Reducing Spreading Processes on Networks to Markov Population Models

Gerrit Gro{\ss}mann; Luca Bortolussi

arxiv: 1906.11508 · v1 · pith:5Y7K7AE7new · submitted 2019-06-27 · 💻 cs.SI

Reducing Spreading Processes on Networks to Markov Population Models

Gerrit Gro{\ss}mann , Luca Bortolussi This is my paper

Pith reviewed 2026-05-25 14:15 UTC · model grok-4.3

classification 💻 cs.SI

keywords epidemic modelsMarkov population modelslumpingcomplex networksspreading processesstochastic processes

0 comments

The pith

Lumping via node partitioning reduces any epidemic model on a network to a Markov population model.

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

The paper shows that stochastic spreading processes on networks can be reduced to Markov population models by partitioning nodes and applying counting abstractions. This creates smaller coarse-grained models that still approximate the original dynamics and inherit analysis tools from the population-model setting. A reader would care because full network state spaces are usually too large for direct computation, so the reduction opens a route to existing approximation methods. Numerical examples examine how the accuracy of these reduced models depends on the lumped state-space size and the abstraction chosen.

Core claim

Lumping can be used to reduce any epidemic model to a Markov Population Model. A novel lumping scheme based on a partitioning of the nodes is proposed. By imposing different types of counting abstractions, coarse-grained Markov models with a natural MPM representation are obtained that approximate the original systems. This transfer makes the rich pool of approximation techniques developed for MPMs available for the computational analysis of complex networks' dynamics.

What carries the argument

A lumping scheme that partitions the nodes of the network and then imposes counting abstractions to produce a Markov population model representation of the process.

If this is right

The reduced models approximate the original network dynamics with controllable error.
Existing approximation techniques for Markov population models become directly applicable to network spreading processes.
Accuracy of the approximation trades off against the size of the lumped state space and the type of counting abstraction used.
Computational analysis of spreading phenomena on large networks becomes feasible through these smaller models.

Where Pith is reading between the lines

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

The same partitioning approach might apply to non-epidemic stochastic processes on networks.
Structure-aware or adaptive partitions could be tested to improve accuracy on heterogeneous graphs.
Moment-closure methods already used for MPMs could be combined with the lumped models for further scaling.

Load-bearing premise

The chosen node partitioning and counting abstraction must preserve enough of the original interaction structure so that the resulting MPM still approximates the true dynamics on the full network.

What would settle it

A concrete network and epidemic process in which trajectories from the reduced Markov population model diverge substantially from direct stochastic simulations of the original full-state process.

Figures

Figures reproduced from arXiv: 1906.11508 by Gerrit Gro{\ss}mann, Luca Bortolussi.

**Figure 1.** Figure 1: The CTMC induced by the SIS model (S: blue, I: magenta, filled) on a toy graph. Only a subset of the CTMC spate space (11 out of 26 = 64 network states) is shown. a node is equal to the sum over its associated neighborhood vector, that is, k = P s∈S m[s]. The set of possible neighborhood vectors is denoted as M = n m ∈ Z |S| ≥0 [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗

**Figure 2.** Figure 2: Illustration of the lumping process. (a): Model. A basic SI-Process where infected nodes (magenta, filled) infect susceptible neighbors (blue) with rate infection λ = 1. The contact graph is divided into two partitions. (b): The underlying CTMC with 24 = 16 states. The graph partition induces the edge-based and node-based lumping. The edge-based lumping refines the node-based lumping and generates one pa… view at source ↗

**Figure 3.** Figure 3: (a) By adding the dummy-node, the edge-based abstraction is able to differentiate the two graphs. Adding the dummy-node ensures that the nodes in each state are counted in the edge-based abstraction. (b) Left: A partitioned network (Zachary’s Karate Club graph from [11]) (S: blue, I: magenta, filled). The network is partitioned into P1 (#-nodes) and P2 (2-nodes). Right: The corresponding counting abstracti… view at source ↗

**Figure 4.** Figure 4: Example of how the neighborhood v influences the update in the edgebased counting abstraction on an example graph. Here, all nodes belong to the same partition (thus, nodes states and species are conceptually the same) and the node states are ordered [S, I, ?]. The population vector y is given in matrix form for the ease of presentation. αr,P,v :Y → R≥0 αr,P,v(y) = 1 L−1(y) X x∈L−1(y) X n∈P f(mx,n)1x(n)=s… view at source ↗

**Figure 5.** Figure 5: Trade of between accuracy and state space size for the node-based (blue) and edge-based (magenta, filled) counting abstraction. Results are shown for node partitions based on the degree (l.), spectral embedding (c.), and random partitioning (r.). The accuracy is measured as the mean (4) and maximal (5) difference between the original and lumped solution over all timepoints. approximation methods for MPM [4… view at source ↗

read the original abstract

Stochastic processes on complex networks, where each node is in one of several compartments, and neighboring nodes interact with each other, can be used to describe a variety of real-world spreading phenomena. However, computational analysis of such processes is hindered by the enormous size of their underlying state space. In this work, we demonstrate that lumping can be used to reduce any epidemic model to a Markov Population Model (MPM). Therefore, we propose a novel lumping scheme based on a partitioning of the nodes. By imposing different types of counting abstractions, we obtain coarse-grained Markov models with a natural MPM representation that approximate the original systems. This makes it possible to transfer the rich pool of approximation techniques developed for MPMs to the computational analysis of complex networks' dynamics. We present numerical examples to investigate the relationship between the accuracy of the MPMs, the size of the lumped state space, and the type of counting abstraction.

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.

Desk Editor's Note private letter to a colleague

The paper gives a concrete node-partition lumping method that maps network epidemic models to Markov population models, which is new and practical but only works cleanly when the partition makes rates depend only on block counts.

read the letter

The main contribution is a lumping scheme that partitions nodes and applies counting abstractions to turn any epidemic process on a network into an MPM. This is new in the specific way it combines partitioning with different abstraction types, and it directly lets existing MPM approximation tools be used on network spreading models. The numerical examples are useful because they show how accuracy trades off against lumped state-space size and the choice of abstraction, giving readers a concrete way to judge when the reduction is worth it. That part is done well and feels honest about the method's behavior. The soft spot is the generality claim. The abstract says the reduction works for any epidemic model once a partitioning is chosen, but the lumped process only stays Markov with rates determined solely by compartment counts if nodes inside each block have identical neighbor counts to other blocks. Arbitrary partitions on general networks will not satisfy this, so the paper must either restrict to equitable partitions or add a closure assumption. If the full text does not state the lumpability condition or give error bounds, the claim that it applies to any model is overstated. The examples probably pick favorable partitions, which is fine for illustration but does not prove broad applicability. This paper is for researchers in computational network epidemiology who already know MPM techniques and want a reduction step to handle larger networks. A reader looking for model-reduction ideas in that subfield will find the scheme and the examples worth reading. It deserves peer review because the reduction is a clear, implementable step forward even if the conditions on partitions need to be spelled out more carefully.

Referee Report

2 major / 0 minor

Summary. The paper claims that lumping via node partitioning and various counting abstractions can reduce any stochastic epidemic process on a network to a Markov Population Model (MPM) that approximates the original dynamics, thereby allowing transfer of MPM analysis techniques to network spreading models. Numerical examples are used to explore how approximation accuracy relates to lumped state-space size and abstraction type.

Significance. If the reduction is accompanied by explicit lumpability conditions and error controls, the approach would enable scalable analysis of large-network epidemic models by connecting them to existing MPM approximation methods. The numerical investigation of accuracy versus granularity is a useful empirical contribution.

major comments (2)

[Abstract] Abstract: the assertion that the scheme reduces 'any epidemic model' to a closed MPM via arbitrary node partitioning plus counting abstraction is not supported by a lumpability condition; in general networks the infection rate for a susceptible node in one block depends on its specific neighbors in infected blocks, so two micro-states with identical block counts can have different total rates unless the partition is equitable.
Proposed lumping scheme (as described after the abstract): no formal statement is given of the conditions under which the aggregated process remains Markov with transition rates depending only on compartment counts per block, nor are error bounds or convergence results supplied for the approximation under different counting abstractions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting important points regarding the formal presentation of the lumping scheme. We address each major comment below.

read point-by-point responses

Referee: [Abstract] Abstract: the assertion that the scheme reduces 'any epidemic model' to a closed MPM via arbitrary node partitioning plus counting abstraction is not supported by a lumpability condition; in general networks the infection rate for a susceptible node in one block depends on its specific neighbors in infected blocks, so two micro-states with identical block counts can have different total rates unless the partition is equitable.

Authors: We agree that exact lumpability holds only for special partitions (e.g., equitable ones). The manuscript already qualifies the reduction as approximate ('approximate the original systems'), but the abstract could be clearer on this point. We will revise the abstract to state explicitly that the resulting MPM is an approximation for arbitrary partitions and add a brief remark on lumpability conditions in Section 3. revision: yes
Referee: [—] Proposed lumping scheme (as described after the abstract): no formal statement is given of the conditions under which the aggregated process remains Markov with transition rates depending only on compartment counts per block, nor are error bounds or convergence results supplied for the approximation under different counting abstractions.

Authors: The core contribution is the construction of approximating MPMs via node partitioning and counting abstractions, supported by numerical examples relating accuracy to state-space size. We will add a formal paragraph stating the (standard) lumpability condition under which the aggregated process is exactly Markov. Theoretical error bounds and convergence rates are not derived in the present work; the paper relies on empirical evaluation. We will expand the discussion to note this limitation and identify it as future work. revision: partial

Circularity Check

0 steps flagged

No significant circularity in the proposed lumping reduction

full rationale

The paper presents a methodological reduction: node partitioning plus counting abstractions are used to obtain approximate coarse-grained Markov models that admit an MPM representation. The abstract and description frame this as an approximation technique that transfers existing MPM analysis tools, without any self-definitional equations, fitted parameters renamed as predictions, or load-bearing self-citations. The derivation relies on standard lumping applied to network epidemic processes and is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only abstract available; no explicit free parameters, axioms, or invented entities are stated. The central claim rests on the unstated assumption that suitable node partitions exist and that the counting abstractions yield valid approximations.

pith-pipeline@v0.9.0 · 5689 in / 1070 out tokens · 14296 ms · 2026-05-25T14:15:49.519050+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean absolute_floor_iff_bare_distinguishability unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

we demonstrate that lumping can be used to reduce any epidemic model to a Markov Population Model (MPM). Therefore, we propose a novel lumping scheme based on a partitioning of the nodes. By imposing different types of counting abstractions...
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

q(y,y′) = 1/|L⁻¹(y)| ∑_{x∈L⁻¹(y)} ∑_{x′∈L⁻¹(y′)} q(x,x′)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

51 extracted references · 51 canonical work pages · 3 internal anchors

[1]

G. E. Allen and C. Dytham. An eﬃcient method for stochastic simulation of biological populations in continuous time. Biosystems, 98(1):37–42, 2009

work page 2009
[2]

Barab´ asi.Network science

A.-L. Barab´ asi.Network science. Cambridge university press, 2016

work page 2016
[3]

Bortolussi

L. Bortolussi. Hybrid behaviour of Markov population models. Information and Computation, 247:37–86, 2016

work page 2016
[4]

Bortolussi, J

L. Bortolussi, J. Hillston, D. Latella, and M. Massink. Continuous approximation of collective system behaviour: A tutorial. Performance Evaluation, 70(5):317–349, 2013

work page 2013
[5]

Buchholz

P. Buchholz. Exact and ordinary lumpability in ﬁnite markov chains. Journal of applied probability, 31(1):59–75, 1994

work page 1994
[6]

Y. Cao, D. T. Gillespie, and L. R. Petzold. Eﬃcient step size selection for the tau-leaping simulation method. The Journal of chemical physics , 124(4)

work page
[7]

Cardelli, M

L. Cardelli, M. Tribastone, M. Tschaikowski, and A. Vandin. Erode: a tool for the evaluation and reduction of ordinary diﬀerential equations. In International Conference on Tools and Algorithms for the Construction and Analysis of Systems , pages 310–328. Springer, 2017

work page 2017
[8]

Cota and S

W. Cota and S. C. Ferreira. Optimized gillespie algorithms for the simulation of markovian epidemic processes on large and heterogeneous networks. Computer Physics Communications, 219:303–312, 2017

work page 2017
[9]

Devriendt and P

K. Devriendt and P. Van Mieghem. Uniﬁed mean-ﬁeld framework for susceptible- infected-susceptible epidemics on networks, based on graph partitioning and the isoperimetric inequality. Physical Review E, 96(5):052314, 2017

work page 2017
[10]

C. Gan, X. Yang, W. Liu, Q. Zhu, and X. Zhang. Propagation of computer virus under human intervention: a dynamical model. Discrete Dynamics in Nature and Society, 2012, 2012

work page 2012
[11]

Girvan and M

M. Girvan and M. E. Newman. Community structure in social and biological networks. Proceedings of the national academy of sciences, 99(12):7821–7826, 2002

work page 2002
[12]

J. P. Gleeson. High-accuracy approximation of binary-state dynamics on networks. Physical Review Letters, 107(6):068701, 2011

work page 2011
[13]

J. P. Gleeson. Binary-state dynamics on complex networks: Pair approximation and beyond. Physical Review X, 3(2):021004, 2013

work page 2013
[14]

Goltsev, F

A. Goltsev, F. De Abreu, S. Dorogovtsev, and J. Mendes. Stochastic cellular automata model of neural networks. Physical Review E, 81(6):061921, 2010

work page 2010
[15]

Goutsias and G

J. Goutsias and G. Jenkinson. Markovian dynamics on complex reaction networks. Physics Reports, 529(2):199–264, 2013

work page 2013
[16]

Goyal and E

P. Goyal and E. Ferrara. Graph embedding techniques, applications, and perfor- mance: A survey. Knowledge-Based Systems, 151:78–94, 2018

work page 2018
[17]

R. Grima. An eﬀective rate equation approach to reaction kinetics in small volumes: Theory and application to biochemical reactions in nonequilibrium steady-state conditions. The Journal of Chemical Physics , 133(3):035101, 2010

work page 2010
[18]

R. Grima. A study of the accuracy of moment-closure approximations for stochastic chemical kinetics. The Journal of chemical physics , 136(15):04B616, 2012

work page 2012
[19]

Großmann, C

G. Großmann, C. Kyriakopoulos, L. Bortolussi, and V. Wolf. Lumping the ap- proximate master equation for multistate processes on complex networks. In Qest, pages 157–172. Springer, 2018

work page 2018
[20]

Rejection-Based Simulation of Stochastic Spreading Processes on Complex Networks

G. Großmann and V. Wolf. Rejection-based simulation of stochastic spreading processes on complex networks. arXiv preprint arXiv:1812.10845 , 2018. Reducing Spreading Processes on Networks to Markov Population Models 17

work page internal anchor Pith review Pith/arXiv arXiv 2018
[21]

Hagberg, P

A. Hagberg, P. Swart, and D. S Chult. Exploring network structure, dynamics, and function using networkx. Technical report, Los Alamos National Lab.(LANL), Los Alamos, NM (United States), 2008

work page 2008
[22]

T. A. Henzinger, M. Mateescu, and V. Wolf. Sliding window abstraction for inﬁnite markov chains. In International Conference on Computer Aided Veriﬁcation, pages 337–352. Springer, 2009

work page 2009
[23]

P. Holme. Shadows of the susceptible-infectious-susceptible immortality transition in small networks. Physical Review E, 92(1):012804, 2015

work page 2015
[24]

M. J. Keeling and P. Rohani. Modeling infectious diseases in humans and animals . Princeton University Press, 2011

work page 2011
[25]

W. R. KhudaBukhsh, A. Auddy, Y. Disser, and H. Koeppl. Approximate lumpabil- ity for markovian agent-based models using local symmetries. arXiv:1804.00910

work page internal anchor Pith review Pith/arXiv arXiv
[26]

I. Z. Kiss, J. C. Miller, and P. L. Simon. Mathematics of epidemics on networks: from exact to approximate models. Forthcoming in Springer TAM series , 2016

work page 2016
[27]

Kitsak, L

M. Kitsak, L. K. Gallos, S. Havlin, F. Liljeros, L. Muchnik, H. E. Stanley, and H. A. Makse. Identiﬁcation of inﬂuential spreaders in complex networks. Nature physics, 6(11):888, 2010

work page 2010
[28]

Kyriakopoulos, G

C. Kyriakopoulos, G. Grossmann, V. Wolf, and L. Bortolussi. Lumping of degree- based mean-ﬁeld and pair-approximation equations for multistate contact pro- cesses. Physical Review E, 97(1):012301, 2018

work page 2018
[29]

Li and H

G. Li and H. Rabitz. A general analysis of approximate lumping in chemical kinetics. Chemical engineering science, 45(4):977–1002, 1990

work page 1990
[30]

L´ opez-Garc´ ıa

M. L´ opez-Garc´ ıa. Stochastic descriptors in an sir epidemic model for heterogeneous individuals in small networks. Mathematical biosciences, 271:42–61, 2016

work page 2016
[31]

Mateescu, V

M. Mateescu, V. Wolf, F. Didier, and T. Henzinger. Fast adaptive uniformisation of the chemical master equation. IET systems biology , 4(6):441–452, 2010

work page 2010
[32]

R. M. May and N. Arinaminpathy. Systemic risk: the dynamics of model banking systems. Journal of the Royal Society Interface , 7(46):823–838, 2009

work page 2009
[33]

Moslonka-Lefebvre, M

M. Moslonka-Lefebvre, M. Pautasso, and M. J. Jeger. Disease spread in small- size directed networks: epidemic threshold, correlation between links to and from nodes, and clustering. Journal of theoretical biology, 260(3):402–411, 2009

work page 2009
[34]

T. W. Ng, G. Turinici, and A. Danchin. A double epidemic model for the sars propagation. BMC Infectious Diseases, 3(1):19, 2003

work page 2003
[35]

Pautasso, M

M. Pautasso, M. Moslonka-Lefebvre, and M. J. Jeger. The number of links to and from the starting node as a predictor of epidemic size in small-size directed networks. Ecological Complexity, 7(4):424–432, 2010

work page 2010
[36]

Porter and J

M. Porter and J. Gleeson. Dynamical systems on networks: A tutorial , volume 4. Springer, 2016

work page 2016
[37]

H. S. Rodrigues. Application of sir epidemiological model: new trends. arXiv:1611.02565, 2016

work page internal anchor Pith review Pith/arXiv arXiv 2016
[38]

H. S. Rodrigues, M. T. T. Monteiro, and D. F. Torres. Dynamics of dengue epi- demics when using optimal control. Mathematical and Computer Modelling , 52(9- 10):1667–1673, 2010

work page 2010
[39]

Schnoerr, G

D. Schnoerr, G. Sanguinetti, and R. Grima. Approximation and inference meth- ods for stochastic biochemical kinetics - a tutorial review. Journal of Physics A , 51:169501, 2018

work page 2018
[40]

P. L. Simon, M. Taylor, and I. Z. Kiss. Exact epidemic models on graphs using graph-automorphism driven lumping. Journal of mathematical biology , 62(4):479– 508, 2011

work page 2011
[41]

Singh and J

A. Singh and J. P. Hespanha. Stochastic hybrid systems for studying biochemical processes. Royal Society A, 368(1930):4995–5011, 2010. 18 G. Großmann and L. Bortolussi

work page 1930
[42]

Soltani, C

M. Soltani, C. A. Vargas-Garcia, and A. Singh. Conditional moment closure schemes for studying stochastic dynamics of genetic circuits. IEEE transactions on biomedical circuits and systems , 9(4):518–526, 2015

work page 2015
[43]

St-Onge, J.-G

G. St-Onge, J.-G. Young, L. H´ ebert-Dufresne, and L. J. Dub´ e. Eﬃcient sampling of spreading processes on complex networks using a composition and rejection algorithm. Computer Physics Communications , 2019

work page 2019
[44]

N. G. Van Kampen. Stochastic processes in physics and chemistry , volume 1. Elsevier, 1992

work page 1992
[45]

Van Mieghem, J

P. Van Mieghem, J. Omic, and R. Kooij. Virus spread in networks. IEEE/ACM Transactions on Networking, 17(1):1–14, 2009

work page 2009
[46]

J. A. Ward and J. Evans. A general model of dynamics on networks with graph automorphism lumping. In International Workshop on Complex Networks and their Applications, pages 445–456. Springer, 2018

work page 2018
[47]

D. J. Watts and P. S. Dodds. Inﬂuentials, networks, and public opinion formation. Journal of consumer research, 34(4):441–458, 2007

work page 2007
[48]

Wei and J

J. Wei and J. C. Kuo. Lumping analysis in monomolecular reaction systems. analysis of the exactly lumpable system. Industrial & Engineering chemistry fun- damentals, 8(1):114–123, 1969

work page 1969
[49]

X. Wei, N. C. Valler, B. A. Prakash, I. Neamtiu, M. Faloutsos, and C. Faloutsos. Competing memes propagation on networks: A network science perspective. IEEE Journal on Selected Areas in Communications , 31(6):1049–1060, 2013

work page 2013
[50]

L. Zhao, H. Cui, X. Qiu, X. Wang, and J. Wang. Sir rumor spreading model in the new media age. Physica A, 392(4):995–1003, 2013

work page 2013
[51]

L. Zhao, J. Wang, Y. Chen, Q. Wang, J. Cheng, and H. Cui. Sihr rumor spreading model in social networks. Physica A, 391(7):2444–2453, 2012. Reducing Spreading Processes on Networks to Markov Population Models 19 Appendix 8 Direct Construction of MPMs Here, we prosper a way of directly deriving the lumped MPMs from the contact network without building the ...

work page 2012

[1] [1]

G. E. Allen and C. Dytham. An eﬃcient method for stochastic simulation of biological populations in continuous time. Biosystems, 98(1):37–42, 2009

work page 2009

[2] [2]

Barab´ asi.Network science

A.-L. Barab´ asi.Network science. Cambridge university press, 2016

work page 2016

[3] [3]

Bortolussi

L. Bortolussi. Hybrid behaviour of Markov population models. Information and Computation, 247:37–86, 2016

work page 2016

[4] [4]

Bortolussi, J

L. Bortolussi, J. Hillston, D. Latella, and M. Massink. Continuous approximation of collective system behaviour: A tutorial. Performance Evaluation, 70(5):317–349, 2013

work page 2013

[5] [5]

Buchholz

P. Buchholz. Exact and ordinary lumpability in ﬁnite markov chains. Journal of applied probability, 31(1):59–75, 1994

work page 1994

[6] [6]

Y. Cao, D. T. Gillespie, and L. R. Petzold. Eﬃcient step size selection for the tau-leaping simulation method. The Journal of chemical physics , 124(4)

work page

[7] [7]

Cardelli, M

L. Cardelli, M. Tribastone, M. Tschaikowski, and A. Vandin. Erode: a tool for the evaluation and reduction of ordinary diﬀerential equations. In International Conference on Tools and Algorithms for the Construction and Analysis of Systems , pages 310–328. Springer, 2017

work page 2017

[8] [8]

Cota and S

W. Cota and S. C. Ferreira. Optimized gillespie algorithms for the simulation of markovian epidemic processes on large and heterogeneous networks. Computer Physics Communications, 219:303–312, 2017

work page 2017

[9] [9]

Devriendt and P

K. Devriendt and P. Van Mieghem. Uniﬁed mean-ﬁeld framework for susceptible- infected-susceptible epidemics on networks, based on graph partitioning and the isoperimetric inequality. Physical Review E, 96(5):052314, 2017

work page 2017

[10] [10]

C. Gan, X. Yang, W. Liu, Q. Zhu, and X. Zhang. Propagation of computer virus under human intervention: a dynamical model. Discrete Dynamics in Nature and Society, 2012, 2012

work page 2012

[11] [11]

Girvan and M

M. Girvan and M. E. Newman. Community structure in social and biological networks. Proceedings of the national academy of sciences, 99(12):7821–7826, 2002

work page 2002

[12] [12]

J. P. Gleeson. High-accuracy approximation of binary-state dynamics on networks. Physical Review Letters, 107(6):068701, 2011

work page 2011

[13] [13]

J. P. Gleeson. Binary-state dynamics on complex networks: Pair approximation and beyond. Physical Review X, 3(2):021004, 2013

work page 2013

[14] [14]

Goltsev, F

A. Goltsev, F. De Abreu, S. Dorogovtsev, and J. Mendes. Stochastic cellular automata model of neural networks. Physical Review E, 81(6):061921, 2010

work page 2010

[15] [15]

Goutsias and G

J. Goutsias and G. Jenkinson. Markovian dynamics on complex reaction networks. Physics Reports, 529(2):199–264, 2013

work page 2013

[16] [16]

Goyal and E

P. Goyal and E. Ferrara. Graph embedding techniques, applications, and perfor- mance: A survey. Knowledge-Based Systems, 151:78–94, 2018

work page 2018

[17] [17]

R. Grima. An eﬀective rate equation approach to reaction kinetics in small volumes: Theory and application to biochemical reactions in nonequilibrium steady-state conditions. The Journal of Chemical Physics , 133(3):035101, 2010

work page 2010

[18] [18]

R. Grima. A study of the accuracy of moment-closure approximations for stochastic chemical kinetics. The Journal of chemical physics , 136(15):04B616, 2012

work page 2012

[19] [19]

Großmann, C

G. Großmann, C. Kyriakopoulos, L. Bortolussi, and V. Wolf. Lumping the ap- proximate master equation for multistate processes on complex networks. In Qest, pages 157–172. Springer, 2018

work page 2018

[20] [20]

Rejection-Based Simulation of Stochastic Spreading Processes on Complex Networks

G. Großmann and V. Wolf. Rejection-based simulation of stochastic spreading processes on complex networks. arXiv preprint arXiv:1812.10845 , 2018. Reducing Spreading Processes on Networks to Markov Population Models 17

work page internal anchor Pith review Pith/arXiv arXiv 2018

[21] [21]

Hagberg, P

A. Hagberg, P. Swart, and D. S Chult. Exploring network structure, dynamics, and function using networkx. Technical report, Los Alamos National Lab.(LANL), Los Alamos, NM (United States), 2008

work page 2008

[22] [22]

T. A. Henzinger, M. Mateescu, and V. Wolf. Sliding window abstraction for inﬁnite markov chains. In International Conference on Computer Aided Veriﬁcation, pages 337–352. Springer, 2009

work page 2009

[23] [23]

P. Holme. Shadows of the susceptible-infectious-susceptible immortality transition in small networks. Physical Review E, 92(1):012804, 2015

work page 2015

[24] [24]

M. J. Keeling and P. Rohani. Modeling infectious diseases in humans and animals . Princeton University Press, 2011

work page 2011

[25] [25]

W. R. KhudaBukhsh, A. Auddy, Y. Disser, and H. Koeppl. Approximate lumpabil- ity for markovian agent-based models using local symmetries. arXiv:1804.00910

work page internal anchor Pith review Pith/arXiv arXiv

[26] [26]

I. Z. Kiss, J. C. Miller, and P. L. Simon. Mathematics of epidemics on networks: from exact to approximate models. Forthcoming in Springer TAM series , 2016

work page 2016

[27] [27]

Kitsak, L

M. Kitsak, L. K. Gallos, S. Havlin, F. Liljeros, L. Muchnik, H. E. Stanley, and H. A. Makse. Identiﬁcation of inﬂuential spreaders in complex networks. Nature physics, 6(11):888, 2010

work page 2010

[28] [28]

Kyriakopoulos, G

C. Kyriakopoulos, G. Grossmann, V. Wolf, and L. Bortolussi. Lumping of degree- based mean-ﬁeld and pair-approximation equations for multistate contact pro- cesses. Physical Review E, 97(1):012301, 2018

work page 2018

[29] [29]

Li and H

G. Li and H. Rabitz. A general analysis of approximate lumping in chemical kinetics. Chemical engineering science, 45(4):977–1002, 1990

work page 1990

[30] [30]

L´ opez-Garc´ ıa

M. L´ opez-Garc´ ıa. Stochastic descriptors in an sir epidemic model for heterogeneous individuals in small networks. Mathematical biosciences, 271:42–61, 2016

work page 2016

[31] [31]

Mateescu, V

M. Mateescu, V. Wolf, F. Didier, and T. Henzinger. Fast adaptive uniformisation of the chemical master equation. IET systems biology , 4(6):441–452, 2010

work page 2010

[32] [32]

R. M. May and N. Arinaminpathy. Systemic risk: the dynamics of model banking systems. Journal of the Royal Society Interface , 7(46):823–838, 2009

work page 2009

[33] [33]

Moslonka-Lefebvre, M

M. Moslonka-Lefebvre, M. Pautasso, and M. J. Jeger. Disease spread in small- size directed networks: epidemic threshold, correlation between links to and from nodes, and clustering. Journal of theoretical biology, 260(3):402–411, 2009

work page 2009

[34] [34]

T. W. Ng, G. Turinici, and A. Danchin. A double epidemic model for the sars propagation. BMC Infectious Diseases, 3(1):19, 2003

work page 2003

[35] [35]

Pautasso, M

M. Pautasso, M. Moslonka-Lefebvre, and M. J. Jeger. The number of links to and from the starting node as a predictor of epidemic size in small-size directed networks. Ecological Complexity, 7(4):424–432, 2010

work page 2010

[36] [36]

Porter and J

M. Porter and J. Gleeson. Dynamical systems on networks: A tutorial , volume 4. Springer, 2016

work page 2016

[37] [37]

H. S. Rodrigues. Application of sir epidemiological model: new trends. arXiv:1611.02565, 2016

work page internal anchor Pith review Pith/arXiv arXiv 2016

[38] [38]

H. S. Rodrigues, M. T. T. Monteiro, and D. F. Torres. Dynamics of dengue epi- demics when using optimal control. Mathematical and Computer Modelling , 52(9- 10):1667–1673, 2010

work page 2010

[39] [39]

Schnoerr, G

D. Schnoerr, G. Sanguinetti, and R. Grima. Approximation and inference meth- ods for stochastic biochemical kinetics - a tutorial review. Journal of Physics A , 51:169501, 2018

work page 2018

[40] [40]

P. L. Simon, M. Taylor, and I. Z. Kiss. Exact epidemic models on graphs using graph-automorphism driven lumping. Journal of mathematical biology , 62(4):479– 508, 2011

work page 2011

[41] [41]

Singh and J

A. Singh and J. P. Hespanha. Stochastic hybrid systems for studying biochemical processes. Royal Society A, 368(1930):4995–5011, 2010. 18 G. Großmann and L. Bortolussi

work page 1930

[42] [42]

Soltani, C

M. Soltani, C. A. Vargas-Garcia, and A. Singh. Conditional moment closure schemes for studying stochastic dynamics of genetic circuits. IEEE transactions on biomedical circuits and systems , 9(4):518–526, 2015

work page 2015

[43] [43]

St-Onge, J.-G

G. St-Onge, J.-G. Young, L. H´ ebert-Dufresne, and L. J. Dub´ e. Eﬃcient sampling of spreading processes on complex networks using a composition and rejection algorithm. Computer Physics Communications , 2019

work page 2019

[44] [44]

N. G. Van Kampen. Stochastic processes in physics and chemistry , volume 1. Elsevier, 1992

work page 1992

[45] [45]

Van Mieghem, J

P. Van Mieghem, J. Omic, and R. Kooij. Virus spread in networks. IEEE/ACM Transactions on Networking, 17(1):1–14, 2009

work page 2009

[46] [46]

J. A. Ward and J. Evans. A general model of dynamics on networks with graph automorphism lumping. In International Workshop on Complex Networks and their Applications, pages 445–456. Springer, 2018

work page 2018

[47] [47]

D. J. Watts and P. S. Dodds. Inﬂuentials, networks, and public opinion formation. Journal of consumer research, 34(4):441–458, 2007

work page 2007

[48] [48]

Wei and J

J. Wei and J. C. Kuo. Lumping analysis in monomolecular reaction systems. analysis of the exactly lumpable system. Industrial & Engineering chemistry fun- damentals, 8(1):114–123, 1969

work page 1969

[49] [49]

X. Wei, N. C. Valler, B. A. Prakash, I. Neamtiu, M. Faloutsos, and C. Faloutsos. Competing memes propagation on networks: A network science perspective. IEEE Journal on Selected Areas in Communications , 31(6):1049–1060, 2013

work page 2013

[50] [50]

L. Zhao, H. Cui, X. Qiu, X. Wang, and J. Wang. Sir rumor spreading model in the new media age. Physica A, 392(4):995–1003, 2013

work page 2013

[51] [51]

L. Zhao, J. Wang, Y. Chen, Q. Wang, J. Cheng, and H. Cui. Sihr rumor spreading model in social networks. Physica A, 391(7):2444–2453, 2012. Reducing Spreading Processes on Networks to Markov Population Models 19 Appendix 8 Direct Construction of MPMs Here, we prosper a way of directly deriving the lumped MPMs from the contact network without building the ...

work page 2012