Stochastic blockmodels and community structure in networks

Karrer, Brian; Newman, M. E. J.

doi:10.1103/physreve.83.016107

Cited by 1,759 publications

(2,111 citation statements)

References 35 publications

(52 reference statements)

Supporting

Mentioning

2,073

Contrasting

Unclassified

Order By: Relevance

“…This approach is similar to the minimization of energy when finding the ground state or stable state of a physical system, and the connection has been widely exploited. A variety of different measures for assigning scores have been proposed, such as the so-called E/I ratio 48 , likelihood-based measures 49 and others 50 , but the most widely used is the measure known as the modularity 18,51 .…”

Section: Optimization Methodsmentioning

confidence: 99%

See 1 more Smart Citation

Communities, modules and large-scale structure in networks

2011

Self Cite

View full text Add to dashboard Cite

Networks, also called graphs by mathematicians, provide a useful abstraction of the structure of many complex systems, ranging from social systems and computer networks to biological networks and the state spaces of physical systems. In the past decade there have been significant advances in experiments to determine the topological structure of networked systems, but there remain substantial challenges in extracting scientific understanding from the large quantities of data produced by the experiments. A variety of basic measures and metrics are available that can tell us about small-scale structure in networks, such as correlations, connections and recurrent patterns, but it is considerably more difficult to quantify structure on medium and large scales, to understand the 'big picture'. Important progress has been made, however, within the past few years, a selection of which is reviewed here.A network is, in its simplest form, a collection of dots joined together in pairs by lines (Fig. 1). In the jargon of the field, a dot is called a 'node' or 'vertex' (plural 'vertices') and a line is called an 'edge'. Networks are used in many branches of science as a way to represent the patterns of connections between the components of complex systems [1][2][3][4][5][6] . Examples include the Internet 7,8 , in which the nodes are computers and the edges are data connections such as optical-fibre cables, food webs in biology 9,10 , in which the nodes are species in an ecosystem and the edges represent predator-prey interactions, and social networks 11,12 , in which the nodes are people and the edges represent any of a variety of different types of social interaction including friendship, collaboration, business relationships or others.In the past decade there has been a surge of interest in both empirical studies of networks 13 and development of mathematical and computational tools for extracting insight from network data 1-6 . One common approach to the study of networks is to focus on the properties of individual nodes or small groups of nodes, asking questions such as, 'Which is the most important node in this network?' or 'Which are the strongest connections?' Such approaches, however, tell us little about large-scale network structure. It is this large-scale structure that is the topic of this paper.The best-studied form of large-scale structure in networks is modular or community structure 14,15 . A community, in this context, is a dense subnetwork within a larger network, such as a close-knit group of friends in a social network or a group of interlinked web pages on the World Wide Web (Fig. 1). Although communities are not the only interesting form of large-scale structure-there are others that we will come to-they serve as a good illustration of the nature and scope of present research in this area and will be our primary focus.Communities are of interest for a number of reasons. They have intrinsic interest because they may correspond to functional units within a networked system, an example of the kind of link...

show abstract

Section: Optimization Methodsmentioning

confidence: 99%

“…Figure 4b shows a different fit to the same network using now a 'degree-corrected' block model that allows for widely varying degrees 49 . As the figure shows, the model now finds a division that corresponds closely to the known division between left-and right-leaning blogs.…”

Section: Block Modelsmentioning

confidence: 99%

Communities, modules and large-scale structure in networks

2011

Self Cite

View full text Add to dashboard Cite

show abstract

“…We then take Kt k=1 H ik,t H jk,t as the expected number of links between nodes i and j at snapshot t. The V ij,t is drawn from a Poisson distribution with the mean Kt k=1 H ik,t H jk,t . It means that the probability of an edge and the expected number of edges are equal in the limitation of a large network, which has been also 210 used in [33] [34].…”

Section: Model Formulizationmentioning

confidence: 99%

Autonomous overlapping community detection in temporal networks: A dynamic Bayesian nonnegative matrix factorization approach

Wang

Jiao

et al. 2016

Knowledge-Based Systems

View full text Add to dashboard Cite

A wide variety of natural or artificial systems can be modeled as time-varying or temporal networks.To understand the structural and functional properties of these time-varying networked systems, it is desirable to detect and analyze the evolving community structure. In temporal networks, the identified communities should reflect the current snapshot network, and at the same time be similar to the communities identified in history or say the previous snapshot networks. Most of the existing approaches assume that the number of communities is known or can be obtained by some heuristic methods. This is unsuitable and complicated for most real world networks, especially temporal networks. In this paper, we propose a Bayesian probabilistic model, named Dynamic Bayesian Nonnegative Matrix Factorization (DBNMF), for automatic detection of overlapping communities in temporal networks. Our model can not only give the overlapping community structure based on the probabilistic memberships of nodes in each snapshot network but also automatically determines the number of communities in each snapshot network based on automatic relevance determination. Thereafter, a gradient descent algorithm is proposed to optimize the objective function of our DBNMF model. The experimental results using both synthetic datasets and real-world temporal networks demonstrate that the DBNMF model has superior performance compared with two widely used methods, especially when the number of communities is unknown and when the network is highly sparse.

show abstract

“…The communities are represented with a stochastic block network model (9), in which contacts between individuals within the same block are far more common than those between blocks. This assumption is essential as it increases the strength of indirect effects within clusters relative to scenarios in which there is more between-cluster transmission (10).…”

Section: Simulated Population Structurementioning

confidence: 99%

Competing Effects of Indirect Protection and Clustering on the Power of Cluster-Randomized Controlled Vaccine Trials

Hitchings

Lipsitch

Wang

et al. 2018

American Journal of Epidemiology

View full text Add to dashboard Cite

Power considerations for trials evaluating vaccines against infectious diseases are complicated by indirect protective effects of vaccination. While cluster-randomized trials (cRCTs) are less statistically efficient than individually randomized trials (iRCT), a cRCT's ability to measure direct and indirect vaccine effects may mitigate the loss of efficiency due to clustering. Within cRCTs, the number and size of clusters affects three determinants of power: the effect size being measured, disease incidence, and intra-cluster correlation. We simulate trials conducted in a collection of small communities to assess how indirect protection and clustering affect the power of cRCTs and iRCTs during an emerging epidemic. Across diverse parameters, we find that within the same trial population, cRCTs are never more powerful than iRCTs, although the difference can be small. We also identify two effects that attenuate the loss of cRCT power traditionally associated with increased cluster size. First, if enrollment of fewer, larger clusters is performed to achieve higher vaccine coverage within vaccinated communities, this increases the effect to be measured and, consequently, power. Second, the greater rate of imported transmission in larger communities may increase the attack rate and similarly mitigate loss of power relative to a trial in many, smaller communities. Cluster-randomized controlled trials (cRCTs) have become an increasingly common method for evaluating interventions for infectious diseases, including vaccines. Compared to individually randomized controlled trials (iRCTs), cRCTs may offer logistical, operational, and acceptability advantages (1), and allow the measurement of direct and indirect effects of vaccination, which are often relevant for policy-makers (2). The statistical theory of cRCT design has largely focused on the effect of clustering, commonly measured by intracluster correlation, on power (3-5). Intracluster correlation arises because outcomes of members of the same cluster are more similar than those from different clusters. Therefore, increasing the number of individuals within a cluster provides less information than would adding the same number of individuals in a new cluster.When the trial outcome is an infectious disease, correlation arises also because each case in a cluster can transmit infection to other cluster members. Thus, trials of vaccines against infectious diseases exhibit a more complicated relationship between statistical power and sample size than in trials for non-infectious outcomes (6, 7); in particular, the total or overall vaccine effect measured by a cRCT is generally larger than the direct effect measured by an iRCT. In principle, this increased effect size in a cRCT might partially or fully offset the loss of power due to within-cluster correlation. Understanding these complexities can aid in vaccine trial design for emerging epidemics. While an important consideration in any clinical trial, maximizing efficiency is particularly crucial in trials during ...

show abstract

Stochastic blockmodels and community structure in networks

Cited by 1,759 publications

References 35 publications

Communities, modules and large-scale structure in networks

Communities, modules and large-scale structure in networks

Autonomous overlapping community detection in temporal networks: A dynamic Bayesian nonnegative matrix factorization approach

Competing Effects of Indirect Protection and Clustering on the Power of Cluster-Randomized Controlled Vaccine Trials

Contact Info

Product

Resources

About