Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-11T01:07:50.508Z Has data issue: false hasContentIssue false
  • English
  • Français

THE FRIENDSHIP PARADOX FOR WEIGHTED AND DIRECTED NETWORKS

Part of: Markov processes Graph theory Mathematical sociology

Published online by Cambridge University Press:  18 September 2018

Kenneth S. Berenhaut  and
Hongyi Jiang
Kenneth S. Berenhaut
Affiliation:
Department of Mathematics and Statistics, Wake Forest University, 1834 Wake Forest Road, Winston Salem, NC, USA E-mails: berenhks@wfu.edu; jianh15@wfu.edu
Hongyi Jiang
Affiliation:
Department of Mathematics and Statistics, Wake Forest University, 1834 Wake Forest Road, Winston Salem, NC, USA E-mails: berenhks@wfu.edu; jianh15@wfu.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper studies the friendship paradox for weighted and directed networks, from a probabilistic perspective. We consolidate and extend recent results of Cao and Ross and Kramer, Cutler and Radcliffe, to weighted networks. Friendship paradox results for directed networks are given; connections to detailed balance are considered.

Keywords

configuration modeldetailed balancedirected networksfriendship paradoxrandom walksweighted networks

MSC classification

Primary: 05C81: Random walks on graphs
Secondary: 05C07: Vertex degrees 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 05C20: Directed graphs (digraphs), tournaments 91D30: Social networks 05C80: Random graphs
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 33 , Issue 1 , January 2019 , pp. 136 - 145
Copyright
Copyright © Cambridge University Press 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Aldous, D. & Fill, J. (2002) Reversible markov chains and random walks on graphs.Google Scholar
2.Barrat, A., Barthelemy, M., Pastor-Satorras, R. & Vespignani, A. (2004). The architecture of complex weighted networks. Proceedings of the National Academy of Sciences of the United States of America 101: 3747–3752.Google Scholar
3.Barthélemy, M., Barrat, A., Pastor-Satorras, R. & Vespignani, A. (2005). Characterization and modeling of weighted networks. Physica A: Statistical Mechanics and its Applications 346: 34–43.Google Scholar
4.Bender, E.A. & Canfield, E.R. (1978). The asymptotic number of labeled graphs with given degree sequences. Journal of Combinatorial Theory, Series A 24: 296–307.Google Scholar
5.Cao, Y. & Ross, S.M. (2016). The friendship paradox. Mathematical Scientist 41: 61–64.Google Scholar
6.Chen, N. & Olvera-Cravioto, M. (2013). Directed random graphs with given degree distributions. Stochastic Systems 3: 147–186.Google Scholar
7.Christakis, N.A. & Fowler, J.H. (2010). Social network sensors for early detection of contagious outbreaks. PLOS ONE 5: e12948.Google Scholar
8.Cohen, R., Havlin, S. & Ben-Avraham, D. (2003). Efficient immunization strategies for computer networks and populations. Physical Review Letters 91: 247901.Google Scholar
9.Eom, Y.-H. & Jo, H.-H. (2014). Generalized friendship paradox in complex networks: The case of scientific collaboration. Scientific Reports 4: 4603.Google Scholar
10.Eom, Y.-H. & Jo, H.-H. (2015). Tail-scope: Using friends to estimate heavy tails of degree distributions in large-scale complex networks. Scientific Reports 5: 09752.Google Scholar
11.Feld, S.L. (1991). Why your friends have more friends than you do. American Journal of Sociology 96: 1464–1477.Google Scholar
12.Fotouhi, B., Momeni, N. & Rabbat, M.G. (2014). Generalized friendship paradox: An analytical approach. In International Conference on Social Informatics. Springer, pp. 339–352.Google Scholar
13.Garcia-Herranz, M., Moro, E., Cebrian, M., Christakis, N.A. & Fowler, J.H. (2014). Using friends as sensors to detect global-scale contagious outbreaks. PLOS ONE 9: e92413.Google Scholar
14.Herrera, J.L., Srinivasan, R., Brownstein, J.S., Galvani, A.P. & Meyers, L.A. (2016). Disease surveillance on complex social networks. PLoS Computational Biology 12: e1004928.Google Scholar
15.Hodas, N.O., Kooti, F. & Lerman, K. (2013). Friendship paradox redux: Your friends are more interesting than you, CoRR abs/ 1304.3480.Google Scholar
16.Jackson, M.O. (2016) The friendship paradox and systematic biases in perceptions and social norms, Available at SSRN.Google Scholar
17.Jo, H.-H. & Eom, Y.-H. (2014). Generalized friendship paradox in networks with tunable degree-attribute correlation. Physical Review E 90: 022809.Google Scholar
18.Kim, D.A., Hwong, A.R., Stafford, D., Hughes, D.A., O'Malley, A.J., Fowler, J.H. & Christakis, N.A. (2015). Social network targeting to maximise population behaviour change: a cluster randomised controlled trial. The Lancet 386: 145–153.Google Scholar
19.Kooti, F., Hodas, N.O. & Lerman, K. (2014). Network weirdness: Exploring the origins of network paradoxes. In Eighth International AAAI Conference on Weblogs and Social Media, pp. 266–274.Google Scholar
20.Kramer, J.B., Cutler, J. & Radcliffe, A. (2016). The multistep friendship paradox. American Mathematical Monthly 123: 900–908.Google Scholar
21.Lerman, K., Yan, X. & Wu, X.-Z. (2016). The “majority illusion” in social networks. PLOS ONE 11: e0147617.Google Scholar
22.Lovász, L. (1993). Random walks on graphs. Combinatorics, Paul Erdos is Eighty 2: 1–46.Google Scholar
23.Milo, R., Kashtan, N., Itzkovitz, S., Newman, M.E. & Alon, U. (2003). On the uniform generation of random graphs with prescribed degree sequences, arXiv preprint cond-mat/ 031 2028.Google Scholar
24.Molloy, M. & Reed, B. (1995). A critical point for random graphs with a given degree sequence. Random Structures & Algorithms 6: 161–180.Google Scholar
25.Molloy, M. & Reed, B. (1998). The size of the giant component of a random graph with a given degree sequence. Combinatorics, Probability and Computing 7: 295–305.Google Scholar
26.Momeni, N. & Rabbat, M.G. (2015). Measuring the generalized friendship paradox in networks with quality-dependent connectivity. In Complex Networks VI. Springer, pp. 45–55.Google Scholar
27.Momeni, N. & Rabbat, M. (2016). Qualities and inequalities in online social networks through the lens of the generalized friendship paradox. PLOS ONE 11: e0143633.Google Scholar
28.Newman, M.E.J. (2004). Analysis of weighted networks. Physical Review E 70: 056131.Google Scholar
29.Opsahl, T., Agneessens, F. & Skvoretz, J. (2010). Node centrality in weighted networks: Generalizing degree and shortest paths. Social Networks 32: 245–251.Google Scholar
30.Pons, P. & Latapy, M. (2005). Computing communities in large networks using random walks. In Computer and Information Sciences-ISCIS 2005. Springer, pp. 284–293.Google Scholar
31.Serrano, M.Á., Boguná, M. & Vespignani, A. (2009). Extracting the multiscale backbone of complex weighted networks. Proceedings of the National Academy of Sciences of the United States of America 106: 6483–6488.Google Scholar
32.Singer, Y. (2016). Influence maximization through adaptive seeding. ACM SIGecom Exchanges 15: 32–59.Google Scholar
33.Solé-Ribalta, A., De Domenico, M., Gómez, S., & Arenas, A. (2016). Random walk centrality in interconnected multilayer networks. Physica D: Nonlinear Phenomena 323: 73–79.Google Scholar
34.Wasserman, S. & Faust, K. (1994) Social network analysis: methods and applications, vol. 8. Cambridge University Press.Google Scholar
35.Wu, X. Z., Percus, A. G. & Lerman, K. (2017). Neighbor-Neighbor Correlations Explain Measurement Bias in Networks. Scientific Reports 7(1): 5576.Google Scholar
36.Yook, S.H., Jeong, H., Barabási, A.-L. & Tu, Y. (2001). Weighted evolving networks. Physical Review Letters 86: 5835–5838.Google Scholar
37.Zhou, H. & Lipowsky, R. (2004). Network brownian motion: A new method to measure vertex-vertex proximity and to identify communities and subcommunities. In Computational Science-ICCS 2004. Springer, pp. 1062–1069.Google Scholar
38.Zhou, T., Ren, J., Medo, M. & Zhang, Y.-C. (2007). Bipartite network projection and personal recommendation. Physical Review E 76: 046115.Google Scholar