Poisson Degree Distribution Research Articles

Modular tipping points: How local network structure impacts critical transitions in networked spin systems

Critical transitions describe a phenomenon where a system abruptly shifts from one stable state to an alternative, often detrimental, stable state. Understanding and possibly preventing the occurrence of a critical transition is thus highly relevant to many ecological, sociological, and physical systems. In this context, it has been shown that the underlying network structure of a system heavily impacts the transition behavior of that system. In this paper, we study a crucial but often overlooked aspect in critical transitions: the modularity of the system’s underlying network topology. In particular, we investigate how the transition behavior of a networked system changes as we alter the local network structure of the system through controlled changes of the degree assortativity. We observe that systems with high modularity undergo cascading transitions, while systems with low modularity undergo more unified transitions. We also observe that networked systems that consist of nodes with varying degrees of connectivity tend to transition earlier in response to changes in a control parameter than one would anticipate based solely on the average degree of that network. However, in rare cases, such as when there is both low modularity and high degree disassortativity, the transition behavior aligns with what we would expected given the network’s average degree. Results are confirmed for a diverse set of degree distributions including stylized two-degree networks, uniform, Poisson, and power-law degree distributions. On the basis of these results, we argue that to understand critical transitions in networked systems, they must be understood in terms of individual system components and their roles within the network structure.

Identifying bias in network clustering quality metrics.

We study potential biases of popular network clustering quality metrics, such as those based on the dichotomy between internal and external connectivity. We propose a method that uses both stochastic and preferential attachment block models construction to generate networks with preset community structures, and Poisson or scale-free degree distribution, to which quality metrics will be applied. These models also allow us to generate multi-level structures of varying strength, which will show if metrics favour partitions into a larger or smaller number of clusters. Additionally, we propose another quality metric, the density ratio. We observed that most of the studied metrics tend to favour partitions into a smaller number of big clusters, even when their relative internal and external connectivity are the same. The metrics found to be less biased are modularity and density ratio.

Integrated analysis of topoisomerase I inhibitors from flavonoids of Scutellaria baicalensis Georgi using bioaffinity ultrafiltration UPLC-TripleTOF MS/MS, molecular docking and target-based multiple complex networks

Scutellaria baicalensis Georgi (SBG) has been widely used as medical plant in East Asia with remarkable anti-cancer activity. However, the underlying mechanisms are still confused. In this study, an integrated analysis was conducted to screen topoisomerase I (topo I) inhibitors from flavonoids of SBG and investigate the anti-cancer mechanisms, containing bioaffinity ultrafiltration UPLC-ESI-TripleTOF-MS/MS, molecular docking, and multiple complex networks. The SBG extract exhibited notable cytotoxic activity on Hela cells. Five flavonoids were identified as potential topo I inhibitors, including skullcapflavone II, wogonin, chrysin, oroxylin A, and tenaxin I. Their ESI-MS/MS spectra showed that RDA reaction and neutral molecule loss were the main fragment patterns. Docking results demonstrated that π–π interaction and the formation of hydrogen bond contributed most to their binding with topo I. The selected compounds, related target proteins and pathways were integrated into target-based multiple complex networks, which consisted of three subnetworks. Statistical and topological analysis of these networks revealed a series of characteristics, including scale-free property with power-law degree distribution, Poisson degree distribution, and small-world property. Chrysin, wogonin, and oroxylin A exhibited as main active components with much higher degree values. Chemical carcinogenesis-receptor activation (hsa05207) was considered as critical pathway due to remarkable centrality indexes. Additionally, potential synergistic effect of wogonin and chrysin was observed on the conversion of supercoiled DNA to relaxed forms. These results improved current understanding of flavonoid-rich plants on the treatment of cancer. Moreover, the multi-disciplinary approach provided a new strategy for the research of natural products from medical plants.

Resilience of supply-chain systems under perturbations: A network approach.

Supply-chain systems (SCSs) are an indispensable part of our daily infrastructures. Note that a small perturbation in a SCS can be amplified, eliciting cascading failures. It is of significant value to ensure a high resilience of SCSs. However, due to the complexity of SCSs, it is quite challenging to study their resilience under conditions of perturbations. In view of this, this paper presents a complex network perspective toward the resilience of SCSs. To achieve this goal, a complex SCS is modeled as a multilayer supply-chain network (SCN) with physical organizations being modeled as nodes and interactions among them as edges. A modeled SCN contains three types of nodes, i.e., suppliers, manufacturers, and retailers. An algorithm is proposed to construct a multilayer SCN. For each layer of a multilayer SCN, two kinds of networks, i.e., networks with Poisson degree distributions and networks with power-law degree distributions, are considered. For a given multilayer SCN, a ripple-effect network model is proposed to analyze its resilience under perturbations. Regarding the perturbations, two scenarios, i.e., random node failures and target node failures, are adopted in this research. In order to validate the effectiveness of the proposed network perspective, simulations on computer-generated SCNs are carried out. Interestingly, it is found that the resilience of SCNs under both random and target perturbations presents a discontinuous phase-change phenomenon, which indicates that SCNs are quite fragile under perturbations. It is further noticed that SCNs with power-law degree distributions are relatively more robust than SCNs with Poisson degree distributions. Although SCNs are found to be fragile, it has been discovered that denser interactions between different system organizations can enhance the network's resilience.

Network resilience of non-hub nodes failure under memory and non-memory based attacks with limited information.

Previous studies on network robustness mainly concentrated on hub node failures with fully known network structure information. However, hub nodes are often well protected and not accessible to damage or malfunction in a real-world networked system. In addition, one can only gain insight into limited network connectivity knowledge due to large-scale properties and dynamic changes of the network itself. In particular, two different aggression patterns are present in a network attack: memory based attack, in which failed nodes are not attacked again, or non-memory based attack; that is, nodes can be repeatedly attacked. Inspired by these motivations, we propose an attack pattern with and without memory based on randomly choosing n non-hub nodes with known connectivity information. We use a network system with the Poisson and power-law degree distribution to study the network robustness after applying two failure strategies of non-hub nodes. Additionally, the critical threshold 1 - p and the size of the giant component S are determined for a network configuration model with an arbitrary degree distribution. The results indicate that the system undergoes a continuous second-order phase transition subject to the above attack strategies. We find that 1 - p gradually tends to be stable after increasing rapidly with n. Moreover, the failure of non-hub nodes with a higher degree is more destructive to the network system and makes it more vulnerable. Furthermore, from comparing the attack strategies with and without memory, the results highlight that the system shows better robustness under a non-memory based attack relative to memory based attacks for n > 1. Attacks with memory can block the system's connectivity more efficiently, which has potential applications in real-world systems. Our model sheds light on network resilience under memory and non-memory based attacks with limited information attacks and provides valuable insights into designing robust real-world systems.

Revisiting Finite Size Effect of Percolation in Degree Correlated Networks

In this study, we investigate bond percolation in networks that have the Poisson degree distribution and a nearest-neighbor degree-degree correlation. Previous numerical studies on percolation critical behaviors of degree-correlated networks remain controversial. We perform finite-size scaling for the peak values of the second-largest cluster size and the mean cluster size and find a large finite-size effect when a network has a strong degree-degree correlation. Evaluating the size dependence of estimated critical exponents carefully, we demonstrate that the bond percolation in the networks exhibits the mean-field critical behavior, independent of the strength of their nearest-neighbor degree correlations.

Seed Investment Bounds for Viral Marketing Under Generalized Diffusion and Selection Guidance

This article attempts to provide viral marketeers guidance in terms of an investment level that could help capture some desired γ percentage of the market share by some target time t with the desired level of confidence. To do this, we first introduce a generalized diffusion model for social networks. A distance-dependent random graph is then considered as a model for the underlying social network, which we use to analyze the proposed diffusion model. Using the fact that vertices degrees have an almost Poisson distribution in distance-dependent random networks, we then provide a lower bound on the probability of the event that the time it takes for an idea (or a product, campaign, disease, and so on) to dominate a prespecified γ percentage of a social network (denoted by R <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">γ</sub> ) is smaller than some preselected target time t > 0, i.e., we find a lower bound on the probability of the event {R <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">γ</sub> ≤ t}. Simulation results performed over a wide variety of networks, including random and real world, are then provided to verify that our bound indeed holds in practice. The Kullback-Leibler divergence measure is used to evaluate the performance of our lower bound over these groups of networks, and as expected, we note that for networks that deviate more from the Poisson degree distribution, our lower bound weakens. In the case where absolute/full domination of the market-share is desired, under the linear threshold diffusion model, a particular case of our generalized diffusion model, an upper bound on the size of the seed set is derived, and a selection algorithm is developed to show its tightness. This is also extended to the partial market-share situation.

Resilience of IoT Systems Against Edge-Induced Cascade-of-Failures: A Networking Perspective

Internet of Things (IoT) is a networking paradigm that interconnects physical systems to the cyber world, to provide automation and intelligence via interdependent links between the two domains. Such interdependence renders IoT systems vulnerable to random failures, e.g., broken communication links or crashed cyber instances, because a single incident in one domain can develop into a cascade-of-failures across domains, which dissolves the network structure, and has devastating consequences. To answer how robust an IoT system is, this paper studies its resilience by examining the impact of edge- and jointly-induced cascades, that is, a sequence of failures caused by randomly broken physical links (and simultaneous failing cyber nodes). Resilience of an IoT system is quantified by two new metrics, the critical edge disconnecting probability $\phi _{cr}$ , i.e., the maximum intensity of random failures the system can withstand, and the cascade length $\tau _{cf}$ , i.e., the lifetime of a cascade. For IoT systems with Poisson degree distributions, we derive exact solutions for the critical disconnecting probability $\phi _{cr}$ , above which an edge-induced cascade will completely fragment the network. We also find that the critical condition $\phi _{cr}$ marks a dichotomy of the expected cascade length $\mathbb {E}(\tau _{cf})$ : for the super-critical ( $\phi > \phi _{cr}$ ) scenario, we obtain $\mathbb {E}(\tau _{cf}) \sim \exp (1-\phi)$ through analysis, while for the subcritical scenario, we observe $\mathbb {E}(\tau _{cf}) \sim \exp (1/1-\phi)$ through simulations. With these results, the final outcome of a cascade can be anticipated upon the initial failures, while the reaction window of time-sensitive countermeasures can be obtained before a cascade fully unfolds.

Influence of Clustering on Cascading Failures in Interdependent Systems

We study the influence of clustering, more specifically triangles, on cascading failures in interdependent networks or systems, in which we model the dependence between comprising systems using a dependence graph. First, we propose a new model that captures how the presence of triangles in the dependence graph alters the manner in which failures transmit from affected systems to others. Unlike existing models, the new model allows us to approximate the failure propagation dynamics using a multi-type branching process, even with triangles. Second, making use of the model, we provide a simple condition that indicates how increasing clustering will affect the likelihood that a random failure triggers a cascade of failures, which we call the probability of cascading failures (PoCF). In particular, our condition reveals an intriguing observation that the influence of clustering on PoCF depends on the vulnerability of comprising systems to an increasing number of failed neighboring systems and the current PoCF, starting with different types of failed systems. Our numerical studies hint that increasing clustering impedes cascading failures under both (truncated) power law and Poisson degree distributions. Furthermore, our finding suggests that, as the degree distribution becomes more concentrated around the mean degree with smaller variance, increasing clustering will have greater impact on the PoCF. A numerical investigation of networks with Poisson and power law degree distributions reflects this finding and demonstrates that increasing clustering reduces the PoCF much faster under Poisson degree distributions in comparison to power law degree distributions.

On the Robustness of Complex Systems With Multipartitivity Structures Under Node Attacks

Complex systems in the real world inevitably suffer from unpredictable perturbations, which can trigger system disasters, wreaking significant economical losses. To exploit the robustness of complex systems in the face of disturbances is of great significance. One of the most useful methods for system robustness analysis comes from the field of complex networks characterized by percolation theories. Many percolation theories, therefore, have been developed by researchers to investigate the robustness of diverse complex networks. Nevertheless, extant percolation theories are primarily devised for multilayer or interdependent networks. Little endeavor is dedicated to systems with multipartitivity structures, that is, multipartite networks, which are an indispensable part of complex networks. This paper fills this research gap by theoretically examining the robustness of multipartite networks under random or target node attacks. The generic percolation theory for robustness analysis of multipartite networks is accordingly put forward. To validate the correctness of the proposed percolation theory, we carry out simulations on computer-generated multipartite networks with Poisson degree distributions. The results yielded by the proposed theory coincide well with the simulations. Both theoretical and simulation results suggest that complex systems with multipartitivity structures could be more robust than those with multilayer structures.

Average hopcount of the shortest path in tree-like components with finite size

An exact formula for computing the average hopcount of the shortest path in finite-size tree-like components of undirected unweighted random networks is proposed. In a tree-like component with size s, there exists virtually only one shortest path between two arbitrary nodes. The summation of hopcounts of all shortest paths can be calculated approximately by the summation of the betweenness of all nodes, and the difference between them is only a constant s(s−1). Therefore, the average hopcount can be calculated by further dividing the summation by the number of all shortest paths. In this paper, we first derive the conditional probability p(k|s) of the degree distribution of finite components with size s and the summation of all nodal betweenness respectively. By means of these results, we obtain the exact formula for calculating the average hopcount. Also, We confirm the proposed formula by simulations for networks with Poisson and power law degree distributions respectively.

Statistical analysis of articulation points in configuration model networks

An articulation point (AP) in a network is a node whose deletion would split the network component on which it resides into two or more components. APs are vulnerable spots that play an important role in network collapse processes, which may result from node failures, attacks or epidemics. Therefore, the abundance and properties of APs affect the resilience of the network to these collapse scenarios. We present analytical results for the statistical properties of APs in configuration model networks. In order to quantify their abundance, we calculate the probability $P(i \in {\rm AP})$, that a random node, i, in a configuration model network with P(K=k), is an AP. We also obtain the conditional probability $P(i \in {\rm AP}|k)$ that a random node of degree k is an AP, and find that high degree nodes are more likely to be APs than low degree nodes. Using Bayes' theorem, we obtain the conditional degree distribution, $P(K=k|{\rm AP})$, over the set of APs and compare it to P(K=k). We propose a new centrality measure based on APs: each node can be characterized by its articulation rank, r, which is the number of components that would be added to the network upon deletion of that node. For nodes which are not APs the articulation rank is $r=0$, while for APs $r \ge 1$. We obtain a closed form expression for the distribution of articulation ranks, P(R=r). Configuration model networks often exhibit a coexistence between a giant component and finite components. To examine the distinct properties of APs on the giant and on the finite components, we calculate the probabilities presented above separately for the giant and the finite components. We apply these results to ensembles of configuration model networks with a Poisson, exponential and power-law degree distributions. The implications of these results are discussed in the context of common attack scenarios and network dismantling processes.

Information filtering by smart nodes in random networks.

Diffusion of information in social networks has drawn extensive attention from various scientific communities, with many contagion models proposed to explain related phenomena. In this paper, we present a simple contagion mechanism, in which a node will change its state immediately if it is exposed to the diffusive information. By considering two types of nodes (smart and normal) and two kinds of information (true and false), we study analytically and numerically how smart nodes influence the spreading of information, which leads to information filtering. We find that for randomly distributed smart nodes, the spreading dynamics over random networks with Poisson degree distribution and power-law degree distribution (with relatively small cutoffs) can both be described by the same approximate mean-field equation. Increasing the heterogeneity of the network may elicit more deviations, but not much. Moreover, we demonstrate that more smart nodes make the filtering effect on a random network better. Finally, we study the efficacy of different strategies of selecting smart nodes for information filtering.

Replica Bounds by Combinatorial Interpolation for Diluted Spin Systems

In two papers Franz, Leone and Toninelli proved bounds for the free energy of diluted random constraints satisfaction problems, for a Poisson degree distribution [5] and a general distribution [6]. Panchenko and Talagrand [16] simplified the proof and generalized the result of [5] for the Poisson case. We provide a new proof for the general degree distribution case and as a corollary, we obtain new bounds for the size of the largest independent set (also known as hard core model) in a large random regular graph. Our proof uses a combinatorial interpolation based on biased random walks [21] and allows to bypass the arguments in [6] based on the study of the Sherrington-Kirkpatrick (SK) model.

Roles of the spreading scope and effectiveness in spreading dynamics on multiplex networks

Comparing with single networks, the multiplex networks bring two main effects on the spreading process among individuals. First, the pathogen or information can be transmitted to more individuals through different layers at one time, which enlarges the spreading scope. Second, through different layers, an individual can also transmit the pathogen or information to the same individuals more than once at one time, which makes the spreading more effective. To understand the different roles of the spreading scope and effectiveness, we propose an epidemic model on multiplex networks with link overlapping, where the spreading effectiveness of each interaction as well as the variety of channels (spreading scope) can be controlled by the number of overlapping links. We find that for Poisson degree distribution, increasing the epidemic scope (the first effect) is more efficient than enhancing epidemic probability (the second effect) to facilitate the spreading process. However, for power-law degree distribution, the effects of the two factors on the spreading dynamics become complicated. Enhancing epidemic probability makes pathogen or rumor easier to outbreak in a finite system. But after that increasing epidemic scopes is still more effective for a wide spreading. Theoretical results along with reasonable explanation for these phenomena are all given in this paper, which indicates that the epidemic scope could play an important role in the spreading dynamics.

The distribution of path lengths of self avoiding walks on Erdős–Rényi networks

We present an analytical and numerical study of the paths of self avoiding walks (SAWs) on random networks. Since these walks do not retrace their paths, they effectively delete the nodes they visit, together with their links, thus pruning the network. The walkers hop between neighboring nodes, until they reach a dead-end node from which they cannot proceed. Focusing on Erdős–Rényi networks we show that the pruned networks maintain a Poisson degree distribution, , with an average degree, , that decreases linearly in time. We enumerate the SAW paths of any given length and find that the number of paths, , increases dramatically as a function of . We also obtain analytical results for the path-length distribution, , of the SAW paths which are actually pursued, starting from a random initial node. It turns out that follows the Gompertz distribution, which means that the termination probability of an SAW path increases with its length.

Increased signaling entropy in cancer requires the scale-free property of protein interaction networks.

One of the key characteristics of cancer cells is an increased phenotypic plasticity, driven by underlying genetic and epigenetic perturbations. However, at a systems-level it is unclear how these perturbations give rise to the observed increased plasticity. Elucidating such systems-level principles is key for an improved understanding of cancer. Recently, it has been shown that signaling entropy, an overall measure of signaling pathway promiscuity, and computable from integrating a sample's gene expression profile with a protein interaction network, correlates with phenotypic plasticity and is increased in cancer compared to normal tissue. Here we develop a computational framework for studying the effects of network perturbations on signaling entropy. We demonstrate that the increased signaling entropy of cancer is driven by two factors: (i) the scale-free (or near scale-free) topology of the interaction network, and (ii) a subtle positive correlation between differential gene expression and node connectivity. Indeed, we show that if protein interaction networks were random graphs, described by Poisson degree distributions, that cancer would generally not exhibit an increased signaling entropy. In summary, this work exposes a deep connection between cancer, signaling entropy and interaction network topology.

Robustness of oscillatory behavior in correlated networks.

Understanding network robustness against failures of network units is useful for preventing large-scale breakdowns and damages in real-world networked systems. The tolerance of networked systems whose functions are maintained by collective dynamical behavior of the network units has recently been analyzed in the framework called dynamical robustness of complex networks. The effect of network structure on the dynamical robustness has been examined with various types of network topology, but the role of network assortativity, or degree–degree correlations, is still unclear. Here we study the dynamical robustness of correlated (assortative and disassortative) networks consisting of diffusively coupled oscillators. Numerical analyses for the correlated networks with Poisson and power-law degree distributions show that network assortativity enhances the dynamical robustness of the oscillator networks but the impact of network disassortativity depends on the detailed network connectivity. Furthermore, we theoretically analyze the dynamical robustness of correlated bimodal networks with two-peak degree distributions and show the positive impact of the network assortativity.

Robustness of network of networks with interdependent and interconnected links

Robustness of network of networks (NON) has been studied only for dependency coupling (Gao et al., 2012) and only for connectivity coupling (Leicht and Souza, arxiv:0907.0894). The case of network of networks with both interdependent and interconnected links is more complicated, and also more close to real-life coupled network systems. Understanding the robustness of NON with interdependent and interconnected links is helpful to design resilient infrastructures. Here we develop a framework to study analytically and numerically the robustness of this system with no-feedback and feedback conditions for the case of starlike NON. When assumed that all degree distributions of the connectivity intra- and inter-links are Poissonian, we find that the system undergoes from second order through hybrid order to first order phase transition as coupling strength q increases. Additionally, for both conditions, the results suggest that increasing density of connectivity links (intra-connectivity links or inter-connectivity links) can increase the robustness of the system, while the interdependency links decrease its robustness. Furthermore, by comparing critical attacking strength under same parameters for both conditions, we also find that feedback condition of dependency links, in contrast to no-feedback condition, makes the system extremely vulnerable. Although our detailed analysis is for Poisson degree distribution, the theory can be applied to any degree distribution and other basic topological structure of network such like tree-like and loop-like NON.

Research on the Public Transport Network Based on Complex Network

The public transport infrastructure of a city is one of the most important indicators of its economic growth and development. Here we investigate the statistical properties of the public transport network in Shenyang to explore its various properties based on complex network theory. The statistical properties of the public transport system consist of the degree of a node, the average shortest path length, the clustering coefficient of a node, the average clustering coefficient, and the degree distribution. In contrast with the small world evolution model, we find that the public transport system of Shenyang, a network of public transportation routes connected by bus links, is a small-world network characterized by a Poisson degree distribution. Simulation results show that the public transport network exhibits small world behavior with N=148.