Local modularity measure for network clusterizations

Many complex networks have an underlying modular structure, i.e., structural subunits (communities or clusters) characterized by highly interconnected nodes. The modularity $Q$ has been introduced as a measure to assess the quality of clusterizations. $Q$ has a global view, while in many real-world networks clusters are linked mainly \emph{locally} among each other (\emph{local cluster-connectivity}). Here, we introduce a new measure, localized modularity $LQ$, which reflects local cluster structure. Optimization of $Q$ and $LQ$ on the clusterization of two biological networks shows that the localized modularity identifies more cohesive clusters, yielding a complementary view of higher granularity.Comment: 5 pages, 4 figures, RevTex4; Changed conten

Spectral centrality measures in complex networks

Complex networks are characterized by heterogeneous distributions of the degree of nodes, which produce a large diversification of the roles of the nodes within the network. Several centrality measures have been introduced to rank nodes based on their topological importance within a graph. Here we review and compare centrality measures based on spectral properties of graph matrices. We shall focus on PageRank, eigenvector centrality and the hub/authority scores of HITS. We derive simple relations between the measures and the (in)degree of the nodes, in some limits. We also compare the rankings obtained with different centrality measures.Comment: 11 pages, 10 figures, 5 tables. Final version published in Physical Review

Finding local community structure in networks

Although the inference of global community structure in networks has recently become a topic of great interest in the physics community, all such algorithms require that the graph be completely known. Here, we define both a measure of local community structure and an algorithm that infers the hierarchy of communities that enclose a given vertex by exploring the graph one vertex at a time. This algorithm runs in time O(d*k^2) for general graphs when $d$ is the mean degree and k is the number of vertices to be explored. For graphs where exploring a new vertex is time-consuming, the running time is linear, O(k). We show that on computer-generated graphs this technique compares favorably to algorithms that require global knowledge. We also use this algorithm to extract meaningful local clustering information in the large recommender network of an online retailer and show the existence of mesoscopic structure.Comment: 7 pages, 6 figure

How to project a bipartite network?

The one-mode projecting is extensively used to compress the bipartite networks. Since the one-mode projection is always less informative than the bipartite representation, a proper weighting method is required to better retain the original information. In this article, inspired by the network-based resource-allocation dynamics, we raise a weighting method, which can be directly applied in extracting the hidden information of networks, with remarkably better performance than the widely used global ranking method as well as collaborative filtering. This work not only provides a creditable method in compressing bipartite networks, but also highlights a possible way for the better solution of a long-standing challenge in modern information science: How to do personal recommendation?Comment: 7 pages, 4 figure

A Method to Find Community Structures Based on Information Centrality

Community structures are an important feature of many social, biological and technological networks. Here we study a variation on the method for detecting such communities proposed by Girvan and Newman and based on the idea of using centrality measures to define the community boundaries (M. Girvan and M. E. J. Newman, Community structure in social and biological networks Proc. Natl. Acad. Sci. USA 99, 7821-7826 (2002)). We develop an algorithm of hierarchical clustering that consists in finding and removing iteratively the edge with the highest information centrality. We test the algorithm on computer generated and real-world networks whose community structure is already known or has been studied by means of other methods. We show that our algorithm, although it runs to completion in a time O(n^4), is very effective especially when the communities are very mixed and hardly detectable by the other methods.Comment: 13 pages, 13 figures. Final version accepted for publication in Physical Review

Finding community structure in very large networks

The discovery and analysis of community structure in networks is a topic of considerable recent interest within the physics community, but most methods proposed so far are unsuitable for very large networks because of their computational cost. Here we present a hierarchical agglomeration algorithm for detecting community structure which is faster than many competing algorithms: its running time on a network with n vertices and m edges is O(m d log n) where d is the depth of the dendrogram describing the community structure. Many real-world networks are sparse and hierarchical, with m ~ n and d ~ log n, in which case our algorithm runs in essentially linear time, O(n log^2 n). As an example of the application of this algorithm we use it to analyze a network of items for sale on the web-site of a large online retailer, items in the network being linked if they are frequently purchased by the same buyer. The network has more than 400,000 vertices and 2 million edges. We show that our algorithm can extract meaningful communities from this network, revealing large-scale patterns present in the purchasing habits of customers

The macro-behavior of agents' opinion under the influence of an external field

In this paper, a model about the evolution of opinion on small world networks is proposed. We studied the macro-behavior of the agents' opinion and the relative change rate as time elapses. The external field was found to play an important role in making the opinion $s(t)$ balance or increase, and without the influence of the external field, the relative change rate $\gamma(t)$ shows a nonlinear increasing behavior as time runs. What's more, this nonlinear increasing behavior is independent of the initial condition, the strength of the external field and the time that we cancel the external field. Maybe the results can reflect some phenomenon in our society, such as the function of the macro-control in China or the Mass Media in our society.Comment: 8 pages, 3 figure

Maximal planar networks with large clustering coefficient and power-law degree distribution

In this article, we propose a simple rule that generates scale-free networks with very large clustering coefficient and very small average distance. These networks are called {\bf Random Apollonian Networks}(RAN) as they can be considered as a variation of Apollonian networks. We obtain the analytic results of power-law exponent $\gamma =3$ and clustering coefficient $C={46/3}-36\texttt{ln}{3/2}\approx 0.74$, which agree very well with the simulation results. We prove that the increasing tendency of average distance of RAN is a little slower than the logarithm of the number of nodes in RAN. Since most real-life networks are both scale-free and small-world networks, RAN may perform well in mimicking the reality. The RAN possess hierarchical structure as $C(k)\sim k^{-1}$ that in accord with the observations of many real-life networks. In addition, we prove that RAN are maximal planar networks, which are of particular practicability for layout of printed circuits and so on. The percolation and epidemic spreading process are also studies and the comparison between RAN and Barab\'{a}si-Albert(BA) as well as Newman-Watts(NW) networks are shown. We find that, when the network order $N$(the total number of nodes) is relatively small(as $N\sim 10^4$), the performance of RAN under intentional attack is not sensitive to $N$, while that of BA networks is much affected by $N$. And the diseases spread slower in RAN than BA networks during the outbreaks, indicating that the large clustering coefficient may slower the spreading velocity especially in the outbreaks.Comment: 13 pages, 10 figure

Efficacy and tolerability of evolocumab vs. ezetimibe in patients with muscle-related statin intolerance: the GAUSS-3 randomized clinical trial

Importance: Muscle-related statin intolerance is reported by 5% to 20% of patients. Objective: To identify patients with muscle symptoms confirmed by statin rechallenge and compare lipid-lowering efficacy for 2 nonstatin therapies, ezetimibe and evolocumab. Design, Setting, and Participants: Two-stage randomized clinical trial including 511 adult patients with uncontrolled low-density lipoprotein cholesterol (LDL-C) levels and history of intolerance to 2 or more statins enrolled in 2013 and 2014 globally. Phase A used a 24-week crossover procedure with atorvastatin or placebo to identify patients having symptoms only with atorvastatin but not placebo. In phase B, after a 2-week washout, patients were randomized to ezetimibe or evolocumab for 24 weeks. Interventions: Phase A: atorvastatin (20 mg) vs placebo. Phase B: randomization 2:1 to subcutaneous evolocumab (420 mg monthly) or oral ezetimibe (10 mg daily). Main Outcome and Measures: Coprimary end points were the mean percent change in LDL-C level from baseline to the mean of weeks 22 and 24 levels and from baseline to week 24 levels. Results: Of the 491 patients who entered phase A (mean age, 60.7 [SD, 10.2] years; 246 women [50.1%]; 170 with coronary heart disease [34.6%]; entry mean LDL-C level, 212.3 [SD, 67.9] mg/dL), muscle symptoms occurred in 209 of 491 (42.6%) while taking atorvastatin but not while taking placebo. Of these, 199 entered phase B, along with 19 who proceeded directly to phase B for elevated creatine kinase (N = 218, with 73 randomized to ezetimibe and 145 to evolocumab; entry mean LDL-C level, 219.9 [SD, 72] mg/dL). For the mean of weeks 22 and 24, LDL-C level with ezetimibe was 183.0 mg/dL; mean percent LDL-C change, −16.7% (95% CI, −20.5% to −12.9%), absolute change, −31.0 mg/dL and with evolocumab was 103.6 mg/dL; mean percent change, −54.5% (95% CI, −57.2% to −51.8%); absolute change, −106.8 mg/dL (P < .001). LDL-C level at week 24 with ezetimibe was 181.5 mg/dL; mean percent change, −16.7% (95% CI, −20.8% to −12.5%); absolute change, −31.2 mg/dL and with evolocumab was 104.1 mg/dL; mean percent change, −52.8% (95% CI, −55.8% to −49.8%); absolute change, −102.9 mg/dL (P < .001). For the mean of weeks 22 and 24, between-group difference in LDL-C was −37.8%; absolute difference, −75.8 mg/dL. For week 24, between-group difference in LDL-C was −36.1%; absolute difference, –71.7 mg/dL. Muscle symptoms were reported in 28.8% of ezetimibe-treated patients and 20.7% of evolocumab-treated patients (log-rank P = .17). Active study drug was stopped for muscle symptoms in 5 of 73 ezetimibe-treated patients (6.8%) and 1 of 145 evolocumab-treated patients (0.7%). Conclusions and Relevance: Among patients with statin intolerance related to muscle-related adverse effects, the use of evolocumab compared with ezetimibe resulted in a significantly greater reduction in LDL-C levels after 24 weeks. Further studies are needed to assess long-term efficacy and safety