Sub-graph entropy has recently been applied to functional brain network analysis for identifying important brain regions associated with different brain states and for discriminating brain networks of subjects with psychiatric disorders from healthy controls. This letter describes two pertinent properties of sub-graph entropy. It is shown that when a graph is divided into multiple smaller graphs, the summation of their sub-graph entropy is always less than a constant. Additionally, this summation is always greater than the corresponding graph entropy. We also demonstrate that node entropy, a special case of sub-graph entropy, is stable. Experiments using both synthetic data and real world brain network data are carried out to further validate these points. Overall, node entropy has better stability compared to other centrality metrics. Furthermore, our results illustrate that, for human functional brain networks with two different induced states, node entropy has relatively higher change in ranking. Altogether these findings pave the way for real-world applications of sub-graph entropy as a centrality metric in graph signals.

中文翻译：

---

子图熵的图论性质

子图熵最近已应用于功能性脑网络分析，以识别与不同大脑状态相关的重要大脑区域，并用于区分患有精神疾病的受试者与健康对照者的大脑网络。这封信描述了子图熵的两个相关属性。结果表明，当一个图被分成多个较小的图时，它们的子图熵的总和总是小于一个常数。此外，这个总和总是大于相应的图熵。我们还证明了节点熵（子图熵的一种特殊情况）是稳定的。使用合成数据和真实世界的大脑网络数据进行实验以进一步验证这些观点。总体而言，与其他中心性指标相比，节点熵具有更好的稳定性。此外，我们的结果表明，对于具有两种不同诱导状态的人类功能性大脑网络，节点熵的排名变化相对较大。总之，这些发现为子图熵作为图信号中心性度量的实际应用铺平了道路。