Variance and Covariance of Distributions on Graphs

Devriendt, Karel; Martin-Gutierrez, Samuel; Lambiotte, Renaud

doi:10.1137/20m1361328

Cited by 12 publications

(9 citation statements)

References 22 publications

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“…Note that the entropy is maximum when ρ is a uniform distribution and then this can be understood as a measure of the diversity of ρ. Other possible metrics, like the diversity index [61], or different measures of variance on graphs [62], give qualitatively similar results. In the figure, we observe how varying m does indeed give place to different behaviours, with a higher level of uniformity obtained for larger values of the exponent.…”

Section: A Comparing Linear and Nonlinear Dynamicsmentioning

confidence: 67%

From random walks on networks to nonlinear diffusion

Falcó¹

2022

Preprint

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Mathematical models of motility are often based on random-walk descriptions of discrete individuals that can move according to certain rules. It is usually the case that large masses concentrated in small regions of space have a great impact on collective movement of the group. For this reason, many models in mathematical biology have incorporated crowding effects and managed to understand their implications. Here, we build on a previously developed framework for random walks on networks to show that in the continuum limit, the underlying stochastic process can be identified with a diffusion partial differential equation. The diffusion coefficient of the emerging equation is in general density-dependent, and can be directly related to the transition probabilities of the random walk. Moreover, the relaxation time of the stochastic process is directly linked to the diffusion coefficient and also to network structure, as it usually happens in the case of linear diffusion. As a specific example, we study the equivalent of a porous-medium type equation on networks, which can be coarse-grained to obtain a known nonlinear equation. These findings also provide insights into reaction-diffusion systems with general diffusion operators, which have appeared recently in some applications.

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Section: A Comparing Linear and Nonlinear Dynamicsmentioning

confidence: 67%

From random walks on networks to nonlinear diffusion

Falcó¹

2022

Preprint

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“…To take into account the complex structure of the network, we calculate the network variance 40 of originality and upvotes, using effective resistance 41 to estimate the distance between nodes—effective resistance is more robust to random graph fluctuations than the simple shortest path lengths, which can vary greatly with the addition/deletion of a single edge 40 . Network variance is high when original posts are scattered in the periphery, and it is low when they concentrate in the core.…”

Section: Resultsmentioning

confidence: 99%

Posts on central websites need less originality to be noticed

Coscia

Vandeweerdt²

2022

Sci Rep

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Information has major consequences for democracy and society. It is important to understand what factors favor its diffusion. The impact of the content of a message on its likelihood of going viral is poorly understood. Some studies say originality is important for a message not to be overlooked. Others give more relevance to paratextual elements—network centrality, timing, human cognitive limits. Here we propose that originality and centrality interact in a nontrivial way, which might explain why originality by itself is not a good predictor of success. We collected data from Reddit on users sharing hyperlinks. We estimated the originality of each post title and the centrality of the website hosting the shared link. We show that the interaction effect exists: if users share content from a central website, originality no longer increases the odds of receiving at least one upvote. The same is not true for the odds of becoming one of the top 10% scoring posts. We show that originality is concentrated in the domain network: domains in the core of the network produce more original content. Our results imply that research on online information virality needs to take into account the nontrivial interaction between originality and prominence.

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“…Affective polarization, which pertains to how people with different opinions interact with each other ( 11 ), is also of great interest as it is the one truly affecting the quality of online discourse. One way that we could approach affective polarization is via network covariance ( 52 ) and/or correlations ( 63 ), since affective polarization should manifest as a correlation on the edges. Specifically, one would look whether the sentiment of a relationship is correlated with the opinion difference between the two individuals.…”

Section: Discussionmentioning

confidence: 99%

“…The former is determined on the basis of social media users’ sharing behavior ( 34 , 51 ), and the latter is determined by downloading their connections such as, e.g., follower relationships on Twitter. We estimate the GE distance ( 24 , 52 ) between two opposing opinions across all the edges of the network. By doing so, we avoid using a local approach, and we do not assume a community structure by default.…”

Section: Introductionmentioning

confidence: 99%

Quantifying ideological polarization on a network using generalized Euclidean distance

2023

Self Cite

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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.

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Variance and Covariance of Distributions on Graphs

Cited by 12 publications

References 22 publications

From random walks on networks to nonlinear diffusion

From random walks on networks to nonlinear diffusion

Posts on central websites need less originality to be noticed

Quantifying ideological polarization on a network using generalized Euclidean distance

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