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Friendship networks and social status

Brian Ball, M. E. J. Newman

arXiv:1205.6822v1cs.SIphysics.soc-ph

TL;DR

Questionnaires produce directed friendship networks, yet many reported friendships are unreciprocated and their meaning is unclear. The paper fits a maximum-likelihood model to infer rankings from both reciprocated and unreciprocated ties, finding that one-way ties usually point from lower to higher rank and that rank correlates with age and popularity.

  • Problem

    Many reported friendship ties are unreciprocated, raising the question of whether their direction is meaningful or should be disregarded.

    Source: Introduction · p. 1

  • Method

    A maximum-likelihood method models reciprocated and unreciprocated ties separately to infer rankings from complete friendship networks.

    Source: Inference of rank from network structure · p. 2, Conclusions · p. 6

  • Results

    In the studied school networks, almost all unreciprocated ties run from lower-ranked to higher-ranked individuals, and rankings correlate with age and popularity.

    Source: Discussion · p. 5, Conclusions · p. 6

  • Takeaways & Limitations

    The inferred rankings may reflect social status, as they correlate with age and overall popularity.

    Source: Conclusions · p. 6

    The analysis has only limited data on participants’ personal characteristics, leaving correlations with other traits untested.

    Source: Conclusions · p. 7

Abstract

from arXiv · show

In empirical studies of friendship networks participants are typically asked, in interviews or questionnaires, to identify some or all of their close friends, resulting in a directed network in which friendships can, and often do, run in only one direction between a pair of individuals. Here we analyze a large collection of such networks representing friendships among students at US high and junior-high schools and show that the pattern of unreciprocated friendships is far from random. In every network, without exception, we find that there exists a ranking of participants, from low to high, such that almost all unreciprocated friendships consist of a lower-ranked individual claiming friendship with a higher-ranked one. We present a maximum-likelihood method for deducing such rankings from observed network data and conjecture that the rankings produced reflect a measure of social status. We note in particular that reciprocated and unreciprocated friendships obey different statistics, suggesting different formation processes, and that rankings are correlated with other characteristics of the participants that are traditionally associated with status, such as age and overall popularity as measured by total number of friends.

Introduction

Questionnaires produce directed friendship networks, where many claimed ties are unreciprocated. Rather than discard their direction, the paper treats these ties as potential signals of community structure and finds a near-universal rank ordering in US school networks.

Introduction

  • In the studied networks, reciprocated ties rarely exceed 50% and can be as low as 30%.Friendships are reported through questionnaires and can run in only one direction.Source: Introduction · p. 1
  • The paper asks what unreciprocated ties mean, contrasting approaches that discard direction with the possibility that their pattern reflects community structure.One common approach treats people as friends whenever a tie runs in either direction.Source: Introduction · p. 1
  • In every studied network, almost all one-way friendship claims run from lower-ranked to higher-ranked individuals.The paper analyzes a large collection of friendship networks from US schools.Source: Introduction · p. 1
  • The proposed rankings are conjectured to reflect social status and are positively correlated with age and total number of friends.Older students rank higher on average, and reciprocated ties are usually between people of similar rank while some one-way ties span large rank differences.Source: Introduction · p. 1

Inference of rank from network structure

A minimum-violations ranking captures the direction of most one-way ties, but it ignores reciprocated friendships that also carry ranking information. The paper instead models both tie types separately and uses maximum likelihood with expectation–maximization to infer ranks and edge probabilities.

Inference of rank from network structure

  • 98% of unreciprocated friendships run from lower to higher ranks under minimum violations rankings, versus about 10% wrong-way edges after direction randomization.The observed rankings have about 2% wrong-way unreciprocated ties; statistical errors are 1% or less.Source: Inference of rank from network structure · p. 2
  • The richer model uses reciprocated friendships because they can make up half of all ties and often connect people of similar rank.Minimum violations rankings focus only on unreciprocated ties and therefore miss network information.Source: Inference of rank from network structure · p. 2
  • The maximum-likelihood model separates reciprocated ties from one-way ties and infers rankings from network structure using distinct edge-probability functions.The symmetric S matrix represents reciprocated ties and asymmetric T represents unreciprocated ties, with A = S + T; the model assumes edge probabilities depend on rank differences.Source: Inference of rank from network structure · p. 2
  • Expectation–maximization alternates between estimating rank probabilities and optimizing edge-probability functions using expected edge counts grouped by rank difference.The procedure iterates until convergence, maximizing the directed- and undirected-edge terms independently.Source: Inference of rank from network structure · pp. 2–3
  • The likelihood assumes no self-edges and a uniform prior over complete rankings.Source: Inference of rank from network structure · p. 2
  • For implementation, α is represented by a centered Gaussian and β by a squared five-term Fourier cosine series plus a Gaussian peak; MCMC approximates the large ranking sum.The reported results are described as robust to the choice of parametrization.Source: Inference of rank from network structure · p. 3

Results

Across these school friendship networks, reciprocated ties cluster among similarly ranked students, while many unreciprocated ties connect lower-ranked claimants to higher-ranked students. The fitted patterns also show scaling with network degree and a similar long-range pattern across schools.

  • 84 school-pair networks were analyzed, drawn from a study of over 90,000 students in grades 7–12.Source: Results · pp. 3–4
  • For a typical network, reciprocated-friendship probability peaks near equal ranks and is fitted by a Gaussian; the pattern indicates that mutual ties mostly join similarly ranked students.Source: Results · p. 4, Figure 1 · p. 4
  • Unreciprocated-friendship probability has a central peak of similar width, plus a positive-rank-difference tail where lower-ranked people claim higher-ranked friends.Source: Results · p. 4, Figure 1 · p. 4
  • Across 78 networks, central-peak widths for both tie types are approximately proportional to average degree, so they remain roughly constant when rank difference is scaled by degree.Six small networks were excluded from the direct comparison of directed-edge peaks because other model parameters could fit them.Source: Results · p. 4, Figure 2 · p. 4
  • Outside the central peak, unreciprocated-friendship probability has a similar shape across networks when rank difference is rescaled from −1 to 1, with fewer claims toward lower ranks.The pattern suggests a roughly constant relative probability of the lowest-ranked student claiming friendship with the highest-ranked student across school sizes.Source: Results · pp. 4–5, Figure 3 · p. 5
  • Reciprocated friendships cluster at similar ranks, while unreciprocated ties include a substantial tail of lower-ranked students claiming higher-ranked friends.Source: Results · pp. 4–5

Discussion

The inferred ranking tracks popularity and grade, supporting—but not establishing—the conjecture that it reflects social status. Different friendship patterns suggest distinct interpretations of one-way ties.

Discussion

  • Most reciprocated friendships cluster near similar rank; central-peak one-way ties may also be mutual relationships omitted by survey listing limits.Respondents could list only five male and five female friends, potentially leaving some friendships unreported.Source: Discussion · p. 5
  • One-way claims toward higher-ranked individuals may be aspirational, and modest rank gaps are more common than large ones.Source: Discussion · p. 5
  • Rank correlates strongly with in-degree and total degree, and more weakly with out-degree; all three correlations have p-values below 0.001.The in-degree and total-degree associations are strong, while claiming many friends alone is less closely associated with rank.Source: Discussion · p. 5, Figure 4 · p. 6
  • Average rank more than doubles from the youngest to oldest students; sex and ethnicity distributions are close to uniform (KS D < 0.08).The grade association is significant at p < 0.001.Source: Discussion · p. 5, Figure 5 · p. 6
  • Average rank does not differ significantly between eighth- and ninth-graders (p > 0.95), possibly because eighth-graders are temporarily the oldest students in feeder junior highs.Source: Discussion · p. 6
  • In the sample network, rank rises with grade, while most undirected ties connect individuals of similar rank.The displayed ranks average over the model's posterior distribution rather than showing only the maximum-likelihood ranking.Source: Figure 6 · p. 6, Discussion · p. 6

Conclusions

Across US high- and junior-high-school friendship networks, inferred rankings align with age and popularity, but appear independent on average of sex and ethnicity. The authors note limited participant-characteristic data and unresolved explanations for one-way ties.

Conclusions

  • Across a large set of US school networks, most one-way friendship claims run from lower- to higher-ranked students, and inferred ranks correlate with age and popularity.The authors conjecture that these rankings may correspond to social status.Source: Conclusions · p. 6
  • Rankings seem essentially independent on average of participants' sex and ethnicity.Source: Conclusions · p. 6
  • Personal-characteristic data are limited, leaving possible correlations with traits such as academic achievement and future career success untested.Source: Conclusions · p. 7
  • One proposed account is that some one-way ties are new or unstable friendships; cited evidence suggests about half later become reciprocated and half disappear.Source: Conclusions · p. 7
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