Centrality measures for graphons: Accounting for uncertainty in networks

As relational datasets modeled as graphs keep increasing in size and their data-acquisition is permeated by uncertainty, graph-based analysis techniques can become computationally and conceptually challenging. In particular, node centrality measures rely on the assumption that the graph is perfectly known -- a premise not necessarily fulfilled for large, uncertain networks. Accordingly, centrality measures may fail to faithfully extract the importance of nodes in the presence of uncertainty. To mitigate these problems, we suggest a statistical approach based on graphon theory: we introduce formal definitions of centrality measures for graphons and establish their connections to classical graph centrality measures. A key advantage of this approach is that centrality measures defined at the modeling level of graphons are inherently robust to stochastic variations of specific graph realizations. Using the theory of linear integral operators, we define degree, eigenvector, Katz and PageRank centrality functions for graphons and establish concentration inequalities demonstrating that graphon centrality functions arise naturally as limits of their counterparts defined on sequences of graphs of increasing size. The same concentration inequalities also provide high-probability bounds between the graphon centrality functions and the centrality measures on any sampled graph, thereby establishing a measure of uncertainty of the measured centrality score. The same concentration inequalities also provide high-probability bounds between the graphon centrality functions and the centrality measures on any sampled graph, thereby establishing a measure of uncertainty of the measured centrality score.Comment: Authors ordered alphabetically, all authors contributed equally. 21 pages, 7 figure

Centrality in children's best friend networks: the role of social behaviour

Centrality is an indicator of an individual's relative importance within a social group. Predictors of centrality in best friendship networks were examined in 146 children (70 boys, 76 girls, Mage= 9.95). Children completed measures of social confidence, social desirability, friendship quality, school liking, and loneliness, and nominated their best friends from within their class at two time points, 3 months apart. Multigroup path analysis revealed gender differences in the antecedents of centrality. Social confidence, social desirability, and friendship quality predicted changes in the indicators of centrality in best friend networks over time. In boys’ social behaviour positively predicted changes in centrality whereas in girls’ social behaviour negatively predicted changes in centrality. Together, these findings suggest that some aspects of social behaviour are influential for centrality in best friend groups

Numerical Investigation of Metrics for Epidemic Processes on Graphs

This study develops the epidemic hitting time (EHT) metric on graphs measuring the expected time an epidemic starting at node $a$ in a fully susceptible network takes to propagate and reach node $b$. An associated EHT centrality measure is then compared to degree, betweenness, spectral, and effective resistance centrality measures through exhaustive numerical simulations on several real-world network data-sets. We find two surprising observations: first, EHT centrality is highly correlated with effective resistance centrality; second, the EHT centrality measure is much more delocalized compared to degree and spectral centrality, highlighting the role of peripheral nodes in epidemic spreading on graphs.Comment: 6 pages, 1 figure, 3 tables, In Proceedings of 2015 Asilomar Conference on Signals, Systems, and Computer