Using Spectral Radius Ratio for Node Degree to Analyze the Evolution of Scale-Free Networks and Small-World Networks

The spectral radius of a network graph is the largest Eigen value of the adjacency matrix of the graph. We hypothesize the spectral radius to be a measure of the variation in the degrees of the nodes. In this pursuit, we define a metric called the spectral radius ratio for node degree as the ratio of the spectral radius to the average node degree. We validate our hypothesis by determining this metric on some of the commonly studied classical large real-world complex network graphs (undirected) for network analysis. Based on the results collected, we observe the spectral radius ratio for node degree to be positively correlated (correlation coefficient: 0.75) to the coefficient of variation in node degree (the ratio of the average node degree to the standard deviation in node degree), thus confirming our hypothesis.

Naming Game with Multiple Hearers

Recently there has been considerable interest in the theoretical and numerical investigation on the reaction segregation phenomena in physics, chemistry, biology, and chemical engineering. It is well known that the binary reaction has been investigated numerically and experimentally on many phenomena. Ovchinnikov and Zeldovich have played an important role to develop a research of segregation phenomenon. The early time behavior for reaction-diffusion process investigated by Taitelbaum et al. shows that both the global reaction rate and the reaction front grow as t at very early times. Cornell et al. studied theoretically the generalized and more complicated nAþ mB ! C reactions under initially separated reactant conditions. Zumofen et al. studied the breakdown phenomena of Ovchinnikov–Zeldovich segregation in the Aþ B ! 0 reaction under Levy mixing which is responsible to anomalous diffusion. They found that the segregation disappear in d 1⁄4 3 dimensions for 0 and 1 < < 2. Yen et al. studied the early time scaling in the ternary reaction-diffusion system with initially separated reactants. On the other hand, the phenomena of small-world and scale-free networks are really shown to be different from the dynamical behavior of the regular lattice system. It is really of fundamental importance to discuss the numerical and analytical result of scale-free network models compared with that of regular and small-world network models. The bimolecular chemical reaction in scale-free networks was studied for the generation of the depletion zone and the segregation of the reactants, and it was found that the reaction-diffusion processes in scale-free networks are different in their nature compared to regular lattice models, due to the small diameter of networks and the existence of hubs. Similarly, Catanzaro et al. have found that the inverse particle density scales linearly as 1= ðtÞ t. From this result, the inverse particle density in an uncorrelated scale-free network is shown to cross over to a linear behavior. Gallos and Argyrakis have also discussed the reaction-diffusion process of two species on the scale-free network between the correlated and the uncorrelated configuration models, and they especially revealed that the two models are identical when 1⁄4 3:0. Our group has studied analytic and numerical model of Aþ Bþ C ! D based on the regular lattice and small world networks, and the global reaction rate has analytically been derived before and after the crossover in the system of Aþ Bþ C ! D. We obtained that the chemical reaction of three species of reactants occurs equivalently in both regular and small-world networks at early time regime, but the decay process on small-world network proceeds slightly faster than in the case of the regular network at long-time regime. The aim of this paper is to examine the result in the annihilation process of three species on Barabasi–Albert (BA) scale-free networks. Classically, the the particle density ðtÞ of the surviving particles in the reactions of Aþ A ! 0 and Aþ B ! 0 type on a d-dimensional regular lattice scales as

In this study, small-world network analysis was performed to identify the similarities and differences between functional brain networks for right- and left-hand motor imageries (MIs). First, Pearson correlation coefficients among the nodes within the functional brain networks from healthy subjects were calculated. Then, small-world network indicators, including the clustering coefficient, the average path length, the global efficiency, the local efficiency, the average node degree, and the small-world index, were generated for the functional brain networks during both right- and left-hand MIs. We identified large differences in the small-world network indicators between the functional networks during MI and in the random networks. More importantly, the functional brain networks underlying the right- and left-hand MIs exhibited similar small-world properties in terms of the clustering coefficient, the average path length, the global efficiency, and the local efficiency. By contrast, the right- and left-hand MI brain networks showed differences in small-world characteristics, including indicators such as the average node degree and the small-world index. Interestingly, our findings also suggested that the differences in the activity intensity and range, the average node degree, and the small-world index of brain networks between the right- and left-hand MIs were associated with the asymmetry of brain functions.

The Extended Directed Friendship Paradox

In order to analyze the topological properties of complex Metro network, this paper takes Shenzhen as an example, choosing 8 metro lines in Shenzhen as of August 2019 and 168 station nodes as sample data, and constructing a complex network model based on adjacent stations using complex network theory. This method takes metro traffic stations as nodes and metro traffic lines between adjacent stations as edges, which makes the network have the topological properties of complex networks. The characteristics of node degree, average degree, agglomeration coefficient, average shortest path and centrality in the network are analyzed. In addition, the importance of these sites is explored by some important nodes deliberately destroying. By calculating, it is found that the nodes of Shenzhen Metro network occupied a large proportion between 14 and 30, which is 84%. The average node degree of Shenzhen metro network is 29.6, and the average node degree is between 14 and 30. It shows that Shenzhen Metro has serious traffic connectivity. After deleting the important seven nodes, the average node degree decreases by 4, the average shortest path increases by 0.3, and the network diameter or clustering coefficient changes relatively small, showing that these seven nodes play an important role in the transfer trip and can effectively reduce the average transfer times. These laws provide a new reference for optimizing Shenzhen metro traffic network and traffic planning development.

Collective decision making is crucial in human organizations and societies. When a collective is working on exploration of problem space and/or ideation for creative solutions, the evolutionary perspective is useful for conceptualizing and modeling collective decision making, where populations of ideas spread and evolve on a social network habitat via continual applications of evolutionary operators by human individuals (Sayama & Dionne 2015). Using an evolutionary approach to model collective decision making, we conducted agent-based simulations to investigate how collective decision making would be affected by the size and topology of social network structure (Sayama, Dionne & Yammarino 2013). In our model, each agent has its own utility function defined over a multi-dimensional problem space, which is marginally different from the “true” utility function that is not accessible from any agent. Each agent can memorize multiple ideas in mind, and iteratively applies evolutionary operators (e.g., replication, mutation, recombination, elimination) to the idea population it has. The outcomes of evolutionary operations are stored in the agent’s mind, and also sent to the neighbors to which the agent is connected. This allows the spread of ideas through social ties. Each simulation was run for a fixed number of iterations, and then the level of idea convergence at a social level and the true utility value of the most supported idea were measured as the performance metrics of collective decision making. The former metric characterizes the ability for the society to form consensus, while the latter characterizes its ability to find the true best solution. The size of the network (number of agents or nodes) was varied from 5 to 640 logarithmically. Three network topologies with the same average node degree were tested: Random, small-world and scale-free. For more details of the model and the simulation settings, see Sayama, Dionne & Yammarino (2013). Simulation results indicated that expanding the size of the social network generally improved the quality of ideas at the cost of decision convergence. Simulations with different social network topologies further revealed that collective decision making on small-world networks with high local clustering tended to achieve highest decision quality more often than on random or scale-free networks (Fig. 1). This can be understood in that local clustering helps agents in different regions in a network maintain their respective focus areas and engage in different local search, possibly enhancing the effective parallelism of collective decision making and therefore resulting in a greater number of successful decisions. In contrast, the links in random and scale-free networks are all “global”, mixing discussions prematurely and therefore reducing the effective parallelism of collective decision making. These observations have an interesting contrast with the fact that random and scale-free networks are highly efficient in information dissemination because of their global connectedness. Our results indicate that such efficiency of information dissemination may not necessarily imply the same for collective decision making. This work may also offer an evolutionary explanation for the nontrivial relationship between network clustering and problem solving, a problem that is actively debated in organizational science (Shore, Bernstein & Lazer 2014).

Collective decision making is crucial in human organizations and societies. When a collective is working on exploration of problem space and/or ideation for creative solutions, the evolutionary perspective is useful for conceptualizing and modeling collective decision making, where populations of ideas spread and evolve on a social network habitat via continual applications of evolutionary operators by human individuals (Sayama & Dionne 2015). Using an evolutionary approach to model collective decision making, we conducted agent-based simulations to investigate how collective decision making would be affected by the size and topology of social network structure (Sayama, Dionne & Yammarino 2013). In our model, each agent has its own utility function defined over a multi-dimensional problem space, which is marginally different from the “true” utility function that is not accessible from any agent. Each agent can memorize multiple ideas in mind, and iteratively applies evolutionary operators (e.g., replication, mutation, recombination, elimination) to the idea population it has. The outcomes of evolutionary operations are stored in the agent’s mind, and also sent to the neighbors to which the agent is connected. This allows the spread of ideas through social ties. Each simulation was run for a fixed number of iterations, and then the level of idea convergence at a social level and the true utility value of the most supported idea were measured as the performance metrics of collective decision making. The former metric characterizes the ability for the society to form consensus, while the latter characterizes its ability to find the true best solution. The size of the network (number of agents or nodes) was varied from 5 to 640 logarithmically. Three network topologies with the same average node degree were tested: Random, small-world and scale-free. For more details of the model and the simulation settings, see Sayama, Dionne & Yammarino (2013). Simulation results indicated that expanding the size of the social network generally improved the quality of ideas at the cost of decision convergence. Simulations with different social network topologies further revealed that collective decision making on small-world networks with high local clustering tended to achieve highest decision quality more often than on random or scale-free networks (Fig. 1). This can be understood in that local clustering helps agents in different regions in a network maintain their respective focus areas and engage in different local search, possibly enhancing the effective parallelism of collective decision making and therefore resulting in a greater number of successful decisions. In contrast, the links in random and scale-free networks are all “global”, mixing discussions prematurely and therefore reducing the effective parallelism of collective decision making. These observations have an interesting contrast with the fact that random and scale-free networks are highly efficient in information dissemination because of their global connectedness. Our results indicate that such efficiency of information dissemination may not necessarily imply the same for collective decision making. This work may also offer an evolutionary explanation for the nontrivial relationship between network clustering and problem solving, a problem that is actively debated in organizational science (Shore, Bernstein & Lazer 2014).

Based on an Ising model proposed by Harras and Sornette, we established an artificial stock market model to describe the interactions among diverse agents. We regard these participants as network nodes and link them with their correlation. Then, we analyze the financial market based on the social network of market participants. We take the random network, scale-free network, and small-world network into consideration, and then build the stock market evolution model according to the characteristics of the investors’ trading behavior under the different network systems. This allows us to macroscopically study the effects of herd behavior on the rate of stock return and price volatility under different network structures . The numerical simulation results show that herd behavior will lead to excessive market volatility. Specifically, the greater the degree of investor’s trust in neighbors and their exchange, the greater the volatility of stock price will be. With different network synchronization capabilities, price fluctuations based on the small-world network are larger than those based on the regular network. Similarly, price fluctuations based on the random network are larger than those based on the small-world network. On the other hand, price fluctuations based on both the random network and the small-world network firstly increase and then decrease with the increase of the average node degree. All of these results illustrate the network topology that has an important impact on the stock market’s price behavior.

The small world network model of the topology structure of the wireless sensor network and analyzes its complex characteristics from the perspective of network science theory is proposed in this paper. Topology structure is the first step for designing and constructing wireless sensor network. The life-time of the whole network can be prolong by a desirable topology. The measurement of wireless sensor network is analyzed in the research. The results show that, in the mesh network, the node degree is uniformly distributed, it has a relatively small average path length and higher cluster coefficient. A comparison with other types of networks, wireless sensor networks is not a regular network, the network is not completely random. It is the random network between the small-world networks, has a certain property of a similar small-world networks. In order to reduce the hops for the network, this paper construct a small-world wireless sensor network by adding some shortcuts to the network, which is subject to the distance constraints between individual nodes. Simulation results shoe that these shortcues will remarkably reduce the average path length. This will improve the energy efficiency of the network.

The small world network model of the topology structure of the wireless sensor network and analyzes its complex characteristics from the perspective of network science theory is proposed in this paper. Topology structure is the first step for designing and constructing wireless sensor network. The life-time of the whole network can be prolong by a desirable topology. The measurement of wireless sensor network is analyzed in the research. The results show that, in the mesh network, the node degree is uniformly distributed, it has a relatively small average path length and higher cluster coefficient. A comparison with other types of networks, wireless sensor networks is not a regular network, the network is not completely random. It is the random network between the small-world networks, has a certain property of a similar small-world networks. In order to reduce the hops for the network, this paper construct a small-world wireless sensor network by adding some shortcuts to the network, which is subject to the distance constraints between individual nodes. Simulation results shoe that these shortcues will remarkably reduce the average path length. This will improve the energy efficiency of the network.

Our primary objective in this paper is to study the distribution of the maximal clique size of the vertices in complex networks. We define the maximal clique size for a vertex as the maximum size of the clique that the vertex is part of and such a clique need not be the maximum size clique for the entire network. We determine the maximal clique size of the vertices using a modified version of a branch-and-bound based exact algorithm that has been originally proposed to determine the maximum size clique for an entire network graph. We then run this algorithm on two categories of complex networks: One category of networks capture the evolution of small-world networks from regular network (according to the wellknown Watts-Strogatz model) and their subsequent evolution to random networks; we show that the distribution of the maximal clique size of the vertices follows a Poisson-style distribution at different stages of the evolution of the small-world network to a random network; on the other hand, the maximal clique size of the vertices is observed to be in-variant and to be very close to that of the maximum clique size for the entire network graph as the regular network is transformed to a small-world network. The second category of complex networks studied are real-world networks (ranging from random networks to scale-free networks) and we observe the maximal clique size of the vertices in five of the six real-world networks to follow a Poisson-style distribution. In addition to the above case studies, we also analyze the correlation between the maximal clique size and clustering coefficient as well as analyze the assortativity index of the vertices with respect to maximal clique size and node degree.

Research on low-carbon diffusion considering the game among enterprises in the complex network context

Scale-free networks are a type of complex networks in which the degree distribution of the nodes is according to the power-law. Centrality of the nodes is a quantitative measure of the importance of the nodes according to the topological structure of the network. The commonly used centrality measures are the degree-based degree centrality and eigenvector centrality and the shortest path-based closeness centrality and betweenness centrality. We use the widely studied Barabasi-Albert (BA) model to simulate the evolution of scale-free networks. The model works by adding new nodes to the network, one at a time, with the new node connected to m of the currently existing nodes. Accordingly, nodes that have been in the network for a longer time have greater chances of acquiring more links and hence a larger degree centrality. While the degree centrality of the nodes has been observed to show a concave down pattern of increase with time; but the time-dependent variation of the other centrality measures has not been analyzed until now. In this paper, we study the time-dependent variation of degree centrality, eigenvector centrality, closeness centrality and betweenness centrality of the nodes during the evolution of a scale-free network according to the BA model

This article deals with complex networks, and in particular small world and scale free networks. Various networks exhibit the small world phenomenon, including social networks and gene expression networks. The local ordering property of small world networks is typically associated with regular networks such as a 2D square lattice. The small world phenomenon can be observed in most scale free networks, but few small world networks are scale free. The article first provides a brief background on small world networks and two models of scale free graphs before describing the replica method and how it can be applied to calculate the spectral densities of the adjacency matrix and Laplacian matrix of a scale free network. It then shows how the effective medium approximation can be used to treat networks with finite mean degree and concludes with a discussion of the local properties of random matrices associated with complex networks.