# RESULTS FROM PROTON-LEAD AND FIXED-TARGET COLLISIONS AT LHCB

DANIELE MARANGOTTO on behalf of the LHCb Collaboration  INFN Milano &
Università degli Studi di Milano, Italy

###### Abstract

The latest results obtained by the LHCb collaboration from proton-lead and
proton-gas fixed-target collisions are presented. Results related to charm and
beauty flavour hadron and quarkonia production in proton-lead collisions,
being clean probes for Cold Nuclear Matter effects, are compatible to
predictions based on initial-state effects. Indications of significant final-
state effects in excited $\Upsilon(nS)$ resonances production are reported.
The first cross-section measurement of $c\bar{c}$ and antiproton production in
fixed-target proton-helium collisions at 100 GeV energy scale are also
presented. More results will come from the latest Run 2 and future Run 3 data.

##  1 Introduction

The LHCb detector, mainly designed to study flavour physics in proton-proton
collisions, is the only LHC experiment fully instrumented in the forward
pseudorapidity region $2<\eta<5$, providing complementary coverage with
respect to the other LHC experiments. LHCb features excellent tracking
performances, momentum resolution and particle identification capabilities.

Besides proton-proton collisions, LHCb also recorded proton-lead and lead-lead
collisions. In asymmetric proton-lead collisions, LHCb is able to probe both
forward and backward nucleon-nucleon centre-of-mass rapidity ($y^{*}$) by
exchanging the proton and lead beam directions: LHCb covers the forward
$1.5<y^{*}<4.0$ rapidity region when the proton beam goes towards the
detector, and the backward $-5.0<y^{*}<2.5$ range when the lead beam is
directed towards LHCb.

LHCb studied the production of charm and beauty quarks in proton-lead
collisions, which is a clean probe for Cold Nuclear Matter (CNM) effects,
comprising the modification of nuclear partonic distributions (nPDFs),
initial-state radiation or coherent energy loss of the heavy quark, gluon
saturation (described by the Colour Glass Condensate theory, CGC) and final-
state hadronic rescatterings. The quantitative understanding of CNM effects is
crucial for a correct interpretation of Quark Gluon Plasma (QGP) signatures in
lead-lead collisions, where heavy quarks constitute special probes for QGP
properties. LHCb provides measurements for CNM effects at low transverse
momentum ($p_{T}$) and forward/backward rapidity, down to very small parton
momentum fraction (Björken $x\sim 10^{-5}-10^{-6}$), where nPDFs are poorly
constrained.

LHCb is also the only LHC experiment able to collect proton-gas fixed-target
collisions thanks to its internal gas target, SMOG, which injects noble gases
at $10^{-7}-10^{-6}$ mbar pressures into the beampipe, and designed to perform
precise luminosity measurements [1]. SMOG allows LHCb to explore fixed-target
collisions at the unprecedented $\sqrt{s_{NN}}=100$ GeV nucleon-nucleon
centre-of-mass energy scale.

##  2 Observables and theoretical predictions

Besides total and differential cross-sections, heavy quark production is
studied by defining cross-section ratios as a function of $p_{T}$ and $y^{*}$,
less sensitive to systematic uncertainties and more easily comparable to
theoretical predictions. The nuclear modification factor $R_{pPb}$ describe
differences in heavy quark production between proton-lead and proton-proton
collisions,

$R_{pPb}(p_{T},y^{*})\equiv\frac{1}{208}\frac{d^{2}\sigma_{pPb}(p_{T},y^{*})/dp_{T}dy^{*}}{d^{2}\sigma_{pp}(p_{T},y^{*})/dp_{T}dy^{*}},$
(1)

which is sensitive to the effects of nuclear interactions. The forward-to-
backward ratio describes differences in heavy quark production for equal but
opposite rapidity,

$R_{FB}(p_{T},y^{*})\equiv\frac{d^{2}\sigma(p_{T},+y^{*})/dp_{T}dy^{*}}{d^{2}\sigma(p_{T},-y^{*})/dp_{T}dy^{*}}.$
(2)

LHCb also measured baryon-to-meson ratios, which are sensitive to the heavy
quark hadronisation mechanism,

$R_{B/M}(p_{T},y^{*})\equiv\frac{d^{2}\sigma_{B}(p_{T},y^{*})/dp_{T}dy^{*}}{d^{2}\sigma_{M}(p_{T},y^{*})/dp_{T}dy^{*}}.$
(3)

Measured ratios are compared to different theoretical predictions. Predictions
based on initial-state (nPDFs) effects are obtained using the HELAC-Onia
generator for collinear factorisation [2] or a perturbative QCD calculation at
fixed-order next-to-leading log (FONLL) [3], based on different nPDF sets:
EPS09 at LO/NLO [4], nCTEQ15 [5] and CT14NLO [6]. Where applicable,
predictions from CGC effective field theory in dilute-dense approximation [7]
and coherent energy loss from soft gluon radiation [8] are employed.

##  3 Heavy quark production in proton-lead collisions

LHCb studied the $J/\psi$ production at nucleon-nucleon centre-of-mass energy
ofs $\sqrt{s_{NN}}=5.02$ and $8.16$ TeV energies, separating prompt quarkonia
produced in the proton-lead interaction from displaced ones coming from $b$
quark decays [9]. A strong suppression of the nuclear modification factor
(down to 0.6) is observed for prompt $J/\psi$ at low $p_{T}$ and forward
$y^{*}$, while it is much less pronounced for backward rapidity. No
significant suppression for $J/\psi$ from $b$ decays is seen. Predictions from
nPDF effects, CGC and coherent energy loss follow the observed ratios, but on
average they tend to overestimate data, indicating less anti-shadowing than
expected. The forward-to-backward ratio is found to rise with $p_{T}$,
especially for prompt $J/\psi$, while a flat rapidity dependence is seen.

The production of the $\Lambda^{+}_{c}$ charm baryon is studied at
$\sqrt{s_{NN}}=5.02$ TeV [10]. The forward-to-backward ratio is found
consistent with HELAC-Onia predictions. The baryon-to-meson ratio
$R_{\Lambda^{+}_{c}/D^{0}}$ agrees with results from proton-proton data but
for forward rapidity and high $p_{T}$, where the $\Lambda^{+}_{c}$ baryon
production seems suppressed.

LHCb studied open beauty hadron production at $\sqrt{s_{NN}}=8.16$ TeV [11],
reconstructing beauty hadrons in the exclusive decay modes
$B^{+}\to\bar{D}^{0}\pi^{+}$, $B^{+}\to J/\psi K^{+}$, $B^{0}\to D^{-}\pi^{+}$
and $\Lambda^{0}_{b}\to\Lambda^{+}_{c}\pi^{-}$. This is the first measurement
in nuclear collisions down to very low $p_{T}$. The measured nuclear
modification factor confirms the suppression pattern seen in $J/\psi$ from $b$
decays production and the $\Lambda^{0}_{b}$-to-meson ratio is consistent with
proton-proton data results.

The production of the three $\Upsilon(nS)$ states is studied at
$\sqrt{s_{NN}}=8.16$ TeV [12]. Thanks to the performances of the LHCb detector
the three $\Upsilon(nS)$ are clearly resolved in the reconstructed
$\mu^{+}\mu^{-}$ invariant mass. The suppression of nuclear modification
factors for excited $\Upsilon(2S)$, $\Upsilon(3S)$ resonances is expected to
be larger than the $\Upsilon(1S)$ state according to the “comover model”
final-state effects [16], especially for backward rapidities. This model
predicts the production of excited $\Upsilon(nS)$ resonances to be reduced due
to the interaction of comoving particles, a possible explanation for the
$\Upsilon(nS)$ suppression observed in lead-lead collisions. The measured
nuclear modification factors show indeed a larger suppression for excited
$\Upsilon(nS)$ quarkonia states, compatible with the “comover” model
predictions, Fig. 1.

Figure 1: (left) Reconstructed $\Upsilon(nS)\to\mu^{+}\mu^{-}$ invariant mass
and nuclear modification factor for (center) $\Upsilon(2S)$ and (right)
$\Upsilon(3S)$ states normalized to the $\Upsilon(1S)$ one.

##  4 Fixed-target p-gas collisions studies

LHCb collected the first charm ($J/\psi$ and $D^{0}$) samples in proton-argon
and proton-helium collisions at $\sqrt{s_{NN}}=110$ and $87$ GeV energy,
respectively [13]. The first $c\bar{c}$ cross-section measurement is performed
using the proton-helium dataset, Fig. 2, for which the integrated luminosity
was determined from the yield of elastically scattered electrons off the
target helium atoms [14]. Predictions from HELAC-Onia generator follow the
shape of the measured differential cross-sections but underestimate the total
by a factor 1.5-1.8. Results are also compared to linear and logarithmic
interpolations from cross-section results at the closest energy available.
From the measured differential cross-section it is possible to infer that
there is no evidence for a significant $c$ quark content in the nPDF functions
in the kinematic region probed by LHCb. [17].

Figure 2: (left) Reconstructed $J/\psi\to\mu^{+}\mu^{-}$ invariant mass and
(center) total and (right) differential $J/\psi$ cross-sections, obtained from
the proton-helium dataset at $\sqrt{s_{NN}}=87$ GeV.

Space-based high-energy cosmic ray experiments AMS-02 and PAMELA have observed
an excess of the antiproton-to-proton ($\bar{p}/p$) ratio at the 100 GeV
energy scale [18]. An enhancement of the $\bar{p}/p$ ratio can be an indirect
signal of dark matter, however the significance of the result is limited by
the large uncertainties related to antiproton production predictions. In
particular, the antiproton production cross-sections in proton-hydrogen and
proton-helium collisions are poorly known. LHCb performed the first
measurement of antiproton production in proton-helium collisions at
$\sqrt{s_{NN}}=110$ GeV [14], providing precious information for more precise
$\bar{p}/p$ ratio predictions. The measured total and differential cross-
sections have been compared to predictions from different generators [19]. The
absolute antiproton yield is found to be underestimated by predictions up to a
factor two, a result tending to disfavour the hypothesis of a dark matter
origin of the measured $\bar{p}/p$ ratio excess in cosmic rays. Generators are
able to reproduce the shape of the measured proton-helium differential cross-
section, but for low momentum antiproton production is found to be lower than
expected.

##  5 Prospects

The presented studies are just the beginning of the LHCb physics program with
heavy ion and fixed-target collisions. Much more analyses are to come, like
Drell-Yan and vector boson production studies, the investigation of more
quarkonia states and dihadron correlations, Bose-Einstein condensates and flow
studies. The latest LHC Run 2 samples of proton-neon, lead-lead and lead-neon
collisions, the LHCb datasets with higher statistics in their respective
configurations, are being analysed. At least a factor ten more integrated
luminosity of proton-lead and lead-lead collisions is planned for the LHC Run
3 data-taking, to be recorded with an upgraded LHCb detector. The LHCb
Collaboration is considering the installation of an upgraded gas injection
system, SMOG2, aiming at increasing the injected gas pressure, therefore the
luminosity of fixed-target collisions, by a factor 100. SMOG2 would also allow
to inject more gas species (not only noble gases) with a precise density
control. The LHCb Collaboration is also considering a Phase II upgrade of the
LHCb detector for the LHC Run 5, for which a detailed physics case has been
presented [15].

A rich physics program with heavy ions and fixed-target is ahead for the LHCb
experiment.

## References

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# Variability of Contact Process in Complex Networks

Kai Gong  Web Sciences Center, University of Electronic Science and Technology
of China, Chengdu 610054, People’s Republic China  Ming Tang
<EMAIL_ADDRESS>Web Sciences Center, University of Electronic
Science and Technology of China, Chengdu 610054, People’s Republic China
Computer Experimental Teaching Center, University of Electronic Science and
Technology of China, Chengdu 610054, People’s Republic of China  Hui Yang  Web
Sciences Center, University of Electronic Science and Technology of China,
Chengdu 610054, People’s Republic China  Mingsheng Shang  Web Sciences Center,
University of Electronic Science and Technology of China, Chengdu 610054,
People’s Republic China

###### Abstract

We study numerically how the structures of distinct networks influence the
epidemic dynamics in contact process. We first find that the variability
difference between homogeneous and heterogeneous networks is very narrow,
although the heterogeneous structures can induce the lighter prevalence.
Contrary to non-community networks, strong community structures can cause the
secondary outbreak of prevalence and two peaks of variability appeared.
Especially in the local community, the extraordinarily large variability in
early stage of the outbreak makes the prediction of epidemic spreading hard.
Importantly, the bridgeness plays a significant role in the predictability,
meaning the further distance of the initial seed to the bridgeness, the less
accurate the predictability is. Also, we investigate the effect of different
disease reaction mechanisms on variability, and find that the different
reaction mechanisms will result in the distinct variabilities at the end of
epidemic spreading.

###### pacs:

05.40.Fb,05.60Cd,89.75.Hc

The variability of outbreaks is defined as the relative variation of the
prevalence. In order to assess the accuracy and the forecasting capabilities
of numerical models, the variability of outbreaks has been investigated in
many studies. In numerical models, many factors such as network structures,
travel flows, and initial conditions can affect the reliability of the
epidemic spreading forecast. Recently, a contact process model with identical
infectivity is proposed to study both dynamical processes and phase
transitions of epidemic spreading in complex networks, but the predictability
of the model is totally overlooked. In this paper, by investigating the
variabilities of contact process in distinct networks, we show numerically
that the bridgeness plays a significant role on the predictability of the
epidemic pattern in community network, meaning the further distance of the
initial seed to the bridgeness, the less accurate the predictability is.
Hopefully, this work will provide us further understanding and new perspective
in the variability of contact process in complex networks.

##  I Introduction

The great threat of epidemic spreading to human society has been strongly
catching scientists’ eyes Bailey:1975 ; Anderson:1992 . In order to realize
the impact of diseases and develop effective strategies for their control and
containment, the accurate mathematical models of epidemic spreading are the
basic conceptual tools Bailey:1975 ; Anderson:1992 ; Diekmann:2000 ;
Dailey:2001 . In mathematical models, the dynamical patterns of epidemic
spreading will be influenced by many different factors such as the age and
social structure of the population, the contact network among individuals, and
the meta-population characteristics Ferguson:2003 . Especially the
heterogeneity of the population network Albert:2000  can result in the absence
of endemic threshold when the population size is infinite and the exponent of
degree distribution $\gamma\leq 3$ Cohen:2000 ; Satorras:2001a ;
Satorras:2001b ; May:2001 ; Lloyd:2001 . With the further study, the local
structures of complex networks (such as degree correlation, clustering
coefficient, community structure and so on) bring quantitative influences on
epidemic spreading Eguiluz:2002 ; Boguna:2002 ; Boguna:2003 . Considering the
complicated local structures in real networks, the forecasting capabilities
(i.e. variability) of current numerical models have been investigated
Barthelemy:2005 . In addition, both the stochastic nature of travel flows
Tang:2006 ; Kishore:2011  and initial conditions can affect the reliability of
the epidemic spreading forecast Colizza:2006 ; Crepey:2006 ; Gautreau:2007 ;
Gautreau:2008 ; Tang:2009a ; Tang:2009b ; Barthelemy:2010 .

In view of this point, Colizza _et al._ have studied the effect of the airline
transportation network on the predictability of the epidemic pattern by means
of the normalized entropy function Colizza:2006 , and found that the
heterogeneous distribution of this network contributes to enhancing the
predictability. In complex networks, many factors can decrease the forecasting
accuracy of epidemic spreading. Crépey _et al._ have found that initial
conditions such as the degree heterogeneity of the seed show a large
variability on the prediction of the epidemic prevalence, and the infection
time of nodes have non-negligible fluctuations caused by the further distance
and the multiplicity of paths to the seed Crepey:2006 . Comparing the scale-
free network (SFN) with community structure Liu:2005 ; Huang:2006 ; Zhou:2007
; Liu:2008 ; Chu:2009 ; Chen:2009  with the random SFN, the predictability of
the prevalence can be found to be better Huang:2007 .

The common assumption in all the aforementioned works is that each node’s
potential infection-activity (infectivity), measured by its possibly maximal
contribution to the propagation process within one time step, is strictly
equal to its degree. However, there are still many real spreading processes
which can not be described well by this assumption Zhou:2006 . Therefore, a
contact process (CP) model with identical infectivity is proposed to study the
epidemic spreading in complex networks Castellano:2006 . Almost all studies in
CP are focused on dynamical processes and phase transitions Zhou:2006 ;
Castellano:2006 ; Ha:2007 ; Castellano:2007 ; Hong:2007 ; Yang:2007 ;
Yang:2008a ; Yang:2008b ; Noh:2009 ; Lee:2009 ; Munoz:2010 , but the
predictability of the model is totally overlooked. To this end, we study how
the structures of distinct networks (i.e. homogeneous, heterogeneous and
community networks) influence the variabilities of epidemic patterns in CP.
Through numerical experiments, we find that the community structures can
remarkably influence the prevalence and its variability, contrary to non-
community networks (i.e. homogeneous and heterogeneous networks). It is worth
noting that it’s hard for the extraordinarily large variability in a local
community to predict the epidemic prevalence.

This paper is organized as follows. In Sec. II, we briefly describe disease
models in CP in complex networks and provide quantitative measurements of the
predictability of epidemic spreading. In Sec. III, we investigate the
prevalence variabilities in both random graph (RG) Erdos:1960  and SFN
Barabasi:1999 . In Sec. IV, we discuss the essential differences of the
prevalence variabilities both in the global network and the local community.
Finally, we draw conclusions in Sec. V.

##  II CP model in complex networks

In our model, three distinct networks, i.e. the homogeneous, heterogeneous and
community networks are adopted to investigate the predictability of epidemic
spreading therein. Firstly, as the mother of all network models, the random
graph of Erdős and Rényi Erdos:1960  is regularly used in the study of complex
networks because networks with a complex topology and unknown organizing
principles often appear randomly Albert:2002 . Random graph is defined as a
graph with $N$ nodes and connection probability $p$, which has a Poisson
distribution. Secondly, since scale-free property is observed in many real
complex systems, dynamics study on scale-free networks have been holding
everyone’s concern. In 1999, Barabási and Albert (BA) put forward the most
classical SFN model which is rooted in two generic mechanisms: growth and
preferential attachment Barabasi:1999 . As there are community structures in
social networks, the last studied structure substrate is community network
Newman:2002 . Here we will adopt a simplified community model proposed by Liu
and Hu, which emphasizes on the community feature in social networks Liu:2005
. For simplicity, two independent random graphs are first produced, and then
two RGs are connected randomly by only one link.

In general, the standard disease models conclude susceptible-infected (SI),
susceptible-infected-susceptible (SIS), and susceptible-infected-refractory
(SIR) epidemiological model. Each node of the network represents an individual
and each link plays as one connection which transmits disease to other node.
In SI (SIS or SIR) model, ’S’, ’I’ and ’R’ represents respectively the
susceptible (healthy), the infected, and the refractory (recovered) state. At
each time step of contact process, each infected node randomly contacts one of
its neighbors, and then the contacted neighboring node will be infected with
probability $\lambda$ if it is in the healthy state, or else its state will
stay the same. At the same time, each infected nodes is cured and becomes
susceptible (refractory) with rate $\mu$ in SIS (SIR) model. To eliminate the
stochastic effect of the disease transmission, we can set $\lambda=1$ and
$\mu=0.2$.

In order to analyze the effect of the underlying network topology on the
predictability of epidemic spreading, the variability of outbreaks is defined
as the relative variation of the prevalence [density of infected individuals
$i(t)$] given by Crepey:2006

$\bigtriangleup[i(t)]=\frac{\sqrt{\langle i(t)^{2}\rangle-{\langle
i(t)\rangle}^{2}}}{\langle i(t)\rangle}.$ (1)

$\bigtriangleup[i(t)]=0$ denotes all independent dynamics realizations are
essentially the same, and the prevalence in the network is deterministic.
Larger $\bigtriangleup[i(t)]$ means worse predictability that a particular
realization is far from average over independent realizations.

##  III Predictability in homogeneous and heterogeneous networks

The first issue of our study is how the heterogeneity of network structures
influences the variability of the prevalence in CP. By using a numerical
approach in this section, we analyze the variabilities of outbreaks generated
by different sets of initial nodes, both for random graphs and scale-free
networks with the same network size and average degree. Considering the fact
that the results of a particular network can be generalized to any instances
of network model Crepey:2006 , the numerical simulations we studied here are
run in one network. In Fig. 1, we show the curves $i(t)$ and
$\bigtriangleup[i(t)]$ computed for the different disease models in both RG
and SFN. For SI model in Fig. 1 (a), the density of infected $i(t)$ in RG
reachs its stationary state faster than that in SFN; for SIS model in Fig. 1
(b), the stationary $i(t)$ in RG is greater than that in SFN; and for SIR
model in Fig. 1 (c), RG has the higher peak prevalence. Contrary to the
results for the case of contacting all neighbors, it is first discovered that
the heterogeneous structure can slow down the prevalence of outbreaks in CP.
Because hubs may be contacted many times by their neighboring nodes at each
time step, the total contact ability (i.e. the actual number of contacting
nodes at one time step) of SFN is reduced further accordingly, as a result,
the hub effect holds back the prevalence of diseases. Meaningfully, owing to
the limited contact ability of CP, the infected densities starting from the
initial infected nodes (seeds) with different degrees are almost the same in
SFN, which is distinct from the results for the case of contacting all
neighbors Crepey:2006 .

As shown in Fig. 1 (d), (e), and (f), there is slightly different between
variabilities in RG and SFN when $t<20$, which implies heterogeneous structure
does not visibly alter the predictability of CP before the outbreak of
disease. An important contribution of this study is to analyze the differences
among the variabilities of three kinds of disease models. From the comparison
among them, we find that different recovery mechanisms can result in distinct
variabilities at the end of epidemic spreading. For SI model in Fig. 1 (d),
the time arriving at $i(t)=1$ varies in a mass of realizations, which can
induce an exponential decay of the variability when $t>30$. For SIS model in
Fig. 1 (e), the variability of the prevalence will keep on a steady value in
stationary state. For SIR model in Fig. 1 (f), due to the different lifetimes
of the epidemics in a mass of independent realizations Colizza:2006 , the
greater and greater variabilities are observed by approaching the end of the
epidemics.

Figure 1: (color online). Evolution of both $i(t)$ and $\bigtriangleup[i(t)]$
for the different disease models where the ”triangles” and ”circles” denote
the cases of SF and RG networks with the random initial seeds. $i(t)$ versus
$t$ for SI model (a), SIS model (b), and SIR model (c), and
$\bigtriangleup[i(t)]$ versus $t$ for SI model (d), SIS model (e), and SIR
model (f). The parameters are chosen as $N=0.5\times 10^{4},\langle
k\rangle=10,\lambda=1$, and $\mu=0.2$. The results are averaged over $2\times
10^{4}$ independent realizations in one network.

##  IV predictability in the global network and the local community

###  IV.1 Global network

Figure 2: (color online). Evolution of both $i(t)$ and $\bigtriangleup[i(t)]$
in community networks where the ”squares”, ”circles”, ”triangleups”,
”triangledowns”, and ”trianglelefts” denote the cases of the bridgeness,
$d=1,2,3$, and $4$, respectively. $i(t)$ versus $t$ for SI model (a), SIS
model (b), and SIR model (c), and $\bigtriangleup[i(t)]$ versus $t$ for SI
model (d), SIS model (e), and SIR model (f). The parameters are chosen as
$N=10^{4},\langle k\rangle=10,\lambda=1$, and $\mu=0.2$. The results are
averaged over $2\times 10^{4}$ independent realizations.

As many social networks combined by several communities, such as Facebook
Facebook , YouTube Youtube , and Xiaonei Xiaonei , information propagation
taking place in community networks Liu:2005 ; Huang:2006 ; Zhou:2007 ;
Liu:2008 ; Chu:2009  is one of the most important subjects studying in complex
networks, but in CP, the related research has been ignored for a long time.
Therefore, in this section, we study the variability of CP in a very simple
community network, where two RGs are connected randomly by only one link.
Obviously, this network has a strong strength of community structure. In order
to normalize the terms of community network, we define the link as weak tie
Onnela:2007 , and call two nodes connected by this link ”bridgeness”
Cheng:2010 . We first investigate the time evolution of epidemics generated by
different seeds staying away from the bridgeness, and it is noted that there
is only one initial seed in each realization. From Fig. 2 (a), (b), and (c),
we can get that the closer the seed to the bridgeness,the epidemic spreads
much faster in the global network, and among all cases, the epidemic starting
at the bridgeness spreads fastest. For SI model, the further distance to the
bridgeness such as $d=3,4$ induces two periods of the quickly rising trend at
the beginning time $t=10,30$, respectively. If the initial seed is far away
from the bridgeness, the disease will be restricted in the first community for
a long time till the bridgeness infected, in which almost all nodes is
infected. As a result, the outbreak in the second community just starts at
that moment the prevalence in the first community get towards the end, which
causes the second outbreak. In the case of SIS and SIR model, the recovery
mechanism reduces this phenomenon occurred, for instance there is the tiny
second peak of the prevalence for $d=3,4$ in Fig. 2 (c). From the above, it is
found that the bridgeness plays a distinctly important role in the rapid
transmission of information in CP.

The variability of prevalence in community network is distinct from that in
the network which has no community structure. As shown in Fig. 2(d), (e), and
(f), the curves $\bigtriangleup[i(t)]$ display two peaks because of the time
delay between two outbreaks occurred in different communities. In addition,
the further distance to the bridgeness makes the second peak occur much later.
In Fig. 2(d), for SI model, the first peak corresponds to the prevalence in
the community with the initial seed, so the variability is almost the same as
that in RG before $t=10$. With the outbreak in the second community, the
second peak occurs. Owing to the greater randomness of the time that disease
first occurs in the second community (see Fig. 3), the second peak of the
variabilities is slightly greater than the first peak. After the infection
density is close to saturated at $t\approx 40$ (see Fig. 2(a)), the
variability will be on exponential decay. As all nodes of community network
are infected in more and more realizations, the variability
$\bigtriangleup[i(t)]$ will rapidly decay to zero. In contrast with the case
of SI model, the second peak in SIS model is less than the first peak, because
the recovery mechanism slows down the propagation velocity of diseases, which
reduces the variability of the prevalence. That is to say, the recovery
mechanism reduces the variability of epidemic spreading. Another extremely
obvious difference is the variability decreases to a steady value at the
stationary state. For SIR model in Fig. 2(f), as time goes by, the epidemics
has the greater and greater variability, which is caused by the different
lifetimes of the epidemics in $2\times 10^{4}$ independent realizations.

###  IV.2 The local community

Figure 3: (color online). The distribution of arrival time of disease in the
second community for SI model (a), SIS model (b), and SIR model (c), where the
”squares”, ”circles”, ”triangleups”, ”triangledowns”, and ”trianglelefts”
denote the cases of the bridgeness, $d=1,2,3$, and $4$, respectively. The
results are averaged over $2\times 10^{4}$ independent realizations. Figure 4:
(color online). Evolution of both $i(t)$ and $\bigtriangleup[i(t)]$ in the
second community where the ”squares”, ”circles”, ”triangleups”,
”triangledowns”, and ”trianglelefts” denote the cases of the bridgeness,
$d=1,2,3$, and $4$, respectively. $i(t)$ versus $t$ for SI model (a), SIS
model (b), and SIR model (c), and $\bigtriangleup[i(t)]$ versus $t$ for SI
model (d), SIS model (e), and SIR model (f). The results are averaged over
$2\times 10^{4}$ independent realizations.

Considering the relative independence of a local community, we should take the
prevalence and variability into account. On the other hand, since disease must
be transmitted through bridgenesses from the first community to the second
community, this study contributes to understand the effect of them on epidemic
spreading Zhao:2010 . In this section, we will specifically analyze the effect
of different distances of seeds (to the bridgeness in the first community) on
epidemic spreading in the second community.

At first, the arrival time of disease is defined as the moment that infectious
individual first occurs in the second community in each realization, thus the
distribution of arrival time is obtained through massive realizations. In Fig.
3, the distribution of arrival time for the different initial seeds is showen.
For SI model, the arrival distribution of the bridgeness as seed $d=0$
strictly obey the distribution $P(t)=\lambda(1-\lambda)^{t-1}$. When disease
seed is the node with one step to bridgeness, the arrival time increases
generally, and the distribution becomes much wider and flatter. With the
further increasing of distance of seed to the bridgeness (such as $d=3,4$),
the distributions are nearly the same. It is understood that due to the finite
size effect of network, the disease is transmitted through weak tie to the
second community till overall outbreak happened in the first community. For
SIS model, as a result of the recovery mechanism, the distributions of arrival
time are much more evenly and smoothly than that for SI model, given the
various initial seeds. Compared with Fig. 3 (a), we can find that there are
two peaks for SIS and SIR model with $d=1$. This is because the bridgeness may
be infected through two basic pathways: the bridgeness may be infected
directly by the initial seed (i.e. its neighboring node) in $t\leq 1/\mu$; the
other route is the transmission of infection from the other neighboring nodes
when the disease outbreak in the first community, thus the second peak occurs
at $t\approx 20$ (see Fig. 2(b)). For SIR model, owing to the recovery
mechanism, the prevalence might well disappear in the first community before
arriving at the weak tie. Consequently, arrival rate (i.e. the area of the
distribution) is less than one, and the peak value of the corresponding
distribution is less than that for both SI and SIS model.

Fig. 4 shows the prevalence and variability in the second community which are
generated from the different initial seeds in the first community. From Fig. 4
(a), (b), and (c),we can easily know that the prevalence of the bridgeness
acting as seed increases much faster than that in the other cases generated by
the further seeds. In particular, for SIR model in Fig. 4 (c), the peak of the
prevalence (for the case of the bridgeness) occurs first, and has the maximum
value. In addition, there are two peaks of the prevalence for the case of
$d=1$, which is attributed to the propagation delay between two communities.
Since there is only one interconnection between two communities, the chance of
infecting one from the other is low. In other words, infection within intra-
community will be much faster than inter-community infection. Therefore, the
first peak reflects the outbreak and extinction of disease inside with
infectious seed, and the second peak emerges after the other community is
infected. Note that the two peaks can be only observed in SIR model because of
the fact there will be no peak if virus does not ”die”. With the increase of
distances $d$, the peak value is greater than that for $d=1$, although the
outbreaks occur later.

In Fig. 4 (d), (e), and (f), it is a surprise that the variabilities in the
second community are distinct from that in the global network. Firstly, the
variabilities in the second community are very large and also much greater
than that in the global network in Fig. 2, which implies the huge
unpredictability of prevalence produced in the local community. In particular,
at the beginning of outbreaks, the variability for the case of $d=4$ reaches
about $50$, which is 100 times the maximum value $0.5$ in the global network.
Secondly, the closer distance of the seed to bridgeness, the lower level of
variability it has in the local community. In particular, the maximum
variability value for the case of the bridgeness is only about $1$, which is
much less than $50$ for the case of $d=4$. Thus, the bridgeness plays a
significant role in enhancing the predictability, that the closer initial seed
to the bridgeness, the more accurate the predictability is. Thirdly, each
curve of variabilities can be divided into four parts: the sudden drop stage,
the relatively stable stage, the slowing-down stage, and the final stage of
outbreaks (i.e. the exponential decay stage for SI model, the steady state
stage for SIS model, and the sharp increase stage for SIR model,
respectively). The first stage is originated from the uncertain arrival time
of disease, with the increase of the arrival rate, the variability decreases.
The second stage ($10<t<20$ in most cases) is induced by the interplay between
the outbreak and the arrival of diseases in massive realizations: on the one
hand, the outbreaks in some realizations upgrade the variability; on the other
hand, the increase of the arrival rate counteracts this effect. As the
infection density is close to saturated, the variability will enter the
slowing-down stage. What is noteworthy is that the minimum variability value
just corresponds to the peak value of prevalence for SIR model. In the end of
epidemic spreading, the variabilities $\bigtriangleup[i(t)]$ display the
distinct phenomena for the different disease models, for instance, SIR model
shows the higher and higher variabilities.

##  V conclusions and discussions

In conclusions, we have studied the variability of CP in complex networks, and
get the clear understanding that the different network structures can
remarkably influence the prevalence and its variability. Firstly, we find that
the variability difference between homogeneous and heterogeneous networks is
very narrow, although the heterogeneous structure induces a lighter
prevalence. Secondly, two peaks of both the prevalence and variability are
shown in the community network. It’s noted that in the local community, the
extraordinarily large variability in early stage of the outbreak makes the
prediction of disease spreading hard. This result is in accordance with Ref.
Yan:2007  in which the networks with strong community structures are of weak
synchronizability, and the amplitudes of the time series in the local
communities are much larger than that in the global networks. Fortunately, the
bridgeness plays a significant role in enhancing the predictability, the
closer initial seed to the bridgeness, the more accurate the predictability
is. This result suggests that bridgenesses may be the ideal detection stations
in community networks. Moreover, the different reaction mechanisms of disease
models can result in the distinct variabilities. Especially for the case of
SIR model, the greater and greater variabilities are observed at the end of
the epidemics for the different lifetimes of the epidemics in various
realizations.

The community network employed in this study is much more simple, but the
actual community networks have complex structures, such as multifarious
communities, many bridgenesses, and the heterogeneous degree distribution in a
local community. Therefore, the further investigation should be focused on the
more complex community networks.

###### Acknowledgements.

This work is supported by the NNSF of China under Grant Nos. 90924011, and the
Sichuan Provincial Science and Technology Department (Grant No. 2010HH0002).

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