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# Rejection-Based Simulation of Non-Markovian Agents on Complex Networks

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## Abstract

Stochastic models in which agents interact with their neighborhood according to a network topology are a powerful modeling framework to study the emergence of complex dynamic patterns in real-world systems. Stochastic simulations are often the preferred-sometimes the only feasible-way to investigate such systems. Previous research focused primarily on Markovian models where the random time until an interaction happens follows an exponential distribution. In this work, we study a general framework to model systems where each agent is in one of several states. Agents can change their state at random, influenced by their complete neighborhood, while the time to the next event can follow an arbitrary probability distribution. Classically, these simulations are hindered by high computational costs of updating the rates of interconnected agents and sampling the random residence times from arbitrary distributions. We propose a rejection-based, event-driven simulation algorithm to overcome these limitations. Our method over-approximates the instantaneous rates corresponding to inter-event times while rejection events counterbalance these over-approximations. We demonstrate the effectiveness of our approach on models of epidemic and information spreading.
Rejection-Based Simulation of Non-Markovian
Agents on Complex Networks
Gerrit Großmann( )1 ,
Luca Bortolussi1,2, and Verena Wolf1
1Saarland University, 66123 Saarbr¨ucken, Germany
{gerrit.grossmann,verena.wolf}@uni-saarland.de
2University of Trieste, Trieste, Italy
luca@dmi.units.it
Abstract. Stochastic models in which agents interact with their neigh-
borhood according to a network topology are a powerful modeling frame-
work to study the emergence of complex dynamic patterns in real-world
systems. Stochastic simulations are often the preferred—sometimes the
only feasible—way to investigate such systems. Previous research focused
primarily on Markovian models where the random time until an interac-
tion happens follows an exponential distribution.
In this work, we study a general framework to model systems where each
agent is in one of several states. Agents can change their state at random,
inﬂuenced by their complete neighborhood, while the time to the next
event can follow an arbitrary probability distribution. Classically, these
simulations are hindered by high computational costs of updating the
rates of interconnected agents and sampling the random residence times
from arbitrary distributions.
We propose a rejection-based, event-driven simulation algorithm to over-
come these limitations. Our method over-approximates the instantaneous
rates corresponding to inter-event times while rejection events counter-
balance these over-approximations. We demonstrate the eﬀectiveness of
our approach on models of epidemic and information spreading.
Keywords: Gillespie Simulation, Complex Networks, Epidemic Model-
ing, Rejection Sampling, Multi-Agent System
1 Introduction
Computational modeling of dynamic processes on complex networks is a thriving
research area [1–3]. Arguably, the most common formalism for spreading pro-
cesses is the continuous-time SIS model and its variants [4–6]. Generally speak-
ing, an underlying contact network speciﬁes the connectivity between nodes (i.e.,
agents) and each agent occupies one of several mutually exclusive (local) states
(or compartments). In the well-known SIS model, these states are susceptible
(S) and infected (I). Infected nodes can recover (become susceptible again) and
propagate their infection to neighboring susceptible nodes.
2 G. Großmann et al.
SIS-type models have shown to be extremely useful for analyzing and pre-
dicting the spread of opinions, rumors, and memes in online social networks [7,
8] as well as the neural activity [9,10], the spread of malware [11], and blackouts
in ﬁnancial institutions [12, 13].
Previous research focused mostly on models where the probability of an event
(e.g. infection or recovery) happening in the next (inﬁnitesimal) time unit is
constant, i.e. independent of the time the agent has already spent in its current
state. We call such agents memoryless and the corresponding stochastic process
Markovian. The semantics of such a model can be described using a so-called
(discrete-state) continuous-time Markov chain (CTMC).
One particularly important consequence of the memoryless property is that
the random time until an agent changes its state, either because of an inter-
action with another agent or because of a spontaneous transition, follows an
exponential distribution. The distribution of this residence time is parameter-
ized by an (interaction-speciﬁc) rate λR0[4]. Each agent has an associated
event-modulated Poisson process whose rate depends on the agent’s state and
the state of its neighbors [14]. For instance, infection of an agent increases the
rate at which its susceptible neighbors switch to the infected state.
However, exponentially distributed residence times are an unrealistic assump-
tion in many real-word systems. This holds in particular for the spread of epi-
demics [15–19], for the diﬀusion of opinions in online social networks [20, 21], and
interspike times of neurons [22] as empirical results show. However, assuming
time delays that can follow non-exponential distributions complicate the anal-
ysis of such processes and typically only allow Monte-Carlo simulations, which
suﬀer from high computational costs.
Recently, the Laplace-Gillespie algorithm (LGA) for the simulation of non-
Markovian dynamics has been introduced by Masuda and Rocha in [14]. It is
based on the non-Markovian Gillespie algorithm by Bogun´a et al (nMGA) [23]
and minimizes the costs of sampling inter-event times. However, both methods
require computationally expensive updating of an agent’s neighborhood in each
simulation step, which renders them ineﬃcient for large-scale networks. In the
context of Markovian processes on networks, it has recently been shown that
rejection-based simulation can overcome this limitation [24–26].
Here, we extend the idea of rejection-based simulation to non-Markovian net-
worked systems, proposing RED-Sim, a rejection-based, event-driven simulation
approach. Speciﬁcally, we combine three ideas to obtain an eﬃcient simulation
of non-Markovian processes: (i) we express the distributions of inter-event times
through time-varying instantaneous rates, (ii) we sample events based on an
over-approximation of these rates and compensate via a rejection step, and (iii)
we use a priority queue to sample the next event. The combination of these ele-
ments makes it possible to reduce the time-complexity of each simulation step.
Speciﬁcally, if an agent changes its state, no update of the rate of neighboring
agents is necessary. This comes at the costs of rejection events to counter-balance
missing information about the neighborhood. However, using a priority queue
renders the computational burden of each rejection event very small.
Rejection-Based Simulation of Non-Markovian Agents 3
The remainder of the paper is organized as follows: We describe our frame-
work for non-Markovian dynamics in Section 2 and provide a review of previous
simulation approaches in Section 3. Next, we propose our rejection-based simula-
tion algorithm in Section 4. Section 5 presents numerical results and we conclude
our work in Section 6.
2 Multi-Agent Model
This section introduces the underlying formalism to express agent-based dy-
namics on networks. Let G= (N,E) be a an undirected, ﬁnite graph without
self-loops, called contact network. Nodes n∈ N are also referred to as agents.
Network State. The current state of a network Gis described by two functions:
S:N → S assigns to each agent na local state S(n)∈ S, where Sis a ﬁnite
set of local states (e.g., S={S,I}for the SIS model);
R:N R0, describes the residence time of each agent, i.e. the time since
the last change of state of the agent.
We say that an agent ﬁres when it changes its state and refer to the remaining
time until it ﬁres as its time delay. The neighborhood state M(n) of an agent n
is a multi-set containing the states of all neighboring agents together with their
respective residence times:
M(n) = nS(n0), R(n0)(n, n0)∈ Eo.
The set of all possible neighborhood-states of all agents in a given system is
denoted by M.
Network Dynamics. The dynamics of the network is described by assigning
to each agent ntwo functions φnand ψn:
φn:S × R0× M R0deﬁnes the instantaneous rate of n, i.e. if
λ=φnS(n), R(n), M (n), then the probability that nﬁres in the next
inﬁnitesimal time interval tis λt;
ψn:S ×R0×M → PSdetermines the state probabilities when a transition
occurs. Here, PSdenotes the set of all probability distributions over S. Hence
if p=ψnS(n), R(n), M (n), then, when agent nﬁres, the next local state
is swith probability p(s).
Note that we do not consider cases of pathological behavior here, e.g. where φn
is deﬁned in such a way that an inﬁnite amount of simulation steps is possible
in ﬁnite time.
A multi-agent network model is completely speciﬁed by a tuple (G,S,{φn},
{ψn}, S0), where S0denotes a function that assigns to each node an initial state.
4 G. Großmann et al.
Example. In the classical SIS model we have S={S,I}and φand ψare the
same for all agents, i.e.,
φn(s, t, m) =
crif s=I
ciP
s0,t0m
I(s0) if s=Sψn(s, t, m) = (Sif s=I
Iif s=S
Here, ci, crR0denote the infection and recovery rate constants, respectively.
Note that the infection rate is proportional to the number of infected neighbors
whereas the rate of recovery is independent from neighboring agents. Moreover,
s:S → {0,1}maps a state s0to one iﬀ s=s0and to zero otherwise. The model
is Markovian as neither φnor ψdepend on the residence time of any agent.
2.1 Semantics
We will specify the semantics of a multi-agent model by describing a stochastic
simulation algorithm that generates trajectories of the system. It is based on a
race condition among agents: each agent picks a random time until it will ﬁre,
but only the one with the shortest time delay wins and changes its state.
Time Delay Density. Assume that tis the time increment of the algorithm.
We deﬁne for each nthe eﬀective rate λn(t) as
λn(t) = φnS(n), R(n) + t, Mt(n),where
Mt(n) = nS(n0), R(n0) + tn, n0∈ Eo,
describes the neighborhood-state of nin ttime units assuming that all agents
remain in their current state. Next we assume that for each node n, the prob-
ability density of the (non-negative) time delay is γn, i.e. γn(t) is the density
of ﬁring after ttime units. Leveraging the theory of renewal processes [27], we
ﬁnd the relationship
λn(t) = γn(t)
1Rt
0γn(t)and γn(t) = λn(t)eRt
0λn(y)dy .(1)
We assume λn(t) to be zero if the denominator is zero. Note that using this
equation, we can derive rate functions from a given time delay distribution (i.e.
uniform, log-normal, gamma, and so on). If it is not possible to derive λnana-
lytically, it can be computed numerically.
For example, a constant rate function λ(t) = ccorresponds to an exponen-
tial time delay distribution γ(t) = cectwith rate c. Fig. 1 (b) illustrates the
rate function when γis the uniform distribution on [1,2].
Rejection-Based Simulation of Non-Markovian Agents 5
(a) (b) (c) (d)
Fig. 1: (a-c) Sampling event times with a rate function 1t[1,2]
2t. (a) Generate a
random variate from the exponential distribution with rate λ= 1, the sample
here is 0.69. (b) We integrate the rate function until the area is 0.69, here tn=
1.5. (c) This is the rate function corresponding to the uniform distribution in
γ(t) = 1t[1,2]. (d) Sampling tnfrom a time-varying rate function using an
upper-bound of c= 1, rejection probabilities shown in red.
Sampling Time Delays. The eﬀective rate λn(t) allows us to sample the
time delay tnafter which agent nﬁres, using the inversion method. First, we
sample an exponential random variate xwith rate 1, then we integrate λn(t)
to ﬁnd tnsuch that Ztn
0
λn(t)dt=x . (2)
In general it is possible to pre-compute the integral [28], but its parameterization
(on states, residence times, etc) renders this diﬃcult.
Another viable approach is to use rejection sampling. Assume that we have
cR0such that λn(t)cfor all t. We start with tn= 0. In each step,
we sample an exponentially distributed random variate t0
nwith rate cand set
tn=tn+t0
n. We accept tnwith probability λn(tn)
c. Otherwise we reject it and
repeat the process. If a reasonable over-approximation can be constructed, this
is typically much faster than the integral approach in (2).
Na¨ıve Simulation Algorithm. The following simulation algorithm generates
statistically correct trajectories of the model. It starts by initializing the global
clock tglobal = 0 and setting R(n) = 0 for all n. The algorithm repeatedly
performs simulation steps until a predeﬁned time horizon or some other stopping
criterion is reached. Each stimulation step is as follows:
1. Generate a random time delay (candidate) tnfor each agent nusing γn.
Identify agent n0with the smallest time delay tn0.
2. Pick the next state s0for n0according to ψn0S(n0), R(n0) + tn0, M (n0)and
set S(n0) = s0. Set R(n0) = 0 and R(n) = R(n) + tn0(n6=n0).
3. Set tglobal =tglobal +tn0to update the global clock and go to Line 1.
Note that this algorithm is very ineﬃcient as it requires an expensive iteration
over all agents and sampling of time delays in each step.
6 G. Großmann et al.
3 Previous Simulation Approaches
Most recent work on non-Markovian dynamics focuses on the mathematical mod-
eling of such processes [29–33]. In particular, research has focused on how spe-
ciﬁc distributions (e.g. constant recovery times) alter the properties of epidemic
spreading such as the epidemic threshold (see [3, 4] for an overview). However,
only few approaches are known for the simulation of non-Markovian dynamics
[23, 14]. We shortly review them in the sequel.
3.1 Non-Markovian Gillespie Algorithm
Bogun´a et al. propose a direct generalization of the Gillespie algorithm to non-
Markovian systems, nMGA, which is statistically exact but computationally ex-
pensive [23]. The algorithm is conceptually similar to our baseline in Section
2.1 but computes the time delay using so-called survival functions. An agent’s
survival function determines the probability that its time delay is larger than a
certain time t. The joint survival function of all agents determines the proba-
bility that all time delays are larger than twhich can be used to sample the
next event time.
The drawback of the nMGA is that it is necessary to iterate over all agents
in each step in order to construct their joint survival function. As a fast ap-
proximation, the authors suggest to only use the current instantaneous rate at
t= 0 i.e., λn(0)and assume all rates remain constant until the next event.
This is correct in the limit of inﬁnite agents, because when the number of agent
approaches inﬁnity, the time until the next ﬁring of any agent approaches zero.
3.2 Laplace-Gillespie Algorithm
The LGA, introduced by Masuda and Rocha in [14], aims at reducing the compu-
tational cost of ﬁnding the next event time compared to nMGA, while remaining
statistically correct. It assumes that the time delay distributions can be expressed
in the form of a weighted average of exponential distributions
γn(t) = Z
0
pn(λ)λeλtdλ ,
where pnis a PDF over the rate λR0. This formulation of γn, while being
very elegant, limits the applicability to cases where the corresponding survival
function is completely monotone [14], which limits the set of possible inter-event
time distributions.
The LGA has two advantages. Firstly, we can sample tnby ﬁrst sampling
λaccording to pnand then, instead of the numerical integration in Eq. (2),
compute tn=ln u/λ where uis uniformly distributed on (0,1). Secondly, we
can assume that the sampled λfor a particular agent remains constant until one
of its neighbors ﬁres. Thus, in each step, it is only necessary to update the rates
of the neighbors of the ﬁring agent, and not of all agents.
Rejection-Based Simulation of Non-Markovian Agents 7
4 Our Method
Rejection sampling for the eﬃcient simulation of Markovian stochastic processes
on complex networks has been proposed recently [24–26, 34], but not for the
non-Markovian case where arbitrary distributions for the inter-event times are
considered.
Here, we proposes the RED-Sim algorithm for the generation of statistically
correct simulations of non-Markovian network models, as described in Section
2. The main idea of RED-Sim is to rely on rejection sampling to reduce the
computational cost, making it unnecessary to update the rates of the neighbors
of a ﬁring agent. Independently from that, rejection sampling can also be utilized
to sample tnwithout numerical integration.
4.1 Rate Over-Approximation
Recall that λn(·) expresses how the instantaneous rate of nchanges over time,
assuming that no neighboring agent changes its state. A key in ingredient of our
method is now b
λn(·) which upper-bounds the instantaneous rate of n, assuming
that all neighbors are allowed to freely change their state as often as possible.
That is, at all times b
λn(t) is an upper-bound of λn(t) taking into consideration
all possible states of the neighborhood.
Consider again the Markovian SIS example. The curing of an infected node
does not depend on an agent’s neighborhood anyway. The rate is always cr,
which is a trivial upper bound. A susceptible node becomes infected with rate ci
times “number of infected neighbors”. Thus, the instantaneous infection rate of
an agent ncan be bounded by b
λn(t) = knciwhere knis the degree of n. Upper-
bounds may also be constant or depend on time. Consider for example a recovery
time that is uniformly distributed on [1,2]. In this case, λn(·) approaches inﬁnity
(cf. Fig. 1b) making a constant upper-bound impossible. For multi-agent models,
a time-dependent upper-bound always exists since we can compute the maximal
instantaneous rate w.r.t. all reachable neighborhood states.
4.2 The RED-Sim Algorithm
For a given multi-agent model speciﬁcation (G,S,{φn},{ψn}, S0) and given
upper-bounds {b
λn}, we propose a statistically exact simulation algorithm, which
is based on two basic data structures:
Labeled Graph
A graph represents the contact network and each agent (node) nis annotated
with its current state S(n) and T(n), the time point of its last state change.
Event Queue
The event queue stores the list of future events, where an event is a tuple
(n, bµ, b
tn). Here, nis the agent that ﬁres, b
tnthe prospective absolute time
point of ﬁring, and bµR0is an over-approximation of the true eﬀective rate
(at time point b
tn). The queue is sorted w.r.t. b
tnand initialized by generating
one event per agent.
8 G. Großmann et al.
A global clock, tglobal, keeps track of the elapsed time since the simulation
started. We use T(n) instead of R(n) to avoid updates for all agents after each
event (i.e., R(n) = tglobal T(n)). We perform simulation steps until some ter-
mination criterion is fulﬁlled, each step is as follows:
1. Take the ﬁrst event (n, bµ, b
tn) from the event queue and update tglobal =b
tn.
2. Evaluate the true instantaneous rate µ=φnS(n), tglobal T(n), M(n)of
nat the current system state.
3. With probability 1 µ
bµ,reject the ﬁring and go to Line 5.
4. Randomly choose the next state s0of naccording to the distribution
ψnS(n), tglobal T(n), M (n). If S(n)6=s0: set S(n) = s0and T(n) = tglobal.
5. Generate a new event for agent nand push it to the event queue.
6. Go to Line 1.
The correctness of RED-Sim can be shown similarly to [26, 24] (see also the
proof sketch in Appendix A). Note that in all approaches evaluating an agent’s
instantaneous rate is linear in the number of its neighbors. In previous ap-
proaches, the rate has to be updated for all neighbors of a ﬁring agent. In
RED-Sim only the rate of the ﬁring agent has to be updated. The key asset
of RED-Sim is that, due to the over-approximation of the rate function, we do
not need to update the neighborhood of the ﬁring agent n, even though the
neighbor’s respective rates might change as a result from the event. We provide
a more detailed analysis of the time-complexity of RED-Sim in Appendix B.
Event Generation. To generate new events in Line 5, we sample a random
time delay tnand set b
tn=tglobal +tn. To sample tnaccording to the over-
approximated rate b
λn(·), we either use the integration approach of Eq. (2) or
sample directly from an upper-bounded the exponential distribution (cf. Fig. 1d).
To sample tnfrom an exponential distribution, we need to be able to ﬁnd an
upper bound that is constant in time b
λn(t) = cfor all t. Hence, we simply set
bµ=cand sample tnfrom an exponential distribution with rate c. Otherwise,
when a constant upper bound either does not exits or is unfeasible to construct,
we use numerical integration over b
λn(·) (see Eq. (2)), and set bµ=b
λn(tn). Al-
ternatively, when b
λn(t) has the required form (cf. Section 3), we can even use
LGA-like approach to sample tn[23] (and also set bµ=b
λn(tn)).
Discussion. We expect RED-Sim to perform poor only in some special cases,
where either the construction of an upper-bound is numerically too expensive
or where the diﬀerence between the upper-bound and the actual average rate is
very large, which would render the number of rejections events too high.
It is easy to extend RED-Sim to diﬀerent types of non-Markovian behavior.
For instance, we might keep track of the number of ﬁrings of an agent and
parameterize φand ψaccordingly to generate the behavior of self-exiting point
processes or to cause correlated ﬁrings among agents [35, 36].
Note that, we can turn the method into a rejection-free approach by gener-
ating a new event for nand all of its neighbors in Line 5while taking the new
Rejection-Based Simulation of Non-Markovian Agents 9
5 Case Studies
We demonstrate the eﬀectiveness our approach on classical epidemic-type pro-
cesses and synthetically generated networks following the conﬁguration model
with a truncated power-law degree distribution [37]. That is P(k)kβfor
3k |N |. We use β∈ {2,2.5}(a smaller βcorresponds to a larger average
degree). The implementation is written in Julia and publicly available3. As a
baseline for comparison, we use the rejection-free variant of the algorithm where
neighbors are updated after an event (as described at the end of Section 4.2).
The evaluation was performed on a 2017 MacBook Pro with a 3.1 GHz Intel
Core i5 CPU and results are shown in Fig. 2.
SIS Model. We consider an SIS model (with ψand φas deﬁned above), but
infected nodes become less infectious over time. That is, the rate at which an
infected agent with residence time t“attacks” its susceptible neighbors is ueut
for u= 0.4. This shifts the exponential distribution to the left. We upper-bound
the infection rate of an agent nwith degree knwith b
λn(t) = uknwhich is
constant in time. Thus, we sample tnusing an exponential distribution. The
time until an infected agent recovers is, independent from its neighborhood,
uniformly distributed in [0,1] (similar to [38]). Hence, we can sample it directly.
Voter Model. The voter model describes the competition of two opinions of
agents in state Aswitch to Band vice versa (i.e. ψis deterministic). The time
until an agent switches follows a Weibull distribution (similar to [23, 39]):
γn(t) = cu(tu)c1e(tu)cand λn(t) = cu(tu)c1, t 0
where we set c=cA= 2.0, u=uAif S(n) = Aand c=cB= 2.05, u=uBif
S(n) = B. We let the fraction of opposing neighbors modulate u, i.e., uA=Bn
kn,
where Bndenotes the number of neighbors currently in state Band knis the
degree of agent n(and analogously for A). Hence, the instantaneous rate depends
on the current residence time and the states of the neighboring agents. To get
an upper-bound for the rate, we set uA=uB= 1 and get b
λn(t) = ctc1. We
use numerical integration to sample tnto show that RED-Sim performs well also
in the case of this more costly sampling. We start with 50% of agents being in
each state.
Discussion. Our results provide strong evidence for the usefulness of rejection
sampling for non-Markovian simulation. As expected, we ﬁnd that the number
of interconnections (edges) and the number of agents inﬂuence the runtime be-
havior. Especially for RED-Sim, the number of edges shows to be much more
relevant than purely the number of agents. Our method consistently outper-
forms the baseline up to several orders of magnitude. The gain (RED-Sim speed
by baseline speed) ranges from 10.2 (103nodes, voter model, β= 2.5) to 674
(105nodes, SIS model, β= 2.0).
3github.com/gerritgr/non-markovian-simulation
10 G. Großmann et al.
(a) (b)
Fig. 2: Computation time of a single simulation step w.r.t. network size and
connectivity of the SIS model (a) and voter model (b). We measure the
CPU time per simulating step by dividing the simulation time by the number of
successful (i.e., non-rejection) steps.
We expect the baseline algorithm to be comparable with LGA as both of
them only update the rates of the relevant agents after an event. Moreover,
in the SIS model, sampling the next event times is very cheap. However, a
detailed statistical comparison remains to be performed (both case-studies could
not straightforwardly be simulated with LGA due to its constraints on the time
delays). Note that, when LGA is applicable, its key asset, the fast sampling of time
delays, can also be used in RED-Sim. We also tested a nMGA-like implementation
where rates are consider to remain constant until the next event. However, the
method was—even though it is only approximate—slower than the baseline.
Note that the SIS model is somewhat unfavorable for RED-Sim as it generates
a large amount of rejection events when only a small fraction of agents are
infected. Consider, for instance, an agent with many neighbours of which only a
few are infected. The over-approximation essentially assumes that all neighbors
are infected to sample the next event time (and, in addition, over-approximates
the rate of each individual neighbor), leading to a high rejection probability.
Nevertheless, the low computational costs of each rejection event overcome this.
6 Conclusions
We presented a rejection-based algorithm for the simulation of non-Markovian
agent models on networks. The key advantage of the rejection-based approach
is that in each simulation step it is no longer necessary to update the rates of
neighboring agents. This greatly reduces the time complexity of each step com-
pared to previous approaches and makes our method viable for the simulation
of dynamical processes on real-world networks. As future work, we plan to au-
tomate the computation of the over-approximation b
λand investigate correlated
time delays [40, 14] and self-exiting point processes [35, 36].
non-Markovian dynamics. This research was been partially funded by the Ger-
man Research Council (DFG) as part of the Collaborative Research Center
“Methods and Tools for Understanding and Controlling Privacy”.
Rejection-Based Simulation of Non-Markovian Agents 11
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Rejection-Based Simulation of Non-Markovian Agents 13
A Correctness
First, consider the rejection-free version of the algorithm:
1. Take the ﬁrst event (n, bµ, b
tn) from the event queue and update tglobal =b
tn.
2. Evaluate the true instantaneous rate µ=φnS(n), tglobal T(n), M(n)of
nat the current system state.
3. With probability 1 µ
bµ,reject the ﬁring and go to Line 5.
4. Randomly choose the next state s0of naccording to the distribution
ψnS(n), tglobal T(n), M (n). If S(n)6=s0: set S(n) = s0and T(n) = tglobal.
5. Generate a new event for agent nand push it to the event queue.
6. For each neighbor n0of n: Remove the event corresponding to n0from the
queue and generate a new event (taking the new state of ninto account).
7. Go to Line 1.
Rejection events are not necessary in this version of the algorithm because all
events in the queue are generated by the “real” rate and are therefore consistent
with the current system state. It is easy to see that the rejection-free version is
a direct event-driven implementation of the Na¨ıve Simulation Algorithm which
speciﬁes the semantics in Section 2.1. The correspondence between Gillespie-
approaches and even-driven simulation exploited in literature, for instance in
[4]. Thus, it is suﬃcient to show that the rejection-free version and RED-Sim
(Section 4.2) are statistically equivalent.
We do this with the following trick: We modify φnand ψninto b
φnand b
ψn,
respectively. When we simulate the rejection-free algorithm, it will admit exactly
the same behavior as RED-Sim. The key to that are so-called shadow-process [24,
26]. A shadow process does not change the state of the corresponding agent but
still ﬁres with a certain rate. They are conceptually similar to self-loops in a
Markov chain. In the end, we can interpret the rejection events not as rejections,
but as the statistically necessary application of the shadow process.
Here, we consider the case of a constant cR0upper-bound exits for
all φn. That is, cφn(s, t, m) for all reachable s, t, m. The case of an time-
dependent upper-bound is, however, analogous. Now, for each n, we deﬁne the
φnas
e
φn(s, t, m) = cφn(s, t, m).
Consequently, for all n, s, t, m:
b
φn(s, t, m) = c=φn(s, t, m) + e
φn(s, t, m)
The only thing remaining is to deﬁne b
has no inﬂuence on the system state. Therefore, we simply trigger a null event
(or self-loop) with the probability proportional to how much of b
φnis induced by
b
ψn(s, t, m) = (p(s) = 1 (self-loop) with probability e
φn
b
φn
ψn(s, t, m) otherwise .
14 G. Großmann et al.
Note that, ﬁrstly, the model speciﬁcation with b
φn,b
ψnor φn, ψnare equiva-
lent, because e
φ, e
ψhas to actual eﬀect on the system. Secondly, simulating the
rejection-free algorithm with b
φn,b
ψndirectly yields RED-Sim. In particular, the
rejections events have the same likelihood as the shadow-process being chosen
in b
ψ. Moreover, updating the rates of all neighbors is redundant because all the
rates remain at c. Whatever the change in φnis, after an event, that shadow
process balances it out, such hat it actually remains constant.
For the case that an upper-bound cdoes not exits, we can still look at the
limit case of c→ ∞. In particular, we truncate all rate functions at cand ﬁnd
that, as capproaches inﬁnity, the simulated model approaches the real model.
B Time-Complexity
Next, we discuss how the runtime of RED-Sim scales with the size of the underly-
ing contact network (and number of agents). Assume that a binary heap is used
to implement the event queue and that the graph structure is implemented using
a hashmap. Each step starts by popping an element from the queue which has
constant time complexity. Next, we compute µ. Therefore, we have to lookup
all neighbors of nin the graph structure iterate over them. We also have to
lookup all states and residence times. This step has linear time-complexity in
the number of neighbors. More precisely, lookups in the hashmaps have constant
time-complexity on average and are linear in the number of agents in the worst
case. Computing the rejection probability has constant time complexity. In the
case of a real event, we update Sand T. Again, this has constant time-complexity
on average. Generating a new event does not depend on the neighborhood of an
agent and has, therefore, constant time-complexity. Note that this step can still
be somewhat expensive when it requires integration to sample tebut not in
an asymptotic sense. Thus, a step in the simulation is linear in the number of
neighbors of the agent under consideration.
In contrast, previous methods require that after each update, the rate of
each neighbor n0is re-computed. The rate of n0, however, depends on the whole
neighborhood of n0. Hence, it is necessary to iterate over all neighbors n00 of
every single neighbor n0of n.
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