Characterization of Complex Networks for Epidemics Modeling

The problem of identifying the influential spreaders - the important nodes - in a real world network is of high importance due to its theoretical interest as well as its practical applications, such as the acceleration of information diffusion, the control of the spread of a disease and the improvement of the resilience of networks to external attacks. In this paper, we propose a graph exploration sampling method that accurately identifies the influential spreaders in a complex network, without any prior knowledge of the original graph, apart from the collected samples/subgraphs. The method explores the graph, following a deterministic selection rule and outputs a graph sample - the set of edges that have been crossed. The proposed method is based on a version of Rank Degree graph sampling algorithm. We conduct extensive experiments in eight real world networks by simulating the susceptible-infected-recovered (SIR) and susceptible-infected-susceptible (SIS) epidemic models which serve as ground truth identifiers of nodes spreading efficiency. Experimentally, we show that by exploring only the 20% of the network and using the degree centrality as well as the k-core measure, we are able to identify the influential spreaders with at least the same accuracy as in the full information case, namely, the case where we have access to the original graph and in that graph, we compute the centrality measures. Finally and more importantly, we present strong evidence that the degree centrality - the degree of nodes in the collected samples - is almost as accurate as the k-core values obtained from the original graph.

Ring vaccination is a mitigation strategy that consists in seeking and vaccinating the contacts of a sick patient, in order to provide immunization and halt the spread of disease. We study an extension of the susceptible-infected-recovered (SIR) epidemic model with ring vaccination in complex and spatial networks. Previously, a correspondence between this model and a link percolation process has been established, however, this is only valid in complex networks. Here, we propose that the SIR model with ring vaccination is equivalent to a mixed percolation process of links and nodes, which offers a more complete description of the process. We verify that this approach is valid in both complex and spatial networks, the latter being built according to the Waxman model. This model establishes a distance-dependent cost of connection between individuals arranged in a square lattice. We determine the epidemic-free regions in a phase diagram based on the wiring cost and the parameters of the epidemic model (vaccination and infection probabilities and recovery time). In addition, we find that for long recovery times this model maps into a pure node percolation process, in contrast to the SIR model without ring vaccination, which maps into link percolation.

Controlling COVID-19 transmission with isolation of influential nodes

Traffic-driven SIR epidemic spread dynamics on scale-free networks

The topological structures of complex networks have been playing an important role on the epidemic spreading. There has been several studies of pairwise epidemic models on adaptive networks with Poisson distribution, all of which have shown that the rewiring behaviors can lead to complex dynamics numerically or analytically. However, the triples approximation formula under Poisson distribution overlooked the degree of center node of triples which has dramatic effects on the structures. Therefore in this paper, through a new moment closure incorporating the effect of center node's degree, we study how the topological structures of adaptive networks influences epidemic dynamics. The SIS pairwise epidemic model is first closed by the new triple approximation formula, then we transform the model into an equivalent nondimensionalized three dimensional system. By the qualitative theory and the stability theory of ordinary differential equations, the basic reproduction number R0 of the model is obtained, the existence and stabilities of the equilibria are analyzed. Moreover, we prove that the model exhibits transcritical forward bifurcation, backward bifurcation, saddle-node bifurcation and Hopf bifurcation using the methods of bifurcation theory. In addition, by a numerical example, the normal form of Hopf bifurcation and the first Lyapunov coefficient are derived, which show that a stable limit cycle can bifurcate from the endemic equilibrium with larger epidemicity. Our study show that the adaptive behavior can lead to rich dynamics on epidemic transmission, including oscillation and bistability. Finally the numerical simulations which is consistent with the analytical results above are given.

Traffic flow prediction provides support for travel management, vehicle scheduling, and intelligent transportation system construction. In this work, a graph space–time network (GSTNCNI), incorporating complex network feature information, is proposed to predict future highway traffic flow time series. Firstly, a traffic complex network model using traffic big data is established, the topological features of traffic road networks are then analyzed using complex network theory, and finally, the topological features are combined with graph neural networks to explore the roles played by the topological features of 97 traffic network nodes. Consequently, six complex network properties are discussed, namely, degree centrality, clustering coefficient, closeness centrality, betweenness centrality, point intensity, and shortest average path length. This study improves the graph convolutional neural network based on the above six complex network properties and proposes a graph spatial–temporal network consisting of a combination of several complex network properties. By comparison with existing baselines containing graph convolutional neural networks, it is verified that GSTNCNI possesses high traffic flow prediction accuracy and robustness. In addition, ablation experiments are conducted for six different complex network features to verify the effect of different complex network features on the model’s prediction accuracy. Experimental analysis indicates that the model with combined multiple complex network features has a higher prediction accuracy, and its performance is improved by 31.46% on average, compared with the model containing only one complex network feature.

Identifying influential spreaders in networks is an essential issue in order to prevent epidemic spreading, or to accelerate information diffusion. Several centrality measures take advantage of various network topological properties to quantify the notion of influence. However, the vast majority of works ignore its community structure while it is one of the main features of many real-world networks. In a recent study, we show that the centrality of a node in a network with non-overlapping communities depends on two features: Its local influence on the nodes belonging to its community, and its global influence on the nodes belonging to the other communities. Using global and local connectivity of the nodes, we introduced a framework allowing to redefine all the classical centrality measures (designed for networks without community structure) to non-overlapping modular networks. In this paper, we extend the so-called “Modular Centrality” to networks with overlapping communities. Indeed, it is a frequent scenario in real-world networks, especially for social networks where nodes usually belong to several communities. The “Overlapping Modular Centrality” is a two-dimensional measure that quantifies the local and global influence of overlapping and non-overlapping nodes. Extensive experiments have been performed on synthetic and real-world data using the Susceptible-Infected-Recovered (SIR) epidemic model. Results show that the Overlapping Modular Centrality outperforms its alternatives designed for non-modular networks. These investigations provide better knowledge on the influence of the various parameters governing the overlapping community structure on the nodes’ centrality. Additionally, two combinations of the components of the Overlapping Modular Centrality are evaluated. Comparative analysis with competing methods shows that they produce more efficient centrality scores.

Link prediction algorithms in complex networks, such as social networks, biological networks, drug-drug interactions, communication networks, and so on, assign scores to predict potential links between two nodes. Link prediction (LP) enables researchers to learn unknown, new as well as future interactions among the entities being modeled in the complex networks. In addition to measures like degree distribution, clustering coefficient, centrality, etc., another metric to characterize structural properties is network assortativity which measures the tendency of nodes to connect with similar nodes. In this paper, we explore metrics that effectively predict the links based on the assortativity profiles of the complex networks. To this end, we first propose an approach that generates networks of varying assortativity levels and utilize three sets of link prediction models combining the similarity of neighborhoods and preferential attachment. We carry out experiments to study the LP accuracy (measured in terms of area under the precision-recall curve) of the link predictors individually and in combination with other baseline measures. Our analysis shows that link prediction models that explore a large neighborhood around nodes of interest, such as CH2-L2 and CH2-L3, perform consistently for assortative as well as disassortative networks. While common neighbor-based local measures are effective for assortative networks, our proposed combination of common neighbors with node degree is a good choice for the LP metric in disassortative networks. We discuss how this analysis helps achieve the best-parameterized combination of link prediction models and its significance in the context of link prediction from incomplete social and biological network data.

In order to provide effective managements of urban bus transport network and serve better for citizen travels, we modelled and simulated the Tai'an city bus lines by space P way, space L way and bus line way with complex network and graph theory. By computing adjacency matrix and complex network topology characters in bus line model, we concluded that the Tai'an city network had small-world character with short average path length and high clustering coefficient. The same results were achieved by complex network characters statistics in space P way and space L way after getting rid of noise. Theoretical analysis and experimental simulation imply that it is feasible and effective for bus line management and citizen travel by modelling and optimisation on bus transport network with complex network characters statistics and analysis.

Efficiently identifying multiple influential nodes is crucial for maximizing information diffusion and minimizing rumor spread in complex networks. Selecting multiple influential seed nodes requires consideration of both their individual influence potential and their spatial dispersion within the network topology to avoid overlapping propagation ranges ("rich-club effect"). Traditional VoteRank method suffer from two key limitations: (1) they assume uniform voting contributions from a node to all its neighbors, neglecting the impact of topological similarity (structural homophily) on voting preferences observed in real-world scenarios, and (2) they employ a homogeneous voting attenuation strategy which inadequately suppresses the propagation range overlap among selected seed nodes. To address these shortcomings, this paper proposes IMVoteRank, an improved VoteRank algorithm featuring dual innovations. First, we introduce a Structural Similarity-Driven Voting Contribution Mechanism. Recognizing that voters (nodes) are more likely to support candidates (neighbors) with whom they share stronger topological ties, we decompose a neighbor's voting contribution into two components: a Direct Connection Contribution and a Structural Similarity Contribution (quantified using common neighbors). A dynamic weight parameter θ, adjusted based on the candidate node's degree, balances these components, significantly refining vote allocation accuracy. Second, we devise a Dynamic Group Isolation Strategy. During each iteration, after selecting the highest-scoring seed node <i>v<sub>max</sub></i>, we identify and isolate a tightly-knit group (OG) centered around it. This involves: (i) forming an initial group based on shared neighbor density with <i>v<sub>max</sub></i>, (ii) expanding it by incorporating nodes with more connections inside the group than outside, and (iii) isolating this group by setting the voting capacity (Va) of all its members to zero and virtually removing their connections from the adjacency matrix. Neighbors of <i>v<sub>max</sub></i> not in OG have their <i>Va</i> halved. This strategy proactively enforces spatial dispersion among seeds. Extensive simulations using the Susceptible-Infected-Recovered (SIR) propagation model on nine diverse real-world networks (ECON-WM3, Facebook-SZ, USAir, Celegans, ASOIAF, Dnc-corecipient, ERIS1176, DNC-emails, Facebook-combined) demonstrate the superior performance of IMVoteRank. Compared to seven benchmark methods (Degree, K-shell, VoteRank, NCVoteRank, VoteRank++, AIGCrank, EWV), IMVoteRank consistently achieves significantly larger final propagation coverage (infected scale) for a given number of seed nodes and transmission probability (<i>β</i>=0.1). Furthermore, seeds selected by IMVoteRank exhibit a consistently larger average shortest path length (<i>Ls</i>) among themselves across most networks, confirming their effective topological dispersion. This combination of high individual influence potential (optimized voting) and low redundancy (group isolation) directly translates to more efficient global information spread, as evidenced by the SIR results. Tests on LFR benchmark networks further validate these advantages, particularly at transmission rates above the epidemic threshold. IMVoteRank effectively overcomes the limitations of traditional voting models by integrating structural similarity into the voting process and employing dynamic group isolation to ensure seed dispersion. It provides a highly effective and physically well-grounded approach for identifying multiple influential nodes in complex networks, optimizing the trade-off between influence strength and spatial coverage. Future work will focus on enhancing computational efficiency for large-scale networks and exploring the impact of meso-scale community structures.

Preferential attachment models have been widely studied in complex networks, because they can explain the formation of many networks like social networks, citation networks, power grids, and biological networks, to name a few. Motivated by the application of key predistribution in wireless sensor networks (WSN), we initiate the study of preferential attachment with degree bound. Our paper has two important contributions to two different areas. The first is a contribution in the study of complex networks. We propose preferential attachment model with degree bound for the first time. In the normal preferential attachment model, the degree distribution follows a power law, with many nodes of low degree and a few nodes of high degree. In our scheme, the nodes can have a maximum degree $d_{\max}$, where $d_{\max}$ is an integer chosen according to the application. The second is in the security of wireless sensor networks. We propose a new key predistribution scheme based on the above model. The important features of this model are that the network is fully connected, it has fewer keys, has larger size of the giant component and lower average path length compared with traditional key predistribution schemes and comparable resilience to random node attacks. We argue that in many networks like key predistribution and Internet of Things, having nodes of very high degree will be a bottle-neck in communication. Thus, studying preferential attachment model with degree bound will open up new directions in the study of complex networks, and will have many applications in real world scenarios.

Higher order assortativity in complex networks

We investigate the dynamics of a susceptible infected recovered (SIR) epidemic model on small networks with different topologies, as a stepping stone to determining how the structure of a contact network impacts the transmission of infection through a population. For an SIR model on a network of$N$nodes, there are$3^{N}$configurations that the network can be in. To simplify the analysis, we group the states together based on the number of nodes in each infection state and the symmetries of the network. We derive analytical expressions for the final epidemic size of an SIR model on small networks composed of three or four nodes with different topological structures. Differential equations which describe the transition of the network between states are also derived and solved numerically to confirm our analysis. A stochastic SIR model is numerically simulated on each of the small networks with the same initial conditions and infection parameters to confirm our results independently. We show that the structure of the network, degree of the initial infectious node, number of initial infectious nodes and the transmission rate all significantly impact the final epidemic size of an SIR model on small networks.

5G smart grid applications rely on its high-performance transmission and bearer network. With the help of complex network theory, this paper first analyzes the complex network characteristic parameters of 5G smart grid, and explains the necessity and supporting significance of network vulnerability analysis for efficient transmission of 5G network. Then the node importance analysis algorithm based on node degree and clustering coefficient (NIDCC) is proposed. According to the results of simulation analysis, the power network has smaller path length and higher clustering coefficient in terms of static parameters, which indicates that the speed and breadth of fault propagation are significantly higher than that of random network. It further shows the necessity of network vulnerability analysis. By comparing with the other two commonly used algorithms, we can see that NIDCC algorithm can more accurately estimate and analyze the weak links of the network. It is convenient to carry out the targeted transformation of the power grid and the prevention of blackout accidents.

Evolution of Fossil Energy International Trade Pattern Based on Complex Network