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arxiv: 2605.14743 · v2 · pith:SED37HU3new · submitted 2026-05-14 · 💻 cs.SI · math.OC

Betweenness Central Nodes Under Uncertainty: An Absorbing Markov Chain Approach

Wencheng Bao , Eleftheria Kontou , Chrysafis Vogiatzis This is my paper

Pith reviewed 2026-05-20 21:07 UTC · model grok-4.3

classification 💻 cs.SI math.OC
keywords betweenness centralitystochastic networksabsorbing Markov chainnetwork uncertaintynode importanceMonte Carlo simulationcentrality robustness
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The pith

Node importance in stochastic networks is measured by the fraction of time an absorbing Markov chain spends at each node before absorption.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a betweenness centrality measure for networks where edges can fail or change weights from one realization to the next, making the single most central node itself random. It models the sequence of reported top nodes as steps of an absorbing Markov chain and defines a node's importance as the share of steps spent at that node prior to absorption. The resulting scores can be obtained by Monte Carlo simulation of the network and the chain. The approach also examines how sensitive the rankings are when the chain's transition probabilities are only approximately known.

Core claim

By modeling the sequence of reported central nodes as an absorbing Markov chain and measuring node importance by the share of pre-absorption time spent at each node, the method produces a centrality score under uncertainty that can be estimated with Monte Carlo simulation, analyzed for robustness under row-wise perturbations, and extended to weighted rewards or restricted candidate sets without altering the underlying chain formulation.

What carries the argument

An absorbing Markov chain on the nodes, with transitions given by the probability that one node is reported as most central immediately after another, where importance equals the expected time spent in each transient state before absorption.

If this is right

  • The method consistently identifies a small set of dominant nodes across Erdős-Rényi, Watts-Strogatz, and real networks with stochastic edges.
  • Row-wise perturbations of the transition kernel reveal which nodes have stable versus easily swapped rankings.
  • Weighted-reward and restricted-candidate extensions follow directly from the same absorbing-chain formulation.
  • Monte Carlo estimation of the chain remains valid even when the exact transition kernel is replaced by an empirical estimate.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same absorption-time measure could be used to rank nodes in transportation or communication networks whose link failures follow known statistical patterns.
  • Comparing absorption-time rankings against classical betweenness on the mean network would quantify how much uncertainty changes the ordering.
  • Extending the state space to include pairs of nodes might allow the framework to capture betweenness rather than just degree-like centrality.

Load-bearing premise

The process of identifying the most central node in successive random realizations of the network can be represented faithfully as a Markov chain whose transition kernel is known or estimable from simulation.

What would settle it

For a given stochastic network, compute the empirical frequency with which each node is the unique highest-betweenness node across thousands of independent realizations and compare those frequencies to the pre-absorption occupancy measures obtained from the fitted absorbing Markov chain; a large mismatch would falsify the modeling assumption.

read the original abstract

We propose a betweenness centrality measure and algorithms for stochastic networks, where edges can fail and weights vary across realizations, making the most central node random. Our approach models the sequence of reported central nodes as an absorbing Markov chain and measures node importance by the share of pre-absorption time spent at each node. This produces a way to study centrality under uncertainty, which can then be estimated with Monte Carlo simulation. We also analyze robustness when the transition kernel is only approximately known, using row-wise perturbations to assess sensitivity and potential ranking changes. The framework further admits extensions to weighted rewards and restricted candidate sets without altering the Markov chain formulation. Experiments on Erd\H{o}s-R\'enyi, Watts-Strogatz, and Les Mis\'erables networks with stochastic edges show that the method identifies a small set of dominant nodes, reveals stable versus sensitive rankings under perturbations, and supports reward-based and structure-constrained variants.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The paper proposes a betweenness centrality measure for stochastic networks (with edge failures and varying weights) by treating the sequence of most-central nodes across realizations as an absorbing Markov chain; node importance is defined as the expected pre-absorption occupancy time in each state. The measure is estimated via Monte Carlo simulation, with additional analysis of robustness under row-wise perturbations of the transition kernel and extensions to weighted rewards or restricted candidate sets. Experiments on Erdős-Rényi, Watts-Strogatz, and Les Misérables networks illustrate identification of dominant nodes and sensitivity of rankings.

Significance. If the central construction is well-defined and order-independent, the approach supplies a principled way to rank nodes under structural uncertainty and to quantify stability of centrality rankings, which is relevant for applications involving unreliable or stochastic networks. The perturbation analysis and reward-based extensions are natural and potentially useful.

major comments (2)
  1. [Abstract / Proposed method] Abstract / Proposed method paragraph: The construction models the sequence of reported central nodes across independent realizations as an absorbing Markov chain whose transition kernel is 'estimable from simulation.' Because realizations are i.i.d., any ordering is arbitrary; the resulting transition probabilities (and therefore the pre-absorption occupancy measures) are artifacts of the chosen Monte Carlo ordering rather than intrinsic properties of the network ensemble. No canonical, order-independent construction (e.g., via direct perturbation of the betweenness functional or a generative edge model) is supplied, so the claimed importance scores are not guaranteed to be uniquely determined by the stochastic network.
  2. [Abstract] Abstract: The abstract states that the chain is Markovian and that the kernel is estimable from simulation, yet supplies no derivation showing that the process of identifying the most central node across realizations satisfies the Markov property, nor any error analysis or validation that the occupancy measure converges to a well-defined limit independent of sampling order.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their insightful comments, which have helped us identify areas for clarification and improvement in the manuscript. We respond to each major comment below and outline the revisions we intend to make in the next version.

read point-by-point responses

  1. Referee: [Abstract / Proposed method] Abstract / Proposed method paragraph: The construction models the sequence of reported central nodes across independent realizations as an absorbing Markov chain whose transition kernel is 'estimable from simulation.' Because realizations are i.i.d., any ordering is arbitrary; the resulting transition probabilities (and therefore the pre-absorption occupancy measures) are artifacts of the chosen Monte Carlo ordering rather than intrinsic properties of the network ensemble. No canonical, order-independent construction (e.g., via direct perturbation of the betweenness functional or a generative edge model) is supplied, so the claimed importance scores are not guaranteed to be uniquely determined by the stochastic network.

    Authors: We appreciate the referee pointing out the potential dependence on the ordering of realizations. In our approach, the sequence is generated from independent Monte Carlo simulations of the stochastic network, and the transition kernel is estimated empirically from consecutive pairs in this sequence. While for a finite number of samples the specific ordering affects the empirical transitions, we will revise the manuscript to emphasize that the underlying model is the distribution over central nodes, and the absorbing chain is constructed using the marginal probabilities as transition probabilities from every state (reflecting independence). This makes the construction order-independent in the limit. We will add a new subsection deriving the pre-absorption occupancy using the fundamental matrix and proving convergence to an order-independent quantity as the number of realizations tends to infinity. We will also discuss an alternative formulation that directly uses the generative model of the stochastic network without relying on a specific simulation ordering. revision: yes

  2. Referee: [Abstract] Abstract: The abstract states that the chain is Markovian and that the kernel is estimable from simulation, yet supplies no derivation showing that the process of identifying the most central node across realizations satisfies the Markov property, nor any error analysis or validation that the occupancy measure converges to a well-defined limit independent of sampling order.

    Authors: We agree that the abstract and main text would benefit from more detail on these aspects. The Markov property holds because the central node in each realization is determined independently of the others due to the i.i.d. nature of the realizations; thus, the next state depends only on the current state through the transition probabilities derived from the ensemble. We will expand the abstract slightly and add a dedicated paragraph in the methods section providing the derivation of the Markov property and the expression for the occupancy times. Furthermore, we will include an error analysis based on Hoeffding's inequality to bound the Monte Carlo estimation error and numerical experiments demonstrating that the occupancy measures stabilize and become insensitive to reordering for large sample sizes. These additions will be incorporated in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

Proposed centrality measure defined directly via Markov chain occupancy; no reduction to inputs by construction

full rationale

The paper proposes modeling sequences of central nodes across stochastic realizations as an absorbing Markov chain, with node importance defined as the share of pre-absorption time spent at each node. This is a modeling choice and definitional construction for a new measure, estimated via Monte Carlo, rather than a derivation claiming to predict or derive a result that reduces to the same inputs or fitted parameters by construction. No self-citations are load-bearing the central claim, no uniqueness theorems are invoked from prior author work, and no ansatz or renaming of known results is presented as a first-principles outcome. The abstract and method description indicate the framework has independent content in its perturbation-based robustness analysis and extensions to rewards and constraints. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the modeling assumption that centrality reports form a Markov process and on the ability to estimate its kernel from simulation; no free parameters or new physical entities are introduced in the abstract.

axioms (1)
  • domain assumption The sequence of reported central nodes across network realizations forms a Markov chain with absorbing states.
    Invoked when the authors state that the sequence is modeled as an absorbing Markov chain.
invented entities (1)
  • Absorbing Markov chain representation of centrality reports no independent evidence
    purpose: To quantify node importance under edge uncertainty via pre-absorption occupancy
    New modeling construct introduced to turn random centrality into a steady-state measure.

pith-pipeline@v0.9.0 · 5693 in / 1349 out tokens · 32164 ms · 2026-05-20T21:07:44.221076+00:00 · methodology

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Reference graph

Works this paper leans on

74 extracted references · 74 canonical work pages

  1. [1]

    Nature Reviews Physics , author=

    Robustness and resilience of complex networks , volume=. Nature Reviews Physics , author=. 2024 , month=. doi:https://doi.org/10.1038/s42254-023-00676-y , number=

    work page doi:10.1038/s42254-023-00676-y 2024

  2. [2]

    Journal of the American Statistical Association , author=

    Probability Inequalities for Sums of Bounded Random Variables , volume=. Journal of the American Statistical Association , author=. 1963 , pages=. doi:https://doi.org/10.2307/2282952 , number=

    work page doi:10.2307/2282952 1963

  3. [3]

    Concentration inequalities : a nonasymptotic theory of independence , ISBN=

    Stéphane Boucheron and Gábor Lugosi and Pascal Massart , year=. Concentration inequalities : a nonasymptotic theory of independence , ISBN=

    work page

  4. [4]

    Matrix Perturbation Theory , publisher=

    Stewart, G W and Sun, Ji-guang , year=. Matrix Perturbation Theory , publisher=

    work page

  5. [5]

    Transportation Research Part B: Methodological , author=

    Expected shortest paths in dynamic and stochastic traffic networks , volume=. Transportation Research Part B: Methodological , author=. 1998 , month=. doi:https://doi.org/10.1016/s0191-2615(98)00016-2 , number=

    work page doi:10.1016/s0191-2615(98)00016-2 1998

  6. [6]

    Transportation Science , author=

    Least Expected Time Paths in Stochastic, Time-Varying Transportation Networks , volume=. Transportation Science , author=. 2000 , month=. doi:https://doi.org/10.1287/trsc.34.2.198.12304 , number=

    work page doi:10.1287/trsc.34.2.198.12304 2000

  7. [7]

    Shortest path problem considering on-time arrival probability , volume=

    Yu (Marco) Nie and Wu, Xing , year=. Shortest path problem considering on-time arrival probability , volume=. doi:https://doi.org/10.1016/j.trb.2009.01.008 , number=

    work page doi:10.1016/j.trb.2009.01.008 2009

  8. [8]

    Multi-class time reliability-based congestion pricing model based on a degradable transportation network , volume=

    Liu, Bing-quan and Huang, Chong-chao , year=. Multi-class time reliability-based congestion pricing model based on a degradable transportation network , volume=. Applied mathematical modelling , publisher=. doi:https://doi.org/10.1016/j.apm.2015.09.052 , number=

    work page doi:10.1016/j.apm.2015.09.052 2015

  9. [9]

    and Zhu, Emily C

    Shen, Liang and Shao, Hu and Wu, Ting and Lam, William H.K. and Zhu, Emily C. , year=. An energy-efficient reliable path finding algorithm for stochastic road networks with electric vehicles , volume=. doi:https://doi.org/10.1016/j.trc.2019.03.020 , journal=

    work page doi:10.1016/j.trc.2019.03.020 2019

  10. [10]

    Measuring urban road network vulnerability to extreme events: An application for urban floods , volume=

    Morelli, André Borgato and Cunha, André Luiz , year=. Measuring urban road network vulnerability to extreme events: An application for urban floods , volume=. doi:https://doi.org/10.1016/j.trd.2021.102770 , journal=

    work page doi:10.1016/j.trd.2021.102770 2021

  11. [11]

    Role of road network features in the evaluation of incident impacts on urban traffic mobility , volume=

    Sun, Chenshuo and Pei, Xin and Hao, Junheng and Wang, Yewen and Zuo, Zhigang and Wong, S C , year=. Role of road network features in the evaluation of incident impacts on urban traffic mobility , volume=. doi:https://doi.org/10.1016/j.trb.2018.08.013 , journal=

    work page doi:10.1016/j.trb.2018.08.013 2018

  12. [12]

    Does transportation network centrality determine housing price? , volume=

    Chakrabarti, Sandip and Kushari, Triparnee and Mazumder, Taraknath , year=. Does transportation network centrality determine housing price? , volume=. doi:https://doi.org/10.1016/j.jtrangeo.2022.103397 , journal=

    work page doi:10.1016/j.jtrangeo.2022.103397 2022

  13. [13]

    Scientific Reports , author=

    Two betweenness centrality measures based on Randomized Shortest Paths , volume=. Scientific Reports , author=. 2016 , month=. doi:https://doi.org/10.1038/srep19668 , number=

    work page doi:10.1038/srep19668 2016

  14. [14]

    Proceedings of the VLDB Endowment , author=

    Shortest paths and centrality in uncertain networks , volume=. Proceedings of the VLDB Endowment , author=. 2021 , month=. doi:https://doi.org/10.14778/3450980.3450988 , number=

    work page doi:10.14778/3450980.3450988 2021

  15. [15]

    An Efficient Approximation of Betweenness Centrality for Uncertain Graphs , year=

    Wang, Chenxu and Lin, Ziyuan , journal=. An Efficient Approximation of Betweenness Centrality for Uncertain Graphs , year=

    work page

  16. [16]

    Social Networks , author=

    A measure of betweenness centrality based on random walks , volume=. Social Networks , author=. 2005 , month=. doi:https://doi.org/10.1016/j.socnet.2004.11.009 , number=

    work page doi:10.1016/j.socnet.2004.11.009 2005

  17. [17]

    Physical Review Letters , author=

    Random Walks on Complex Networks , volume=. Physical Review Letters , author=. 2004 , month=. doi:https://doi.org/10.1103/physrevlett.92.118701 , number=

    work page doi:10.1103/physrevlett.92.118701 2004

  18. [18]

    Networks , volume=

    A survey on optimization studies of group centrality metrics , author=. Networks , volume=. 2024 , publisher=

    work page 2024

  19. [19]

    HEART , year=

    Fast Computation of Betweenness Centrality to enable Real-time Resilience Assessment and Improvement of Complex Transport Networks , author=. HEART , year=

    work page

  20. [20]

    PloS one , volume=

    Identification of critical links in a large-scale road network considering the traffic flow betweenness index , author=. PloS one , volume=. 2020 , publisher=

    work page 2020

  21. [21]

    Journal of anthropological research , volume=

    An information flow model for conflict and fission in small groups , author=. Journal of anthropological research , volume=. 1977 , publisher=

    work page 1977

  22. [22]

    Social network analysis : methods and applications , ISBN=

    Wasserman, Stanley and Faust, Katherine , year=. Social network analysis : methods and applications , ISBN=

    work page

  23. [23]

    ACM Transactions on Database Systems , author=

    Uncertain Graph Processing through Representative Instances , volume=. ACM Transactions on Database Systems , author=. 2015 , month=. doi:https://doi.org/10.1145/2818182 , number=

    work page doi:10.1145/2818182 2015

  24. [24]

    Optimization Letters , author=

    Multi-criteria approximation schemes for the resource constrained shortest path problem , volume=. Optimization Letters , author=. 2017 , month=. doi:https://doi.org/10.1007/s11590-017-1212-z , number=

    work page doi:10.1007/s11590-017-1212-z 2017

  25. [25]

    Classical and Quantum Random-Walk Centrality Measures in Multilayer Networks , volume=

    Böttcher, Lucas and Porter, Mason A , year=. Classical and Quantum Random-Walk Centrality Measures in Multilayer Networks , volume=. SIAM Journal on Applied Mathematics , publisher=. doi:https://doi.org/10.1137/20m1385998 , number=

    work page doi:10.1137/20m1385998

  26. [26]

    Random Walks on Dense Graphs and Graphons , volume=

    Petit, Julien and Lambiotte, Renaud and Carletti, Timoteo , year=. Random Walks on Dense Graphs and Graphons , volume=. SIAM Journal on Applied Mathematics , publisher=. doi:https://doi.org/10.1137/20m1339246 , number=

    work page doi:10.1137/20m1339246

  27. [27]

    Random Walks on Simplicial Complexes and the Normalized Hodge 1-Laplacian , volume=

    Schaub, Michael T and Benson, Austin R and Horn, Paul and Lippner, Gabor and Jadbabaie, Ali , year=. Random Walks on Simplicial Complexes and the Normalized Hodge 1-Laplacian , volume=. SIAM review , publisher=. doi:https://doi.org/10.1137/18m1201019 , number=

    work page doi:10.1137/18m1201019

  28. [28]

    SIAM Journal on Matrix Analysis and Applications , author=

    On the Limiting Behavior of Parameter-Dependent Network Centrality Measures , volume=. SIAM Journal on Matrix Analysis and Applications , author=. 2015 , month=. doi:https://doi.org/10.1137/130950550 , number=

    work page doi:10.1137/130950550 2015

  29. [29]

    Physical Review E , author=

    Absorbing random walks interpolating between centrality measures on complex networks , volume=. Physical Review E , author=. 2020 , month=. doi:https://doi.org/10.1103/physreve.101.012302 , number=

    work page doi:10.1103/physreve.101.012302 2020

  30. [30]

    Adjustable reach in a network centrality based on current flows , volume=

    Gurfinkel, Aleks J and Per Arne Rikvold , year=. Adjustable reach in a network centrality based on current flows , volume=. Physical review. E , publisher=. doi:https://doi.org/10.1103/physreve.103.052308 , number=

    work page doi:10.1103/physreve.103.052308

  31. [31]

    Computer Networks and ISDN Systems , author=

    The anatomy of a large-scale hypertextual Web search engine , volume=. Computer Networks and ISDN Systems , author=. 1998 , month=. doi:https://doi.org/10.1016/s0169-7552(98)00110-x , number=

    work page doi:10.1016/s0169-7552(98)00110-x 1998

  32. [32]

    Worst-Case and Average-Case Approximations by Simple Randomized Search Heuristics , DOI=

    Witt, Carsten , year=. Worst-Case and Average-Case Approximations by Simple Randomized Search Heuristics , DOI=. Lecture notes in computer science , publisher=

    work page

  33. [33]

    , volume=

    Random Walks and Electric Networks. , volume=. The American Mathematical Monthly , author=. 1987 , month=. doi:https://doi.org/10.2307/2322439 , number=

    work page doi:10.2307/2322439 1987

  34. [34]

    SIAM Journal on Matrix Analysis and Applications , author=

    An Edge Centrality Measure Based on the Kemeny Constant , volume=. SIAM Journal on Matrix Analysis and Applications , author=. 2023 , month=. doi:https://doi.org/10.1137/22m1486728 , number=

    work page doi:10.1137/22m1486728 2023

  35. [35]

    Random Walks, Conductance, and Resistance for the Connection Graph Laplacian , volume=

    Cloninger, Alexander and Gal Mishne and Oslandsbotn, Andreas and Robertson, Sawyer J and Wan, Zhengchao and Wang, Yusu , year=. Random Walks, Conductance, and Resistance for the Connection Graph Laplacian , volume=. SIAM Journal on Matrix Analysis and Applications , publisher=. doi:https://doi.org/10.1137/23m1595400 , number=

    work page doi:10.1137/23m1595400

  36. [36]

    Scientific collaboration networks. II. Shortest paths, weighted networks, and centrality , volume=. Physical Review E , author=. 2001 , month=. doi:https://doi.org/10.1103/physreve.64.016132 , number=

    work page doi:10.1103/physreve.64.016132 2001

  37. [37]

    Nature , author=

    Error and attack tolerance of complex networks , volume=. Nature , author=. 2000 , month=. doi:https://doi.org/10.1038/35019019 , number=

    work page doi:10.1038/35019019 2000

  38. [38]

    Seneta , year=

    Darroch, J N and E. Seneta , year=. On Quasi-Stationary distributions in absorbing discrete-time finite Markov chains , volume=. Journal of Applied Probability , publisher=. doi:https://doi.org/10.2307/3211876 , number=

    work page doi:10.2307/3211876

  39. [39]

    The Annals of Statistics , author=

    Large Sample Theory of Empirical Distributions in Biased Sampling Models , volume=. The Annals of Statistics , author=. 1988 , month=. doi:https://doi.org/10.1214/aos/1176350948 , number=

    work page doi:10.1214/aos/1176350948 1988

  40. [40]

    Problems and results in combinatorial analysis and graph theory , volume=

    Erdös, Paul , year=. Problems and results in combinatorial analysis and graph theory , volume=. Discrete mathematics , publisher=. doi:https://doi.org/10.1016/0012-365x(88)90196-3 , number=

    work page doi:10.1016/0012-365x(88)90196-3

  41. [41]

    small-world

    Collective dynamics of “small-world” networks , volume=. Nature , author=. 1998 , month=

    work page 1998

  42. [42]

    , title =

    Knuth, Donald E. , title =. 1993 , publisher =

    work page 1993

  43. [43]

    Operations Research , author=

    Robust Control of Markov Decision Processes with Uncertain Transition Matrices , volume=. Operations Research , author=. 2005 , month=. doi:https://doi.org/10.1287/opre.1050.0216 , number=

    work page doi:10.1287/opre.1050.0216 2005

  44. [44]

    SIAM Review , author=

    PageRank Beyond the Web , volume=. SIAM Review , author=. 2015 , month=. doi:https://doi.org/10.1137/140976649 , number=

    work page doi:10.1137/140976649 2015

  45. [45]

    SIAM Journal on Matrix Analysis and Applications , author=

    Uniform Stability of Markov Chains , volume=. SIAM Journal on Matrix Analysis and Applications , author=. 1994 , month=. doi:https://doi.org/10.1137/s0895479892237562 , number=

    work page doi:10.1137/s0895479892237562 1994

  46. [46]

    Integral Probability Metrics and Their Generating Classes of Functions , volume=

    Müller, Alfred , year=. Integral Probability Metrics and Their Generating Classes of Functions , volume=. Advances in Applied Probability , publisher=. doi:https://doi.org/10.2307/1428011 , number=

    work page doi:10.2307/1428011

  47. [47]

    SIAM Journal on Matrix Analysis and Applications , author=

    Ordinal Ranking for Google's PageRank , volume=. SIAM Journal on Matrix Analysis and Applications , author=. 2009 , month=. doi:https://doi.org/10.1137/070698129 , number=

    work page doi:10.1137/070698129 2009

  48. [48]

    SIAM Journal on Scientific Computing , author=

    Sensitivity and Stability of Ranking Vectors , volume=. SIAM Journal on Scientific Computing , author=. 2011 , month=. doi:https://doi.org/10.1137/090772745 , number=

    work page doi:10.1137/090772745 2011

  49. [49]

    IEEE Trans

    Stability and Continuity of Centrality Measures in Weighted Graphs , volume=. IEEE Transactions on Signal Processing , author=. 2016 , month=. doi:https://doi.org/10.1109/tsp.2015.2486740 , number=

    work page doi:10.1109/tsp.2015.2486740 2016

  50. [50]

    Matrix methods in health demography: a new approach to the stochastic analysis of healthy longevity and DALYs , volume=

    Caswell, Hal and Zarulli, Virginia , year=. Matrix methods in health demography: a new approach to the stochastic analysis of healthy longevity and DALYs , volume=. Population Health Metrics , publisher=. doi:https://doi.org/10.1186/s12963-018-0165-5 , number=

    work page doi:10.1186/s12963-018-0165-5

  51. [51]

    Data Mining and Knowledge Discovery , author=

    Fast approximation of betweenness centrality through sampling , volume=. Data Mining and Knowledge Discovery , author=. 2015 , month=. doi:https://doi.org/10.1007/s10618-015-0423-0 , number=

    work page doi:10.1007/s10618-015-0423-0 2015

  52. [52]

    Science , author=

    Network Motifs: Simple Building Blocks of Complex Networks , volume=. Science , author=. 2002 , month=. doi:https://doi.org/10.1126/science.298.5594.824 , number=

    work page doi:10.1126/science.298.5594.824 2002

  53. [53]

    Benson, David F

    Higher-order organization of complex networks , volume=. Science , author=. 2016 , month=. doi:https://doi.org/10.1126/science.aad9029 , number=

    work page doi:10.1126/science.aad9029 2016

  54. [54]

    Nature , author=

    Uncovering the overlapping community structure of complex networks in nature and society , volume=. Nature , author=. 2005 , month=. doi:https://doi.org/10.1038/nature03607 , number=

    work page doi:10.1038/nature03607 2005

  55. [55]

    PageRank Computation, with Special Attention to Dangling Nodes , volume=

    Ilse and Selee, Teresa M , year=. PageRank Computation, with Special Attention to Dangling Nodes , volume=. SIAM Journal on Matrix Analysis and Applications , publisher=. doi:https://doi.org/10.1137/060664331 , number=

    work page doi:10.1137/060664331

  56. [56]

    Journal of the ACM (JACM) , author=

    An Algorithm for Subgraph Isomorphism , volume=. Journal of the ACM (JACM) , author=. 1976 , month=. doi:https://doi.org/10.1145/321921.321925 , number=

    work page doi:10.1145/321921.321925 1976

  57. [57]

    Markov chains and mixing times , ISBN=

    David Asher Levin and Y Peres and Wilmer, Elizabeth L and Propp, James and Wilson, David B , year=. Markov chains and mixing times , ISBN=

    work page

  58. [58]

    Mitrophanov, A. Yu. , year=. Sensitivity and convergence of uniformly ergodic Markov chains , volume=. Journal of Applied Probability , publisher=. doi:https://doi.org/10.1239/jap/1134587812 , number=

    work page doi:10.1239/jap/1134587812

  59. [59]

    Journal of Discrete Algorithms , author=

    Choose the damping, choose the ranking? , volume=. Journal of Discrete Algorithms , author=. 2010 , month=. doi:https://doi.org/10.1016/j.jda.2009.11.001 , number=

    work page doi:10.1016/j.jda.2009.11.001 2010

  60. [60]

    Linear Algebra and its Applications , author=

    Sensitivity analysis of discrete Markov chains via matrix calculus , volume=. Linear Algebra and its Applications , author=. 2013 , month=. doi:https://doi.org/10.1016/j.laa.2011.07.046 , number=

    work page doi:10.1016/j.laa.2011.07.046 2013

  61. [61]

    The Role of the Group Generalized Inverse in the Theory of Finite Markov Chains , volume=

    Meyer, Carl D , year=. The Role of the Group Generalized Inverse in the Theory of Finite Markov Chains , volume=. Siam Review , publisher=. doi:https://doi.org/10.1137/1017044 , number=

    work page doi:10.1137/1017044

  62. [62]

    Using Centrality Measures to Rank the Importance of the Components of a Complex Network Infrastructure , ISBN=

    Cadini, Francesco and Zio, Enrico and Petrescu, Cristina-Andreea , year=. Using Centrality Measures to Rank the Importance of the Components of a Complex Network Infrastructure , ISBN=. doi:https://doi.org/10.1007/978-3-642-03552-4_14 , journal=

    work page doi:10.1007/978-3-642-03552-4_14

  63. [63]

    PLoS ONE , author=

    Network Centrality of Metro Systems , volume=. PLoS ONE , author=. 2012 , month=. doi:https://doi.org/10.1371/journal.pone.0040575 , number=

    work page doi:10.1371/journal.pone.0040575 2012

  64. [64]

    Physical Review E , author=

    Ranking and clustering of nodes in networks with smart teleportation , volume=. Physical Review E , author=. 2012 , month=. doi:https://doi.org/10.1103/physreve.85.056107 , number=

    work page doi:10.1103/physreve.85.056107 2012

  65. [65]

    and Lambiotte, Renaud , year=

    Masuda, Naoki and Porter, Mason A. and Lambiotte, Renaud , year=. Random walks and diffusion on networks , volume=. doi:https://doi.org/10.1016/j.physrep.2017.07.007 , journal=

    work page doi:10.1016/j.physrep.2017.07.007 2017

  66. [66]

    Management Science , author=

    Robust Solutions of Optimization Problems Affected by Uncertain Probabilities , volume=. Management Science , author=. 2013 , month=. doi:https://doi.org/10.1287/mnsc.1120.1641 , number=

    work page doi:10.1287/mnsc.1120.1641 2013

  67. [67]

    Identifying effective scenarios in distributionally robust stochastic programs with total variation distance , volume=

    Rahimian, Hamed and Güzi̇n Bayraksan and Tito Homem-de-Mello , year=. Identifying effective scenarios in distributionally robust stochastic programs with total variation distance , volume=. Mathematical Programming , publisher=. doi:https://doi.org/10.1007/s10107-017-1224-6 , number=

    work page doi:10.1007/s10107-017-1224-6

  68. [68]

    Mathematical Programming , author =

    Data-driven distributionally robust optimization using the Wasserstein metric: performance guarantees and tractable reformulations , volume=. Mathematical Programming , author=. 2017 , month=. doi:https://doi.org/10.1007/s10107-017-1172-1 , number=

    work page doi:10.1007/s10107-017-1172-1 2017

  69. [69]

    IEEE Transactions on Network Science and Engineering , volume=

    Centrality measures for graphons: Accounting for uncertainty in networks , author=. IEEE Transactions on Network Science and Engineering , volume=. 2018 , publisher=

    work page 2018

  70. [70]

    Networks , volume=

    Novel centrality metrics for studying essentiality in protein-protein interaction networks based on group structures , author=. Networks , volume=. 2022 , publisher=

    work page 2022

  71. [71]

    INFORMS Journal on Computing , volume=

    Identification of essential proteins using induced stars in protein--protein interaction networks , author=. INFORMS Journal on Computing , volume=. 2019 , publisher=

    work page 2019

  72. [72]

    European Journal of Operational Research , volume=

    Finding clique clusters with the highest betweenness centrality , author=. European Journal of Operational Research , volume=. 2018 , publisher=

    work page 2018

  73. [73]

    Optimization Letters , volume=

    An integer programming approach for finding the most and the least central cliques , author=. Optimization Letters , volume=. 2015 , publisher=

    work page 2015

  74. [74]

    INFORMS Journal on Computing , year=

    Centrality of shortest paths: Algorithms and complexity results , author=. INFORMS Journal on Computing , year=

    work page