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This paper discusses the reliability of a graph in which the links are perfectly reliable but the nodes may fail with certain probability p. Calculating graph node reliability is an NP-Hard problem. We introduce an efficient and accurate Monte Carlo method and a stochastic approximation for the node reliability polynomial based solely on the degree distribution. We provide the formulas for the node reliability polynomial of both Erdos-Renyi graphs and Random Geometric graphs. The phase transition in the node reliability of Erdos-Renyi graphs is also discussed. Additionally, we propose two increasingly accurate upper bounds for the node reliability polynomial solely based on the graph's degree distributions. The advantages and disadvantages of these two upper bounds are thoroughly compared. Beyond the computation of node reliability polynomials, we also estimate the number of cut sets and present a solution to the reliability-based network enhancement problem.

We study human mobility networks through timeseries of contacts between individuals. Our proposed Random Walkers Induced temporal Graph (RWIG) model generates temporal graph sequences based on independent random walkers that traverse an underlying graph in discrete time steps. Co-location of walkers at a given node and time defines an individual-level contact. RWIG is shown to be a realistic model for temporal human contact graphs, which may place RWIG on a same footing as the Erdos-Renyi (ER) and Barabasi-Albert (BA) models for fixed graphs. Moreover, RWIG is analytically feasible: we derive closed form solutions for the probability distribution of contact graphs.

Based on data sets provided by Statistics Netherlands and the International Institute of Social History, we investigate the Dutch municipality merging process and the survivability of municipalities over the period 1830-2019. We examine the dynamics of the population and area per municipality and how their distributions evolved during the researched period. We apply a Network Science approach, where each node represents a municipality and the links represent the geographical interconnections between adjacent municipalities via roads, railways, bridges or tunnels which were available in each specific yearly network instance. Over the researched period, we find that the distributions of the logarithm of both the population and area size closely follow a normal and a logistic distribution respectively. The tails of the population distributions follow a power-law distribution, a phenomenon observed in community structures of many real-world networks. The dynamics of the area distribution are mainly determined by the merging process, while the population distribution is also driven by the natural population growth and migration across the municipality network. Finally, we propose a model of the Dutch Municipality Network that captures population increase, population migration between municipalities and the process of municipality merging. Our model allows for predictions of the population and area distributions over time., Comment: 48 pages, 26 figures

We extend the N-Intertwined Mean-Field Approximation (NIMFA) for the Susceptible-Infectious-Susceptible (SIS) epidemiological process to time-varying networks. Processes on time-varying networks are often analysed under the assumption that the process and network evolution happen on different timescales. This approximation is called timescale separation. We investigate timescale separation between disease spreading and topology updates of the network. We introduce the transition times $\mathrm{\underline{T}}(r)$ and $\mathrm{\overline{T}}(r)$ as the boundaries between the intermediate regime and the annealed (fast changing network) and quenched (static network) regimes, respectively, for a fixed accuracy tolerance $r$. By analysing the convergence of static NIMFA processes, we analytically derive upper and lower bounds for $\mathrm{\overline{T}}(r)$. Our results provide insights/bounds on the time of convergence to the steady state of the static NIMFA SIS process. We show that, under our assumptions, the upper-transition time $\mathrm{\overline{T}}(r)$ is almost entirely determined by the basic reproduction number $R_0$ of the network. The value of the upper-transition time $\mathrm{\overline{T}}(r)$ around the epidemic threshold is large, which agrees with the current understanding that some real-world epidemics cannot be approximated with the aforementioned timescale separation., Comment: 20 pages, 13 figures

Epidemic forecasts are only as good as the accuracy of epidemic measurements. Is epidemic data, particularly COVID-19 epidemic data, clean and devoid of noise? Common sense implies the negative answer. While we cannot evaluate the cleanliness of the COVID-19 epidemic data in a holistic fashion, we can assess the data for the presence of reporting delays. In our work, through the analysis of the first COVID-19 wave, we find substantial reporting delays in the published epidemic data. Motivated by the desire to enhance epidemic forecasts, we develop a statistical framework to detect, uncover, and remove reporting delays in the infectious, recovered, and deceased epidemic time series. Our framework can uncover and analyze reporting delays in 8 regions significantly affected by the first COVID-19 wave. Further, we demonstrate that removing reporting delays from epidemic data using our statistical framework may decrease the error in epidemic forecasts. While our statistical framework can be used in combination with any epidemic forecast method that intakes infectious, recovered, and deceased data, to make a basic assessment, we employed the classical SIRD epidemic model. Our results indicate that the removal of reporting delays from the epidemic data may decrease the forecast error by up to 50. We anticipate that our framework will be indispensable in the analysis of novel COVID-19 strains and other existing or novel infectious diseases., Comment: 10 pages, 4 figures

Interpreting natural language is an increasingly important task in computer algorithms due to the growing availability of unstructured textual data. Natural Language Processing (NLP) applications rely on semantic networks for structured knowledge representation. The fundamental properties of semantic networks must be taken into account when designing NLP algorithms, yet they remain to be structurally investigated. We study the properties of semantic networks from ConceptNet, defined by 7 semantic relations from 11 different languages. We find that semantic networks have universal basic properties: they are sparse, highly clustered, and many exhibit power-law degree distributions. Our findings show that the majority of the considered networks are scale-free. Some networks exhibit language-specific properties determined by grammatical rules, for example networks from highly inflected languages, such as e.g. Latin, German, French and Spanish, show peaks in the degree distribution that deviate from a power law. We find that depending on the semantic relation type and the language, the link formation in semantic networks is guided by different principles. In some networks the connections are similarity-based, while in others the connections are more complementarity-based. Finally, we demonstrate how knowledge of similarity and complementarity in semantic networks can improve NLP algorithms in missing link inference.

We propose a linear clustering process on a network consisting of two opposite forces: attraction and repulsion between adjacent nodes. Each node is mapped to a position on a one-dimensional line. The attraction and repulsion forces move the nodal position on the line, depending on how similar or different the neighbourhoods of two adjacent nodes are. Based on each node position, the number of clusters in a network, together with each node's cluster membership, is estimated. The performance of the proposed linear clustering process is benchmarked on synthetic networks against widely accepted clustering algorithms such as modularity, the Louvain method and the non-back tracking matrix. The proposed linear clustering process outperforms the most popular modularity-based methods, such as the Louvain method, while possessing a comparable computational complexity.

The $k$-th power of the adjacency matrix of a simple undirected graph represents the number of walks with length $k$ between pairs of nodes. As a walk where no node repeats, a path is a walk where each node is only visited once. The set of paths constitutes a relatively small subset of all possible walks. We introduce three types of walks, representing subsets of all possible walks. Considered types of walks allow for deriving an analytic solution for the number of paths of a certain length between node pairs in a matrix form. Depending on the path length, different approaches possess the lowest computational complexity. We also propose a recursive algorithm for determining all paths in a graph, which can be generalised to directed (un)weighted networks.

Despite many studies on the transmission mechanism of the Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), it remains still challenging to efficiently reduce mortality. In this work, we apply a two-population Susceptible-Infected-Removed (SIR) model to investigate the COVID-19 spreading when contacts between elderly and non-elderly individuals are reduced due to the high mortality risk of elderly people. We discover that the reduction of connections between two populations can delay the death curve but cannot well reduce the final mortality. We propose a merged SIR model, which advises elderly individuals to interact less with their non-elderly connections at the initial stage but interact more with their non-elderly relationships later, to reduce the final mortality. Finally, immunizing elderly hub individuals can also significantly decrease mortality., Comment: 16 pages, 12 figures

Infectious diseases typically spread over a contact network with millions of individuals, whose sheer size is a tremendous challenge to analysing and controlling an epidemic outbreak. For some contact networks, it is possible to group individuals into clusters. A high-level description of the epidemic between a few clusters is considerably simpler than on an individual level. However, to cluster individuals, most studies rely on equitable partitions, a rather restrictive structural property of the contact network. In this work, we focus on Susceptible-Infected-Susceptible (SIS) epidemics, and our contribution is threefold. First, we propose a geometric approach to specify all networks for which an epidemic outbreak simplifies to the interaction of only a few clusters. Second, for the complete graph and any initial viral state vectors, we derive the closed-form solution of the nonlinear differential equations of the N-Intertwined Mean-Field Approximation (NIMFA) of the SIS process. Third, by relaxing the notion of equitable partitions, we derive low-complexity approximations and bounds for epidemics on arbitrary contact networks. Our results are an important step towards understanding and controlling epidemics on large networks.

Initially emerged in the Chinese city Wuhan and subsequently spread almost worldwide causing a pandemic, the SARS-CoV-2 virus follows reasonably well the SIR (Susceptible-Infectious-Recovered) epidemic model on contact networks in the Chinese case. In this paper, we investigate the prediction accuracy of the SIR model on networks also for Italy. Specifically, the Italian regions are a metapopulation represented by network nodes and the network links are the interactions between those regions. Then, we modify the network-based SIR model in order to take into account the different lockdown measures adopted by the Italian Government in the various phases of the spreading of the COVID-19. Our results indicate that the network-based model better predicts the daily cumulative infected individuals when time-varying lockdown protocols are incorporated in the classical SIR model.

Several exact expansions as well as lower and upperbounds of the Kermack and McKendrick SIR equations are presented.

Currently, many countries are considering the introduction of tracing software on mobile smartphones with the main purpose to inform and alarm the mobile app user. Here, we demonstrate that, in addition to alarming and informing, mobile tracing can detect nearly all users that are infected by SARS-CoV-2. Our algorithm BETIS (Bayesian Estimation for Tracing Infection States) makes use of self-reports of the user's health status. Then, BETIS guarantees that almost all SARS-CoV-2 infections of the group of users can be detected. Furthermore, BETIS estimates the virus prevalence in the whole population, consisting of users and non-users. BETIS is based on a hidden Markov epidemic model and recursive Bayesian filtering. The potential that mobile tracing apps, in addition to medical testing and quarantining, can eradicate COVID-19 may persuade citizens to trade-off privacy against public health.

Processes on networks consist of two interdependent parts: the network topology, consisting of the links between nodes, and the dynamics, specified by some governing equations. This work considers the prediction of the future dynamics on an unknown network, based on past observations of the dynamics. For a general class of governing equations, we propose a prediction algorithm which infers the network as an intermediate step. Inferring the network is impossible in practice, due to a dramatically ill-conditioned linear system. Surprisingly, a highly accurate prediction of the dynamics is possible nonetheless: Even though the inferred network has no topological similarity with the true network, both networks result in practically the same future dynamics.

We review the function theoretical properties of the Mittag-Leffler function $E_{a,b}\left( z\right) $ in a self-contained manner, but also add new results; more than half is new!, Comment: Some errors are corrected and citation added

At the moment of writing (12 February, 2020), the future evolution of the 2019-nCoV virus is unclear. Predictions of the further course of the epidemic are decisive to deploy targeted disease control measures. We consider a network-based model to describe the 2019-nCoV epidemic in the Hubei province. The network is composed of the cities in Hubei and their interactions (e.g., traffic flow). However, the precise interactions between cities is unknown and must be inferred from observing the epidemic. We propose a network-based method to predict the future prevalence of the 2019-nCoV virus in every city. Our results indicate that network-based modelling is beneficial for an accurate forecast of the epidemic outbreak., Comment: Ordering of cities in Table 1 has been corrected; Prediction considers more recent observations

The symmetric nonnegative inverse eigenvalue problem (SNIEP) asks which sets of numbers (counting multiplicities) can be the eigenvalues of a symmetric matrix with nonnegative entries. While examples of such matrices are abundant in linear algebra and various applications, this question is still open for matrices of dimension $N\geq 5$. One of the approaches to solve the SNIEP was proposed by George W. Soules, relying on a specific type of eigenvectors (Soules vectors) to derive sufficient conditions for this problem. Elsner et al. later showed a canonical way to construct all Soules vectors, based on binary rooted trees. While Soules vectors are typically treated as a totally ordered set of vectors, we propose in this article to consider a relaxed alternative: a partially ordered set of Soules vectors. We show that this perspective enables a more complete characterization of the sufficient conditions for the SNIEP. In particular, we show that the set of eigenvalues that satisfy these sufficient conditions is a convex cone, with symmetries corresponding to the automorphisms of the binary rooted tree from which the Soules vectors were constructed. As a second application, we show how Soules vectors can be used to construct graph Laplacian matrices with a given spectrum and describe a number of interesting connections with the concepts of hierarchical random graphs, equitable partitions and effective resistance.

We study the electrodynamic impedance of percolating conductors with a pre-defined network topology using a scanning microwave impedance microscope (sMIM) at GHz frequencies. For a given percolation number we observe strong spatial variations across a sample which correlate with the connected regions (clusters) in the network when the resistivity is low such as in Aluminum. For the more resistive material NbTiN the impedance becomes dominated by the local structure of the percolating network (connectivity). The results can qualitatively be understood and reproduced with a network current spreading model based on the pseudo-inverse Laplacian of the underlying network graph., Comment: Main text (4 pages, 4 figures) + Supp. Info. (14 pages, 13 figures)

In networked spreading models, each node can infect its neighbors and cure spontaneously. The curing is assumed to occur uniformly over time. A pulse immunization/curing strategy is more efficient and broadly applied to suppressing spreading processes. We model the epidemic process by the basic Susceptible-Infected (SI) process with a pulse curing and incorporate the underlying contact network. The mean-field epidemic threshold of the pulse SI model is shown to be $\frac{1}{\lambda_1}\ln\frac{1}{1-p}$, where $\lambda_1$ and $p$ are the largest eigenvalue of the adjacency matrix and the fraction of nodes covered by each curing, respectively. Compared to the extensively studied uniform curing process, we show that the pulse curing strategy saves about $36.8$\%, i.e. $p\approx 0.632$, of the number of curing operations invariant to the network structure. Our results may help related policy makers to estimate the cost of controlling spreading processes., Comment: 8 pages, 1 figure

The majority of research on epidemics relies on models which are formulated in continuous-time. However, real-world epidemic data is gathered and processed in a digital manner, which is more accurately described by discrete-time epidemic models. We analyse the discrete-time NIMFA epidemic model on directed networks with heterogeneous spreading parameters. In particular, we show that the viral state is increasing and does not overshoot the steady-state, the steady-state is globally exponentially stable, and we provide linear systems that bound the viral state evolution. Thus, the discrete-time NIMFA model succeeds to capture the qualitative behaviour of a viral spread and provides a powerful means to study real-world epidemics.

Predicting the viral dynamics of an epidemic process requires the knowledge of the underlying contact network. However, the network is not known for most applications and has to be inferred from observing the viral state evolution instead. We propose a polynomial-time network reconstruction algorithm for the discrete-time NIMFA model based on a basis pursuit formulation. Given only few initial viral state observations, the network reconstruction method allows for an accurate prediction of the further viral state evolution of every node provided that the network is sufficiently sparse.

To shed light on the disease localization phenomenon, we study a bursty susceptible-infected-susceptible (SIS) model and analyze the model under the mean-field approximation. In the bursty SIS model, the infected nodes infect all their neighbors periodically, and the near-threshold steady-state prevalence is non-constant and maximized by a factor equal to the largest eigenvalue $\lambda_1$ of the adjacency matrix of the network. We show that the maximum near-threshold prevalence of the bursty SIS process on a localized network tends to zero even if $\lambda_1$ diverges in the thermodynamic limit, which indicates that the burst of infection cannot turn a localized spreading into a delocalized spreading. Our result is evaluated both on synthetic and real networks., Comment: 11 pages (6 pages for main text and 5 pages for appendix), 9 figures (2 figures in main text and 7 figures in appendix) and 1 table in appendix; Modifications made in the main text: simulations added (Sec. 4.2; In Fig.2(b), the original networks with $\gamma=3$ in Version 1 around 10^2 and 10^3 have lost and this two networks are regenerated), references added. No changes in the appendix

Networks are often made up of several layers that exhibit diverse degrees of interdependencies. A multilayer interdependent network consists of a set of graphs $G$ that are interconnected through a weighted interconnection matrix $ B $, where the weight of each inter-graph link is a non-negative real number $ p $. Various dynamical processes, such as synchronization, cascading failures in power grids, and diffusion processes, are described by the Laplacian matrix $ Q $ characterizing the whole system. For the case in which the multilayer graph is a multiplex, where the number of nodes in each layer is the same and the interconnection matrix $ B=pI $, being $ I $ the identity matrix, it has been shown that there exists a structural transition at some critical coupling, $ p^* $. This transition is such that dynamical processes are separated into two regimes: if $ p > p^* $, the network acts as a whole; whereas when $ p

The knowledge of the network topology is imperative to precisely describing the viral dynamics of an SIS epidemic process. In scenarios for which the network topology is unknown, one resorts to reconstructing the network from observing the viral state trace. This work focusses on the impact of the viral state observations on the computational complexity of the resulting network reconstruction problem. We propose a novel method of constructing a specific class of viral state traces from which the inference of the presence or absence of links is either easy or difficult. In particular, we use this construction to prove that the maximum-likelihood SIS network reconstruction is NP-hard. The NP-hardness holds for any adjacency matrix of a graph which is connected.

Graphs are a central object of study in various scientific fields, such as discrete mathematics, theoretical computer science and network science. These graphs are typically studied using combinatorial, algebraic or probabilistic methods, each of which highlights the properties of graphs in a unique way. Here, we discuss a novel approach to study graphs: the simplex geometry (a simplex is a generalized triangle). This perspective, proposed by Miroslav Fiedler, introduces techniques from (simplex) geometry into the field of graph theory and conversely, via an exact correspondence. We introduce this graph-simplex correspondence, identify a number of basic connections between graph characteristics and simplex properties, and suggest some applications as example.

Designing an optimal network topology while balancing multiple, possibly conflicting objectives like cost, performance, and resiliency to viruses is a challenging endeavor, let alone in the case of decentralized network formation. We therefore propose a game-formation technique where each player aims to minimize its cost in installing links, the probability of being infected by a virus and the sum of hopcounts on its shortest paths to all other nodes. In this article, we (1) determine the Nash Equilibria and the Price of Anarchy for our novel network formation game, (2) demonstrate that the Price of Anarchy (PoA) is usually low, which suggests that (near-)optimal topologies can be formed in a decentralized way, and (3) give suggestions for practitioners for those cases where the PoA is high and some centralized control/incentives are advisable., Comment: accepted for publication in IEEE Transactions on Control of Network Systems

We propose a new approximation framework that unifies and generalizes a number of existing mean-field approximation methods for the SIS epidemic model on complex networks. We derive the framework, which we call the Universal Mean-Field Framework (UMFF), as a set of approximations of the exact Markovian SIS equations. Our main novelty is that we describe the mean-field approximations from the perspective of the isoperimetric problem, an insight which results in bounds on the UMFF approximation error. These new bounds provide insight in the accuracy of existing mean-field methods, such as the widely-used N-Intertwined Mean-Field Approximation (NIMFA) and Heterogeneous Mean-Field method (HMF). Additionally, the geometric perspective of the isoperimetric problem enables the UMFF approximation accuracy to be related to the regularity notions of Szemer\'edi's regularity lemma, which yields a prediction about the behavior of the SIS process on large graphs.

Epidemic models are increasingly used in real-world networks to understand diffusion phenomena (such as the spread of diseases, emotions, innovations, failures) or the transport of information (such as news, memes in social on-line networks). A new analysis of the prevalence, the expected number of infected nodes in a network, is presented and physically interpreted. The analysis method is based on spectral decomposition and leads to a universal, analytic curve, that can bound the time-varying prevalence in any finite time interval. Moreover, that universal curve also applies to various types of Susceptible-Infected-Susceptible (SIS) (and Susceptible-Infected-Removed (SIR)) infection processes, with both homogenous and heterogeneous infection characteristics (curing and infection rates), in temporal and even disconnected graphs and in SIS processes with and without self-infections. The accuracy of the universal curve is comparable to that of well-established mean-field approximations.

An accurate approximate formula of the die-out probability in a SIS epidemic process on a network is proposed. The formula contains only three essential parameters: the largest eigenvalue of the adjacency matrix of the network, the effective infection rate of the virus, and the initial number of infected nodes in the network. The die-out probability formula is compared with the exact die-out probability in complete graphs, Erd\H{o}s-R\'enyi graphs, and a power-law graph. Furthermore, as an example, the formula is applied to the $N$-Intertwined Mean-Field Approximation, to explicitly incorporate the die-out., Comment: Version2: 10 figures, 11 pagers. Corrected typos; simulation results of ER graphs and a power-law graph are added. Accepted by the 5th International Workshop on Complex Networks and their Applications, November 30 - December 02, 2016, Milan, Italy

In this paper, we are analyzing the interactivity time, defined as the duration between two consecutive tasks such as sending emails, collecting friends and followers and writing comments in online social networks (OSNs). The distributions of these times are heavy tailed and often described by a power-law distribution. However, power-law distributions usually only fit the heavy tail of empirical data and ignore the information in the smaller value range. Here, we argue that the durations between writing emails or comments, adding friends and receiving followers are likely to follow a lognormal distribution. We discuss the similarities between power-law and lognormal distributions, show that binning of data can deform a lognormal to a power-law distribution and propose an explanation for the appearance of lognormal interactivity times. The historical debate of similarities between lognormal and power-law distributions is reviewed by illustrating the resemblance of measurements in this paper with the historical problem of income and city size distributions.

The design of an efficient curing policy, able to stem an epidemic process at an affordable cost, has to account for the structure of the population contact network supporting the contagious process. Thus, we tackle the problem of allocating recovery resources among the population, at the lowest cost possible to prevent the epidemic from persisting indefinitely in the network. Specifically, we analyze a susceptible-infected-susceptible epidemic process spreading over a weighted graph, by means of a first-order mean-field approximation. First, we describe the influence of the contact network on the dynamics of the epidemics among a heterogeneous population, that is possibly divided into communities. For the case of a community network, our investigation relies on the graph-theoretical notion of equitable partition; we show that the epidemic threshold, a key measure of the network robustness against epidemic spreading, can be determined using a lower-dimensional dynamical system. Exploiting the computation of the epidemic threshold, we determine a cost-optimal curing policy by solving a convex minimization problem, which possesses a reduced dimension in the case of a community network. Lastly, we consider a two-level optimal curing problem, for which an algorithm is designed with a polynomial time complexity in the network size., Comment: to be published on Journal of Complex Networks

Cascading failures are one of the main reasons for blackouts in electrical power grids. Stable power supply requires a robust design of the power grid topology. Currently, the impact of the grid structure on the grid robustness is mainly assessed by purely topological metrics, that fail to capture the fundamental properties of the electrical power grids such as power flow allocation according to Kirchhoff's laws. This paper deploys the effective graph resistance as a metric to relate the topology of a grid to its robustness against cascading failures. Specifically, the effective graph resistance is deployed as a metric for network expansions (by means of transmission line additions) of an existing power grid. Four strategies based on network properties are investigated to optimize the effective graph resistance, accordingly to improve the robustness, of a given power grid at a low computational complexity. Experimental results suggest the existence of Braess's paradox in power grids: bringing an additional line into the system occasionally results in decrease of the grid robustness. This paper further investigates the impact of the topology on the Braess's paradox, and identifies specific sub-structures whose existence results in Braess's paradox. Careful assessment of the design and expansion choices of grid topologies incorporating the insights provided by this paper optimizes the robustness of a power grid, while avoiding the Braess's paradox in the system., Comment: 7 pages, 13 figures conference

Interconnected networks are mathematical representation of systems where two or more simple networks are coupled to each other. Depending on the coupling weight between the two components, the interconnected network can function in two regimes: one where the two networks are structurally distinguishable, and one where they are not. The coupling threshold--denoting this structural transition--is one of the most crucial concepts in interconnected networks. Yet, current information about the coupling threshold is limited. This letter presents an analytical expression for the exact value of the coupling threshold and outlines network interrelation implications.

In recent decades, a number of centrality metrics describing network properties of nodes have been proposed to rank the importance of nodes. In order to understand the correlations between centrality metrics and to approximate a high-complexity centrality metric by a strongly correlated low-complexity metric, we first study the correlation between centrality metrics in terms of their Pearson correlation coefficient and their similarity in ranking of nodes. In addition to considering the widely used centrality metrics, we introduce a new centrality measure, the degree mass. The m order degree mass of a node is the sum of the weighted degree of the node and its neighbors no further than m hops away. We find that the B_{n}, the closeness, and the components of x_{1} are strongly correlated with the degree, the 1st-order degree mass and the 2nd-order degree mass, respectively, in both network models and real-world networks. We then theoretically prove that the Pearson correlation coefficient between x_{1} and the 2nd-order degree mass is larger than that between x_{1} and a lower order degree mass. Finally, we investigate the effect of the inflexible antagonists selected based on different centrality metrics in helping one opinion to compete with another in the inflexible antagonists opinion model. Interestingly, we find that selecting the inflexible antagonists based on the leverage, the B_{n}, or the degree is more effective in opinion-competition than using other centrality metrics in all types of networks. This observation is supported by our previous observations, i.e., that there is a strong linear correlation between the degree and the B_{n}, as well as a high centrality similarity between the leverage and the degree., Comment: 20 pages