Baxter G J, da Costa R A, Dorogovtsev S N, Mendes J F F
Department of Physics, University of Aveiro de & I3N, Campus Universitário de Santiago, 3810-193 Aveiro, Portugal.
Entropy (Basel). 2020 Oct 13;22(10):1149. doi: 10.3390/e22101149.
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.
压缩、过滤、加密以及复杂系统的采样，都可被视为信息处理。一个庞大的初始配置或输入空间被以非平凡的方式映射到一组更小的输出或最终状态。作为复杂系统中此类信息处理的一个易于处理的示例，我们研究了在一些确定性和随机图上简单模式的过滤统计。在这个问题中，多个输入映射到相同的输出，并且过滤统计由这种简并性的分布来表示。对于环上的一些简单过滤模式，我们得到了该问题的精确解，并对更复杂的过滤设置进行了数值描述。对于每个过滤模式和网络，我们找到了三个关键数字，它们本质上描述了过滤统计，并对不同网络进行了比较。我们针对具有不同架构的网络所得到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的统计结果比确定性图中的丰富得多，这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，而在最小度数不为2时达到最大值。对节点邻域有强烈依赖的过滤模式对这些影响更为敏感。
