Solution of the 2-star model of a network

The p-star model or exponential random graph is among the oldest and best-known of network models. Here we give an analytic solution for the particular case of the 2-star model, which is one of the most fundamental of exponential random graphs. We derive expressions for a number of quantities of interest in the model and show that the degenerate region of the parameter space observed in computer simulations is a spontaneously symmetry broken phase separated from the normal phase of the model by a conventional continuous phase transition.Comment: 5 pages, 3 figure

Emerging and scripted roles in computer-supported collaborative learning

Emerging and scripted roles pose an intriguing approach to analysing and facilitating CSCL. The concept of emerging roles provides a perspective on how learners structure and self-regulate their CSCL processes. Emerging roles appear to be dynamic over longer periods of time in relation to learners’ advancing knowledge, but are often unequally distributed in ad hoc CSCL settings, e.g. a learner being the ‘typist’ and another being the ‘thinker’. Empirical findings show that learners benefit from structuring or scripting CSCL. Scripts can specify roles and facilitate role rotation for learners to equally engage in relevant learning roles and activities. Scripted roles can, however, collide with emerging roles and therefore need to be carefully attuned to the advancing capabilities of the learners

What the eye does not see: visualizations strategies for the data collection of personal networks

The graphic representation of relational data is one of the central elements of social network analysis. In this paper, the author describe the use of visualization in interview-based data collection procedures designed to obtain personal networks information, exploring four main contributions. First, the author shows a procedure by which the visualization is integrated with traditional name generators to facilitate obtaining information and reducing the burden of the interview process. Second, the author describes the reactions and qualitative interpretation of the interviewees when they are presented with an analytical visualization of their personal network. The most frequent strategies consist in identifying the key individuals, dividing the personal network in groups and classifying alters in concentric circles of relative importance. Next, the author explores how the visualization of groups in personal networks facilitates the enumeration of the communities in which individuals participate. This allows the author to reflect on the role of social circles in determining the structure of personal networks. Finally, the author compares the graphic representation obtained through spontaneous, hand-drawn sociograms with the analytical visualizations elicited through software tools. This allows the author to demonstrate that analytical procedures reveal aspects of the structure of personal networks that respondents are not aware of, as well as the advantages and disadvantages of using both modes of data collection. For this, the author presents findings from a study of highly skilled migrants living in Spain (n = 95) through which the author illustrates the challenges, in terms of data reliability, validity and burden on both the researcher and the participants

Sampling motif-constrained ensembles of networks

The statistical significance of network properties is conditioned on null models which satisfy spec- ified properties but that are otherwise random. Exponential random graph models are a principled theoretical framework to generate such constrained ensembles, but which often fail in practice, either due to model inconsistency, or due to the impossibility to sample networks from them. These problems affect the important case of networks with prescribed clustering coefficient or number of small connected subgraphs (motifs). In this paper we use the Wang-Landau method to obtain a multicanonical sampling that overcomes both these problems. We sample, in polynomial time, net- works with arbitrary degree sequences from ensembles with imposed motifs counts. Applying this method to social networks, we investigate the relation between transitivity and homophily, and we quantify the correlation between different types of motifs, finding that single motifs can explain up to 60% of the variation of motif profiles.Comment: Updated version, as published in the journal. 7 pages, 5 figures, one Supplemental Materia