Identification of important nodes based on dynamic evolution of inter-layer isomorphism rate in temporal networks

Abstract

The identification of important nodes can not only improve the research about the structure and function of the network, but also encourage people to widely promote the application fields such as in infectious disease prevention, power grid fault detection, information dissemination control, etc. Currently, numerous conclusions have been proved on the identification of important nodes based on the static-network, which may lead the general property to be weakened as resistivity and conductivity experience the dynamic evolution of the relationship between network nodes with time. Temporal network analysis can more accurately describe the change of interaction order and interaction relationship of network nodes in the process of spatio-temporal evolution, and establish an appropriate temporal network model, as well as provide scientific theoretical support for the identification of important nodes. In this paper, we pay attention to considering the intensity of adjacent and cross-layer coupling, and propose a super-adjacency matrix (ISAM) method based on inter-layer isomorphism rate to represent the temporal networks and measure the importance of nodes. And at the same time, it is given that the temporal network <i>G</i> has <i>N</i> nodes and <i>T</i> time layers, and the ISAM is a super adjacency matrix composed of intra-layer and inter-layer relationships of adjacent and cross-layer networks, and its size is <i>NT</i> × <i>NT</i>. We focus on the study of the coupling between adjacent and cross-layer networks. The traditional method (SAM) considers the isomorphism rate of adjacent layers as a constant. In the improved method (SSAM), the connection between layers is described by a neighbor topological overlap coefficient. In this paper, the concept of the compatible similarity between cross-layer networks is given first, and then, by combining the projection value of vectors in <i>n</i>-dimensional real space and the contribution value of node neighbors, the inter-layer approximation relation coefficient of temporal network is inferred and analyzed. Generally speaking, it ensures the difference in coupling degree among different nodes in the inter-layer relationship. We calculate the importance of nodes based on eigenvector centrality in temporal network, which presents the importance of node <i>i</i> progressing with time. Simultaneously, the robustness of temporal network is studied by making use of the difference in temporal global efficiency. In the end, the operator of Kendall correlation coefficient is used to evaluate the node ranking effect of different time layers between the eigenvector-based centrality and the difference of temporal global efficiency. According to the experimental results of ISAM, SSAM and SAM on Workspace and Email-eu-core data sets, the average Kendall <i>τ</i> of both ISAM methods considering adjacent and cross-layer network isomorphism rate can be increased by 8.37% and 2.99% respectively. The conclusions show that the measurement method of temporal network inter-layer isomorphism rate is reliable and effective.

Node Importance Identification for Temporal Networks Based on Optimized Supra-Adjacency Matrix

“…Existing methods [ 9 , 10 , 21 , 22 ] have made a series of improvements to inter-layer relationships, but the network in each time slice is only represented by an adjacency matrix; this ignores the impact of intra-layer relationships on the importance of nodes in temporal networks. A node has edges connecting to each neighbor, but the strengths of all edges are not the same; this situation is consistent with the real world, in which a person has different levels of closeness to each friend.…”

“…Identifying important nodes plays a significant role in analyzing the characteristics of the net system and understanding the network structure and function [ 3 ]. There are many methods to identify the importance of nodes: degree centrality (DC) [ 4 ], betweenness centrality (BC) [ 5 ], closeness centrality (CC) [ 6 ], k-shell [ 7 ], Entropy Variation [ 2 ], and so on [ 8 , 9 ]. These methods are widely used in static networks, but they cannot be directly applied in temporal networks.…”

“…Yang et al [ 10 ] believed that the parameter of interlayer relations in SAM ignores discrepancies in the connections of nodes in different layers, so they proposed the similarity-based supra-adjacency (SSAM) method to construct models of temporal networks. Hu et al [ 9 ] constructed an isomorphism rate-based SAM (ISAM) that expresses the coupling relationship between nonadjacent networks. Jiang et al [ 22 ] introduced a time attenuation factor to more accurately describe the coupling strengths of nodes in different time layers to express different inter-layer relationships, and the attenuation-based SAM (ASAM) temporal network model was established.…”

“…Although the modeling and results of these aforementioned methods [ 9 , 10 , 21 , 22 ] are more realistic and detailed, the representation of intra-layer relationships is relatively simple and does not assess the global importance of nodes. To address these challenges, in this paper, an optimized supra-adjacency matrix (OSAM) temporal network modeling method was established.…”

Importance evaluation method of complex network nodes based on information entropy and iteration factor

“…出提高的混合排序方法 [23] ，该方法结合了两个中心性--扩展的邻居中心性 (C nc+ )和h指数中心性，该方法在排序准确性上具有很好的性能。阮逸润 [24] 等人提 出了 基于结构洞引力模型的改进算法, 综合考虑节点h指数、节点核数以及节点 的结构洞位置。 除此之外，在动态网络中，网络拓扑随着时间的推移而不断变化，导致动态网 络中关键节点的识别变成一项艰巨的任务。研究者们通过不同时间段的网络快照信 息对节点进行排名 [29][30] ，Qu等人 [31] 提出时态网络中的时态信息收集(TIG)过程，将 TIG过程作为节点的重要性度量，可用于节点重要性排名。胡钢 [32] 等人依托复杂网 络的层间时序关联耦合关系、层内连接关系和层间逼近关系构建时序网络超邻接矩 阵，提出了基于时序网络层间同构率动态演化的超邻接矩阵建模的重要节点辨识方…”