Real Temporal Networks Research Articles

Influence Maximization in Temporal Social Networks with the Mixed K-Shell Method

The study of influence maximizing in temporal social networks (IMT) is an important aspect of influence maximization (IM) research. Currently, two main types of algorithms can solve the IMT problem: greedy-based algorithms and heuristic-based algorithms. However, the greedy-based algorithm is too time-consuming to be used in practice, and most existing heuristic methods do not consider the attributes of nodes, resulting in these methods being unable to solve the IMT problem. Therefore, this paper proposes a mixed k-shell (MKS) algorithm, which considers nodes’ local and global attributes to characterize their influence and select seed nodes. At the local level, we consider the degree centrality of nodes, and at the global level, we propose the temporal k-shell decomposition (TKS) algorithm. Ultimately, the influence of a node is determined by combining the influence of itself and its neighbors. Experiments on four real temporal social networks show that MKS performs better in effectiveness than other heuristic baselines and can maintain a balance between effectiveness and efficiency, providing a useful solution for solving the IMT problem.

An Efficient Dynamic Programming Algorithm for Finding Group Steiner Trees in Temporal Graphs

The computation of a group Steiner tree (GST) in various types of graph networks, such as social network and transportation network, is a fundamental graph problem in graphs, with important applications. In these graphs, time is a common and necessary dimension, for example, time information in social network can be the time when a user sends a message to another user. Graphs with time information can be called temporal graphs. However, few studies have been conducted on GST in terms of temporal graphs. This study analyzes the computation of GST for temporal graphs, i.e., the computation of temporal GST (TGST), which is shown to be an NP-hard problem. We propose an efficient solution based on a dynamic programming algorithm for our problem. This study adopts new optimization techniques, including graph simplification, state pruning, and A ∗ search, are adopted to dramatically reduce the algorithm search space. Moreover, we consider three extensions for our problem, namely the TGST with unspecified tree root, the progressive search of TGST, and the top-N search of TGST. Results of the experimental study performed on real temporal networks verify the efficiency and effectiveness of our algorithms.

Inferring Patient Zero on Temporal Networks via Graph Neural Networks

The world is currently seeing frequent local outbreaks of epidemics, such as COVID-19 and Monkeypox. Preventing further propagation of the outbreak requires prompt implementation of control measures, and a critical step is to quickly infer patient zero. This backtracking task is challenging for two reasons. First, due to the sudden emergence of local epidemics, information recording the spreading process is limited. Second, the spreading process has strong randomness. To address these challenges, we tailor a gnn-based model to establish the inverse statistical association between the current and initial state implicitly. This model uses contact topology and the current state of the local population to determine the possibility that each individual could be patient zero. We benchmark our model on data from important epidemiological models on five real temporal networks, showing performance significantly superior to previous methods. We also demonstrate that our method is robust to missing information about contact structure or current state. Further, we find the individuals assigned higher inferred possibility by model are closer to patient zero in terms of core number and the activity sequence recording the times at which the individual had contact with other nodes.

Dynamics of cold random hyperbolic graphs with link persistence.

We consider and analyze a dynamic model of random hyperbolic graphs with link persistence. In the model, both connections and disconnections can be propagated from the current to the next snapshot with probability ω∈[0,1). Otherwise, with probability 1-ω, connections are reestablished according to the random hyperbolic graphs model. We show that while the persistence probability ω affects the averages of the contact and intercontact distributions, it does not affect the tails of these distributions, which decay as power laws with exponents that do not depend on ω. We also consider examples of real temporal networks, and we show that the considered model can adequately reproduce several of their dynamical properties. Our results advance our understanding of the realistic modeling of temporal networks and of the effects of link persistence on temporal network properties.

ITeM: Independent temporal motifs to summarize and compare temporal networks

Networks are a fundamental and flexible way of representing various complex systems. Many domains such as communication, citation, procurement, biology, social media, and transportation can be modeled as a set of entities and their relationships. Temporal networks are a specialization of general networks where every relationship occurs at a discrete time. The temporal evolution of such networks is as important to understand as the structure of the entities and relationships. We present the Independent Temporal Motif (ITeM) to characterize temporal graphs from different domains. ITeMs can be used to model the structure and the evolution of the graph. In contrast to existing work, ITeMs are edge-disjoint directed motifs that measure the temporal evolution of ordered edges within the motif. For a given temporal graph, we produce a feature vector of ITeM frequencies and the time it takes to form the ITeM instances. We apply this distribution to measure the similarity of temporal graphs. We show that ITeM has higher accuracy than other motif frequency-based approaches. We define various ITeM-based metrics that reveal salient properties of a temporal network. We also present importance sampling as a method to efficiently estimate the ITeM counts. We present a distributed implementation of the ITeM discovery algorithm using Apache Spark and GraphFrame. We evaluate our approach on both synthetic and real temporal networks.

Directed percolation in temporal networks

Connectivity and reachability on temporal networks, which can describe the spreading of a disease, decimation of information or the accessibility of a public transport system over time, have been among the main contemporary areas of study in complex systems for the last decade. However, while isotropic percolation theory successfully describes connectivity in static networks, a similar description has not been yet developed for temporal networks. Here address this problem and formalize a mapping of the concept of temporal network reachability to percolation theory. We show that the limited-waiting-time reachability, a generic notion of constrained connectivity in temporal networks, displays directed percolation phase transition in connectivity. Consequently, the critical percolation properties of spreading processes on temporal networks can be estimated by a set of known exponents characterising the directed percolation universality class. This result is robust across a diverse set of temporal network models with different temporal and topological heterogeneities, while by using our methodology we uncover similar reachability phase transitions in real temporal networks too. These findings open up an avenue to apply theory, concepts and methodology from the well-developed directed percolation literature to temporal networks.

Dynamics of hot random hyperbolic graphs.

We derive the most basic dynamical properties of random hyperbolic graphs (the distributions of contact and intercontact durations) in the hot regime (network temperature T>1). We show that for sufficiently large networks the contact distribution decays as a power law with exponent 2+T>3 for durations t>T, while for t<T it exhibits exponential-like decays. This result holds irrespective of the expected degree distribution, as long as it has a finite Tth moment. Otherwise, the contact distribution depends on the expected degree distribution and we show that if the latter is a power law with exponent γ∈(2,T+1], then the former decays as a power law with exponent γ+1>3. However, the intercontact distribution exhibits power-law decays with exponent 2-T∈(0,1) for T∈(1,2), while for T>2 it displays linear decays with a slope that depends on the observation interval. This result holds irrespective of the expected degree distribution as long as it has a finite Tth moment if T∈(1,2), or a finite second moment if T>2. Otherwise, the intercontact distribution depends on the expected degree distribution and if the latter is a power law with exponent γ∈(2,3), then the former decays as a power law with exponent 3-γ∈(0,1). Thus, hot random hyperbolic graphs can give rise to contact and intercontact distributions that both decay as power laws. These power laws, however, are unrealistic for the case of the intercontact distribution, as their exponent is always less than one. These results mean that hot random hyperbolic graphs are not adequate for modeling real temporal networks, in stark contrast to cold random hyperbolic graphs (T<1). Since the configuration model emerges at T→∞, these results also suggest that this is not an adequate null temporal network model.

Temporal Networks Based on Human Mobility Models: A Comparative Analysis With Real-World Networks

Mobility is a critical element for understanding human contact networks. In many studies, the researchers use random processes to model human mobility. However, people do not move randomly in their environment. Their interactions do not depend only on spatial constraints but on their temporal, social, economic, and cultural activities. The topological structure of the physical and/or proximity contact networks depends, therefore, entirely on the mobility patterns. This paper performs an extensive comparative analysis of real-world temporal contact networks and synthetic networks based on influential mobility models. Results show that the various topological properties of most of the synthetic datasets depart from those observed in real-world contact networks because the randomness of some mobility parameters tends to move away from human contact properties. However, it appears that data generated using Spatio-Temporal Parametric Stepping (STEPS) mobility model reveals similarities with real temporal contact networks such as heavy-tailed distribution of contact duration, frequency of pairs of contacts, and the bursty phenomenon. These results pave the way for further improvement of mobility models to generate meaningful artificial contact networks.

Identifying critical nodes in temporal networks by network embedding

Critical nodes in temporal networks play more significant role than other nodes on the structure and function of networks. The research on identifying critical nodes in temporal networks has attracted much attention since the real-world systems can be illustrated more accurately by temporal networks than static networks. Considering the topological information of networks, the algorithm MLI based on network embedding and machine learning are proposed in this paper. we convert the critical node identification problem in temporal networks into regression problem by the algorithm. The effectiveness of proposed methods is evaluated by SIR model and compared with well-known existing metrics such as temporal versions of betweenness, closeness, k-shell, degree deviation and dynamics-sensitive centralities in one synthetic and five real temporal networks. Experimental results show that the proposed method outperform these well-known methods in identifying critical nodes under spreading dynamic.

Predictability of real temporal networks.

Links in most real networks often change over time. Such temporality of links encodes the ordering and causality of interactions between nodes and has a profound effect on network dynamics and function. Empirical evidence has shown that the temporal nature of links in many real-world networks is not random. Nonetheless, it is challenging to predict temporal link patterns while considering the entanglement between topological and temporal link patterns. Here, we propose an entropy-rate-based framework, based on combined topological–temporal regularities, for quantifying the predictability of any temporal network. We apply our framework on various model networks, demonstrating that it indeed captures the intrinsic topological–temporal regularities whereas previous methods considered only temporal aspects. We also apply our framework on 18 real networks of different types and determine their predictability. Interestingly, we find that, for most real temporal networks, despite the greater complexity of predictability brought by the increase in dimension, the combined topological–temporal predictability is higher than the temporal predictability. Our results demonstrate the necessity for incorporating both temporal and topological aspects of networks in order to improve predictions of dynamical processes.

Temporal Network Pattern Identification by Community Modelling

Temporal network mining tasks are usually hard problems. This is because we need to face not only a large amount of data but also its non-stationary nature. In this paper, we propose a method for temporal network pattern representation and pattern change detection following the reductionist approach. The main idea is to model each stable (durable) state of a given temporal network as a community in a sampled static network and the temporal state change is represented by the transition from one community to another. For this purpose, a reduced static single-layer network, called a target network, is constructed by sampling and rearranging the original temporal network. Our approach provides a general way not only for temporal networks but also for data stream mining in topological space. Simulation results on artificial and real temporal networks show that the proposed method can group different temporal states into different communities with a very reduced amount of sampled nodes.

Detecting and Tracking Community Structure in Temporal Networks: A Low-Rank + Sparse Estimation Based Evolutionary Clustering Approach

Networks provide a powerful tool to model complex systems where the different entities in the system are presented by nodes and their interactions by edges. With the availability of network-type data, different community detection algorithms have been proposed to investigate the organization of the nodes within these networks. In particular, numerous graph-based community detection algorithms have been developed for static networks. However, most real complex systems vary with time. Consequently, it is important to develop graph-based community detection techniques for temporal networks. In this paper, a new low-rank + sparse estimation based evolutionary spectral clustering approach is proposed to detect and track the community structure in temporal networks. The proposed method decomposes the network into low-rank and sparse parts and obtains smooth cluster assignments by minimizing the subspace distance between consecutive time points. The extracted low-rank adjacency matrix is then used for clustering and the subspaces are defined through spectral embedding. The introduced framework is robust to noise and outliers and can detect the community structure in both binary and weighted temporal networks efficiently without making any prior assumptions about the network structure. The proposed approach is evaluated on several weighted and binary simulated and real temporal networks. The results show that the proposed algorithm can detect and track the correct community structure over time efficiently and outperforms state-of-the-art algorithms.

Navigating temporal networks

Navigation on graphs is the problem how an agent walking on the graph can get from a source to a target with limited information about the graph. The information and the way to exploit it can vary. In this paper, we study navigation on temporal networks—networks where we have explicit information about the time of the interaction, not only who interacts with whom. We contrast a type of greedy navigation – where agents follow paths that would have worked well in the past – with two strategies that do not exploit the additional information. We test these on empirical temporal network data sets. The greedy navigation finds the targets faster and more reliably than the reference strategies, meaning that there are correlations in the real temporal networks that can be exploited. We find that both topological and temporal structures affect the navigation.

Constrained information flows in temporal networks reveal intermittent communities.

Many real-world networks represent dynamic systems with interactions that change over time, often in uncoordinated ways and at irregular intervals. For example, university students connect in intermittent groups that repeatedly form and dissolve based on multiple factors, including their lectures, interests, and friends. Such dynamic systems can be represented as multilayer networks where each layer represents a snapshot of the temporal network. In this representation, it is crucial that the links between layers accurately capture real dependencies between those layers. Often, however, these dependencies are unknown. Therefore, current methods connect layers based on simplistic assumptions that do not capture node-level layer dependencies. For example, connecting every node to itself in other layers with the same weight can wipe out dependencies between intermittent groups, making it difficult or even impossible to identify them. In this paper, we present a principled approach to estimating node-level layer dependencies based on the network structure within each layer. We implement our node-level coupling method in the community detection framework Infomap and demonstrate its performance compared to current methods on synthetic and real temporal networks. We show that our approach more effectively constrains information inside multilayer communities so that Infomap can better recover planted groups in multilayer benchmark networks that represent multiple modes with different groups and better identify intermittent communities in real temporal contact networks. These results suggest that node-level layer coupling can improve the modeling of information spreading in temporal networks and better capture intermittent community structure.

Propagation-Based Temporal Network Summarization

Modern networks are very large in size and also evolve with time. As their sizes grow, the complexity of performing network analysis grows as well. Getting a smaller representation of a temporal network with similar properties will help in various data mining tasks. In this paper, we study the novel problem of getting a smaller diffusion-equivalent representation of a set of time-evolving networks. We first formulate a well-founded and general temporal-network condensation problem based on the so-called system-matrix of the network. We then propose NetCondense , a scalable and effective algorithm which solves this problem using careful transformations in sub-quadratic running time, and linear space complexities. Our extensive experiments show that we can reduce the size of large real temporal networks (from multiple domains such as social, co-authorship, and email) significantly without much loss of information. We also show the wide-applicability of NetCondense by leveraging it for several tasks: for example, we use it to understand, explore, and visualize the original datasets and to also speed-up algorithms for the influence-maximization and event detection problems on temporal networks.

Graph regularized nonnegative matrix factorization for temporal link prediction in dynamic networks

Many networks derived from society and nature are temporal and incomplete. The temporal link prediction problem in networks is to predict links at time T+1 based on a given temporal network from time 1 to T, which is essential to important applications. The current algorithms either predict the temporal links by collapsing the dynamic networks or collapsing features derived from each network, which are criticized for ignoring the connection among slices. to overcome the issue, we propose a novel graph regularized nonnegative matrix factorization algorithm (GrNMF) for the temporal link prediction problem without collapsing the dynamic networks. To obtain the feature for each network from 1 to t, GrNMF factorizes the matrix associated with networks by setting the rest networks as regularization, which provides a better way to characterize the topological information of temporal links. Then, the GrNMF algorithm collapses the feature matrices to predict temporal links. Compared with state-of-the-art methods, the proposed algorithm exhibits significantly improved accuracy by avoiding the collapse of temporal networks. Experimental results of a number of artificial and real temporal networks illustrate that the proposed method is not only more accurate but also more robust than state-of-the-art approaches.

The temporal event graph

Temporal networks are increasingly being used to model the interactions of complex systems. Most studies require the temporal aggregation of edges (or events) into discrete time steps to perform analysis. In this article we describe a static, lossless, and unique representation of a temporal network, the temporal event graph (TEG). The TEG describes the temporal network in terms of both the inter-event time and two-event temporal motif distributions. By considering these distributions in unison we provide a new method to characterise the behaviour of individuals and collectives in temporal networks as well as providing a natural decomposition of the network. We illustrate the utility of the TEG by providing examples on both synthetic and real temporal networks.

Nonnegative matrix factorization algorithms for link prediction in temporal networks using graph communicability

Networks derived from many disciplines, such as social relations, web contents, and cancer progression, are temporal and incomplete. Link prediction in temporal networks is of theoretical interest and practical significance because spurious links are critical for investigating evolving mechanisms. In this study, we address the temporal link prediction problem in networks, i.e. predicting links at time T+1 based on a given temporal network from time 1 to T. To address the relationships among matrix decomposition-based algorithms, we prove the equivalence between the eigendecomposition and nonnegative matrix factorization (NMF) algorithms, which serves as the theoretical foundation for designing NMF-based algorithms for temporal link prediction. A novel NMF-based algorithm is proposed based on such equivalence. The algorithm factorizes each network to obtain features using graph communicability, and then collapses the feature matrices to predict temporal links. Compared with state-of-the-art methods, the proposed algorithm exhibits significantly improved accuracy by avoiding the collapse of temporal networks. Experimental results of a number of artificial and real temporal networks illustrate that the proposed method is not only more accurate but also more robust than state-of-the-art approaches.

Empirical study on structural properties in temporal networks under different time scales

Many network analyzing methods are usually based on static networks. However, temporal networks should be considered so as to investigate real complex systems deeply since some dynamics on these systems cannot be described by static networks accurately. In this paper, four structural properties in temporal networks are empirically studied, including degree, clustering coefficient, adjacent correlation, and connected component. Three real temporal networks with different time scales are analyzed in this paper, including short message, telephone, and router networks. Moreover, structural properties of these temporal networks are compared with that of corresponding static aggregation networks in the whole time window. Some essential differences of structural properties between temporal and static networks are achieved through empirical analysis. Finally, the effect of structural properties on spreading dynamics under different time scales is investigated. Some interesting results such as turning point of structure evolving time scale corresponding to certain spreading dynamics time scale from the point of view of infected scale are achieved.

Temporal Networks: Slowing Down Diffusion by Long Lasting Interactions

Interactions among units in complex systems occur in a specific sequential order, thus affecting the flow of information, the propagation of diseases, and general dynamical processes. We investigate the Laplacian spectrum of temporal networks and compare it with that of the corresponding aggregate network. First, we show that the spectrum of the ensemble average of a temporal network has identical eigenmodes but smaller eigenvalues than the aggregate networks. In large networks without edge condensation, the expected temporal dynamics is a time-rescaled version of the aggregate dynamics. Even for single sequential realizations, diffusive dynamics is slower in temporal networks. These discrepancies are due to the noncommutability of interactions. We illustrate our analytical findings using a simple temporal motif, larger network models, and real temporal networks.