Quantifying ideological polarization on a network using generalized Euclidean distance
Open accessAbstract
An intensely debated topic is whether political polarization on social media is on the rise. We can investigate this question only if we can quantify polarization, by taking into account how extreme the opinions of the people are, how much they organize into echo chambers, and how these echo chambers organize in the network. Current polarization estimates are insensitive to at least one of these factors: They cannot conclusively clarify the opening question. Here, we propose a measure of ideological polarization that can capture the factors we listed. The measure is based on the generalized Euclidean distance, which estimates the distance between two vectors on a network, e.g., representing people’s opinion. This measure can fill the methodological gap left by the state of the art and leads to useful insights when applied to real-world debates happening on social media and to data from the U.S. Congress.
The claims and statistics of this paper haven’t been checked yet.
Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.
Integrity signals
Registry
Cites
Statistics
Claims the paper makes
The claims and statistics of this paper haven’t been checked yet.
Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.
Claims other papers make about this one
Each is checked against this paper’s own text: does it say what the citing paper says it does?
Recent structural polarization measures such as Betweenness Centrality Controversy, Boundary Polarization, Dipole Polarization, Krackhardt E/I Ratio, and Random Walk Controversy can produce high polarization scores even for random networks, indicating sensitivity to elementary network features such as average degree and network size.
“Other more recent measures, such as Betweenness Centrality Controversy [15], Boundary Polarization [16], Dipole Polarization [29], Krackhardt E/I Ratio [22], and Random Walk Controversy [15], can produce high polarization scores even for random networks, indicating an undesirable sensitivity to elementary network features such as average degree and network size [38].”
From DSP: A Statistically-Principled Structural Polarization Measure (2026) · cites this paper as [15]
Split into 2 atoms — 0 confirmed, 1 partially supported, 0 disputed, 0 misattributed, 1 unverifiable. Expand the claim to see per-atom verdicts.
Experimental — accuracy not yet measuredRWC measures the relative exposure of a user to influential members of their own community versus those of an opposing community.
“At its core, RWC measures the relative exposure of a user to influential members of their own community versus those of an opposing community.”
From DSP: A Statistically-Principled Structural Polarization Measure (2026) · cites this paper as [15]
The Results section identifies RWC as a random-walk controversy measure, and the Synthetic data section says it bisects a network into two communities and detects structural separation. Our copy does not explain RWC as measuring an individual user’s relative exposure to influential members of their own versus the opposing community; it refers calculation details to section S1. (LLM, conf=0.77)
Experimental — accuracy not yet measuredDSP removes the biases found in the original Random Walk Controversy by eliminating the problematic role of predefined influencers.
“Our design explicitly removes the biases found in the original Random Walk Controversy (RWC) [15], by eliminating the problematic role of predefined influencers.”
From DSP: A Statistically-Principled Structural Polarization Measure (2026) · cites this paper as [15]
The Results > Synthetic data section reports a limitation of Random Walk Controversy (RWCG): it misses changes in opinion polarization because it bisects the network into two communities. Our copy does not identify predefined influencers as the source of that limitation or say that removing them eliminates bias. The citing authors’ statement about their own design is not a finding attributed to this source. (LLM, conf=0.78)
Experimental — accuracy not yet measuredRWC is a widely-used structural polarization measure that informs the present work.
“RWC [12, 15] is a widely-used structural polarization measure that informs the present work.”
From DSP: A Statistically-Principled Structural Polarization Measure (2026) · cites this paper as [15]
Self-check
Percentages, sums and reported differences re-computed from the paper’s own numbers.
The self-check needs the paper’s full text.
Cites
Modularity and community structure in networks
Newman, M. E. J. (2006) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.0601602103 · Cited by 12,256 · see citing works
The Evolution of Random Graphs
Bollobás, B. (1986) · Regional conference series in mathematics · doi:10.1090/cbms/062/06 · Cited by 5,719 · see citing works
Epidemic processes in complex networks
Pastor-Satorras, R., Castellano, C., Van Mieghem, P., & Vespignani, A. (2015) · Reviews of Modern Physics · doi:10.1103/revmodphys.87.925 · Cited by 3,697 · see citing works
The Origins and Consequences of Affective Polarization in the United States
Iyengar, S., Lelkes, Y., Levendusky, M., Malhotra, N., & Westwood, S. J. (2019) · Annual Review of Political Science · doi:10.1146/annurev-polisci-051117-073034 · Cited by 3,317 · cited 3× · see citing works
“The political science literature commonly distinguishes between two types of polarization: ideological and affective polarization ( 11 , 12 ).”
Newman, M. (2002) · Physical review. E, Statistical, nonlinear, and soft matter physics · doi:10.1103/physreve.67.026126 · Cited by 3,130 · cited 1× · see citing works
“The assortativity coefficient, for instance, quantifies to what extent individuals are directly linked to like-minded others ( 30 – 32 ).”
Stochastic blockmodels: First steps
Holland, P. W., Laskey, K. B., & Leinhardt, S. (1983) · Social Networks · doi:10.1016/0378-8733(83)90021-7 · Cited by 2,877 · see citing works
The political blogosphere and the 2004 U.S. election: divided they blog
Adamic, L. A., & Glance, N. (2005) · Proceedings of the 3rd international workshop on Link discovery · doi:10.1145/1134271.1134277 · Cited by 2,619 · cited 1× · see citing works
“51. K. Garimella, G. De Francisci Morales, A. Gionis, M. Mathioudakis, Political discourse on social media: Echo chambers, gatekeepers, and the price of bipartisanship, in Proceedings of the 2018 World Wide Web Conference (2018), Lyon, France, 23 to 27 April 2018, pp.”
The echo chamber effect on social media
Cinelli, M., De Francisci Morales, G., Galeazzi, A., Quattrociocchi, W., & Starnini, M. (2021) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.2023301118 · Cited by 2,092 · see citing works
FILTER BUBBLE: Wie wir im Internet entmündigt werden
Pariser, E. (2012) · doi:10.3139/9783446431164 · Cited by 1,849 · see citing works
Political Polarization in the American Public
Fiorina, M. P., & Abrams, S. J. (2008) · Annual Review of Political Science · doi:10.1146/annurev.polisci.11.053106.153836 · Cited by 1,798 · see citing works
Klein, D. J., & Randić, M. (1993) · Journal of Mathematical Chemistry · doi:10.1007/bf01164627 · Cited by 1,384 · see citing works
Abramowitz, A. I., & Saunders, K. L. (2008) · The Journal of Politics · doi:10.1017/s0022381608080493 · Cited by 1,308 · cited 3× · see citing works
“1A): Traditional political science studies of ideological polarization consider whether and how people ’ s ideological leanings diverge ( 11 , 14 , 15 ): If opinions cluster in the moderate center, then polarization is low (example on the left).”
Friedkin, N. E., & Johnsen, E. C. (1990) · Journal of Mathematical Sociology · doi:10.1080/0022250x.1990.9990069 · Cited by 1,168 · see citing works
Partisans without Constraint: Political Polarization and Trends in American Public Opinion
Baldassarri, D., & Gelman, A. (2008) · American Journal of Sociology · doi:10.1086/590649 · Cited by 874 · cited 1× · see citing works
“Second, by analyzing a debate at a time, we disregard the role of ideological consistency ( 13 , 66 – 69 ).”
Political Polarization on Twitter
Conover, M., Ratkiewicz, J., Francisco, M., Goncalves, B., Menczer, F., & Flammini, A. (2021) · Proceedings of the International AAAI Conference on Web and Social Media · doi:10.1609/icwsm.v5i1.14126 · Cited by 626 · see citing works
What do we measure when we measure affective polarization?
Druckman, J. N., & Levendusky, M. S. (2019) · Public Opinion Quarterly · doi:10.1093/poq/nfz003 · Cited by 615 · see citing works
The role of (social) media in political polarization: A systematic review
Kubin, E., & von Sikorski, C. (2021) · Annals of the International Communication Association · doi:10.1080/23808985.2021.1976070 · Cited by 610 · cited 4× · see citing works
“The political science literature commonly distinguishes between two types of polarization: ideological and affective polarization ( 11 , 12 ).”
InfluentialMass polarization: Manifestations and measurements
Lelkes, Y. (2016) · Public Opinion Quarterly · doi:10.1093/poq/nfw005 · Cited by 591 · cited 4× · see citing works
“Affective polarization describes in-group favoritism and out-group hostility, and it is thus concerned with the affective attitude toward others depending on their opinions ( 12 , 13 , 16 ).”
InfluentialBoxell, L., Gentzkow, M., & Shapiro, J. M. (2017) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.1706588114 · Cited by 467 · see citing works
Modeling echo chambers and polarization dynamics in social networks
Baumann, F., Lorenz-Spreen, P., Sokolov, I. M., & Starnini, M. (2020) · Physical Review Letters · doi:10.1103/physrevlett.124.048301 · Cited by 427 · cited 1× · see citing works
“1B): More recent approaches have emphasized the role of social connections, and especially homo-phily, i.e., the connections between like-minded individuals ( 23 , 53 – 56 ).”
Dynamic Public Opinion: Communication Effects over Time
Chong, D., & Druckman, J. N. (2010) · American Political Science Review · doi:10.1017/s0003055410000493 · Cited by 396 · see citing works
Quattrociocchi, W., Scala, A., & Sunstein, C. R. (2016) · SSRN Electronic Journal · doi:10.2139/ssrn.2795110 · Cited by 389 · see citing works
Quantifying controversy on social media
Garimella, K., Morales, G. D. F., Gionis, A., & Mathioudakis, M. (2018) · ACM Transactions on Social Computing · doi:10.1145/3140565 · Cited by 348 · see citing works
The Nature of Affective Polarization: Disentangling Policy Disagreement from Partisan Identity
Dias, N., & Lelkes, Y. (2022) · American Journal of Political Science · doi:10.1111/ajps.12628 · Cited by 338 · cited 1× · see citing works
“Although the two types of polarization can be mutually reinforcing ( 17 – 19 ), ideological polarization and affective polarization are distinct concepts, both in terms of theory and empirical measurement ( 11 , 12 ).”
Political Discourse on Social Media: Echo Chambers, Gatekeepers, and the Price of Bipartisanship
Garimella, K., De Francisci Morales, G., Gionis, A., & Mathioudakis, M. (2018) · Proceedings of the 2018 World Wide Web Conference on World Wide Web - WWW '18 · doi:10.1145/3178876.3186139 · Cited by 335 · see citing works
Why Can't We All Just Get Along? The Reality of a Polarized America
Abramowitz, A. I., & Saunders, K. L. (2005) · The Forum · doi:10.2202/1540-8884.1076 · Cited by 271 · see citing works
The Polarization of Contemporary American Politics
Hare, C., & Poole, K. T. (2014) · Polity · doi:10.1057/pol.2014.10 · Cited by 247 · cited 1× · see citing works
“Second, by analyzing a debate at a time, we disregard the role of ideological consistency ( 13 , 66 – 69 ).”
Efficient Monte Carlo and greedy heuristic for the inference of stochastic block models
Peixoto, T. P. (2014) · Physical Review E · doi:10.1103/physreve.89.012804 · Cited by 235 · see citing works
Measuring political polarization: Twitter shows the two sides of Venezuela
Morales, A. J., Borondo, J., Losada, J. C., & Benito, R. M. (2015) · Chaos An Interdisciplinary Journal of Nonlinear Science · doi:10.1063/1.4913758 · Cited by 219 · see citing works
Link recommendation algorithms and dynamics of polarization in online social networks
Santos, F. P., Yphtach Lelkes, B., & Levin, S. A. (2021) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.2102141118 · Cited by 216 · cited 1× · see citing works
“1B): More recent approaches have emphasized the role of social connections, and especially homo-phily, i.e., the connections between like-minded individuals ( 23 , 53 – 56 ).”
The Policy Basis of Measured Partisan Animosity in the United States
Orr, L., & Huber, G. A. (2019) · American Journal of Political Science · doi:10.1111/ajps.12498 · Cited by 208 · see citing works
The Rise of Partisanship and Super-Cooperators in the U.S. House of Representatives
Andris, C., Lee, D., Hamilton, M. J., Martino, M., Gunning, C. E., & Selden, J. A. (2015) · PLoS ONE · doi:10.1371/journal.pone.0123507 · Cited by 197 · see citing works
A Measure of Polarization on Social Media Networks Based on Community Boundaries
Guerra, P., Meira, W., Cardie, C., & Kleinberg, R. (2021) · Proceedings of the International AAAI Conference on Web and Social Media · doi:10.1609/icwsm.v7i1.14421 · Cited by 188 · cited 1× · see citing works
“Alternative methods explicitly divide the network into two communities to determine how well they are separated from each other ( 36 – 41 ).”
Measuring and moderating opinion polarization in social networks
Matakos, A., Terzi, E., & Tsaparas, P. (2017) · Data Mining and Knowledge Discovery · doi:10.1007/s10618-017-0527-9 · Cited by 164 · cited 1× · see citing works
“This measure assumes that each individual has both an internal opinion and an expressed opinion that is determined by their own internal opinion and the surrounding opinions in the social network ( 44 , 45 ).”
Quantifying echo chamber effects in information spreading over political communication networks
Cota, W., Ferreira, S. C., Pastor‐Satorras, R., & Starnini, M. (2019) · LA Referencia (Red Federada de Repositorios Institucionales de Publicaciones Científicas) · Cited by 152 · see citing works
Quantifying social organization and political polarization in online platforms
Waller, I., & Anderson, A. (2021) · Nature · doi:10.1038/s41586-021-04167-x · Cited by 149 · cited 1× · see citing works
“Some methods consider exclusively opinions ( 25 ) or reduce the complexity of the structure ( 26 ), which we think does not allow to properly capture what we understand as polarization.”
Quantifying echo chamber effects in information spreading over political communication networks
Cota, W., Ferreira, S. C., Pastor-Satorras, R., & Starnini, M. (2019) · EPJ Data Science · doi:10.1140/epjds/s13688-019-0213-9 · Cited by 146 · see citing works
Minimizing Polarization and Disagreement in Social Networks
Musco, C., Musco, C., & Tsourakakis, C. E. (2018) · International World Wide Web Conference (WWW) · doi:10.1145/3178876.3186103 · Cited by 132 · see citing works
Falling into the Echo Chamber: The Italian Vaccination Debate on Twitter
Cossard, A., De Francisci Morales, G., Kalimeri, K., Mejova, Y., Paolotti, D., & Starnini, M. (2020) · Proceedings of the International AAAI Conference on Web and Social Media · doi:10.1609/icwsm.v14i1.7285 · Cited by 126 · cited 1× · see citing works
“Alternative methods explicitly divide the network into two communities to determine how well they are separated from each other ( 36 – 41 ).”
Policy over party: comparing the effects of candidate ideology and party on affective polarization
Yphtach Lelkes, B. (2019) · Political Science Research and Methods · doi:10.1017/psrm.2019.18 · Cited by 118 · see citing works
Echo Chambers in Parliamentary Twitter Networks: The Catalan Case
Valle, M. E. D., & Borge, R. (2018) · DOAJ (DOAJ: Directory of Open Access Journals) · Cited by 111 · see citing works
A sign of the times? Weak and strong polarization in the U.S. Congress, 1973–2016
Neal, Z. P. (2020) · Social Networks · doi:10.1016/j.socnet.2018.07.007 · Cited by 109 · see citing works
Multiscale mixing patterns in networks
Peel, L., Delvenne, J., & Lambiotte, R. (2018) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.1713019115 · Cited by 104 · cited 1× · see citing works
“However, local measures are myopic to the overall structure and would return the same estimations even if opinions are distributed in radically different mesoscale structures such as communities ( 35 ).”
Polarization and tipping points
Macy, M. W., Ma, M., Tabin, D. R., Gao, J., & Szymanski, B. K. (2021) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.2102144118 · Cited by 102 · see citing works
Characterizing polarization in online vaccine discourse—A large-scale study
Mønsted, B., & Lehmann, S. (2022) · PLoS ONE · doi:10.1371/journal.pone.0263746 · Cited by 95 · see citing works
Quantifying Political Polarity Based on Bipartite Opinion Networks
Akoglu, L. (2014) · Proceedings of the International AAAI Conference on Web and Social Media · doi:10.1609/icwsm.v8i1.14524 · Cited by 79 · see citing works
Lx = b Laplacian Solvers and Their Algorithmic Applications
Vishnoi, N. K. (2013) · Foundations and Trends® in Theoretical Computer Science · doi:10.1561/0400000054 · Cited by 77 · cited 2× · see citing works
“We plan to fix this issue in future work by using Laplacian solvers ( 71 , 72 ).”
Discovering Polarized Communities in Signed Networks
Bonchi, F., Galimberti, E., Gionis, A., et al. (2019) · Institutional Research Information System University of Turin (University of Turin) · doi:10.1145/3357384.3357977 · Cited by 76 · cited 2× · see citing works
“While the measure of ideological polarization relies on data about the opinions of people, affective polarization also requires information about the valence of their relationships ( 20 – 22 ).”
Modelling how social network algorithms can influence opinion polarization
de Arruda, H. F., Cardoso, F. M., de Arruda, G. F., Hernández, A. R., Costa, L. D. F., & Moreno, Y. (2021) · Information Sciences · doi:10.1016/j.ins.2021.12.069 · Cited by 75 · see citing works
The Effect of People Recommenders on Echo Chambers and Polarization
Cinus, F., Minici, M., Monti, C., & Bonchi, F. (2022) · Proceedings of the International AAAI Conference on Web and Social Media · doi:10.1609/icwsm.v16i1.19275 · Cited by 70 · see citing works
Is Public Opinion Stable? Resolving the Micro/Macro Disconnect in Studies of Public Opinion
Druckman, J. N., & Leeper, T. J. (2012) · Daedalus · doi:10.1162/daed_a_00173 · Cited by 62 · see citing works
Quantifying and Minimizing Risk of Conflict in Social Networks
Chen, X., Lijffijt, J., & Bie, T. D. (2018) · ACM SIGKDD Conference on Knowledge Discovery and Data Mining (KDD) · doi:10.1145/3219819.3220074 · Cited by 59 · see citing works
The emergence and perils of polarization
Baldassarri, D., & Page, S. E. (2021) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.2116863118 · Cited by 50 · cited 1× · see citing works
“1B): More recent approaches have emphasized the role of social connections, and especially homo-phily, i.e., the connections between like-minded individuals ( 23 , 53 – 56 ).”
Valle, M. E. D., Broersma, M., & Ponsioen, A. (2021) · Social Science Computer Review · doi:10.1177/0894439320987569 · Cited by 50 · see citing works
Automatic controversy detection in social media: A content-independent motif-based approach
Coletto, M., Garimella, K., Gionis, A., & Lucchese, C. (2017) · Online Social Networks and Media · doi:10.1016/j.osnem.2017.10.001 · Cited by 49 · see citing works
Easy and difficult objective functions for max cut
McCormick, S. T., Rao, M. R., & Rinaldi, G. (2003) · Mathematical Programming · doi:10.1007/s10107-002-0328-8 · Cited by 49 · see citing works
Lu, H., Caverlee, J., & Niu, W. (2015) · doi:10.1145/2806416.2806573 · Cited by 45 · see citing works
Pearson correlations on complex networks
Coscia, M. (2021) · Journal of Complex Networks · doi:10.1093/comnet/cnab036 · Cited by 35 · see citing works
Self-citationDepolarization of echo chambers by random dynamical nudge
Currin, C. B., Vera, S. V., & Khaledi‐Nasab, A. (2022) · Scientific Reports · doi:10.1038/s41598-022-12494-w · Cited by 32 · see citing works
RePBubLik: Reducing Polarized Bubble Radius with Link Insertions
Haddadan, S., Menghini, C., Riondato, M., & Upfal, E. (2021) · ACM International Conference on Web Search and Data Mining (WSDM) · doi:10.1145/3437963.3441825 · Cited by 31 · see citing works
Huang, Z., Silva, A., & Singh, A. K. (2022) · ACM International Conference on Web Search and Data Mining (WSDM) · doi:10.1145/3488560.3498454 · Cited by 27 · see citing works
Quantifying Polarization on Twitter: The Kavanaugh Nomination
Darwish, K. M. (2019) · Lecture notes in computer science · doi:10.1007/978-3-030-34971-4_13 · Cited by 25 · see citing works
Variance and Covariance of Distributions on Graphs
Devriendt, K., Martín-Gutíerrez, S., & Lambiotte, R. (2022) · SIAM Review · doi:10.1137/20m1361328 · Cited by 25 · see citing works
Self-citationCoscia, M., & Rossi, L. (2022) · PLoS ONE · doi:10.1371/journal.pone.0263184 · Cited by 24 · see citing works
Self-citationA Motif-Based Approach for Identifying Controversy
Coletto, M., Garimella, K., Gionis, A., & Lucchese, C. (2017) · Proceedings of the International AAAI Conference on Web and Social Media · doi:10.1609/icwsm.v11i1.14949 · Cited by 23 · cited 1× · see citing works
“Other approaches rely on local network measures to evaluate the structure of interactions in a network ( 27 – 29 ).”
Emamgholizadeh, H., Nourizade, M., Tajbakhsh, M. S., Hashminezhad, M., & Hasheminezha, F. N. E. M. (2020) · Social Network Analysis and Mining · doi:10.1007/s13278-020-00703-1 · Cited by 21 · see citing works
Controversy Detection: A Text and Graph Neural Network Based Approach
Benslimane, S., Azé, J., Bringay, S., et al. (2021) · Lecture notes in computer science · doi:10.1007/978-3-030-90888-1_26 · Cited by 13 · see citing works
Generalized Euclidean Measure to Estimate Network Distances
Coscia, M. (2020) · Proceedings of the International AAAI Conference on Web and Social Media · doi:10.1609/icwsm.v14i1.7284 · Cited by 9 · see citing works
Self-citationDiscovering communities of community discovery
Coscia, M. (2019) · IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining (ASONAM) · doi:10.1145/3341161.3342860 · Cited by 8 · see citing works
Self-citationThe Node Vector Distance Problem in Complex Networks
Coscia, M., Gómez-Liévano, A., McNerney, J., et al. (2020) · ACM Computing Surveys · doi:10.1145/3416509 · Cited by 8 · see citing works
Self-citation2020 United States Presidential Election
Wrubel, L., & Kerchner, D. (2020) · Harvard Dataverse · doi:10.7910/dvn/ucjuuz · Cited by 1 · see citing works
Flagged first, then by citations
Research works printed but not linked: 11
- Y. Mejova A. X. Zhang N. Diakopoulos C. Castillo Controversy and sentiment in online news. arXiv:1409.8152 [cs.CY] (29 September 2014). · arXiv
- M. Gentzkow Toulouse Network for Information Technology Whitepaper (2016) pp. 1–23.
- R.-C. Tzeng B. Ordozgoiti A. Gionis Discovering conflicting groups in signed networks in Advances in Neural Information Processing Systems 33 (NeurIPS 2020) virtual 6 to 12 December 2020.
- C. Musco I. Ramesh J. Ugander R. T. Witter How to quantify polarization in models of opinion dynamics. arXiv:2110.11981 [cs.SI] (26 October 2021). · arXiv
- M. Coscia The atlas for the aspiring network scientist. arXiv:2101.00863 [cs.CY] (8 February 2021). · arXiv
- K. Garimella “Polarization on Social Media ” thesis Aalto University Finland (2018); http://urn.fi/URN:ISBN:978-952-60-7833-5
- J. Zaller The Nature and Origins of Mass Opinion (Cambridge Univ. Press 2012).
- K. T. Poole H. Rosenthal Congress: A Political-Economic History of Roll Call Voting (Oxford Univ. Press on Demand 2000).
- A. Abramowitz The Disappearing Center: Engaged Citizens Polarization and American Democracy (Yale Univ. Press 2010).
- M. Coscia ACM Transactions on Knowledge Discovery from Data (TKDD) (2022).
- R. I. Kondor J. Lafferty Diffusion Kernels on Graphs and Other Discrete Structures in Proceedings of the 19th International Conference on Machine Learning (ICML’2002) Sydney Australia 8 to 12 July 2002 vol. 2002 pp. 315–322.
Cited by
Sorted by citations
S3: Social-network Simulation System with Large Language Model-Empowered Agents
Gao, C., Lan, X., Lu, Z., Mao, J., Piao, J., Wang, H., Jin, D., & Li, Y. (2023) · doi:10.2139/ssrn.4607026 · Cited by 113 · see citing works
A review of deep learning in dentistry
Huang, C., Wang, J., Wang, S., & Zhang, Y. (2023) · Neurocomputing · doi:10.1016/j.neucom.2023.126629 · Cited by 112 · see citing works
Multidimensional political polarization in online social networks
Peralta, A. F., Morales, P. R., Kertész, J., & Íñiguez, G. (2024) · Physical Review Research · doi:10.1103/physrevresearch.6.013170 · Cited by 26 · see citing works
“Producing estimates of positions ϕ via an embedding with CA has the advantage introducing only linear transformations, in contrast with non-linear embedding methods such as UMAP of t-SNE [9].”
Plikynas, D., Rizgelienė, I., & Korvel, G. (2025) · IEEE Access · doi:10.1109/access.2025.3530688 · Cited by 28 · see citing works
“Studies propose methodologies for identifying narratives, estimating underlying dynamics, and quantifying polarization levels in social networks, considering opinion variations, community assortativity, and the interplay between opinions and network structures [119].”
InfluentialEffect of collaborative-filtering-based recommendation algorithms on opinion polarization
Bellina, A., Castellano, C., Pineau, P., Iannelli, G., & Marzo, G. D. (2023) · Physical review. E · doi:10.1103/physreve.108.054304 · Cited by 24 · see citing works
Polarization and multiscale structural balance in signed networks
Talaga, S., Stella, M., Swanson, T., et al. (2023) · Communications Physics · doi:10.1038/s42005-023-01467-8 · Cited by 24 · see citing works
“It can result in a sharp divergence of popular beliefs or attitudes (ideological polarization) as well as in-group favouritism and out-group hostility (affective polarization) [22].”
BackgroundQuantum-Mechanical Modelling of Asymmetric Opinion Polarisation in Social Networks
Maksymov, I. S., & Pogrebna, G. (2024) · Information · doi:10.3390/info15030170 · Cited by 23 · see citing works
Maksymov, I. S., & Pogrebna, G. (2024) · Information · doi:10.3390/info15070413 · Cited by 14 · see citing works
Depolarization of opinions on social networks through random nudges
Pal, R., Kumar, A., & Santhanam, M. S. (2023) · Physical review. E · doi:10.1103/physreve.108.034307 · Cited by 12 · see citing works
Quantifying polarization in online political discourse
Muñoz, P., Bellogín, A., Barba-Rojas, R., & Díez, F. (2024) · EPJ Data Science · doi:10.1140/epjds/s13688-024-00480-3 · Cited by 19 · see citing works
“[51] introduce the Generalized Euclidean (GE) algorithm, based on a generalization of the Euclidean Distance metric to measure the distance between nodes and offer a polarization index based on such distance (for users belonging to only two communities, thus this algorithm is also limited as some of…”
MethodSaeedian, M., Tu, C., Menegazzo, F., D’Odorico, P., Azaele, S., & Suweis, S. S. (2024) · New Journal of Physics · doi:10.1088/1367-2630/ad67fe · Cited by 11 · see citing works
“Quantifying the degree of social polarization is a key issue [30, 31, 52, 53] we have tackled in this work.”
Polarization in Decentralized Online Social Networks
La Cava, L., Mandaglio, D., & Tagarelli, A. (2024) · ACM Web Science Conference · doi:10.1145/3614419.3644013 · Cited by 11 · see citing works
Charting multidimensional ideological polarization across demographic groups in the USA
Ojer, J., Carcamo, D. P., Pastor‐Satorras, R., & Starnini, M. (2025) · Nature Human Behaviour · doi:10.1038/s41562-025-02251-0 · Cited by 14 · see citing works
“To discern these underlying factors, researchers from different disciplines, ranging from political and social science to physics and computer science, have recently embarked on the endeavour of quantifying ideological polarization 33–35 .”
SGS: SqueezeNet-guided Gaussian-kernel SVM for COVID-19 Diagnosis
Shi, F., Wang, J., & Govindaraj, V. (2024) · Mobile Networks and Applications · doi:10.1007/s11036-023-02288-3 · Cited by 9 · see citing works
“The Gaussian kernel formula [33] is as follows: In this formula, G () represents the Gaussian kernel, x and x ′ are the input feature vectors, || x − x � | 2 denotes the squared Euclidean distance [34] between the vectors, and σ controls the width or spread of the kernel.”
K-Means Based Bee Colony Optimization for Clustering in Heterogeneous Sensor Network
Modey, P., Abdul-Salaam, G., Freeman, E., et al. (2024) · Sensors · doi:10.3390/s24237603 · Cited by 7 · see citing works
Competitive interactions shape brain dynamics and computation across speciesPreprint
Luppi, A. I., Perl, Y. S., Vohryzek, J., Mediano, P. A. M., Rosas, F. E., Milisav, F., Suárez, L. E., Gini, S., Gutierrez‐Barragan, D., Gozzi, A., et al. (2024) · bioRxiv (Cold Spring Harbor Laboratory) · doi:10.1101/2024.10.19.619194 · Cited by 7 · see citing works
Lyua, S., & Jiao, Z. (2024) · IEEE Internet of Things Journal · doi:10.1109/jiot.2024.3486714 · Cited by 6 · see citing works
“The distance calculation in the K -means algorithm uses the Euclidean distance, as shown in (2) [33] ( In (2), x i represents the i th data point and μ j represents the j th cluster center.”
Zhang, C., Wang, X., Min, Y., Yu, S., Wu, Y., Xuan, Q., & Fu, C. (2025) · Physica A Statistical Mechanics and its Applications · doi:10.1016/j.physa.2025.130487 · Cited by 6 · see citing works
Neural discovery of balance-aware polarized communities
Gullo, F., Mandaglio, D., & Tagarelli, A. (2024) · Machine Learning · doi:10.1007/s10994-024-06581-4 · Cited by 6 · see citing works
“…and affective polarization (Lelkes, 2016): the former refers to increased ideological divergence and reduced dialogue among individuals with differing views, whereas the latter focuses on affective attitude that individuals show toward others based on their opinions (Hohmann et al., 2023).”
Echo chamber formation sharpened by priority users
Ferraz de Arruda, H., Andrade Oliveira, K., & Moreno, Y. (2024) · iScience · doi:10.1016/j.isci.2024.111098 · Cited by 6 · see citing works
Unveiling the drivers of active participation in social media discourse
Baqir, A., Chen, Y., Diaz-Diaz, F., Kiyak, S., Louf, T., Morini, V., Pansanella, V., Torricelli, M., & Galeazzi, A. (2025) · Scientific Reports · doi:10.1038/s41598-025-88117-x · Cited by 5 · see citing works
Depolarizing power of anticonformity
Lipiecki, A., & Sznajd-Weron, K. (2025) · Expert Systems with Applications · doi:10.1016/j.eswa.2025.127879 · Cited by 5 · see citing works
Quante, L., Stechemesser, A., Hödtke, D., & Levermann, A. (2024) · Humanities and Social Sciences Communications · doi:10.1057/s41599-024-03359-w · Cited by 5 · see citing works
“Our fi ndings align with empirical evidence for opinion drift that leads to stronger societal polarisation such as the political polarisation of the United States in the last decades (Geiger, 2014; Hohmann et al., 2023).”
BackgroundInfluentialCompetitive interactions shape mammalian brain network dynamics and computation
Luppi, A. I., Perl, Y. S., Vohryzek, J., Ali, H., Mediano, P. A. M., Rosas, F. E., Milisav, F., Suárez, L. E., Gini, S., Gutierrez‐Barragan, D., et al. (2026) · Nature Neuroscience · doi:10.1038/s41593-026-02205-3 · Cited by 5 · see citing works
Tian, X., Yuan, B., & Zheng, P. (2025) · Electric Power Systems Research · doi:10.1016/j.epsr.2025.111774 · Cited by 4 · see citing works
Node attribute analysis for cultural data analytics: a case study on Italian XX–XXI century music
Coscia, M. (2024) · Applied Network Science · doi:10.1007/s41109-024-00669-5 · Cited by 3 · see citing works
“…Hristova 2016; Karjus et al. 2023); to sociology—where different ideas and opinions distribute over a social network as node attributes (Bail 2014; Hohmann et al. 2023); to linguistics—with different people in a social network producing content in different languages (Ronen et al. 2014); to…”
BackgroundSelf-citationMulti‐Robot Autonomous Exploration in Unknown Environments With Dynamic Obstacles
Chu, J., Lv, X., Yue, Q., et al. (2025) · IET Cyber-Systems and Robotics · doi:10.1049/csy2.70019 · Cited by 3 · see citing works
Zhang, X., Wang, Q., Song, F., Wang, F., & Qu, C. (2026) · IEEE Transactions on Network Science and Engineering · doi:10.1109/tnse.2026.3654942 · Cited by 3 · see citing works
“Instead of using the variance of opinions, we follow Hohmann [67] and define polarization in terms of pairwise opinion differences to better reflect interaction-driven dynamics: where A ij indicates the presence of an edge between nodes i and j , and | E | represents the total number of edges in the…”
Mohliver, A. C., & Raines, G. W. (2024) · SSRN Electronic Journal · doi:10.2139/ssrn.4923070 · Cited by 3 · see citing works
Key nodes of misinformation source inference: A message-passing-based approach
Yu, X., Nie, Y., Li, W., Lin, T., Chen, Y., Gao, F., & Wang, W. (2024) · International Journal of Modern Physics C · doi:10.1142/s0129183124501468 · Cited by 2 · see citing works
KGRec: A knowledge graph attention-based model for recommender system
Hoan, T. D., & Hùng, B. T. (2026) · PLoS ONE · doi:10.1371/journal.pone.0344585 · Cited by 2 · see citing works
“For each triple ( h , r , t ) , we compute its plausibility score using the squared Euclidean distance [27].”
Quantifying metadata relevance to network block structure using description length
Mangold, L., & Roth, C. (2024) · Communications Physics · doi:10.1038/s42005-024-01819-y · Cited by 2 · see citing works
Social networks and experienced inequality
Mamunuru, S. M., Shrivastava, A., & Jayadev, A. (2024) · Journal of Economic Behavior & Organization · doi:10.1016/j.jebo.2024.106799 · Cited by 2 · see citing works
Dutta, S., Goswami, S., Debnath, S., Adhikary, S., & Majumder, A. (2025) · Computers in Biology and Medicine · doi:10.1016/j.compbiomed.2025.109795 · Cited by 2 · see citing works
Polaris: Sampling from the Multigraph Configuration Model with Prescribed Color Assortativity
Preti, G., Riondato, M., Gionis, A., & Morales, G. D. F. (2025) · ACM International Conference on Web Search and Data Mining (WSDM) · doi:10.1145/3701551.3703560 · Cited by 2 · see citing works
How opinion variation among in-groups can skew perceptions of ideological polarization
Steiglechner, P., Smaldino, P. E., & Merico, A. (2025) · PNAS Nexus · doi:10.1093/pnasnexus/pgaf184 · Cited by 2 · see citing works
“Factors, such as social segregation (62, 63), interrupted or biased flow of information (64, 65), cognitive biases (66), or affective polarization (67 – 69), might further amplify perceptions of polarization (70), but they are not necessarily required for the phenomenon to emerge.”
Network analysis reveals news press landscape and asymmetric user polarization
Lee, B., Ryu, H.-S., Lee, J. K., Jeong, H., & Kim, B. J. (2025) · Physica A Statistical Mechanics and its Applications · doi:10.1016/j.physa.2025.130842 · Cited by 5 · see citing works
Estimating affective polarization on a social network
Hohmann, M., & Coscia, M. (2025) · PLoS ONE · doi:10.1371/journal.pone.0328210 · Cited by 2 · see citing works
“We rely on the recently introduced Generalized Euclidean (GE) polarization measure 𝛿 G , o [23] to quantify this component.”
Self-citationFrom Divide to Debate: A Methodological Approach to Dismantling Polarization in Social NetworksPreprint
Mustafa, Z. U., Amir, M., Mustafa, M., & Anwar, M. A. (2023) · Preprints.org · doi:10.20944/preprints202309.1350.v1 · Cited by 2 · see citing works
Negative ties highlight hidden extremes in social media polarization
Candellone, E., Babul, S., Togay, Ö., Bovet, A., & Garcia‐Bernardo, J. (2025) · Network Science · doi:10.1017/nws.2025.10006 · Cited by 1 · see citing works
“…information spread plays a key role in facilitating an increased polarization around controversial issues and amplifying political divisions and conflicts ( garimella2017reducing ; adamic2005political ; conover2011political ; hohmannQuantifyingIdeologicalPolarization2023a ;…”
Analyzing polarization among Spanish political elites using machine learning techniques
Ansia, D., Folgado, M. G., & Veronica Sanz, B. (2025) · Journal of Computational Social Science · doi:10.1007/s42001-025-00442-3 · Cited by 1 · see citing works
Bassolas, A., Massachs, J., Cozzo, E., & Vicens, J. (2025) · Royal Society Open Science · doi:10.1098/rsos.241974 · Cited by 1 · see citing works
Ipekci, A. M., Filsinger, M., Buitrago‐García, D., Betancourt, C. I. K., Liwanag, H. J., Frahsa, A., & Low, N. N. (2025) · SSM - Population Health · doi:10.1016/j.ssmph.2025.101891 · Cited by 1 · see citing works
“Second, the way that the measures are derived for example from text analyses or network analyses of Twitter/X posts (Hohmann et al., 2023), cannot be compared with self-report or aggregated responses from surveys or elections.”
A survey on opinion dynamics: From rule-based models to data-driven and hybrid approaches
Wang, J., Zhang, J., Dong, Z., & Jin, L. (2026) · Neurocomputing · doi:10.1016/j.neucom.2026.133705 · Cited by 1 · see citing works
Quantifying opinion homophily in online social networks from a bounded confidence perspective
Luan, Y., Ancona, C., Bernardo, C., Pansanella, V., Iudice, F. L., Rossetti, G., Vasca, F., Wu, X., & Altafini, C. (2026) · Communications Physics · doi:10.1038/s42005-026-02760-y · Cited by 1 · see citing works
“In the topic-speci fi c follow graphs analyzed here, we observe zero reciprocity (no mutual fol-lowerpairs);thisisapropertyoftherestrictedgraphsinducedbythedataset construction 32 and should not be interpreted as a general statement about reciprocity on Twitter(see Supplementary Note 6).”
Anatomy of elite and mass polarization in social networks
Salloum, A., Chen, T. H. Y., & Kivelä, M. (2025) · Network Science · doi:10.1017/nws.2025.10010 · Cited by 1 · see citing works
“Polarization is often assumed to significantly impact social network structures, giving rise to two internally tightly-knit groups with sparse connections between them [22, 27].”
Water Level Prediction of Firewater System Based on Improved Hybrid LSTM Algorithm
Li, W., & Gao, T. (2024) · IEEE Access · doi:10.1109/access.2024.3444189 · Cited by 2 · see citing works
Optimizing Diverse Information Exposure in Social Graphs
Colin, J., & Maniu, S. (2024) · IEEE International Conference on Big Data (BigData) · doi:10.1109/bigdata62323.2024.10825032 · Cited by 1 · see citing works
S3: Social-network Simulation System with Large Language Model-Empowered AgentsPreprint
Gao, C., Lan, X., Lu, Z., et al. (2025) · SSRN Electronic Journal · doi:10.2139/ssrn.5604712 · Cited by 1 · see citing works
Maksymov, I. S., & Pogrebna, G. (2023) · doi:10.31234/osf.io/dnmtv
Algorithmic Intervention: Addressing Social Media Polarization Via Genetic AlgorithmsPreprint
Mustafa, Z. U. (2024) · SSRN Electronic Journal · doi:10.2139/ssrn.4814800
Matsui, A., Miyazaki, K., & Murayama, T. (2024) · Proceedings of the International AAAI Conference on Web and Social Media · doi:10.1609/icwsm.v18i1.31370
“…examine the dynamic behavior or trends of political information supply, while the recent literature highlights the significance of comprehending political information consumption in digital platforms such as political prioritization (Hohmann, Devriendt, and Coscia 2023; Waller and Anderson 2021).”
BackgroundDeep graph clustering network by community prediction-guided augmentation
Zheng, Z., Shang, Y., Sun, C., et al. (2024) · International Symposium on Computer Applications and Information Systems (ISCAIS) · doi:10.1117/12.3035234
“Before refining our selection with community pseudo-labels, we rigorously assess the potential triangles using cosine similarity[22] and Euclidean distance[3] metrics among the nodes.”
Li, Y., Zhang, X., Liang, Y., & Li, Q. (2025) · Entropy · doi:10.3390/e27050473
“Hohmann et al. [29] proposed a method to quantify ideological polarization using the generalized Euclidean distance, which estimates the dif‑ ficulty of traversing a network from one opinion to another, taking into account both opin‑ ion divergence and community structure.”
Art Intelligence System (AIS) Based on Edge Information Visualization and Image Reconstruction
Wu, J. H. (2025) · International Conference on Inventive Computing and Informatics (ICICI) · doi:10.1109/icici65870.2025.11069574
Yu, J. (2025) · International Conference on Inventive Research in Computing Applications (ICIRCA) · doi:10.1109/icirca65293.2025.11089737
“For the similarity metric of motion vectors, this study chooses to use the Euclidean distance [14]-[15] measure.”
Lee, H., & Rojas, H. (2025) · Social Media + Society · doi:10.1177/20563051251399007 · Cited by 1 · see citing works
“Social media, by creating echo chambers and facilitating interactions among individuals with similar attitudes and ideologies, can allow exposure to extreme viewpoints and further endorse them (Flaxman et al., 2016; Hohmann et al., 2023; McPherson et al., 2001; Qureshi et al., 2020; Sunstein, 2018).”
Additional file 2 of Quantifying polarization in online political discourse
Muñoz, P., Bellogín, A., Barba-Rojas, R., & Díez, F. (2024) · Figshare · doi:10.6084/m9.figshare.26729962
Additional file 1 of Quantifying polarization in online political discourse
Muñoz, P., Bellogín, A., Barba-Rojas, R., & Díez, F. (2024) · Figshare · doi:10.6084/m9.figshare.26729959
Xing, Y., Zhang, Z., Chang, H., Shang, D., & Gupta, B. (2026) · Journal of Knowledge Management · doi:10.1108/jkm-08-2025-1196
Disconnect between the public face and the voting behavior of political representatives
Andersen, C. I., & Coscia, M. (2026) · Applied Network Science · doi:10.1007/s41109-026-00781-8
Self-citationFulay, S. P., & Roy, D. K. (2026) · Scientific Reports · doi:10.1038/s41598-026-43342-w · Cited by 2 · see citing works
“While much attention has been devoted to studying the rhetoric of politicians on social media 9,10,13,14 , recent interest has expanded to include a broader spectrum of influential figures 24 .”
Causal Interpretation of DBSCAN Algorithm: A Dynamic Modeling for Epsilon Estimation
Garcia-Sanchez, K., Pérez-Ramos, J.-L., Ramírez-Rosales, S., et al. (2026) · Entropy · doi:10.3390/e28040452
“First, Euclidean distance is defined as a metric that estimates the distance between two points within a system in R 2 [21], represented in Equation (1): Secondly, the Nearest Neighbors algorithm can be defined as follows: given a finite dataset S in a Euclidean space E d and a query q , Nearest…”
Unveiling camouflaged malicious accounts via unsupervised behavior-affinity dual-graph learning
Wang, Z., Ji, L., & Liu, S. (2026) · Journal of King Saud University - Computer and Information Sciences · doi:10.1007/s44443-026-00762-y
“Recent studies and surveys further indicate that malicious online activity is increas-0123456789 ingly coordinated, adaptive, and intertwined with broader information manipulation and polarized discourse (Cinelli et al. 2022; Liu et al. 2025; Thee et al. 2020; Hohmann et al. 2023).”
Wang, X., Liu, Y., Wu, S., Zhao, Z., Hu, Y., Li, W., & Bai, Q. (2026) · Neurocomputing · doi:10.1016/j.neucom.2026.134333 · Cited by 1 · see citing works
Imprecision of Human Preferences
Maksymov, I. S. (2026) · Springer eBooks · doi:10.1007/978-3-032-25965-3_5
Top-k Diverse Polarized Communities in Signed Networks
Gullo, F., Mandaglio, D., & Tagarelli, A. (2026) · Machine Learning · doi:10.1007/s10994-026-07098-8
“Polarization is a pervasive phenomenon in both online and offline social systems, where individuals tend to form groups that are cohesive internally and antagonistic externally (Conover et al., 2011 ; Coscia & Rossi, 2022; Hohmann et al., 2023; Garimella et al., 2017, 2018; Kumar et al., 2018).”
Measuring polarization in epistemic social networks: The case of the US 2024 presidential elections
Zaytsev, D., Kuskova, V. V., Khvatsky, G. S., & Chawla, N. V. (2026) · Network Science · doi:10.1017/nws.2026.10039
Wen, C., Jia, H., & Rao, H. (2026) · Information · doi:10.3390/info17080773
Zhang, J., Liu, J., & Zhao, H. (2026) · Array · doi:10.1016/j.array.2026.101108
Temporal Social Influence Design for Polarized Evolving OpinionsPreprint
Kooshkaki, A. R., Mohebbi, A., & Bolouki, S. (2026) · SSRN Electronic Journal · doi:10.2139/ssrn.7428621
Wang, L. (2026) · International Conference Computational Vision and Bio Inspired Computing · doi:10.1109/iccvbic71195.2026.11689182
Xu, J., An, L., Zhou, P., et al. (2026) · Journal of Instrumentation · doi:10.1088/1748-0221/21/10/p10025
“…of dissimilar features, this paper constructs a dual evaluation index based on Euclidean distance, intra-class dispersion and inter-class distance were calculated at the output of the embedding module and the input of the decoupling classifier for the model decoupling performance [37, 38].”
DSP: A Statistically-Principled Structural Polarization Measure
Preti, G., Riondato, M., Gionis, A., & De Francisci Morales, G. (2026) · Proceedings of the Nineteenth ACM International Conference on Web Search and Data Mining · doi:10.1145/3773966.3777942
- Authors
- Hohmann, M., Devriendt, K., & Coscia, M.
- Affiliations
- University of Copenhagen (DK)
- University of Oxford (GB)
- IT University of Copenhagen (DK)
- Corresponding author
- Hohmann, Marilena
- Devriendt, Karel
- Coscia, Michele
- Venue
- Science Advances
- Published
- 2023-03-03
- Volume
- 9
- Issue
- 9
- Pages
- eabq2044-eabq2044
- ISSN
- 2375-2548 (ISSN-L)
- PMC
- PMC9977176
- OpenAlex
- W4322754429
- Semantic Scholar
- 08fadf837b85434de90931d502f5c7821de81616
- Cited by
- 74
- References
- 71
- Citations by year
- 2026: 19 · 2025: 23 · 2024: 24 · 2023: 8
- FWCI
- 7.34
- Type
- journal article
- Language
- en
- Open access
- gold
- doi.org
- Licence
- cc-by
- Locations
- Science Advances (publishedVersion, cc-by)
- PubMed Central (publishedVersion, cc-by)
- PubMed (publishedVersion)
- Research at the University of Copenhagen (University of Copenhagen) (publishedVersion, cc-by)
- IT University Of Copenhagen (IT University of Copenhagen) (acceptedVersion, other-oa)
- Funders
- Alan Turing Institute (N510129)
- Engineering and Physical Sciences Research Council (N510129)
- Topics
- Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Opinion Dynamics and Social Influence
- Social Sciences › Communication › Social Media and Politics
- Social Sciences › Sociology and Political Science › Media Influence and Politics
- Keywords
- ideological polarization, polarimetry, echo chambers, social media, extreme opinions, network distance
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
Are you an author, or do you see a problem with this page? Request a review
