Abstract
We construct a connected network of 3.9 million nodes from mobile phone call records, which can be regarded as a proxy for the underlying human communication network at the societal level. We assign two weights on each edge to reflect the strength of social interaction, which are the aggregate call duration and the cumulative number of calls placed between the individuals over a period of 18 weeks. We present a detailed analysis of this weighted network by examining its degree, strength, and weight distributions, as well as its topological assortativity and weighted assortativity, clustering and weighted clustering, together with correlations between these quantities. We give an account of motif intensity and coherence distributions and compare them to a randomized reference system. We also use the concept of link overlap to measure the number of common neighbours any two adjacent nodes have, which serves as a useful local measure for identifying the interconnectedness of communities. We report a positive correlation between the overlap and weight of a link, thus providing strong quantitative evidence for the weak ties hypothesis, a central concept in social network analysis. The percolation properties of the network are found to depend on the type and order of removed links, and they can help understand how the local structure of the network manifests itself at the global level. We hope that our results will contribute to modelling weighted large-scale social networks, and believe that the systematic approach followed here can be adopted to study other weighted networks.
Highlights
DEUTSCHE PHYSIKALISCHE GESELLSCHAFT strong quantitative evidence for the weak ties hypothesis, a central concept in social network analysis
Modern technologies enable the study of social networks of unprecedented size
A number of such investigations have appeared recently ranging from exploring email communication networks [6]–[8], [47] to identifying groups and strategies in an electronic marketplace [48]–[50]
Summary
We start inspecting the network by showing a small sample of it in figure 1. Social networks are expected to be assortative: people with many friends are connected to others who have many friends This gives rise to degree–degree correlations in the network, meaning that the degrees of two adjacent nodes are not independent. Are written as knNn,i = (1/siN ) j∈N (vi) wNij kj and knDn,i = (1/siD) j∈N (vi) wDij kj, corresponding to the two weighting schemes Averaging these over the network gives knn|k , knNn|k and knDn|k , which measure the effective affinity to connect with neighbours of a given degree while taking the magnitude of the interactions into account [22]. The two weights, number of calls and aggregate call duration, are strongly correlated, and both yield steep strength distributions for nodes This can be understood in light of the only slightly sublinear dependence of strength on degree, governed by the exponent α. The weight of a given link is almost independent of the product of the degrees of adjacent nodes as governed by the almost vanishing exponent γ, but depends on the geometric mean of the strengths of the adjacent nodes as indicated by the value of exponent δ
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