Abstract:
By considering the eigenratio of the Laplacian of the connection graph as synchronizability measure, we propose a procedure for weighting dynamical networks to enhance theirsynchronizability. The method is based on node and edge betweenness centrality measures and is tested on artificially const ructed scale-free, Watts-Strogatz and random networks as well as on some real-world graphs. It is also numerically shown that the same procedure could be used to enhance the phase synchronizability of networks of nonidentical oscillators.

Abstract:
While shorter characteristic path length has in general been believed to enhance synchronizability of a coupled oscillator system on a complex network,the suppressing tendency of the heterogeneity of the degree distribution, even for shorter characteristic path length, has also been reported. To see this, we investigate the effects of various factors such as the degree, characteristic path length, heterogeneity, and betweenness centrality on synchronization, and find a consistent trend between the synchronization and the betweenness centrality. The betweenness centrality is thus proposed as a good indicator for synchronizability.

Abstract:
In this paper, the investigation is first motivated by showing two examples of simple regular symmetrical graphs, which have the same structural parameters, such as average distance, degree distribution and node betweenness centrality, but have very different synchronizabilities. This demonstrates the complexity of the network synchronizability problem. For a given network with identical node dynamics, it is further shown that two key factors influencing the network synchronizability are the network inner linking matrix and the eigenvalues of the network topological matrix. Several examples are then provided to show that adding new edges to a network can either increase or decrease the network synchronizability. In searching for conditions under which the network synchronizability may be increased by adding edges, it is found that for networks with disconnected complementary graphs, adding edges never decreases their synchronizability. This implies that better understanding and careful manipulation of the complementary graphs are important and useful for enhancing the network synchronizability. Moreover, it is found that an unbounded synchronized region is always easier to analyze than a bounded synchronized region. Therefore, to effectively enhance the network synchronizability, a design method is finally presented for the inner linking matrix of rank 1 such that the resultant network has an unbounded synchronized region, for the case where the synchronous state is an equilibrium point of the network.

Abstract:
We investigate the optimization of synchronizability in multiplex networks and demonstrate that the interlayer coupling strength is the deciding factor for the efficiency of optimization. The optimized networks have homogeneity in the degree as well as in the betweenness centrality. Additionally, the interlayer coupling strength crucially affects various properties of individual layers in the optimized multiplex networks. We provide an understanding to how the emerged network properties are shaped or affected when the evolution renders them better synchronizable.

Abstract:
A class of centrality measures called betweenness centralities reflects degree of participation of edges or nodes in communication between different parts of the network. The original shortest-path betweenness centrality is based on counting shortest paths which go through a node or an edge. One of shortcomings of the shortest-path betweenness centrality is that it ignores the paths that might be one or two steps longer than the shortest paths, while the edges on such paths can be important for communication processes in the network. To rectify this shortcoming a current flow betweenness centrality has been proposed. Similarly to the shortest path betwe has prohibitive complexity for large size networks. In the present work we propose two regularizations of the current flow betweenness centrality, \alpha-current flow betweenness and truncated \alpha-current flow betweenness, which can be computed fast and correlate well with the original current flow betweenness.

Abstract:
One of the most fundamental problems in large scale network analysis is to determine the importance of a particular node in a network. Betweenness centrality is the most widely used metric to measure the importance of a node in a network. In this paper, we present a randomized parallel algorithm and an algebraic method for computing betweenness centrality of all nodes in a network. We prove that any path-comparison based algorithm cannot compute betweenness in less than O(nm) time.

Abstract:
There are several centrality measures that have been introduced and studied for real world networks. They account for the different vertex characteristics that permit them to be ranked in order of importance in the network. Betweenness centrality is a measure of the influence of a vertex over the flow of information between every pair of vertices under the assumption that information primarily flows over the shortest path between them. In this paper we present betweenness centrality of some important classes of graphs.

Abstract:
We consider the incremental computation of the betweenness centrality of all vertices in a large complex network modeled as a graph G = (V, E), directed or undirected, with positive real edge-weights. The current widely used algorithm to compute the betweenness centrality of all vertices in G is the Brandes algorithm that runs in O(mn + n^2 log n) time, where n = |V| and m = |E|. We present an incremental algorithm that updates the betweenness centrality score of all vertices in G when a new edge is added to G, or the weight of an existing edge is reduced. Our incremental algorithm runs in O(m' n + n^2) time, where m' is the size of a certain subset of E*, the set of edges in G that lie on a shortest path. We achieve the same bound for the more general incremental update of a vertex v, where the edge update can be performed on any subset of edges incident to v. Our incremental algorithm is the first algorithm that is asymptotically faster on sparse graphs than recomputing with the Brandes algorithm. Our algorithm is also likely to be much faster than the Brandes algorithm on dense graphs since m*, the size of E*, is often close to linear in n. Our incremental algorithm is very simple and the only data structures it uses are arrays, lists, and stack. We give an efficient cache-oblivious implementation that incurs O(scan(n^2) + n sort(m')) cache misses, where scan and sort are well-known measures for efficient caching. We also give a static algorithm for computing betweenness centrality of all vertices that runs in time O(m* n + n^2 log n), which is faster than the Brandes algorithm on any graph with n log n = o(m) and m* = o(m).

Abstract:
In addition to science citation indicators of journals like impact and immediacy, social network analysis provides a set of centrality measures like degree, betweenness, and closeness centrality. These measures are first analyzed for the entire set of 7,379 journals included in the Journal Citation Reports of the Science Citation Index and the Social Sciences Citation Index 2004, and then also in relation to local citation environments which can be considered as proxies of specialties and disciplines. Betweenness centrality is shown to be an indicator of the interdisciplinarity of journals, but only in local citation environments and after normalization because otherwise the influence of degree centrality (size) overshadows the betweenness-centrality measure. The indicator is applied to a variety of citation environments, including policy-relevant ones like biotechnology and nanotechnology.

Abstract:
The Betweenness Centrality index is a very important centrality measure in the analysis of a large number of networks. Despite its significance in a lot of interdisciplinary applications, its computation is very expensive. The fastest known algorithm presently is by Brandes which takes O(|V || E|) time for computation. In real life scenarios, it happens very frequently that a single vertex or a set of vertices is sequentially removed from a network. The recomputation of Betweenness Centrality on removing a single vertex becomes expensive when the Brandes algorithm is repeated. It is to be understood that as the size of the network increases, Betweenness Centrality calculation becomes more and more expensive and even a decrease in running time by a small fraction results in a phenomenal decrease in the actual running time. The algorithm introduced in this paper achieves the same in a significantly lesser time than repetition of the Brandes algorithm. The algorithm can also be extended to a general case.