Searching for Heavy-Tailed Probability Distributions for Modeling Real-World Complex Networks

Robustness and Optimization of Complex Networks: Reconstructability, Algorithms and Modeling

Whether real-world complex networks are scale free or not has long been controversial. Recently, in Broido and Clauset [A. D. Broido, A. Clauset, Nat. Commun. 10, 1017 (2019)], it was claimed that the degree distributions of real-world networks are rarely power law under statistical tests. Here, we attempt to address this issue by defining a fundamental property possessed by each link, the degree-degree distance, the distribution of which also shows signs of being power law by our empirical study. Surprisingly, although full-range statistical tests show that degree distributions are not often power law in real-world networks, we find that in more than half of the cases the degree-degree distance distributions can still be described by power laws. To explain these findings, we introduce a bidirectional preferential selection model where the link configuration is a randomly weighted, two-way selection process. The model does not always produce solid power-law distributions but predicts that the degree-degree distance distribution exhibits stronger power-law behavior than the degree distribution of a finite-size network, especially when the network is dense. We test the strength of our model and its predictive power by examining how real-world networks evolve into an overly dense stage and how the corresponding distributions change. We propose that being scale free is a property of a complex network that should be determined by its underlying mechanism (e.g., preferential attachment) rather than by apparent distribution statistics of finite size. We thus conclude that the degree-degree distance distribution better represents the scale-free property of a complex network.

Self-similar scaling of density in complex real-world networks

We study the property of certain complex networks of being both sparse and highly connected, which is known as “good expansion” (GE). A network has GE properties if every subset S of nodes (up to 50% of the nodes) has a neighborhood that is larger than some “expansion factor” φ multiplied by the number of nodes in S. Using a graph spectral method we introduce here a new parameter measuring the good expansion character of a network. By means of this parameter we are able to classify 51 real-world complex networks — technological, biological, informational, biological and social — as GENs or non-GENs. Combining GE properties and node degree distribution (DD) we classify these complex networks in four different groups, which have different resilience to intentional attacks against their nodes. The simultaneous existence of GE properties and uniform degree distribution contribute significantly to the robustness in complex networks. These features appear solely in 14% of the 51 real-world networks studied here. At the other extreme we find that ∼40% of all networks are very vulnerable to targeted attacks. They lack GE properties, display skewed DD — exponential or power-law — and their topologies are changed more dramatically by targeted attacks directed at bottlenecks than by the removal of network hubs.

A key challenge in the social network community is the problem of network generation -- that is, how can we create synthetic networks that match characteristics traditionally found in most real world networks? Important characteristics that are present in social networks include a power law degree distribution, small diameter, and large amounts of clustering. However, most current network generators, such as the Chung Lu and Kronecker models, largely ignore the clustering present in a graph and focus on preserving other network statistics, such as the power law distribution. Models such as the exponential random graph model have a transitivity parameter that can capture clustering, but they are computationally difficult to learn, making scaling to large real world networks intractable. In this work, we propose an extension to the Chung Lu random graph model, the Transitive Chung Lu (TCL) model, which incorporates the notion transitive edges. Specifically, it combines the standard Chung Lu model with edges that are formed through transitive closure (e.g., by connecting a 'friend of a friend'). We prove TCL's expected degree distribution is equal to the degree distribution of the original input graph, while still providing the ability to capture the clustering in the network. The single parameter required by our model can be learned in seconds on graphs with millions of edges, networks can be generated in time that is linear in the number of edges. We demonstrate the performance of TCL on four real-world social networks, including an email dataset with hundreds of thousands of nodes and millions of edges, showing TCL generates graphs that match the degree distribution, clustering coefficients and hop plots of the original networks.

Although the origin of the fat-tail characteristic of the degree distribution in complex networks has been extensively researched, the underlying cause of the degree distribution characteristic across the complete range of degrees remains obscure. Here, we propose an evolution model that incorporates only two factors: the node’s weight, reflecting its innate attractiveness (nature), and the node’s degree, reflecting the external influences (nurture). The proposed model provides a good fit for degree distributions and degree ratio distributions of numerous real-world networks and reproduces their evolution processes. Our results indicate that the nurture factor plays a dominant role in the evolution of social networks. In contrast, the nature factor plays a dominant role in the evolution of non-social networks, suggesting that whether nodes are people determines the dominant factor influencing the evolution of real-world networks.

Robustness of multipartite networks in face of random node failure

We present new and improved methods for efficient shortest-path query processing. Our methods are tailored to work for two specific classes of graphs: graphs with small tree-width and complex networks. Seemingly unrelated at first glance, these two classes of graphs have some commonalities: complex networks are known to have a core--fringe structure with a dense core and a tree-like fringe.Our main contributions are efficient algorithms and data structures on three different levels. First, we provide two new methods for graphs with small but not necessarily constant tree-width. Our methods achieve new tradeoffs between space and query time. Second, we present an improved tree-decomposition-based method for complex networks, utilizing the methods for graphs with small tree-width. Third, we extend our method to handle the highly inter-connected core with existing exact and approximate methods.We evaluate our algorithms both analytically and experimentally. We prove that our algorithms for low-tree-width graphs achieve improved tradeoffs between space and query time. Our experiments on several real-world complex networks further confirm the efficiency of our methods: Both the exact and the hybrid method have faster preprocessing and query times than existing methods. The hybrid method in particular provides an improved tradeoff between space and accuracy.

Node similarity is a significant basis for analyzing features in complex network. For complex network with directed weighted edge, the complexity of the relationship among nodes and the diversity of relationship weights make measure node similarity complicated from the huge amounts of nodes. Therefore, a novel node similarity measure is proposed based on the design of node nearest neighbor local network relative weighted entropy. Firstly, degree and strength based directed weighted complex network model is constructed, on the basis of that, the Node Nearest Neighbor Local network is defined. The structural features of each node in the local network are quantized into a set of decision probabilities with multiple indicators. Furthermore, Node Connection Tightness is given to compute the influence of structural complex relationship in local network on node similarity. And then, to evaluate how the outward & inward degree and strength of nodes in the local network affect node similarity, the concept of Node Nearest Neighbor Local Network Relative Weighted Entropy is designed to define Node Relative Difference for measuring the structural difference between any two nodes. Accordingly, a novel node similarity measure is designed to measure the similarity between any pair of nodes in directed weighted complex network, and then the Similar Node Mining Algorithm is proposed to obtain most similar nodes. To clarify the availability and effectiveness of the proposed measure, two sets of experiments were conducted on real-world complex networks. The results show that the measure can not only mine nodes with the most similarity in the same module, but also mine the most similarity nodes from different modules.

Community detection can exhibit the aggregation behavior of complex networks. Network motifs are the fundamental building blocks which can reveal the higher-order structure of complex networks. Label propagation algorithm has the advantage of approximately linear time complexity, unfortunately, the randomness of label update is a major but unsolved issue. For these reasons, this paper proposes a novel community detection method, named motif-based embedding label propagation algorithm (MELPA). First, complex network topology is reconstructed by merging higher-order topology with lower-order connectivity features, where higher-order topology is captured by mining network motifs. Second, We design a label propagation characteristic model according to nodes influence, then a new label update rule is formulated based on reconstructed weighted network, the rule integrates frequency among neighbor labels, influence of nodes, propagation characteristics and closeness of nodes to update the node label, the purpose is to overcome the randomness of label selection and identify a better and more stable community structure. Finally, extensive experiments on synthetic networks and real-world complex networks are conducted to verify the effectiveness of MELPA, especially for the complex networks with unobvious community structure, MELPA will get unexpected results.

The issue of mining influential nodes in complex networks is a topic of immense interest. Recently, many methods have been proposed, but they suffer from certain limitations. In this article, a novel centrality measure based on local fuzzy information centrality (LFIC) is proposed. LFIC puts forward the concept that the inner structure of a node’s box contains information about the node’s importance. LFIC uses the amount of information contained in the node’s box as a measure of its importance. In LFIC, the uncertainty of information contained in nodes’ boxes is measured by the improved Shannon entropy. Most importantly, fuzzy logic is applied to deal with the uncertainty of neighbor nodes’ contributions to the center node’s importance, which is neglected by most existing methods. To verify the effectiveness of our proposed method, six existing methods are used for comparison and five experiments are conducted using six real-world complex networks. The experimental results indicate that the influential nodes identified by LFIC can cause a wider scope of infection in networks and have a larger effect on the network connectivity, thereby proving the effectiveness and accuracy of LFIC. The correlation between nodes’ LFIC values and their real infection ability is highly positive according to Kendall’s tau coefficient, proving LFIC’s credibility and superiority. The extension of LFIC, namely the bi-directional local fuzzy information centrality, is also proposed to explore its feasibility in weighted directed complex networks.

Entropy has been widely measured the complexity of complex networks. At present, many measures about entropies were defined based on the directed connection of nodes. The similarity of nodes can better represent the relationship among all nodes in complex networks. In the definition of similarity of nodes, the importance of a node in the network depends not only on the degree of the node itself, but also on the extent of dependence of neighboring nodes on the node. In this paper, we proposed a new structure entropy based on nonextensive statistical mechanics and similarity of nodes. In the proposed method, the similarity of nodes and the betweenness of nodes are both quantified. The proposed method considers the extent of dependence between neighbouring nodes. For some complex networks, the proposed structure entropy can distinguish complexity of that while other entropies can not be. Meanwhile, we construct five ER random networks and small-world networks and some real-world complex networks such as the US Air Lines networks, the GD'01-GD Proceedings Self-Citing networks, the Science Theory networks, the Centrality Literature networks and the Yeast networks are measured by the proposed method. The results illustrated our method for quantifying the complexity of complex networks is effective.

An increasing attention has been recently devoted to uncovering community structure in directed graphs which widely exist in real-world complex networks such as social networks, citation networks, World Wide Web, email networks, etc. A two-stage framework for detecting clusters is an effective way for clustering directed graphs while the first stage is to symmetrize the directed graph using some similarity measures. Any state-of-the-art clustering algorithms for undirected graphs can be leveraged in the second stage. Hence, both stages are important to the effectiveness of the clustering result. However, existing symmetrization methods only consider about the direction of edges but ignore the weights of nodes. In this paper, we first attempt to connect link analysis in directed graph clustering. This connection not only takes into consideration the directionality of edges but also uses node ranking scores such as authority and hub score to explicitly capture in-link and out-link similarity. We also demonstrate the generality of our proposed method by showing that existing state-of-the-art symmetrization methods can be derived from our method. Empirical validation shows that our method can find communities effectively in real world networks.

This study presents a proposition for describing the dynamics of real-world networks under the general framework of complex networks. Outward behaviors of complex networks are the manifestation of the coupled dynamics at the macroscopic level and the individual dynamics at the microscopic level. At the macroscopic level a law of coupling governs the interactions of network constituents. At the microscopic level, the dynamics of individual constituent is defined by energy that follows normal distribution. Constituent dynamics are bounded by physical constraints. Consequently, network dynamics can be quantified using information entropy which is a function of constituent energy. In real-world networks, differences between individual constituents exist due to differing mechanical properties and dynamics. Consequently, network dynamics are of different layers and hierarchies. Construct of network governing equations formulated under the general framework of complex networks are demonstrated using two real-world networks — a brain network and a lymph node network. Brain network is constructed by the neurons that each connected by the synapse. Brain network dynamics is composed by the law of coupling defined by the synaptic dynamics through the transmitting of neurotransmitters that couples the individual neuron dynamics. Since different classifications exist among neurotransmitters and neurons, the post synaptic neuron can present either inhibitory or excitatory action. The inhibitory and excitatory behavior of the neurons changes the mechanical properties of each neuron and further alters the brain network dynamics. Consequently, the brain network emerges dynamics with different layers. Lymph node network drains fluid from blood vessels, filter the lymph (the interstitial fluid lymphatic system collects from the blood circulation) through lymph nodes, and transport the lymph back to the blood circulation. Lymph node dynamics is composed by the dynamics of lymph transportation along the lymph node network and the individual lymph node dynamics that involves lymphocytes-pathogens interactions (adaptive immune response). In each lymph node, lymphocytes fight off the pathogens which also emerges a network dynamics such as the interaction between T cells and HIV viruses. Finally, the lymph is collected from each lymph nodes and drained back to the blood circulation. As a result, the lymph node network has the dynamics of different hierarchies where the lymphocytes-pathogens dynamics exists within each lymph node at the lower hierarchy is further under the influence of the lymph transportation dynamics among the whole lymph node network on the higher hierarchy. Since the constituent dynamics of the brain network and lymph node network can be defined by energy that follows normal distribution and both are bounded by physical constraints, the network dynamics of both cases can be quantified through information entropy. Features pertaining to the global as well as individual constituent dynamics of the networks are identified that are insightful to the control of such complex networks.

A complex network can crash down due to disturbances which significantly reduce the network’s robustness. It is of great significance to study on how to improve the robustness of complex networks. In the literature, the network rewire mechanism is one of the most widely adopted methods to improve the robustness of a given network. Existing network rewire mechanism improves the robustness of a given network by re-connecting its nodes but keeping the total number of edges or by adding more edges to the given network. In this work we propose a novel yet efficient network rewire mechanism which is based on multiobjective optimization. The proposed rewire mechanism simultaneously optimizes two objective functions, i.e., maximizing network robustness and minimizing edge rewire operations. We further develop a multiobjective discrete partite swarm optimization algorithm to solve the proposed mechanism. Compared to existing network rewire mechanisms, the developed mechanism has two advantages. First, the proposed mechanism does not require specific constraints on the rewire mechanism to the studied network, which makes it more feasible for applications. Second, the proposed mechanism can suggest a set of network rewire choices each of which can improve the robustness of a given network, which makes it be more helpful for decision makings. To validate the effectiveness of the proposed mechanism, we carry out experiments on computer-generated Erdős–Rényi and scale-free networks, as well as real-world complex networks. The results demonstrate that for each tested network, the proposed multiobjective optimization based edge rewire mechanism can recommend a set of edge rewire solutions to improve its robustness.