图 1  网络高阶聚类系数计算示意图　(a)节点$ v_1 $及它的邻居形成的网络; (b)去除节点$ v_1 $后的网络; (c)图1(b)所示网络中节点$ v_1 $的邻居之间的距离矩阵; (d)节点$ v_1 $的高阶聚类系数分布

Fig. 1.  Illustration of the calculation of the higher-order clustering coefficient: (a) A network formed by node $ v_1 $ and its neighbors; (b) network after removing node $ v_1 $; (c) distance matrix between neighbors of node $ v_1 $ in the network shown in panel (b); (d) the higher-order clustering coefficient distribution of node $ v_1 $.

图 2  基于高阶信息的网络比较方法计算流程示意图　(a)给定两个拥有 11个节点的网络G和$ G' $, 其中G有 14条边, $ G' $有12条边; (b)如何计算基于高阶信息的网络相似性的示例, 包含了节点高阶聚类系数分布和节点距离分布; (c)网络相似值的计算, 其中$ \beta = 0.5 $

Fig. 2.  Schematic diagram of calculation flow of network comparison method based on high-order information: (a) Given two networks G and $ G' $ with 11 nodes, G has 14 edges and $ G' $ has 12 edges; (b) an illustration of how to compute the network similarity based on higher-order information, including the distribution of node higher-order clustering coefficients and node distance distribution; (c) calculation of the network similarity value $ D_{{\mathrm{HC}}} $, where $ \beta = 0.5 $.

图 3  人工合成网络下(WS, BA)的参数敏感性分析　(a)不同参数β下$ N = 1000 $的WS网络与$ N=[1500, 5000] $, 间隔为500, 重连概率$ p=0.3 $的WS网络之间的相似性; (b)不同参数β下$ N = 1000 $的BA网络与$ N=[1500, 5000] $, 间隔为500的BA网络之间的相似性, 其中每个BA网络每一步加边数$ m=5 $; (c)不同参数γ下WS网络之间的相似性, 参数与(a)图一样; (d)不同参数γ下BA网络之间的相似性, 参数与(b)图一样. 所有的结果均基于100次实验的平均值

Fig. 3.  Parameter sensitivity analysis of synthetic networks generated by the WS and BA model: (a) Similarity between the WS network of $ N = 1000 $ and the WS networks of $ N=[1500, 5000] $ with the interval is 500 under different parameters β, where the probability of rewiring $ p=0.3 $; (b) similarity between the BA network of $ N = 1000 $ and the BA networks of $ N=[1500, 5000] $ with an interval of 500 under different parameters β, where each BA network adds edges at each step with number of $ m=5 $; (c) similarity between WS networks under different parameters γ, the parameters are the same as with those in panel (a); (d) similarity between BA networks under different parameters γ, the parameters are the same as those in panel (b). All results are based on an average of 100 realizations.

图 4  四种相似性方法在人工合成网络上的效果评估(网络规模均为$ N=1000 $)　(a)—(d)不同重连概率p下ER模型生成的每对网络的相似性, 其中相似性方法分别为$ D_{{\mathrm{HC}}} $, $ D_{{\mathrm{SP}}} $, $ D_{{{C}}} $以及$ D_{{{M}}} $; (e)—(h)不同重连概率p, 平均度为10下WS模型生成的每对网络的相似性, 其中相似性方法分别为$ D_{{\mathrm{HC}}} $, $ D_{{\mathrm{SP}}} $, $ D_{{{C}}} $以及$ D_{{{M}}} $; (i)—(l)不同加边数$ m\in\{2, 3, 4, 5, 6\} $下BA模型生成的每对网络的相似性, 其中相似性方法分别为$ D_{{\mathrm{HC}}} $, $ D_{{\mathrm{SP}}} $, $ D_{{{C}}} $以及$ D_{{{M}}} $. 所有的结果均基于100次实验的平均值

Fig. 4.  Effectiveness of four similarity methods in comparing synthetic networks. The network size is set to $ N=1000 $: (a)–(d) Similarity between each pair of networks generated by the ER model under different rewiring probabilities p, where the network comparing methods are $ D_{{\mathrm{HC}}} $, $ D_{{\mathrm{SP}}} $, $ D_{{{C}}} $ and $ D_{{{M}}} $; (e)–(h) similarity between each pair of networks generated by the WS model with different rewiring probabilities p and an average degree of 10, where the network comparing methods are $ D_{{\mathrm{HC}}} $, $ D_{{\mathrm{SP}}} $, $ D_{{{C}}} $ and $ D_{{{M}}} $; (i)–(l) similarity between each pair of networks generated by the BA model under different edge numbers $ m\in\{2, 3, 4, 5, 6\} $ added at each time step, where the similarity methods are $ D_{{\mathrm{HC}}} $, $ D_{{\mathrm{SP}}} $, $ D_{{{C}}} $ and $ D_{{{M}}} $. All results are based on an average of 100 realizations.

图 5  分别使用$ D_{{\mathrm{HC}}} $, $ D_{{\mathrm{SP}}} $, $ D_{{{C}}} $和$ D_{{{M}}} $这4种方法对4种人工合成网络(K-regular, WSC, WSK和BA)进行相互比较. 所有的结果均基于100次实验的平均值

Fig. 5.  Comparison of the four synthetic networks, i.e., K-regular, WSC, WSK, and BA, by using four methods of $ D_{{\mathrm{HC}}} $, $ D_{{\mathrm{SP}}} $, $ D_{{{C}}} $ and $ D_{{{M}}} $. All results are based on an average of 100 realizations.

图 6  真实网络与其零模型生成的网络相似性. 考虑了具有不同k值(1.0, 2.0和2.5)的$ Dk $零模型, 图中的值表示$ D_{{\mathrm{HC}}} $的值的大小. 所有的结果均基于100次实验的平均值

Fig. 6.  Similarity between real networks and their null-models. We considered the $ Dk $ null model with different values k (1.0, 2.0, and 2.5), and the values in the figure indicate the value of $ D_{{\mathrm{HC}}} $. All results are based on an average of 100 realizations.

图 7  原始真实网络和经过扰动后生成的网络之间的相似性, 其中f的负值对应于给定比例的边的随机删除过程, 正值表示随机增边的过程. 所有的结果均基于100次实验的平均值.

Fig. 7.  Similarity between the original real network and the network after perturbation, where negative values of f correspond to the deletion of $ |f| $ fraction of edges and positive values of f indicate the addition of f fraction of edges. All results are based on an average of 100 realizations.