Nodal spreading influence is the capability of a node to activate the rest of the network when it is the seed of spreading. Combining nodal properties (centrality metrics) derived from local and global topological information respectively has been shown to better predict nodal influence than using a single metric. In this work, we investigate to what extent local and global topological information around a node contributes to the prediction of nodal influence and whether relatively local information is sufficient for the prediction. We show that by leveraging the iterative process used to derive a classical nodal centrality such as eigenvector centrality, we can define an iterative metric set that progressively incorporates more global information around the node. We propose to predict nodal influence using an iterative metric set that consists of an iterative metric from order 1 to K produced in an iterative process, encoding gradually more global information as K increases. Three iterative metrics are considered, which converge to three classical node centrality metrics, respectively. In various real-world networks and synthetic networks with community structures, we find that the prediction quality of each iterative based model converges to its optimal when the metric of relatively low orders (K∼4) are included and increases only marginally when further increasing K. This fast convergence of prediction quality with K is further explained by analyzing the correlation between the iterative metric and nodal influence, the convergence rate of each iterative process and network properties. The prediction quality of the best performing iterative metric set with K=4 is comparable with the benchmark method that combines seven centrality metrics: their prediction quality ratio is within the range [91%,106%] across all three quality measures and networks. In two spatially embedded networks with an extremely large diameter, however, iterative metric of higher orders, thus a large K, is needed to achieve comparable prediction quality with the benchmark.

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Temporal networks are networks like physical contact networks whose topology changes over time. Predicting future temporal network is crucial e.g., to forecast and mitigate the spread of epidemics and misinformation on the network. Most existing methods for temporal network prediction are based on machine learning algorithms, at the expense of high computational costs and limited interpretation of the underlying mechanisms that form the networks. This motivates us to develop network-based models to predict the temporal network at the next time step based on the network observed in the past. Firstly, we investigate temporal network properties to motivate our network prediction models and to explain how the performance of these models depends on the temporal networks. We explore the similarity between the network topology (snapshot) at any two time steps with a given time lag/interval. We find that the similarity is relatively high when the time lag is small and decreases as the time lag increases. Inspired by such time-decaying memory of temporal networks and recent advances, we propose two models that predict a link’s future activity (i.e., connected or not), based on the past activities of the link itself or also of neighboring links, respectively. Via seven real-world physical contact networks, we find that our models outperform in both prediction quality and computational complexity, and predict better in networks that have a stronger memory. Beyond, our model also reveals how different types of neighboring links contribute to the prediction of a given link’s future activity, again depending on properties of temporal networks.

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Human social interactions are typically recorded as time-specific dyadic interactions, and represented as evolving (temporal) networks, where links are activated/deactivated over time. However, individuals can interact in groups of more than two people. Such group interactions can be represented as higher-order events of an evolving network. Here, we propose methods to characterize the temporal-topological properties of higher-order events to compare networks and identify their (dis)similarities. We analyzed 8 real-world physical contact networks, finding the following: (a) Events of different orders close in time tend to be also close in topology; (b) Nodes participating in many different groups (events) of a given order tend to involve in many different groups (events) of another order; Thus, individuals tend to be consistently active or inactive in events across orders; (c) Local events that are close in topology are correlated in time, supporting observation (a). Differently, in 5 collaboration networks, observation (a) is almost absent; Consistently, no evident temporal correlation of local events has been observed in collaboration networks. Such differences between the two classes of networks may be explained by the fact that physical contacts are proximity based, in contrast to collaboration networks. Our methods may facilitate the investigation of how properties of higher-order events affect dynamic processes unfolding on them and possibly inspire the development of more refined models of higher-order time-varying networks.

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This paper aims to understand to what extent the amount of drug (e.g., cocaine) trafficking per country can be explained and predicted using the global shipping network. We propose three distinct network approaches, based on topological centrality metrics, Susceptible-Infected-Susceptible spreading process and a flow optimization model of drug trafficking on the shipping network, respectively. These approaches derive centrality metrics, infection probability, and inflow of drug traffic per country respectively, to estimate the amount of drug trafficking. We use the amount of drug seizure as an approximation of the amount of drug trafficking per country to evaluate our methods. Specifically, we investigate to what extent different methods could predict the ranking of countries in drug seizure (amount). Furthermore, these three approaches are integrated by a linear regression method in which we combine the nodal properties derived by each method to build a comprehensive model for the cocaine seizure data. Our analysis finds that the unweighted eigenvector centrality metric combined with the inflow derived by the flow optimization method best identifies the countries with a large amount of drug seizure (e.g., rank correlation 0.45 with the drug seizure). Extending this regression model with two extra features, the distance of a country from the source of cocaine production and a country’s income group, increases further the prediction quality (e.g., rank correlation 0.79). This final model provides insights into network derived properties and complementary country features that are explanatory for the amount of cocaine seized. The model can also be used to identify countries that have no drug seizure data but are possibly susceptible to cocaine trafficking.

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Compared to the burgeoning literature discussing the importance of agglomeration externalities for development, limited attention has been given to network externalities. This is largely due to limited data availability. We propose a general measure to proxy city network externalities based on toponym co-occurrences that indicate the relatedness between cities. This paper extracts intercity relationships based on the co-occurrence of Chinese place names on 2.5 billion webpages. We calculate and map absolute and relative network positions, which we use to explain urban labour productivity. We found that a stronger embeddedness in networks of cities is significantly and positively associated with urban productivity. Smaller cities benefit comparatively more from being well embedded in city networks, suggesting that these relations can compensate for a lack of agglomeration externalities. We also compare the importance for urban performance of city network externalities vis-à-vis agglomeration externalities. City network externalities turn out to be more important in explaining urban performance than agglomeration externalities. This calls for new theorizing on a relational approach to urban and regional development. Rather than stimulating further concentration of urbanization, our findings suggest that fostering relationships between cities is a viable alternative urban development strategy. We conclude with suggestions for a research agenda that delves deeper into city network externalities.

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Temporal networks are networks whose topology changes over time. Two nodes in a temporal network are connected at a discrete time step only if they have a contact/interaction at that time. The classic temporal network prediction problem aims to predict the temporal network one time step ahead based on the network observed in the past of a given duration. This problem has been addressed mostly via machine learning algorithms, at the expense of high computational costs and limited interpretation of the underlying mechanisms that form the networks. Hence, we propose to predict the connection of each node pair one step ahead based on the connections of this node pair itself and of node pairs that share a common node with this target node pair in the past. The concrete design of our two prediction models is based on the analysis of the memory property of real-world physical networks, i.e., to what extent two snapshots of a network at different times are similar in topology (or overlap). State-of-the-art prediction methods that allow interpretation are considered as baseline models. In seven real-world physical contact networks, our methods are shown to outperform the baselines in both prediction accuracy and computational complexity. They perform better in networks with stronger memory. Importantly, our models reveal how the connections of different types of node pairs in the past contribute to the connection estimation of a target node pair. Predicting temporal networks like physical contact networks in the long-term future beyond short-term i.e., one step ahead is crucial to forecast and mitigate the spread of epidemics and misinformation on the network. This long-term prediction problem has been seldom explored. Therefore, we propose basic methods that adapt each aforementioned prediction model to address classic short-term network prediction problem for long-term network prediction task. The prediction quality of all adapted models is evaluated via the accuracy in predicting each network snapshot and in reproducing key network properties. The prediction based on one of our models tends to have the highest accuracy and lowest computational complexity.

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Community detection of temporal (time-evolving) bipartite networks is challenging because it can be performed either on the temporal bipartite network, or on various projected networks, composed of only one type of nodes, via diverse community detection algorithms. In this paper, we aim to systematically design detection methods addressing both network choices and community detection algorithms, and to compare the community structures detected by different methods. We illustrate our methodology by using a telecommunications network as an example. We find that three methods proposed identify evident community structures: one is performed on each snapshot of the temporal network, and the other two, in temporal projections. We characterise the community structures detected by each method by an evaluation network in which the nodes are the services of the telecommunications network, and the weight of the links between them are the number of snapshots that both services were assigned to the same community. Analysing the evaluation networks of the three methods reveals the similarity and difference among these methods in identifying common node pairs or groups of nodes that often belong to the same community. We find that the two methods that are based on the same projected network identify consistent community structures, whereas the method based on the original temporal bipartite network complements this vision of the community structure. Moreover, we found a non-trivial number of node pairs that belong consistently to the same community in all the methods applied.

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Multiple network embedding algorithms have been proposed to perform the prediction of missing or future links in complex networks. However, we lack the understanding of how network topology affects their performance, or which algorithms are more likely to perform better given the topological properties of the network. In this paper, we investigate how the clustering coefficient of a network, i.e., the probability that the neighbours of a node are also connected, affects network embedding algorithms’ performance in link prediction, in terms of the AUC (area under the ROC curve). We evaluate classic embedding algorithms, i.e., Matrix Factorisation, Laplacian Eigenmaps and node2vec, in both synthetic networks and (rewired) real-world networks with variable clustering coefficient. Specifically, a rewiring algorithm is applied to each real-world network to change the clustering coefficient while keeping key network properties. We find that a higher clustering coefficient tends to lead to a higher AUC in link prediction, except for Matrix Factorisation, which is not sensitive to the change of clustering coefficient. To understand such influence of the clustering coefficient, we (1) explore the relation between the link rating (probability that a node pair is the missing link) derived from the aforementioned algorithms and the number of common neighbours of the node pair, and (2) evaluate these embedding algorithms’ ability to reconstruct the original training (sub)network. All the network embedding algorithms that we tested tend to assign higher likelihood of connection to node pairs that share an intermediate or high number of common neighbours, independently of the clustering coefficient of the training network. Then, the predicted networks will have more triangles, thus a higher clustering coefficient. As the clustering coefficient increases, all the algorithms but Matrix Factorisation could also better reconstruct the training network. These two observations may partially explain why increasing the clustering coefficient improves the prediction performance.

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Temporal networks refer to networks like physical contact networks whose topology changes over time. Predicting future temporal network is crucial e.g., to forecast the epidemics. Existing prediction methods are either relatively accurate but black-box, or white-box but less accurate. The lack of interpretable and accurate prediction methods motivates us to explore what intrinsic properties/mechanisms facilitate the prediction of temporal networks. We use interpretable learning algorithms, Lasso Regression and Random Forest, to predict, based on the current activities (i.e., connected or not) of all links, the activity of each link at the next time step. From the coefficients learned from each algorithm, we construct the prediction backbone network that presents the influence of all links in determining each links future activity. Analysis of the backbone, its relation to the link activity time series and to the time aggregated network reflects which properties of temporal networks are captured by the learning algorithms. Via six real-world contact networks, we find that the next step activity of a particular link is mainly influenced by (a) its current activity and (b) links strongly correlated in the time series to that particular link and close in distance (in hops) in the aggregated network.

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Cities relate to other cities in many ways, and much scholarly effort goes into uncovering those relationships. Building on the principle that strongly related cities will co-occur frequently in texts, we propose a novel method to classify those toponym co-occurrences using a lexicon-based text-mining method. Millions of webpages are analysed to retrieve how 293 Chinese cities are related in terms of six types: industry, information technology, finance, research, culture and government. Each class displays different network patterns, and this multiplexity is mapped and analysed. Further refinement of this lexicon-based approach can revolutionize the study of inter-urban relationships.

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Progress has been made in how to suppress epidemic spreading on temporal networks via blocking all contacts of targeted nodes or node pairs. In this work, we develop contact blocking strategies that remove a fraction of contacts from a temporal (time evolving) human contact network to mitigate the spread of a Susceptible-Infected-Recovered epidemic. We define the probability that a contact c(i, j, t) is removed as a function of a given centrality metric of the corresponding link l(i, j) in the aggregated network and the time t of the contact. The aggregated network captures the number of contacts between each node pair. A set of 12 link centrality metrics have been proposed and each centrality metric leads to a unique contact removal strategy. These strategies together with a baseline strategy (random removal) are evaluated in empirical contact networks via the average prevalence, the peak prevalence and the time to reach the peak prevalence. We find that the epidemic spreading can be mitigated the best when contacts between node pairs that have fewer contacts and early contacts are more likely to be removed. A strategy tends to perform better when the average number contacts removed from each node pair varies less. The aggregated pruned network resulted from the best contact removal strategy tends to have a large largest eigenvalue, a large modularity and probably a small largest connected component size.@en

Small and medium enterprises (SME) are crucial for economy and have a higher exposure rate to default than large corporates. In this work, we address the problem of predicting the default of an SME. Default prediction models typically only consider the previous financial situation of each analysed company. Thus, they do not take into account the interactions between companies, which could be insightful as SMEs live in a supply chain ecosystem in which they constantly do business with each other. Thereby, we present a novel method to improve traditional default prediction models by incorporating information about the insolvency situation of customers and suppliers of a given SME, using a graph-based representation of SME supply chains. We analyze its performance and illustrate how this proposed solution outperforms the traditional default prediction approaches.

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In this work, we explore the possibility of using a heterogeneous Susceptible-Infected-Susceptible SIS spreading process on an airline network to model airport congestion contagion with the objective to reproduce airport vulnerability. We derive the vulnerability of each airport from the US Airport Network data as the congestion probability of each airport. In order to capture diverse flight features between airports, e.g. frequency and duration, we construct three types of airline networks. The infection rate of each link in the SIS spreading process is proportional to its corresponding weight in the underlying airline network constructed. The recovery rate of each node is also heterogeneous, dependent on its node strength in the underlying airline network, which is the total weight of the links incident to the node. Such heterogeneous recovery rate is motivated by the fact that large airports may recover fast from congestion due to their well-equipped infrastructures. The nodal infection probability in the meta-stable state is used as a prediction of the vulnerability of the corresponding airport. We illustrate that our model could reproduce the distribution of nodal vulnerability and rank the airports in vulnerability evidently better than the SIS model whose recovery rate is homogeneous. The vulnerability is the largest at airports whose strength in the airline network is neither too large nor too small. This phenomenon can be captured by our heterogeneous model, but not the homogeneous model where a node with a larger strength has a higher infection probability. This explains partially the out-performance of the heterogeneous model. This proposed congestion contagion model may shed lights on the development of strategies to identify vulnerable airports and to mitigate global congestion by e.g. congestion reduction at selected airports.

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In this paper, we aim to effectively suppress the spread of epidemic/information via blocking/removing a given fraction of the contacts in a temporal (time evolving) human contact network. We consider the SI (Susceptible- Infected) spreading process, on a temporal contact network to illustrate our methodology: an infected node infects a susceptible node with a probability β when a contact happens between the two nodes. We address the question: which contacts should be blocked in order to minimize the average prevalence over time. We firstly propose systematically a set of link properties (centrality metrics) based on the aggregated network of a temporal network, that captures the number of contacts between each node pair. Furthermore, we define the probability that a contact c(i, j, t) is removed as a function of the centrality of the corresponding link l(i, j) in the aggregated network as well as the time t of the contact. Each of the centrality metrics proposed can be thus regarded as a contact removal strategy. Empirical results on six temporal contact networks show that the epidemic can be better suppressed if contacts between node pairs that have fewer contacts are more likely to be removed and if contacts happened earlier are likely removed. A strategy tends to perform better when the average number contacts removed per node pair has a lower variance. Strategies that lead to a lower largest eigenvalue of the aggregated network after contact removal do not mitigate the spreading better. This contradicts the finding in static networks, that a network with a small largest eigenvalue tends to be robust against epidemic spreading, illustrating the complexity introduced by the underlying temporal networks.

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Public transport networks constitute critical infrastructure in urban systems. Public transport networks are characterised by their hierarchical structure, yet methods to quantify their underlying hierarchy are lacking. We propose a metric for quantifying the hierarchy in public transport networks which incorporates topological as well as passenger flow information. Our proposed metric consists of three components which jointly define the relative hierarchical position of nodes across the network while the distribution of hierarchy defines the hierarchy of the network itself. We apply the metric to the case studies of Amsterdam and Rotterdam to demonstrate its usefulness in comparing different network states both within and across networks. Using this metric, we identify different patterns in network structures for network states and different spatial distributions of hierarchy between networks. Furthermore, by dividing the network into functional levels, we identify a multilayer hierarchical structure that describes the functionality of the network. The potential application of this metric relates to the assessment of network development scenarios, evaluating bottlenecks and analysing the network vulnerability. Furthermore, the metric is potentially suitable for assessing different network structures such as aviation or maritime networks.

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We model airport congestion contagion as an SIS spreading process on an airport transportation network to explain airport vulnerability. The vulnerability of each airport is derived from the US Airport Network data as its congestion probability. We construct three types of airline networks to capture diverse features such as the frequency and duration of flights. The weight of each link augments its infection rate in SIS spreading process. The nodal infection probability in the meta-stable state is used as estimate the vulnerability of the corresponding airport. We illustrate that our model could reasonably capture the distribution of nodal vulnerability and rank airports in vulnerability evidently better than the random ranking, but not significantly better than using nodal network properties. Such congestion contagion model not only allows the identification of vulnerable airports but also opens the possibility to reduce global congestion via congestion reduction in few airports.

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Link prediction can be used to extract missing information, identify spurious interactions as well as forecast network evolution. Network embedding is a methodology to assign coordinates to nodes in a low-dimensional vector space. By embedding nodes into vectors, the link prediction problem can be converted into a similarity comparison task. Nodes with similar embedding vectors are more likely to be connected. Classic network embedding algorithms are random-walk-based. They sample trajectory paths via random walks and generate node pairs from the trajectory paths. The node pair set is further used as the input for a Skip-Gram model, a representative language model that embeds nodes (which are regarded as words) into vectors. In the present study, we propose to replace random walk processes by a spreading process, namely the susceptible-infected (SI) model, to sample paths. Specifically, we propose two susceptible-infected-spreading-based algorithms, i.e., Susceptible-Infected Network Embedding (SINE) on static networks and Temporal Susceptible-Infected Network Embedding (TSINE) on temporal networks. The performance of our algorithms is evaluated by the missing link prediction task in comparison with state-of-the-art static and temporal network embedding algorithms. Results show that SINE and TSINE outperform the baselines across all six empirical datasets. We further find that the performance of SINE is mostly better than TSINE, suggesting that temporal information does not necessarily improve the embedding for missing link prediction. Moreover, we study the effect of the sampling size, quantified as the total length of the trajectory paths, on the performance of the embedding algorithms. The better performance of SINE and TSINE requires a smaller sampling size in comparison with the baseline algorithms. Hence, SI-spreading-based embedding tends to be more applicable to large-scale networks.

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In this paper, we explore how to effectively suppress the diffusion of (mis)information via blocking/removing the temporal contacts between selected node pairs. Information diffusion can be modelled as, e.g., an SI (Susceptible-Infected) spreading process, on a temporal social network: an infected (information possessing) node spreads the information to a susceptible node whenever a contact happens between the two nodes. Specifically, the link (node pair) blocking intervention is introduced for a given period and for a given number of links, limited by the intervention cost. We address the question: which links should be blocked in order to minimize the average prevalence over time? We propose a class of link properties (centrality metrics) based on the information diffusion backbone [19], which characterizes the contacts that actually appear in diffusion trajectories. Centrality metrics of the integrated static network have also been considered. For each centrality metric, links with the highest values are blocked for the given period. Empirical results on eight temporal network datasets show that the diffusion backbone based centrality methods outperform the other metrics whereas the betweenness of the static network, performs reasonably well especially when the prevalence grows slowly over time.

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Progress has been made in understanding how temporal network features affect the percentage of nodes reached by an information diffusion process. In this work, we explore further: which node pairs are likely to contribute to the actual diffusion of information, i.e., appear in a diffusion trajectory? How is this likelihood related to the local temporal connection features of the node pair? Such deep understanding of the role of node pairs is crucial to tackle challenging optimization problems such as which kind of node pairs or temporal contacts should be stimulated in order to maximize the prevalence of information spreading. We start by using Susceptible-Infected (SI) model, in which an infected (information possessing) node could spread the information to a susceptible node with a given infection probability β whenever a contact happens between the two nodes, as the information diffusion process. We consider a large number of real-world temporal networks. First, we propose the construction of an information diffusion backbone G B (β) for a SI spreading process with an infection probability β on a temporal network. The backbone is a weighted network where the weight of each node pair indicates how likely the node pair appears in a diffusion trajectory starting from an arbitrary node. Second, we investigate the relation between the backbones with different infection probabilities on a temporal network. We find that the backbone topology obtained for low and high infection probabilities approach the backbone G B (β → 0) and G B (β = 1), respectively. The backbone G B (β → 0) equals the integrated weighted network, where the weight of a node pair counts the total number of contacts in between. Finally, we explore node pairs with what local connection features tend to appear in G B (β = 1), thus actually contribute to the global information diffusion. We discover that a local connection feature among many other features we proposed, could well identify the (high-weight) links in G B (β = 1). This local feature encodes the time that each contact occurs, pointing out the importance of temporal features in determining the role of node pairs in a dynamic process. @en

A signed network represents how a set of nodes are connected by two logically contradictory types of links: positive and negative links. In a signed products network, two products can be complementary (purchased together) or substitutable (purchased instead of each other). Such contradictory types of links may play dramatically different roles in the spreading process of information, opinion, behaviour etc. In this work, we propose a self-avoiding pruning (SAP) random walk on a signed network to model e.g. a user's purchase activity on a signed products network. A SAP walk starts at a random node. At each step, the walker moves to a positive neighbour that is randomly selected, the previously visited node is removed and each of its negative neighbours are removed independently with a pruning probability r. We explored both analytically and numerically how signed network topological features influence the key performance of a SAP walk: the evolution of the pruned network resulted from the node removals, the length of a SAP walk and the visiting probability of each node. These findings in signed network models are further verified in two real-world signed networks. Our findings may inspire the design of recommender systems regarding how recommendations and competitions may influence consumers' purchases and products' popularity.@en