Undirected Scale-free Networks Research Articles

Random Walks on Networks with Centrality-Based Stochastic Resetting

We introduce a refined way to diffusely explore complex networks with stochastic resetting where the resetting site is derived from node centrality measures. This approach differs from previous ones, since it not only allows the random walker with a certain probability to jump from the current node to a deliberately chosen resetting node, rather it enables the walker to jump to the node that can reach all other nodes faster. Following this strategy, we consider the resetting site to be the geometric center, the node that minimizes the average travel time to all the other nodes. Using the established Markov chain theory, we calculate the Global Mean First Passage Time (GMFPT) to determine the search performance of the random walk with resetting for different resetting node candidates individually. Furthermore, we compare which nodes are better resetting node sites by comparing the GMFPT for each node. We study this approach for different topologies of generic and real-life networks. We show that, for directed networks extracted for real-life relationships, this centrality focused resetting can improve the search to a greater extent than for the generated undirected networks. This resetting to the center advocated here can minimize the average travel time to all other nodes in real networks as well. We also present a relationship between the longest shortest path (the diameter), the average node degree and the GMFPT when the starting node is the center. We show that, for undirected scale-free networks, stochastic resetting is effective only for networks that are extremely sparse with tree-like structures as they have larger diameters and smaller average node degrees. For directed networks, the resetting is beneficial even for networks that have loops. The numerical results are confirmed by analytic solutions. Our study demonstrates that the proposed random walk approach with resetting based on centrality measures reduces the memoryless search time for targets in the examined network topologies.

Dense power-law networks and simplicial complexes.

There is increasing evidence that dense networks occur in on-line social networks, recommendation networks and in the brain. In addition to being dense, these networks are often also scale-free, i.e., their degree distributions follow P(k)∝k^{-γ} with γ∈(1,2]. Models of growing networks have been successfully employed to produce scale-free networks using preferential attachment, however these models can only produce sparse networks as the numbers of links and nodes being added at each time step is constant. Here we present a modeling framework which produces networks that are both dense and scale-free. The mechanism by which the networks grow in this model is based on the Pitman-Yor process. Variations on the model are able to produce undirected scale-free networks with exponent γ=2 or directed networks with power-law out-degree distribution with tunable exponent γ∈(1,2). We also extend the model to that of directed two-dimensional simplicial complexes. Simplicial complexes are generalization of networks that can encode the many body interactions between the parts of a complex system and as such are becoming increasingly popular to characterize different data sets ranging from social interacting systems to the brain. Our model produces dense directed simplicial complexes with power-law distribution of the generalized out-degrees of the nodes.

Degree correlations in directed scale-free networks.

Scale-free networks, in which the distribution of the degrees obeys a power-law, are ubiquitous in the study of complex systems. One basic network property that relates to the structure of the links found is the degree assortativity, which is a measure of the correlation between the degrees of the nodes at the end of the links. Degree correlations are known to affect both the structure of a network and the dynamics of the processes supported thereon, including the resilience to damage, the spread of information and epidemics, and the efficiency of defence mechanisms. Nonetheless, while many studies focus on undirected scale-free networks, the interactions in real-world systems often have a directionality. Here, we investigate the dependence of the degree correlations on the power-law exponents in directed scale-free networks. To perform our study, we consider the problem of building directed networks with a prescribed degree distribution, providing a method for proper generation of power-law-distributed directed degree sequences. Applying this new method, we perform extensive numerical simulations, generating ensembles of directed scale-free networks with exponents between 2 and 3, and measuring ensemble averages of the Pearson correlation coefficients. Our results show that scale-free networks are on average uncorrelated across directed links for three of the four possible degree-degree correlations, namely in-degree to in-degree, in-degree to out-degree, and out-degree to out-degree. However, they exhibit anticorrelation between the number of outgoing connections and the number of incoming ones. The findings are consistent with an entropic origin for the observed disassortativity in biological and technological networks.

Analysis of critical and redundant nodes in controlling directed and undirected complex networks using dominating sets

Recent studies have drawn attention to the problem of how complex networks can be controlled through a small number of controller nodes. Here, we develop an algorithmic procedure and mathematical tools to compute and evaluate the critical and redundant nodes in controlling directed and undirected scale-free networks using the minimum dominating set (MDS) approach. Because there are multiple MDS configurations that control the entire network, we can classify the nodes depending on the condition whether a node is part of all (critical), some but not all (intermittent), or does not participate in any (redundant) possible MDS. The presented mathematical analysis predicts the probability of finding a critical node in undirected scale-free networks with |$k^{-\gamma }$|⁠, where |$k$| is a node degree, as a function of the scaling exponent |$\gamma $| and its node degree. Critical nodes tend to have high degree and are more abundant in undirected scale-free networks with high |$\gamma $|⁠. In addition, analytical expressions of lower bounds for the number of critical nodes for both undirected and directed networks are also derived. By applying the MDS control model, we find that undirected networks can be controlled with relatively fewer nodes than those engaged in controlling directed networks. On the other hand, our computational experiments also show that the MDS is unimodal with varying the average degree |$\langle k \rangle $| in both directed and undirected networks. In particular, by increasing |$\langle k \rangle $|⁠, the fraction of nodes engaged in control becomes smaller, which highlights a centralized control mode. The analysis of a set of undirected and directed real-world networks confirms the findings shown in theoretical analysis and simulation experiments.

Epidemic spreading in annealed directed networks: Susceptible-infected-susceptible model and contact process

We investigate epidemic spreading in annealed directed scale-free networks with the in-degree (k) distribution P(in)(k)~k(-γ(in)) and the out-degree (ℓ) distribution, P(out)(ℓ)~ℓ(-γ(out)). The correlation <kl> of each node on the networks is controlled by the probability r(0≤r≤1) in two different algorithms, the so-called k and ℓ algorithms. For r=1, the k algorithm gives <kl>=<k(2)>, whereas the ℓ algorithm gives <kl>=<ℓ(2)>. For r=0, <kl>=<k><ℓ> for both algorithms. As the prototype of epidemic spreading, the susceptible-infected-susceptible model and contact process on the networks are analyzed using the heterogeneous mean-field theory and Monte Carlo simulations. The directedness of links and the correlation of the network are found to play important roles in the spreading, so that critical behaviors of both models are distinct from those on undirected scale-free networks.

Convergence speed of consensus problems over undirected scale-free networks

Scale-free networks and consensus behaviour among multiple agents have both attracted much attention. To investigate the consensus speed over scale-free networks is the major topic of the present work. A novel method is developed to construct scale-free networks due to their remarkable power-law degree distributions, while preserving the diversity of network topologies. The time cost or iterations for networks to reach a certain level of consensus is discussed, considering the influence from power-law parameters. They are both demonstrated to be reversed power-law functions of the algebraic connectivity, which is viewed as a measurement on convergence speed of the consensus behaviour. The attempts of tuning power-law parameters may speed up the consensus procedure, but it could also make the network less robust over time delay at the same time. Large scale of simulations are supportive to the conclusions.

ROBUSTNESS IN SCALE-FREE NETWORKS: COMPARING DIRECTED AND UNDIRECTED NETWORKS

In this paper, scale-free networks and their functional robustness with respect to structural perturbations of the network are studied. Two types of perturbations are distinguished: random perturbations and attacks. The robustness of directed and undirected scale-free networks is studied numerically for two different measures and the obtained results are compared. For random perturbations, the results indicate that the strength of the perturbation plays a crucial role. In general, directed scale-free networks are more robust than undirected scale-free networks.

Kauffman Boolean model in undirected scale-free networks

We investigate analytically and numerically the critical line in undirected random Boolean networks with arbitrary degree distributions, including the scale-free topology of connections P(k) ~ k(-gamma). We explain that the unattainability of the critical line in numerical simulations of classical random graphs is due to percolation phenomena. We suggest that recent findings of discrepancy between simulations and theory in directed random Boolean networks might have the same reason. We also show that in infinite scale-free networks the transition between frozen and chaotic phases occurs for 3<gamma<3.5 . Since most critical phenomena in scale-free networks reveal their nontrivial character for gamma<3 , the position of the critical line in the Kauffman model seems to be an important exception to the rule.

Scale-free networks generated by random walkers

We present a simple mechanism for generating undirected scale-free networks using random walkers, where the network growth is determined by choosing parent vertices by sequential random walks. We show that this mechanism produces scale-free networks with degree exponent γ=3 and clustering coefficients depending on random walk length. The mechanism can be interpreted in terms of preferential attachment without explicit knowledge of node degrees.