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Poster Abstract: Birth-Death Processes Reproduce the Infection Footprint of Complex Networks

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Poster Abstract on how birth-death processes can reproduce SIS-type epidemic dynamics on (random) complex networks.
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Poster: Birth-Death Processes Reproduce the
Infection Footprint of Complex Networks
Michael Backenk¨ohler?and Gerrit Großmann( )?
Saarland University, 66123 Saarbr¨ucken, Germany
{michael.backenkoehler,gerrit.grossmann}@uni-saarland.de
1 Introduction and Problem Setting
The Susceptible-Infected-Susceptible (SIS) paradigm is a powerful tool to study
stochastic dynamics happening on complex networks. A contact network specifies
the connectivity between nodes (or agents) that are either infected (I) or suscep-
tible (S). Infected nodes (randomly) propagate their infection to their susceptible
neighbors and can spontaneously recover (i.e., become susceptible again). Here,
we consider a continuous-time model where the waiting times between events
follow an exponential distribution [3].
A crucial value for any SIS model is the effective infection rate λ[4], de-
fined as the ratio of infection rate and recovery rate. It determines the affinity
of an infecting to die out (early) or to become endemic. Formally, we define the
infection footprint τ(λ) to be the expected fraction of infected nodes in equi-
librium as our value of interest. For technical reasons (i.e., to get rid of a trap
state [1]), we include self-infections (SI) with rate , where is much smaller
than λ. Our goal is to estimate τ(·) efficiently. To this end, we propose a model
reduction (or lumping) technique that projects the complicated dynamics of the
underlying model to a simple birth-death process. The construction of the re-
duced model and its analysis are computationally fast and can be performed for
contact networks with thousands of nodes with ease.
2 Method
Given a contact network and λ, we build two birth-death-processes with n+ 1
states each (cf. Fig. 1b., nbeing the number of nodes in the network). A birth-
death process is a special kind of continuous-time Markov chain where the state-
space is a subset of Z0and in each step one can only go from state ito i+ 1
(birth) or to i1 (death). The state iof our birth-death process corresponds
to the number of infected nodes in the original model. In the original model,
the aggregated rate for a new infection is determined by the number of SI-edges
in the current network configuration. In the reduced model, we only know the
number of infected nodes, which might correspond to a wide range of possible SI-
edge counts. Therefore, we construct two birth-death-processes (with footprints
?equal contribution
I
II
I I
I
I
II
I I
I
I
I
I
I
I
I
I
I
I I
I
I
II
I
I
I I
II
1
λ
2λ
0 1 2 33 4
[λ,2λ] [λ, 3λ] [λ, 2λ]
1 2 3 4
Markov Graph
Birth-Death Approximation
(a) Reduction (b) Results
Fig. 1: (a) Original (top) and reduced model (bottom) without self-infections.
Susceptible (infected) nodes are shown in blue (red). (b) Results for: Erd˝os-
enyi, Barab´asi–Albert, Newman-Watts-Strogatz random graphs (top to bot-
tom) with 102(left) and 104(right) nodes (= 0.001) and a simulated baseline.
τmin(·), τmax (·)), one where we under-approximate the number of SI-edges for a
given number of infected nodes and one where we over-approximate it. We can
efficiently determine the equilibria of those two processes and have that τmin(λ)
τ(λ)τmax(λ)λ > 0 . To make our method computationally efficient we use
a simple greedy approach to estimate the number of SI-edges. Empirically, we
never witness the true τ(λ) falling out of our approximated bounds. Our method
is closely related to [2] where we only used the mean number of SI-edges.
3 Results
We tested many different synthetic contact networks. Specifically, we investi-
gated how the network size and the network density influenced the quality of
the approximated bounds. Results are shown in Fig. 1b.
References
1. van de Bovenkamp, R., Van Mieghem, P.: Time to metastable state in sis epidemics
on graphs. In: 2014 Tenth International Conference on Signal-Image Technology and
Internet-Based Systems. pp. 347–354. IEEE (2014)
2. Großmann, G., Bortolussi, L.: Reducing spreading processes on networks to markov
population models. In: Qest. pp. 292–309. Springer (2019)
3. Kiss, I.Z., Miller, J.C., Simon, P.L., et al.: Mathematics of epidemics on networks.
Cham: Springer 598 (2017)
4. Prakash, B.A., Chakrabarti, D., Valler, N.C., Faloutsos, M., Faloutsos, C.: Thresh-
old conditions for arbitrary cascade models on arbitrary networks. Knowledge and
information systems 33(3), 549–575 (2012)
ResearchGate has not been able to resolve any citations for this publication.
Chapter
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.
Time to metastable state in sis epidemics on graphs
  • R Van De Bovenkamp
  • P Van Mieghem
van de Bovenkamp, R., Van Mieghem, P.: Time to metastable state in sis epidemics on graphs. In: 2014 Tenth International Conference on Signal-Image Technology and Internet-Based Systems. pp. 347-354. IEEE (2014)
Threshold conditions for arbitrary cascade models on arbitrary networks
  • B A Prakash
  • D Chakrabarti
  • N C Valler
  • M Faloutsos
  • C Faloutsos
Prakash, B.A., Chakrabarti, D., Valler, N.C., Faloutsos, M., Faloutsos, C.: Threshold conditions for arbitrary cascade models on arbitrary networks. Knowledge and information systems 33(3), 549-575 (2012)