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 metrics and localization in core-periphery networks

Two concepts of centrality have been defined in complex networks. The first considers the centrality of a node and many different metrics for it has been defined (e.g. eigenvector centrality, PageRank, non-backtracking centrality, etc). The second is related to a large scale organization of the network, the core-periphery structure, composed by a dense core plus an outlying and loosely-connected periphery. In this paper we investigate the relation between these two concepts. We consider networks generated via the Stochastic Block Model, or its degree corrected version, with a strong core-periphery structure and we investigate the centrality properties of the core nodes and the ability of several centrality metrics to identify them. We find that the three measures with the best performance are marginals obtained with belief propagation, PageRank, and degree centrality, while non-backtracking and eigenvector centrality (or MINRES}, showed to be equivalent to the latter in the large network limit) perform worse in the investigated networks.Comment: 15 pages, 8 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

Kirchhoff Index As a Measure of Edge Centrality in Weighted Networks: Nearly Linear Time Algorithms

Most previous work of centralities focuses on metrics of vertex importance and methods for identifying powerful vertices, while related work for edges is much lesser, especially for weighted networks, due to the computational challenge. In this paper, we propose to use the well-known Kirchhoff index as the measure of edge centrality in weighted networks, called $\theta$-Kirchhoff edge centrality. The Kirchhoff index of a network is defined as the sum of effective resistances over all vertex pairs. The centrality of an edge $e$ is reflected in the increase of Kirchhoff index of the network when the edge $e$ is partially deactivated, characterized by a parameter $\theta$. We define two equivalent measures for $\theta$-Kirchhoff edge centrality. Both are global metrics and have a better discriminating power than commonly used measures, based on local or partial structural information of networks, e.g. edge betweenness and spanning edge centrality. Despite the strong advantages of Kirchhoff index as a centrality measure and its wide applications, computing the exact value of Kirchhoff edge centrality for each edge in a graph is computationally demanding. To solve this problem, for each of the $\theta$-Kirchhoff edge centrality metrics, we present an efficient algorithm to compute its $\epsilon$-approximation for all the $m$ edges in nearly linear time in $m$. The proposed $\theta$-Kirchhoff edge centrality is the first global metric of edge importance that can be provably approximated in nearly-linear time. Moreover, according to the $\theta$-Kirchhoff edge centrality, we present a $\theta$-Kirchhoff vertex centrality measure, as well as a fast algorithm that can compute $\epsilon$-approximate Kirchhoff vertex centrality for all the $n$ vertices in nearly linear time in $m$

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

From Centrality to Temporary Fame: Dynamic Centrality in Complex Networks

We develop a new approach to the study of the dynamics of link utilization in complex networks using records of communication in a large social network. Counter to the perspective that nodes have particular roles, we find roles change dramatically from day to day. "Local hubs" have a power law degree distribution over time, with no characteristic degree value. Our results imply a significant reinterpretation of the concept of node centrality in complex networks, and among other conclusions suggest that interventions targeting hubs will have significantly less effect than previously thought.Comment: 11 pages, 4 figure