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

Identifying influential spreaders is an important issue for capturing the dynamics of information diffusion in temporal networks. Most of the identification of influential spreaders in previous researches were focused on analysing static networks, rarely highlighted on dynamics. However, those measures which are proposed for static topologies only, unable to faithfully capture the effect of temporal variations on the importance of nodes. In this paper, a shortest temporal path algorithm is proposed for calculating the minimum time that information interaction between nodes. This algorithm can effectively find out the shortest temporal path when considering the network integrity. On the basis of this, the temporal efficiency centrality (TEC) algorithm in temporal networks is proposed, which identify influential nodes by removing each node and taking the variation of the whole network into consideration at the same time. To evaluate the effectiveness of this algorithm, we conduct the experiment on four real-world temporal networks for Susceptible-Infected-Recovered (SIR) model. By employing the imprecision and the Kendall’s au coefficient, The results show that this algorithm can effectively evaluate the importance of nodes in temporal networks.

Many real-world systems can be expressed in temporal networks with nodes playing different roles in structure and function, and edges representing the relationships between nodes. Identifying critical nodes can help us control the spread of public opinions or epidemics, predict leading figures in academia, conduct advertisements for various commodities and so on. However, it is rather difficult to identify critical nodes, because the network structure changes over time in temporal networks. In this paper, considering the sequence topological information of temporal networks, a novel and effective learning framework based on the combination of special graph convolutional and long short-term memory network (LSTM) is proposed to identify nodes with the best spreading ability. The special graph convolutional network can embed nodes in each sequential weighted snapshot and LSTM is used to predict the future importance of timing-embedded features. The effectiveness of the approach is evaluated by a weighted Susceptible-Infected-Recovered model. Experimental results on four real-world temporal networks demonstrate that the proposed method outperforms both traditional and deep learning benchmark methods in terms of the Kendall τ coefficient and top k hit rate.

Identifying important nodes in complex networks is a fundamental problem in network analysis. Although a plethora of measures has been proposed to identify important nodes in static (i.e., time-invariant) networks, there is a lack of tools in the context of temporal networks (i.e., networks whose connectivity dynamically changes over time). The aim of this paper is to propose a system-theoretic approach for identifying important nodes in temporal networks. In this direction, we first propose a generalization of the popular Katz centrality measure to the family of Markovian temporal networks using tools from the theory of Markov jump linear systems. We then show that Katz centrality in Markovian temporal networks can be efficiently computed using linear programming. Finally, we propose a convex program for optimizing the Katz centrality of a given node by tuning the weights of the temporal network in a cost-efficient manner. Numerical simulations illustrate the effectiveness of the obtained results.

Dynamic centrality metrics provide a quantitative assessment of the strength of communication between nodes in temporal networks, as well as the overall capacity of the network for the efficient transmission of information. In this article, the behaviours of two variants of the ‘communicability’ metric are examined in simple null models of uncorrelated temporal networks. Analysis of the long-time behaviour of the null models reveals a simple trade-off in the role of the parameters of the metric, suggesting methods to calibrate parameters and to adapt to temporal variations in the network properties. The null models introduced address two main classes of temporal networks (contact sequences and interval graphs), and their predictions are compared and contrasted with results coming from real-world telecommunications data.

Importance evaluation of power network nodes based on community division and characteristics of coupled network

In order to meet passengers’ demands for increasing travel efficiency, the structure of China’s airport network will become more and more complex, and its operating environment has become more complex, so, it is important to measure the robustness of China’s airport network under the influence of complex disturbance factors. This paper proposes an airport network robustness analysis method based on the multi-layer temporal complex network model (MTNM) by introducing the time attribute. Firstly, we establish the network structure characteristics such as temporal degree, temporal closeness centrality (TC), and temporal betweenness centrality (TB) of airport network nodes, and take temporal network efficiency and temporal maximum connected subgraph as indicators; then inflection points of loss fitting curves for temporal network characteristics metrics under random attacks and deliberate attacks can be identified to evaluate network robustness. The results show that compared with static network, using multi-layer temporal network model can describe the performance characteristics of the network more accurately.

Accurate measurement of important nodes in complex networks has great practical and theoretical significance. Mining important nodes should not only consider the core nodes, but also take into account the locations of the nodes in the network. Despite much research on assessing important nodes, the importance of nodes in the structural holes is still easily ignored. Therefore, a local measuring method is proposed, which evaluates the nodes importance by the total constraints caused by the lack of primary structural holes and secondary structural holes around the nodes. This method simultaneously considers both the centrality and the bridging property of the nodes' first-order and second-order neighbours. Further to prove the accuracy of the method TCM, we carry out deliberate attack simulations through selective deletion in a certain proportion of network nodes. Then, we calculate the decreased ratio of network efficiency in the before-and-after attacks. Experiment results show that the average effect of the TCM in four real networks are improved by 50.64% and 14.92% compared to the clustering coefficient index and the k-shell decomposition method, respectively. Obviously, the TCM is more accurate to mine important nodes than other two methods, and it is suitable for quantitative analysis in large-scale networks.

Evaluating the importance of nodes in complex networks is an important topic in the research of network characteristics. Its relevant research has a wide range of applications, such as network supervision and rumor control. At present, many methods have been proposed to evaluate the importance of nodes in complex networks, but most of them have the deficiency of one-sided evaluation or too high time complexity. In order to break through the limitations of existing methods, in this paper a novel method of evaluating the importance of complex network nodes is proposed based on Tsallis entropy. This method takes into account both the local and global topological information of the node. It considers the structural hole characteristics and K-shell centrality of the node and fully takes into account the influence of the node itself and its neighboring nodes. To illustrate the effectiveness and applicability of this method, eight real networks are selected from different fields and five existing methods of evaluating node importance are used as comparison methods. On this basis, the monotonicity index, SIR (susceptible-infectious-recovered) model, and Kendall correlation coefficient are used to illustrate the superiority of this method and the relationship among different methods. Experimental results show that this method can effectively and accurately evaluate the importance of nodes in complex networks, distinguish the importance of different nodes significantly, and can show good accuracy of evaluating the node importance under different proportions of nodes. In addition, the time complexity of this method is <inline-formula><tex-math id="M2">\begin{document}$ O({n^2}) $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="21-20210979_M2.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="21-20210979_M2.png"/></alternatives></inline-formula>, which is suitable for large-scale complex networks.

In network analysis, the betweenness centrality of a node informally captures\nthe fraction of shortest paths visiting that node. The computation of the\nbetweenness centrality measure is a fundamental task in the analysis of modern\nnetworks, enabling the identification of the most central nodes in such\nnetworks. Additionally to being massive, modern networks also contain\ninformation about the time at which their events occur. Such networks are often\ncalled temporal networks. The temporal information makes the study of the\nbetweenness centrality in temporal networks (i.e., temporal betweenness\ncentrality) much more challenging than in static networks (i.e., networks\nwithout temporal information). Moreover, the exact computation of the temporal\nbetweenness centrality is often impractical on even moderately-sized networks,\ngiven its extremely high computational cost. A natural approach to reduce such\ncomputational cost is to obtain high-quality estimates of the exact values of\nthe temporal betweenness centrality. In this work we present ONBRA, the first\nsampling-based approximation algorithm for estimating the temporal betweenness\ncentrality values of the nodes in a temporal network, providing rigorous\nprobabilistic guarantees on the quality of its output. ONBRA is able to compute\nthe estimates of the temporal betweenness centrality values under two different\noptimality criteria for the shortest paths of the temporal network. In\naddition, ONBRA outputs high-quality estimates with sharp theoretical\nguarantees leveraging on the \\emph{empirical Bernstein bound}, an advanced\nconcentration inequality. Finally, our experimental evaluation shows that ONBRA\nsignificantly reduces the computational resources required by the exact\ncomputation of the temporal betweenness centrality on several real world\nnetworks, while reporting high-quality estimates with rigorous guarantees.\n

Temporal networks come with a wide variety of heterogeneities, from burstiness of event sequences to correlations between timings of node and link activations. In this paper, we set to explore the latter by using greedy walks as probes of temporal network structure. Given a temporal network (a sequence of contacts), greedy walks proceed from node to node by always following the first available contact. Because of this, their structure is particularly sensitive to temporal-topological patterns involving repeated contacts between sets of nodes. This becomes evident in their small coverage per step as compared to a temporal reference model -- in empirical temporal networks, greedy walks often get stuck within small sets of nodes because of correlated contact patterns. While this may also happen in static networks that have pronounced community structure, the use of the temporal reference model takes the underlying static network structure out of the equation and indicates that there is a purely temporal reason for the observations. Further analysis of the structure of greedy walks indicates that burst trains, sequences of repeated contacts between node pairs, are the dominant factor. However, there are larger patterns too, as shown with non-backtracking greedy walks. We proceed further to study the entropy rates of greedy walks, and show that the sequences of visited nodes are more structured and predictable in original data as compared to temporally uncorrelated references. Taken together, these results indicate a richness of correlated temporal-topological patterns in temporal networks.

Real networks are heterogeneous structures, with edges unevenly distributed among nodes, presenting community structure, motifs, transitivity, rich clubs, and other kinds of topological patterns. Consequently, the roles played by nodes in a network can differ greatly. For example, some nodes may be connectors between parts of the network, others may be central or peripheral, etc. The objective of this chapter is to describe how we can find the most important nodes in networks. The idea is to define a centrality measure for each node in the network, sort the nodes according to their centralities, and fix our attention to the first ranked nodes, which can be considered as the most relevant ones with respect to this centrality measure.

Group centralization of network indices

Network modeling aims to learn the latent representations of nodes such that the representations preserve both network structures and node attribute information. This problem is fundamental due to its prevalence in numerous domains. However, existing approaches either target the static networks or struggle to capture the complicated temporal dependency, while most real-world networks evolve over time and the success of network modeling hinges on the understanding of how entities are temporally connected. In this paper, we present TRRN, a transformer-style relational reasoning network with dynamic memory updating, to deal with the above challenges. TRRN employs multi-head self-attention to reason over a set of memories, which provides a multitude of shortcut paths for information to flow from past observations to the current latent representations. By utilizing the policy networks augmented with differentiable binary routers, TRRN estimates the possibility of each memory being activated and dynamically updates the memories at the time steps when they are most relevant. We evaluate TRRN with the tasks of node classification and link prediction on four real temporal network datasets. Experimental results demonstrate the consistent performance gains for TRRN over the leading competitors.

Quantifying the importance of specific types of higher-order structures in complex networks

An important feature of all real-world networks is that the network structure changes over time. Due to this dynamic nature, it becomes difficult to propose suitable growth models that can explain the various important characteristic properties of these networks. In fact, in many application oriented studies only knowing these properties is sufficient. We, in this paper show that even if the network structure at a future time point is not available one can still manage to estimate its properties. We propose a novel method to map a temporal network to a set of time series instances, analyze them and using a standard forecast model of time series, try to predict the properties of a temporal network at a later time instance. We mainly focus on the temporal network of human face- to-face contacts and observe that it represents a stochastic process with memory that can be modeled as ARIMA. We use cross validation techniques to find the percentage accuracy of our predictions. An important observation is that the frequency domain properties of the time series obtained from spectrogram analysis could be used to refine the prediction framework by identifying beforehand the cases where the error in prediction is likely to be high. This leads to an improvement of 7.96% (for error level <= 20%) in prediction accuracy on an average across all datasets. As an application we show how such prediction scheme can be used to launch targeted attacks on temporal networks.