Higher order clustering coefficients in Barabasi-Albert networks

Higher order clustering coefficients $C(x)$ are introduced for random networks. The coefficients express probabilities that the shortest distance between any two nearest neighbours of a certain vertex $i$ equals $x$, when one neglects all paths crossing the node $i$. Using $C(x)$ we found that in the Barab\'{a}si-Albert (BA) model the average shortest path length in a node's neighbourhood is smaller than the equivalent quantity of the whole network and the remainder depends only on the network parameter $m$. Our results show that small values of the standard clustering coefficient in large BA networks are due to random character of the nearest neighbourhood of vertices in such networks.Comment: 10 pages, 4 figure

The time-evolution of bias

We study the evolution of the bias factor b and the mass-galaxy correlation coefficient r in a simple analytic model for galaxy formation and the gravitational growth of clustering. The model shows that b and r can be strongly time-dependent, but tend to approach unity even if galaxy formation never ends as the gravitational growth of clustering debiases the older galaxies. The presence of random fluctuations in the sites of galaxy formation relative to the mass distribution can cause large and rapidly falling bias values at high redshift.Comment: 4 pages, with 2 figures included. Typos corrected to match published ApJL version. Color figure and links at http://www.sns.ias.edu/~max/bias.html or from [email protected]

Interest Clustering Coefficient: a New Metric for Directed Networks like Twitter

We study here the clustering of directed social graphs. The clustering coefficient has been introduced to capture the social phenomena that a friend of a friend tends to be my friend. This metric has been widely studied and has shown to be of great interest to describe the characteristics of a social graph. In fact, the clustering coefficient is adapted for a graph in which the links are undirected, such as friendship links (Facebook) or professional links (LinkedIn). For a graph in which links are directed from a source of information to a consumer of information, it is no more adequate. We show that former studies have missed much of the information contained in the directed part of such graphs. We thus introduce a new metric to measure the clustering of a directed social graph with interest links, namely the interest clustering coefficient. We compute it (exactly and using sampling methods) on a very large social graph, a Twitter snapshot with 505 million users and 23 billion links. We additionally provide the values of the formerly introduced directed and undirected metrics, a first on such a large snapshot. We exhibit that the interest clustering coefficient is larger than classic directed clustering coefficients introduced in the literature. This shows the relevancy of the metric to capture the informational aspects of directed graphs.Comment: 15 pages, 9 figure

Complex-network description of seismicity

The seismic data taken in California and Japan are mapped to growing random networks. It is shown in the undirected network picture that these earthquake networks are scale-free and small-work networks with the power-law connectivity distributions, the large values of the clustering coefficient, and the small values of the average path length. It is demonstrated how the present network approach reveals complexity of seismicity in a novel manner

Clustering of random scale-free networks

We derive the finite size dependence of the clustering coefficient of scale-free random graphs generated by the configuration model with degree distribution exponent $2<\gamma<3$. Degree heterogeneity increases the presence of triangles in the network up to levels that compare to those found in many real networks even for extremely large nets. We also find that for values of $\gamma \approx 2$, clustering is virtually size independent and, at the same time, becomes a {\it de facto} non self-averaging topological property. This implies that a single instance network is not representative of the ensemble even for very large network sizes

The Aha! Experience of Spatial Reorientation

The experience of spatial re-orientation is investigated as an instance of the wellknown phenomenon of the Aha! moment. The research question is: What are the visuospatial conditions that are most likely to trigger the spatial Aha! experience? The literature suggests that spatial re-orientation relies mainly on the geometry of the environment and a visibility graph analysis is used to quantify the visuospatial information. Theories from environmental psychology point towards two hypotheses. The Aha! experience may be triggered by a change in the amount of visual information, described by the isovist properties of area and revelation, or by a change in the complexity of the visual information associated with the isovist properties of clustering coefficient and visual control. Data from participants’ exploratory behaviour and EEG recordings are collected during wayfinding in virtual reality urban environments. Two types of events are of interest here: (a) sudden changes of the visuospatial information preceding subjects' response to investigate changes in EEG power; and (b) participants brain dynamics (Aha! effect) just before the response to examine differences in isovist values at this location. Research on insights, time-frequency analysis of the P3 component and findings from navigation and orientation studies suggest that the spatial Aha! experience may be reflected by: a parietal alpha power decrease associated with the switch of the representation and a frontocentral theta increase indexing spatial processing during decision-making. Single-trial time-frequency analysis is used to classify trials into two conditions based on the alpha/theta power differences between a 3s time-period before participants’ response and a time-period of equal duration before that. Behavioural results show that participants are more likely to respond at locations with low values of clustering coefficient and high values of visual control. The EEG analysis suggests that the alpha decrease/theta increase condition occurs at locations with significantly lower values of clustering coefficient and higher values of visual control. Small and large decreases in clustering coefficient, just before the response, are associated with significant differences in delta/theta power. The values of area and revelation do not show significant differences. Both behavioural and EEG results suggest that the Aha! experience of re-orientation is more likely to be triggered by a change in the complexity of the visual-spatial environment rather than a change in the amount, as measured by the relevant isovist properties

The effect of aging on network structure

In network evolution, the effect of aging is universal: in scientific collaboration network, scientists have a finite time span of being active; in movie actors network, once popular stars are retiring from stage; devices on the Internet may become outmoded with techniques developing so rapidly. Here we find in citation networks that this effect can be represented by an exponential decay factor, $e^{-\beta \tau}$, where $\tau$ is the node age, while other evolving networks (the Internet for instance) may have different types of aging, for example, a power-law decay factor, which is also studied and compared. It has been found that as soon as such a factor is introduced to the Barabasi-Albert Scale-Free model, the network will be significantly transformed. The network will be clustered even with infinitely large size, and the clustering coefficient varies greatly with the intensity of the aging effect, i.e. it increases linearly with $\beta$ for small values of $\beta$ and decays exponentially for large values of $\beta$. At the same time, the aging effect may also result in a hierarchical structure and a disassortative degree-degree correlation. Generally the aging effect will increase the average distance between nodes, but the result depends on the type of the decay factor. The network appears like a one-dimensional chain when exponential decay is chosen, but with power-law decay, a transformation process is observed, i.e., from a small-world network to a hypercubic lattice, and to a one-dimensional chain finally. The disparities observed for different choices of the decay factor, in clustering, average node distance and probably other aspects not yet identified, are believed to bear significant meaning on empirical data acquisition.Comment: 8 pages, 9 figures,V2, accepted for publication in Phys. Rev.