine an analytical expression for the occupation probability of the walkers at equilibrium in the most general case and under different level of network congestions. Then we study different types of synthetic and real-world networks, presenting numerical and analytical results for the entropy rate, a proxy for the network exploration capacities of the walkers. We find that, for each level of the nonlinear bias, there is an optimal crowding that maximizes the entropy rate in a given network topology. The analysis suggests that a large fraction of real-world networks are organized in such a way as to favor exploration under congested conditions. Our work provides a general and versatile framework to model nonlinear stochastic processes whose transition probabilities vary in time depending on the current state of the system.

中文翻译：

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非线性助步器和拥塞网络的有效探索

随机游走是探索或搜索图形的最简单方法，并且揭示了一种非常有用的工具，可以研究和表征现实世界中复杂网络的结构特性。例如，它们已被用来识别给定网络的模块，其最中心的节点和路径，或确定达到目标的典型时间。尽管已经提出了各种类型的随机行走，其运动偏向于节点的属性（例如，度数），这些随机行走仍适用于解析解，但大多数（如果不是全部）依赖于行走者的线性和独立性的假设。在这项工作中，我们引入了一类非线性随机过程，该过程描述了一个相互作用的随机步行者在具有有限节点容量的网络上移动的系统。决定我们模型中步行者运动的过渡概率由目标节点处可用空间的非线性函数调制，并带有一个偏差参数，该参数允许调整步行者避开其他步行者所占据的节点的趋势。首先，我们推导了控制系统动力学的主方程，并确定了在最一般情况下以及网络拥挤程度不同的情况下，步行者处于平衡状态的占用概率的解析表达式。然后，我们研究了不同类型的合成和现实世界网络，并给出了熵率的数值和分析结果，可作为步行者网络探索能力的代表。我们发现，对于非线性偏置的每个级别，在给定的网络拓扑中，有一个最佳的拥挤可以使熵率最大化。分析表明，现实世界中的大部分网络都是以拥挤条件下的勘探为导向的。我们的工作提供了一个通用且通用的框架来建模非线性随机过程，其过渡概率随系统当前状态的变化而变化。