BACKGROUND To understand neurophysiological mechanisms underlying cognitive dysfunction in low-grade glioma (LGG) patients by evaluating the spatial structure of 'resting-state' brain networks with graph theory. METHODS Standardized tests measuring 6 neurocognitive domains were administered in 17 LGG patients and 17 healthy controls. Magnetoencephalography (MEG) recordings were conducted duri...

Let P be a graph property. A graph G is said to be locally P (closed locally P) if the subgraph induced by the open neighbourhood (closed neighbourhood, respectively) of every vertex in G has property P. The clustering coefficient of a vertex is the proportion of pairs of its neighbours that are themselves neighbours. The minimum clustering coefficient of G is the smallest clustering coefficien...

In recent years, researchers have investigated a growing number of weighted networks where ties are differentiated according to their strength or capacity. Yet, most network measures do not take weights into consideration, and thus do not fully capture the richness of the information contained in the data. In this paper, we focus on a measure originally defined for unweighted networks: the glob...

Remark 1. The conditions P (X, f(X)) = 0 and f(0) = 0 imply that P (0, 0) = 0. As P ′ Y (0, 0) is also 0, the sums in both expressions of [X]f are finite. Remark 2. When P (X,Y ) = Xφ(Y ), where φ(X) ∈ K[[X ]] and φ(0) 6= 0, we obtain the Lagrange inversion formula [X]f = [Y ](φ(X) − Y φ(X)φ(X)). If the characteristic of K is 0, we also have the following form [X]f = [Y ]φ(Y ). Remark 3. When P...

A fundamental property of complex networks is the tendency for edges to cluster. The extent of the clustering is typically quantified by the clustering coefficient, which is the probability that a length-2 path is closed, i.e., induces a triangle in the network. However, higher-order cliques beyond triangles are crucial to understanding complex networks, and the clustering behavior with respect...

Probabilistic networks display a wide range of high average clustering coefficients independent of the number of nodes in the network. In particular, the local clustering coefficient decreases with the degree of the subtending node in a complicated manner not explained by any current models. While a number of hypotheses have been proposed to explain some of these observed properties, there are ...

A complex network is characterized by its degree distribution and clustering coefficient. Given a scale-free network, we propose a node-reconnection algorithm that can alter the clustering coefficient of the network while keeping the degree of each node unchanged. Results are shown when the algorithm is applied to reconnect the nodes of scale-free networks constructed using the Barabási-Albert ...

We study the timetable conflict graphs produced by an artificial generator of student enrollments. We find correlations of their chromatic number with their density and clustering coefficient. The work gives evidence that the clustering coefficient is a useful measure of a graph.

Two general random intersection graph models (active and passive) were introduced by Godehardt and Jaworski (Exploratory Data Analysis in Empirical Research, Springer, Berlin, Heidelberg, New York, pp. 68–81, 2002). Recently the models have been shown to have wide real life applications. The two most important ones are: non-metric data analysis and real life network analysis. Within both contex...

In this paper, a random clique network model to mimic the large clustering coefficient and the modular structure that exist in many real complex networks, such as social networks, artificial networks, and protein interaction networks, is introduced by combining the random selection rule of the Erdös and Rényi (ER) model and the concept of cliques. We find that random clique networks having a sm...