Compression, filtering, and cryptography, as well as the sampling of complex systems, can be seen as processing information. A large initial configuration or input space is nontrivially mapped to a smaller set of output or final states. We explored the statistics of filtering of simple patterns on a number of deterministic and random graphs as a tractable example of such information processing in complex systems. In this problem, multiple inputs map to the same output, and the statistics of filtering is represented by the distribution of this degeneracy. For a few simple filter patterns on a ring, we obtained an exact solution of the problem and numerically described more difficult filter setups. For each of the filter patterns and networks, we found three key numbers that essentially describe the statistics of filtering and compared them for different networks. Our results for networks with diverse architectures are essentially determined by two factors: whether the graphs structure is deterministic or random and the vertex degree. We find that filtering in random graphs produces much richer statistics than in deterministic graphs, reflecting the greater complexity of such graphs. Increasing the graph’s degree reduces this statistical richness, while being at its maximum at the smallest degree not equal to two. A filter pattern with a strong dependence on the neighbourhood of a node is much more sensitive to these effects.

中文翻译：

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过滤网络统计信息

压缩、过滤和密码学，以及复杂系统的采样，都可以看作是处理信息。一个大的初始配置或输入空间被非平凡地映射到一个较小的输出或最终状态集。我们探索了在许多确定性和随机图上过滤简单模式的统计数据，作为复杂系统中此类信息处理的一个易处理的例子。在这个问题中，多个输入映射到同一个输出，过滤的统计量由这种退化的分布表示。对于环上的一些简单滤波器模式，我们获得了问题的精确解，并以数值方式描述了更困难的滤波器设置。对于每个过滤器模式和网络，我们找到了三个主要描述过滤统计数据的关键数字，并针对不同的网络对它们进行了比较。我们对具有不同架构的网络的结果基本上由两个因素决定：图结构是确定性的还是随机的，以及顶点度。我们发现随机图中的过滤产生比确定性图中更丰富的统计数据，反映了此类图的更大复杂性。增加图的度数会降低这种统计丰富度，同时在不等于 2 的最小度数处达到最大值。对节点邻域有很强依赖性的滤波器模式对这些影响要敏感得多。图结构是确定性的还是随机的以及顶点度。我们发现随机图中的过滤产生比确定性图中更丰富的统计数据，反映了此类图的更大复杂性。增加图的度数会降低这种统计丰富度，同时在不等于 2 的最小度数处达到最大值。对节点邻域有很强依赖性的滤波器模式对这些影响要敏感得多。图结构是确定性的还是随机的以及顶点度。我们发现随机图中的过滤产生比确定性图中更丰富的统计数据，反映了此类图的更大复杂性。增加图的度数会降低这种统计丰富度，同时在不等于 2 的最小度数处达到最大值。对节点邻域有很强依赖性的滤波器模式对这些影响要敏感得多。