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Statistical analysis for populations of networks is widely applicable, but challenging, as networks have strongly non-Euclidean behaviour. Graph space is an exhaustive framework for studying populations of unlabelled networks that are weighted or unweighted, uni- or multilayered, directed or undirected. Viewing graph space as the quotient of a Euclidean space with respect to a finite group action, we show that it is not a manifold, and that its curvature is unbounded from above. Within this geometrical framework we define generalized geodesic principal components, and we introduce the align-all-and-compute algorithms, all of which allow for the computation of statistics on graph space. The statistics and algorithms are compared with existing methods and empirically validated on three real datasets, showcasing the potential utility of the framework. The whole framework is implemented within the geomstats Python package. [ABSTRACT FROM AUTHOR]

A topic of primary importance for organizations is the ability to identify and appraise Social Media Influencers (SMIs), given their key role in affecting conversations and interactions on social media. According to the current research in this area, influencers make up a single category of social media users, but only limited attention has been paid concerning the extent to which they can exert their influence. In this study, the quantification and classification of SMIs is addressed by proposing an advanced methodology based on social network analysis - K-shell decomposition - together with a discussion on the relationship between the different SMI categories and the effect of each type of influencer on the public relation activity of an organization. The developed methodology was tested through an action research project conducted at the Teatro alla Scala of Milan, and the results were then discussed with the management of the opera house. The main finding of this work is that SMIs can be split into writers, authorities or spreaders on the basis of the kind of influence they exert, thereby delivering a precisely focused typology of SMIs. These findings enhance our academic knowledge on analytics applied to social science, while also providing a real case situation where managers make practical use of analytics.

Every brain is unique, having its structural and functional organization shaped by both genetic and environmental factors over the course of its development. Brain image studies tend to produce results by averaging across a group of subjects, under the common assumption that it is possible to subdivide the cortex into homogeneous areas while maintaining a correspondence across subjects. We investigate this assumption: can the structural properties of a specific region of an atlas be assumed to be the same across subjects? This question is addressed by looking at the network representation of the brain, with nodes corresponding to brain regions and edges to their structural relationships. Using an unsupervised graph matching strategy, we align the structural connectomes of a set of healthy subjects, considering parcellations of different granularity, to understand the connectivity misalignment between regions. First, we compare the obtained permutations with four different algorithm initializations: Spatial Adjacency, Identity, Barycenter, and Random. Our results suggest that applying an alignment strategy improves the similarity across subjects when the number of parcels is above 100 and when using Spatial Adjacency and Identity initialization (the most plausible priors). Second, we characterize the obtained permutations, revealing that the majority of permutations happens between neighbors parcels. Lastly, we study the spatial distribution of the permutations. By visualizing the results on the cortex, we observe no clear spatial patterns on the permutations and all the regions across the context are mostly permuted with first and second order neighbors. [ABSTRACT FROM AUTHOR]