Betweenness Centrality of Fractal and Non-Fractal Scale-Free Model Networks and Tests on Real Networks

We study the betweenness centrality of fractal and non-fractal scale-free network models as well as real networks. We show that the correlation between degree and betweenness centrality $C$ of nodes is much weaker in fractal network models compared to non-fractal models. We also show that nodes of both fractal and non-fractal scale-free networks have power law betweenness centrality distribution $P(C)\sim C^{-\delta}$. We find that for non-fractal scale-free networks $\delta = 2$, and for fractal scale-free networks $\delta = 2-1/d_{B}$, where $d_{B}$ is the dimension of the fractal network. We support these results by explicit calculations on four real networks: pharmaceutical firms (N=6776), yeast (N=1458), WWW (N=2526), and a sample of Internet network at AS level (N=20566), where $N$ is the number of nodes in the largest connected component of a network. We also study the crossover phenomenon from fractal to non-fractal networks upon adding random edges to a fractal network. We show that the crossover length $\ell^{*}$, separating fractal and non-fractal regimes, scales with dimension $d_{B}$ of the network as $p^{-1/d_{B}}$, where $p$ is the density of random edges added to the network. We find that the correlation between degree and betweenness centrality increases with $p$.Comment: 19 pages, 6 figures. Submitted to PR

Betweenness Centrality in Large Complex Networks

We analyze the betweenness centrality (BC) of nodes in large complex networks. In general, the BC is increasing with connectivity as a power law with an exponent $\eta$. We find that for trees or networks with a small loop density $\eta=2$ while a larger density of loops leads to $\eta<2$. For scale-free networks characterized by an exponent $\gamma$ which describes the connectivity distribution decay, the BC is also distributed according to a power law with a non universal exponent $\delta$. We show that this exponent $\delta$ must satisfy the exact bound $\delta\geq (\gamma+1)/2$. If the scale free network is a tree, then we have the equality $\delta=(\gamma+1)/2$.Comment: 6 pages, 5 figures, revised versio

Benchmarking Measures of Network Influence

Identifying key agents for the transmission of diseases (ideas, technology, etc.) across social networks has predominantly relied on measures of centrality on a static base network or a temporally flattened graph of agent interactions. Various measures have been proposed as the best trackers of influence, such as degree centrality, betweenness, and $k$-shell, depending on the structure of the connectivity. We consider SIR and SIS propagation dynamics on a temporally-extruded network of observed interactions and measure the conditional marginal spread as the change in the magnitude of the infection given the removal of each agent at each time: its temporal knockout (TKO) score. We argue that the exhaustive approach of the TKO score makes it an effective benchmark measure for evaluating the accuracy of other, often more practical, measures of influence. We find that none of the common network measures applied to the induced flat graphs are accurate predictors of network propagation influence on the systems studied; however, temporal networks and the TKO measure provide the requisite targets for the hunt for effective predictive measures

Structure and Dynamics of Brain Lobes Functional Networks at the Onset of Anesthesia Induced Loss of Consciousness

Anesthetic agents are neurotropic drugs able to induce dramatic alterations in the thalamo-cortical system, promoting a drastic reduction in awareness and level of consciousness. There is experimental evidence that general anesthesia impacts large scale functional networks leading to alterations in the brain state. However, the way anesthetics affect the structure assumed by functional connectivity in different brain regions have not been reported yet. Within this context, the present study has sought to characterize the functional brain networks respective to the frontal, parietal, temporal and occipital lobes. In this experiment, electro-physiological neural activity was recorded through the use of a dense ECoG-electrode array positioned directly over the cortical surface of an old world monkey of the species Macaca fuscata. Networks were serially estimated over time at each five seconds, while the animal model was under controlled experimental conditions of an anesthetic induction process. In each one of the four cortical brain lobes, prominent alterations on distinct properties of the networks evidenced a transition in the networks architecture, which occurred within about one and a half minutes after the administration of the anesthetics. The characterization of functional brain networks performed in this study represents important experimental evidence and brings new knowledge towards the understanding of neural correlates of consciousness in terms of the structure and properties of the functional brain networks.Comment: 41 pages; 30 figures; 30 tables. arXiv admin note: substantial text overlap with arXiv:1604.0000

Characterization of Large Scale Functional Brain Networks During Ketamine-Medetomidine Anesthetic Induction

Several experiments evidence that specialized brain regions functionally interact and reveal that the brain processes and integrates information in a specific and structured manner. Networks can be used to model brain functional activities constituting a way to characterize and quantify this structured form of organization. Reports state that different physiological states or even diseases that affect the central nervous system may be associated to alterations on those networks, that might reflect in graphs of different architectures. However, the relation of their structure to different states or conditions of the organism is not well comprehended. Thus, experiments that involve the estimation of functional neural networks of subjects exposed to different controlled conditions are of great relevance. Within this context, this research has sought to model large scale functional brain networks during an anesthetic induction process. The experiment was based on intra-cranial recordings of neural activities of an old world macaque of the species Macaca fuscata. Neural activity was recorded during a Ketamine-Medetomidine anesthetic induction process. Networks were serially estimated in time intervals of five seconds. Changes were observed in various networks properties within about one and a half minutes after the administration of the anesthetics. These changes reveal the occurrence of a transition on the networks architecture. During general anesthesia a reduction in the functional connectivity and network integration capabilities were verified in both local and global levels. It was also observed that the brain shifted to a highly specific and dynamic state. The results bring empirical evidence and report the relation of the induced state of anesthesia to properties of functional networks, thus, they contribute for the elucidation of some new aspects of neural correlates of consciousness.Comment: 28 pages , 9 figures, 7 tables; - English errors were corrected; Figures 1,3,4,5,6,8 and 9 were replaced by (exact the same)figures of higher resolution; Three(3) references were added on the introduction sectio

Modeling Minneapolis Skyway Network

Adopting an agent-based approach, this paper explores the topological evolution of the Minneapolis Skyway System from a microscopic perspective. Under a decentralized decision-making mechanism, skyway segments are built by self-interested building owners. We measure the accessibility for the blocks from 1962 to 2002 using the size of office space in each block as an indicator of business opportunities. By building skyway segments, building owners desire to increase their buildingsÕ value of accessibility, and thus potential business revenue. The skyway network in equilibrium generated from the agent model displays similarity to the actual skyway system. The network topology is evaluated by multiple centrality measures (e.g., degree centrality, closeness centrality, and betweenness centrality) and a measure of road contiguity, roadness. Sensitivity tests such parameters as distance decay parameter and construction cost per unit length of segments are performed. Our results disclose that the accessibility- based agent model can provide unique insights for the dynamics of the skyway network growth.skyway network, network growth, agent-based modeling

Generalized Erdos Numbers for network analysis

In this paper we consider the concept of `closeness' between nodes in a weighted network that can be defined topologically even in the absence of a metric. The Generalized Erd\H{o}s Numbers (GENs) satisfy a number of desirable properties as a measure of topological closeness when nodes share a finite resource between nodes as they are real-valued and non-local, and can be used to create an asymmetric matrix of connectivities. We show that they can be used to define a personalized measure of the importance of nodes in a network with a natural interpretation that leads to a new global measure of centrality and is highly correlated with Page Rank. The relative asymmetry of the GENs (due to their non-metric definition) is linked also to the asymmetry in the mean first passage time between nodes in a random walk, and we use a linearized form of the GENs to develop a continuum model for `closeness' in spatial networks. As an example of their practicality, we deploy them to characterize the structure of static networks and show how it relates to dynamics on networks in such situations as the spread of an epidemic