We study the application of spectral clustering, prediction and visualization methods to graphs with negatively weighted edges. We show that several characteristic matrices of graphs can be extended to graphs with positively and negatively weighted edges, giving signed spectral clustering methods, signed graph kernels and network visualization methods that apply to signed graphs. In particular, we review a signed variant of the graph Laplacian. We derive our results by considering random walks, graph clustering, graph drawing and electrical networks, showing that they all result in the same formalism for handling negatively weighted edges. We illustrate our methods using examples from social networks with negative edges and bipartite rating graphs.

This paper discusses and illustrates various approaches for the longitudinal analysis of personal networks (multilevel analysis, regression analysis, and SIENA). We combined the different types of analyses in a study of the changing personal networks of immigrants. Data were obtained from 25 Argentineans in Spain, who were interviewed twice in a two-year interval. Qualitative interviews were used to estimate the amount of measurement error and to isolate important predictors. Quantitative analyses showed that the persistence of ties was explained by tie strength, network density, and alters' country of origin and residence. Furthermore, transitivity appeared to be an important tendency, both for acquiring new contacts and for the relationships among alters. At the network level, immigrants' networks were remarkably stable in composition and structure despite the high turnover. Clustered graphs have been used to illustrate the results. The results are discussed in light of adaptation to the host society.

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In this paper we give models and algorithms to describe and analyze the collaboration among authors of Wikipedia from a network analytical perspective. The edit network encodes who interacts how with whom when editing an article; it significantly extends previous network models that code author communities in Wikipedia. Several characteristics summarizing some aspects of the organization process and allowing the analyst to identify certain types of authors can be obtained from the edit network. Moreover, we propose several indicators characterizing the global network structure and methods to visualize edit networks. It is shown that the structural network indicators are correlated with quality labels of the associated Wikipedia articles.

We describe the early spread of the novel coronavirus (COVID-19) and the first human-to-human transmission networks, in Romania. We profiled the first 147 cases referring to sex, age, place of residence, probable country of infection, return day to Romania, COVID-19 confirmation date and the probable modes of COVID-19 transmissions. Also, we analysed human-to-human transmission networks and explored their structural features and time dynamics. In Romania, local cycles of transmission were preceded by imported cases, predominantly from Italy. We observed an average of 4.8 days (s.d. = 4.0) between the arrival to a Romanian county and COVID-19 confirmation. Furthermore, among the first 147 COVID-19 patients, 88 were imported cases (64 carriers from Italy), 54 were domestic cases, while for five cases the source of infection was unknown. The early human-to-human transmission networks illustrated a limited geographical dispersion, the presence of super-spreaders and the risk of COVID-19 nosocomial infections. COVID-19 occurred in Romania through case importation from Italy. The largest share of the Romanian diaspora is concentrated especially in the northern parts of Italy, heavily affected by COVID-19. Human mobility (including migration) accounts for the COVID-19 transmission and it should be given consideration while tailoring prevention measures.

We propose a method to visually summarize collections of networks on which a clustering of the vertices is given. Our method allows for efficient comparison of individual networks, as well as for visualizing the average composition and structure of a set of networks. As a concrete application we analyze a set of several hundred personal networks of migrants. On the individual level the network images provide visual hints for assessing the mode of acculturation of the respondent. On the population level they show how cultural integration varies with specific characteristics of the migrants such as country of origin, years of residence, or skin color.

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We assess the reliability of relational event model parameters estimated under two sampling schemes:(1) uniform sampling from the observed events and (2) case-control sampling which samples non-events, or null dyads ("controls"), from a suitably defined risk set. We experimentally determine the variability of estimated parameters as a function of the number of sampled events and controls per event, respectively. Results suggest that relational event models can be reliably fitted to networks with more than 12 million nodes connected by more than 360 million dyadic events by analyzing a sample of some tens of thousands of events and a small number of controls per event. Using data that we collected on the Wikipedia editing network, we illustrate how network effects commonly included in empirical studies based on relational event models need widely different sample sizes to be estimated reliably. For our analysis we use an open-source software which implements the two sampling schemes, allowing analysts to fit and analyze relational event models to the same or other data that may be collected in different empirical settings, varying sample parameters or model specification.hampered by runtime complexity (Foucault Welles et al. 2014) -a problem that is rooted in the likelihood function which normalizes event rates on dyads that experience events by the rates on all dyads that could have possibly experienced an event at the given point in time. The corresponding risk set can be huge, since its size is often quadratic in the number of nodes. For example, in Wikipedia -the setting for the data that we examine in the empirical part of this paper -more than 6 million users contribute to one or several of more than 5 million encyclopedic articles, giving rise to more than 360 million dyadic events. Thus, at given points in time a dyadic event could occur on more than 30 trillion (3 × 10 13 ) dyads. Computing explanatory variables (statistics) for each of these dyads is unfeasible even for a single observed event -much less so for all 360 million events. Such problems are rather typical since -as mentioned above -large quantities of information on relational events are often collected with automatic data-collection technologies which in turn can easily yield large, or very large networks.Empirical studies applying REM, therefore, have often been limited to networks with a rather small number of nodes (Foucault Welles et al. 2014;Lerner et al. 2013;Vu et al. 2017). Exceptions include studies exploiting a "sparsity condition," namely that a single observed event changes the values of some statistics not for all dyads in the risk set but only for a limited subset (Perry and Wolfe 2013;Hunter et al. 2011). For instance, an observed event on a dyad (a, b) changes the out-degree statistic only for all dyads with source aimplying update costs that are linear, rather than quadratic in the number of nodes. This approach, however, requires to develop specific estimation algorithms for different model statistics. Moreover, it does not...