Neighbor-Neighbor Correlations Explain Measurement Bias in Networks

Abstract

In numerous physical models on networks, dynamics are based on interactions that exclusively involve properties of a node’s nearest neighbors. However, a node’s local view of its neighbors may systematically bias perceptions of network connectivity or the prevalence of certain traits. We investigate the strong friendship paradox, which occurs when the majority of a node’s neighbors have more neighbors than does the node itself. We develop a model to predict the magnitude of the paradox, showing that it is enhanced by negative correlations between degrees of neighboring nodes. We then show that by including neighbor-neighbor correlations, which are degree correlations one step beyond those of neighboring nodes, we accurately predict the impact of the strong friendship paradox in real-world networks. Understanding how the paradox biases local observations can inform better measurements of network structure and our understanding of collective phenomena.

Analytical approach to the generalized friendship paradox in networks with correlated attributes

“…where p x is the probability that a neighbor of the focal node has an attribute bigger than that of the focal node, hence A similar approach has been taken for the FP when the degrees of neighboring nodes are correlated [37]. In the case with large k, the median-based peer pressure in Eq.…”

The degree-wise effect of a second step for a random walk on a graph

“…For further recent work related to the friendship paradox, see, e.g. [13], [16], [19], [21], [23], [25], [32], [33], [36], and for further information on random walks on graphs, see also [1], [8], [26], and [27].…”

The transsortative structure of networks

“…However, networks also exhibit substantial nonlocal structure as manifested by large numbers of connected triplets and bigger motifs [20], as well as degree [21] or attribute [22] correlations of two-hop, or even more distant, neighbours. This higher-order structure is necessary to explain effects such as the strong friendship paradox [23], where the majority of a node's neighbours have higher degree than the node itself [24].…”