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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 (S→I) 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 Z≥0and in each step one can only go from state ito i+ 1

(birth) or to i−1 (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-

R´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)