Distinct sites visited by random walkers on scale-free networks

Abstract

Random walks on discrete lattices are fundamental models that form the basis for our understanding of transport and diffusion processes. For a single random walker on complex networks, many of its properties such as the mean first passage time and cover time are known. However, recent applications such as search engines and recommender systems involve multiple random walkers on complex networks. In this work, based on numerical simulations, we show that the fraction of nodes of scale-free network not visited by W random walkers in time t has a stretched exponential form independent of the details of the network and number of walkers. This leads to a power-law relation between nodes not visited by W walkers and by one walker. The problem of finding the distinct nodes visited by W walkers, effectively, can be reduced to that of a single walker. The robustness of the results is demonstrated by verifying them on four different real-world networks that approximately display scale-free structure.

Full text

Distinct sites visited by random walkers on scale-free networks
Aanjaneya Kumar1and M. S. Santhanam1
1Indian Institute of Science Education and Research,
Dr. Homi Bhabha Road, Pune 411008, India.
Random walks on discrete lattices are fundamental models that form the basis for our under-
standing of transport and diffusion processes. For a single random walker on complex networks,
many of its properties such as the mean first passage time and cover time are known. However, re-
cent applications such as search engines and recommender systems involve multiple random walkers
on complex networks. In this work, based on numerical simulations, we show that the fraction of
nodes of scale-free network not visited by Wrandom walkers in time thas a stretched exponential
form independent of the details of the network and number of walkers. This leads to a power-law
relation between nodes not visited by Wwalkers and by one walker. The problem of finding the
distinct nodes visited by Wwalkers, effectively, can be reduced to that of a single walker. The
robustness of the results is demonstrated by verifying them on four different real-world networks
that approximately display scale-free structure.
PACS numbers: 05.45.-a, 03.67.Mn, 05.45.Mt
Random walks were introduced more than a century
ago and have formed the basis for our understanding of
diffusion processes in physical systems [1]. As a funda-
mental stochastic process, it is relevant for many fields
ranging from physics and computer sciences [4] to biol-
ogy [2] and economics [3]. Several problems including
animal foraging and migration [5, 6], intracellular molec-
ular transport [7], proteins binding with DNA sequences
[8], for structual information about macromolecules [9]
are based on the dynamics of a single random walker on
regular lattice or its variants.
On the other hand, many recent applications involve
dynamics of multiple random walkers on a disordered lat-
tice, e.g., complex network with non-local edges connect-
ing the nodes. For instance, cellular signal transduction
[10], exciton transport in molecular crystals, web search
algorithms [4], a class of image segmentation algorithms
[11] and recommender systems [12] widely used for per-
sonalization in popular websites are based on the idea
of many random walkers exploring a topology of discrete
nodes connected through their edges.
As a statistical physics problem, in comparison to the
well-studied problem of the dynamics of a single random
walker on regular lattice [13] or complex network [14, 15],
the case of multiple walkers in a network setting has
not attracted sufficient research attention. In random
walk with Wnon-interacting walkers, some results are
a straightforward generalisation of that for single walker
dynamics. For instance, on a complex network, occupa-
tion probability of a single walker on a node with de-
gree kis proportional to k, whereas for Wwalkers it is
∝W k. However, in many cases, the results for multiple
walker dynamics is not a trivial generalization of that
for a single walker. One such statistical quantity of in-
terest is the mean number of distinct sites SW(t) visited
by Wrandom walkers in tdiscrete time steps on a net-
work with Nnodes. This is relevant for problems related
to (mis-)information and contagion spreading and search
problems on networks [16].
Distinct sites visited in t-steps by a random walker
was studied in Refs. [17] and its generalization to W
walkers was considered in Refs. [18, 19]. On regular d-
dimensional lattices and for short times, the mean num-
ber of distinct sites visited by Wwalkers is hSW(t)i ∝ td
and asymptotically hSW(t)i ∝ W t for d > 3. In general,
Ref. [18] identifies three distinct time scales with differ-
ent behaviours for hSW(t)iand limited analytical support
is presented in Ref. [20]. Recently, an exact asymptotic
result for the distribution of number of distinct and com-
mon sites visited by Wwalkers on a regular 1dlattice was
obtained [21] by transforming it as a problem of extreme
value statistics.
In spite of these developments for regular lattices, very
few results are known for multiple random walkers on
complex networks. For a random walker on a Bethe lat-
tice with coordination number z,S1(t) = ((z−2)/(z−
1))t, for z≥3 [22]. Clearly, S1(t)∝nwith a pref-
actor that depends on the local topology of the lattice.
On a random network, a formal relation for the gener-
ating function corresponding to S1(t) has been obtained
in terms of the generating functions for the first passage
probabilities [23]. For a walker on scale-free network, it
was numerically shown that, for short times, S1(t) = t
and as t→ ∞, S1(t)→1 due to finite size of network
[24]. In a small world network, S1(t) displays a cross-
over from √tto linear behaviour depending on whether
the walker has managed to hit a short-cut in the small
world network or not [25].
To the best of our knowledge, exact closed form re-
sult for hSW(t)ion an arbitrary network with Nnodes
is not yet known. Even as this gap continues to exist,
in this work, new results primarily based on numerical
simulations are presented that effectively relate SW(t) to
S1(t) on static networks. In particular, for the class of
scale-free networks, it is shown that the number of nodes
sW(t) not yet visited until time thas a stretched expo-
nential form, with exponent β, depending on the spe-
cific network structure but independent of the number of

2
-8 -6 -4 -2 0 2 4
log (Wt/N)
-15
-10
-5
0
log (SW(t)/N)
W=1, N=2500
W=10, N=5000
W=50, N=10000
W=100, N=10000
Fit
0510 15 20
(Wt/N)β
-15
-10
-5
0
05000 10000 15000 20000
t
0.2
0.3
0.4
0.5
log(sw1
)/log(sw2
)
0.543
0.380
0.295
0.242
0.206
r =1/6
r =1/5
r =1/2
r =1/4
r =1/3
FIG. 1. (a) Fraction of nodes not reachable by random walkers
on Barabasi-Albert scale-free network plotted as a function
of x=W t/N. Symbols are from random walk simulations
and were averaged over 1000 realizations. The solid (red)
curve is the best fit line. Note the excellent scaling collapse.
(Inset) shows the same data (in semi-log scale) as a function
of (W t/N)β. The best fit (solid) line with slope β= 0.693 is
given a vertical offset for easier comparison with simulation
data. (b) The ratio u(t) shown as a function of t. Validity
of scaling can be inferred from the flat lines obtained from
simulations. In this, r=W1/W2and the value of rβis also
indicated.
walkers. As shown below, the results are consistent with
sW(t)∝exp(Wβs1(t)). Thus, effectively, the problem
of finding SW(t) can be reduced to a relatively simpler
problem of finding S1(t) on a scale-free network.
Distinct sites visited by multiple walkers SW(t) can
also be thought of as a statistical relaxation process es-
pecially if the initial position of the walkers is far from
equilibrium distribution. On a network, this is easily
achieved by placing all the walkers on the same node at
t= 0. In this garb, sW(t) = N(1 −SW(t)/N) represents
a relaxation process and the results presented here indi-
cate scaling of this process as a function of Wand N. If
SW(t) denotes the unique ’territory’ covered by Wwalk-
ers, then sW(t) represents its complementary part, the
territory unreachable in time t. Thus sW(t) + SW(t)=1
for all tand we have chosen to present results for sW(t)
in the rest of the paper.
Random walks on complex networks are a straightfor-
ward generalisation of random walks on regular lattices.
In this work, independent and multiple walkers randomly
walk on a connected, scale-free network with Nnodes and
Eedges generated using Barabasi-Albert (BA) [26] and
configuration models [27]. Each node has an associated
degree ki, i = 1,2...N, indicating the number of edges.
The degree distribution of the network is P(k)∼k−γ,
where γis the exponent. A walker at i-th node can hop
to any of its connected neighbours with probability 1/ki.
The information on the edges in the network are encoded
in the adjacency matrix Aof order N, where the element
Aij = 1 if nodes iand jare connected by an edge, and
Aij = 0 if they are not connected.
In Fig. 1(a), the fraction of nodes not reached in time t,
sW(t), is shown for BA scale-free networks NBA(N, E, γ ).
Each data point is averaged over 1000 random walk real-
izations. At time t= 0, all the Wwalkers are all placed
on a randomly chosen node designated as zeroth node,
i.e.,w0(t= 0) = PiW δi,0. This figure shows results
for four different values of Nand W. Remarkably, in
all the cases, the scaled parameter can be identified as
x=W t/N. With this choice, scaling is evident from the
excellent data collapse observed in Fig. 1(a). It can be
inferred (from the fitted solid line) that as W→ ∞ and
N→ ∞, the fraction of unreachable nodes is consistent
with a stretched exponential function of the form,
sW(t) = N−1
Ne−A x(t)β,(1)
in which the parameters Aand βare estimated through
a regression procedure. For the simulations shown in
Fig. 1, A≈0.75 and β≈0.88. The scaled time
x=W t/N can also be expressed in units of mean re-
laxation time as x=Cβt/τr, where τr=CβN/W and
Cβ=AΓ(2/β)
βΓ(1+1/β). For single walker dynamics, W= 1
and scaled time reduces to x=t/N, in agreement with
the results in Ref. [24]. The inset in Fig. 1(a) displays
the same data as in the main figure as a function of xβ
and its linearity suggests Eq. 1. For x1, Eq. 1 be-
comes sW(t)≈1−Axβ. Thus, the fraction of distinct
sites visited is SW(t)≈Axβ.
In connected and finite size networks, unreachable
nodes is a finite time effect since as t→ ∞ all the
nodes are eventually reached. Then, we can expect the
scaling relation in Eq. 1 to hold good in the timescale
τr< t tcov, where tcov is the cover time for all the
nodes to be visited at least once. For a single walker on
a scale-free network, tcov ≤Nlog N, though a similar
result for multiple walkers is not yet known [28]. Since
multiple walkers are known to improve efficiency of cov-
ering network [28], in this case, Nlog Nwill essentially
be a loose upper bound.
Based on Eq. 1, the central result of this paper can
be recast in the form of a scaling relation as N→ ∞,

3
02500 5000 7500 10000 12500
t
0.5
0.52
0.54
0.56
log(sw1
)/log(sw2
)
W1=1, W2=2
W1=10, W2=20
W1=20, W2=40
W1=30, W2=60
W1=50, W2=100
FIG. 2. The deviations from scaling relation in Eq. 2 arise
due to finite number of walkers. As W1, W2→ ∞, keeping
the ratio W1/W 2 = 0.5 a constant, the agreement with the
expected scaling relation (shown as dashed horizontal line)
gets better. This simulations were performed on BA network
with N= 10000 (same as the one of networks used in Fig.
1(a)).
W1, W2→ ∞ and it is of the form
log sW2(t) = W2
W1β
log sW1(t).(2)
This is valid for W1, W20 random walkers on a given
scale-free network Ns(N, E, γ ). Remarkably, the relation
between sW1and sW2depends only on the ratio r=
W1/W2and information about network enters through
the exponent β. This is verified in Figure 1(b) by plotting
the ratio u(t) = log sW1(t)/log sW1(t) as a function of
time t. In this form, presence of scaling is inferred from
horizontal lines such that u(t) = φ=rβ, a constant. In
particular, simulations confirm that φis identical for any
choice of W1and W2such that ris a constant.
The deviations observed in Fig. 1(b) in the vicinity of
t= 0 arise due to the finite number of walkers on the
network. As shown in Fig. 2, as W→ ∞ the agreement
with the scaling curve in Eq. 2 gets better. On the other
hand, the deviations observed in Fig. 1(b) for t >> 1
arise due to the finite size of the network. In finite size
networks, as t→ ∞ nearly all the nodes are ultimately
reached and hence there is no further ’territory’ to be ex-
plored leading to deviations from Eq. 1. Notice also that
if the agreement with scaling relation is reached faster,
as in the case of W1= 50, W2= 100 in Fig. 2, the devi-
ations for t >> 1 also happen earlier in comparison with
the case of, say, W1= 20 and W2= 40. Physically, this
happens because more the number of walkers, agreement
with scaling curve is reached faster and the all the nodes
are visited quickly (than for smaller number of walkers),
and hence the deviation for t >> 1 also appears quickly.
In Fig. 3, sw(t) is shown for a scale-free network ob-
tained from the configuration model. For this case too,
scaled time x(t) = W t/N leads to an excellent data col-
lapse and Eq. 1 fits the data. The parameters A= 0.1
-4 -3 -2 -1 0 1 2
log (Wt/N)
-14
-12
-10
-8
-6
-4
-2
0
log (sw(t)/N)
W=50, N=935
W=1, N=1917
W=100, N=1924
W=10, N=9644
Fit
0 1000 2000 3000 4000 5000
t
0.2
0.4
0.6
u(t)
FIG. 3. Fraction of nodes not reachable by random walkers on
configuration model of scale-free networks plotted as a func-
tion of x=W t/N. Symbols are obtained from random walk
simulations averaged over 1000 realizations. The solid (red)
line represents Eq. 1 with A= 0.601 and β= 0.653 obtained
through regression. Note the excellent scaling collapse. (In-
set) demonstrates the scaling of sw(t) (Eq. 2) for different
number of walkers w1and w2. The solid (horizontal) lines
are the expected value, namely, (W2/W1)βfor W2/W1ratios
1/2, 2/5, 1/3 and 1/4.
and β= 0.2 were estimated through regression. The in-
set in Fig. 3 shows the validity of the scaling relation in
Eq. 2 for the random walks on configuration model for
several choices of W1and W2.
Next, we study the distinct sites visited by random
walkers on four real-life networks, namely, (a) the Enron
email (EE) network and (b) protein-protein interaction
network of a yeast, (c) network of autonomous systems of
the Internet connected with each other from the CAIDA
project and (d) scientific collaboration network of cond-
mat papers. We perform simulation of random walks
on these networks with Wwalkers and the results are
presented in Figs. 4-6. The data sets for (a,c,d) are
obtained from Stanford network database [30] and for (b)
is obtained from Pajek database [31]. All these networks
were extensively studied for their topological properties
and, in particular, their degree distribution is known to
display a power-law form, P(k)∼k−γ. For Enron email
network γ≈1.76 [32], for yeast network γ≈2.5 [33],
for network of autonomous systems γ≈2.09 [32] and for
the network of cond-mat papers γ≈2.81 [32]. For the
purposes of this work, the largest connected component
of these networks was considered to ensure that isolated
nodes do not exist.
Figure 4(a) shows random walk simulation results for
the fraction of nodes not visited until time ton the En-
ron email communication network. Random walk simu-
lations were performed with different number of walkers
W. As this figure reveals, the simulation results are in
good agreement with the postulated relation in Eq. 1,
with A≈0.403 and β≈0.652. In this case as well,
x=W t/N is the scaled time and as seen in Fig. 4(a), an
excellent data collapse is observed for number of walkers

4
AAAA
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
AAA
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
-10 -5 05
log (Wt/N)
-10
-5
0
log (sw(t)/N)
W=1
W=2
W=5
W=10
W=15
W=20
W=30
W=40
W=50
W=60
W=70
W=80
A
W=90
A
W=100
Fit
0510 15 20 25 30
(W t/N)β
-10
-5
0
0 10000 20000 30000 40000 50000 60000
t
0.3
0.4
0.5
0.6
log(sw1
)/log(sw2
)
0.636
0.488
0.405
0.350
0.310
r = 1/2
r = 1/3
r = 1/4
r = 1/5
r = 1/6
FIG. 4. (a) For Enron email network, the fraction of nodes not
reached by random walkers sW(t) as a function of scaled pa-
rameter x(in log-log scale) for several values of W. The data
collapse points to an agreement with Eq. 1 with A= 0.403
and β= 0.652. (Inset) shows that the same data becomes
linear in semi-log plot if plotted as a function of (W t/N)β.
(b) Demonstrates of the scaling relation in Eq. 2 for various
choices of r=W1/W2.
ranging from 1 to 100. The inset in Fig. 4(a) shows the
same data as a function of (W t/N)βand the resulting
straight line supports Eq. 1. Further, Fig. 4(b) shows
the validity of scaling relation in Eq.2 for various ratio
of walkers r=W1/W2. In Fig. 5(a), sW(t) is shown as
a function of scaled parameter for various choices of of
Win log-log scale. As expected, an excellent data col-
lapse is observed in agreement with Eq. 1. The inset to
this figure further confirms the temporal decay of sW(t)
is indeed stretched exponential in form. As would be ex-
pected, a good agreement with scaling relation in Eq. 2 is
seen in Fig. 5(b) for various ratio of walkers r=W1/W2.
The scaling results from random walker simulations on
a network of autonomous systems and author collabora-
tion networks from cond-mat are displayed in Fig. 6. In
these cases too, the simulation results for sw(t) display
an excellent data collapse when plotted as a function of x
(not shown here). The values of parameters Aand βes-
timated through regression is summarised in Table I for
the networks, including the ones corresponding to Fig. 6,
discussed in this work. A good agreement with the scal-
ing form in Eq. 2 is shown in Fig. 6. For t>τr, the ratio
log(sw1)/log(sw2) tends to a constant dependent on the
value of β,W1and W2.
In summary, we have studied the problem of distinct
number of sites visited by multiple walkers on a scale-free
Network A β
Barabsi-Albert 0.724 0.882
Model
Configuration Model (γ= 2.2) 0.601 0.653
Enron Email Network 0.403 0.652
Yeast Network 0.564 0.622
Autonomous Systems 0.601 0.712
Scientific Collaboration 0.273 0.582
Network
TABLE I. The values of Aand βin Eq. 1 obtained through
regression for various scale-free network models.
A
AAAA
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
AA
A
AA
A
AA
A
A
A
AAAA
AAAA
A
AAAA
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
AA
AA
A
A
AAA
AA
AAAAA
AAAA
AAAAAA
-8 -6 -4 -2 0 2 4 6
log (Wt/N)
-15
-10
-5
0
log (sw(t)/N)
W=1
W=2
W=5
W=10
W=15
W=20
W=30
W=40
W=50
W=60
W=70
W=80
A
W=90
A
W=100
0 10 20
(Wt/N)β
-15
-10
-5
0
0 1000 2000 3000 4000
t
0.3
0.4
0.5
0.6
0.7
log(sw1
)/log(sw2
)
0.650
0.505
0.422
0.368
0.328
r = 1/2
r = 1/3
r = 1/4
r = 1/5
r = 1/6
FIG. 5. (a) Fraction of nodes not reached by Wrandom
walkers in the network of protein-protein interaction in yeast.
The decay follows Eq. 1 with A= 0.534 and β= 0.635.
(b) The number of nodes not reached by W1and W2walkers
follows the scaling relation in Eq. 2.
network. Through numerical simulations, we have shown
that the mean number of sites not reached until time t
represented by a stretched exponential function in Eq. 1
with x=W t/N being the scaled parameter. Using this,
we have displayed the results in the form of a scaling re-
lation (Eq. 2) between the nodes not reached in time t
by W1and W2walkers on the same network. Thus, ef-
fectively, the problem of finding the distinct sites visited
by Wwalkers on scale-free networks is related to that
of one walker, effectively simplifying the problem. Thus,
for scale-free networks, exact results for sw(t) for one
walker would also help solve the problem for many walk-
ers. This results gets better as the size of the network N
and number of walkers Wtend to larger values. We have

5
0 2000 4000 6000 8000 10000 12000
t
0.2
0.3
0.4
0.5
0.6
0.7
log(sw1
)/log(sw2
)
0.610
0.457
0.373
0.279
r=1/2
r=1/3
r=1/4
r=1/6
05000 10000 15000 20000 25000 30000
t
0.3
0.4
0.5
0.6
0.7
log(sw1
)/log(sw2
)
0.668
0.527
0.446
0.352
r=1/2
r=1/3
r=1/4
r=1/6
FIG. 6. The ratio u(t) as a function of time for random walk simulations performed on (a) a network of autonomous systems
and (b) the scientific collaboration network of papers in cond-mat. In (a) and (b), the value of (W1, W2) for each red curve
is (5,10) for circles, (5,15) for up-triangles, (5,20) for squares, (5,30) for plus symbols. Corresponding values for blue curves
are (10,20), (10,30), (10,40) and (10,60) respectively. The horizontal black lines correspond to (W1/W2)β. In all the cases,
numerical random walk simulations tends to this constant after the relaxation time τr.
numerically demonstrated these results for random walk
dynamics on Barabasi-Albert scale-free network and that
constructed using configuration model. Finally, we ver-
ified all our results by simulating random walks on four
different real world scale-free networks and showing that
the scaling holds. It turns out that the scale-free net-
work is somewhat special as far as this scaling relations
are concerned because we did not observe such scaling re-
lations for the other popular classes of networks, namely,
small-world and Erdos-Renyi random networks.
While the stretched exponential function is ubiquitous
in the study of relaxation processes in statistical physics,
there are very few models in which stretched exponential
decay of some observable occurs naturally. We propose
that this simple model of random walks on scale-free net-
works with unreachability as the observable can be used
as a model to investigate relaxation dynamics. We have
seen that the stretching exponent βvaries, though not
systematically, for different values of the power law ex-
ponent γof the degree distribution of the scale-free net-
works. This shows that βhas a dependence on the finer
details of the structure of the network. An interesting
and promising direction would be obtain analytical jus-
tifications for the scaling relation reported in this work.
[1] S. Chandrasekhar, Rev. Mod. Phys. 15, 1 (1943).
[2] E. A. Codling, M. J. Plank, and S. Benhamou, J. R. Soc.,
Interface 5, 813 (2008).
[3] Fama, Eugene F., American Economic Review 104, 1467
(2014).
[4] F.R.K.Chung and W.Zhao, Bolyai Soc. Math. Stud, 20,
43 (2010).
[5] G. M. Viswanathan, M. G. E. da Luz, E. P. Raposo, H. E.
Stanley, The Physics of Foraging, ( Cambridge University
Press, New York, 2011).
[6] G. M. Viswanathan, E. P. Raposo and M. G. E. da
Luz, Physics of Life Reviews 5, 133 (2008); G. M.
Viswanathan, et. al., Nature 381, 413 (1996).
[7] S. M. Ali Tabei et. al., PNAS 110, 4911 (2013).
[8] Leonid Mirny et. al., J. Phys. A: Math. Theor. 42 ,
434013 (2009); C. Loverdo et. al., Phys. Rev. Lett. 102,
188101 (2009).
[9] R. Phillips, J. Kondev, J. Theriot, and H. Garcia, Phys-
ical Biology of the Cell, (Garland Science, New York,
2013).
[10] Lu, T., Shen, T., Zong, C., Hasty, J., Wolynes, P. G.,
PNAS 103, 16752. (2006)
[11] L. Grady, IEEE Transactions on Pattern Analysis and
Machine Intelligence 28, 1768 (2006).
[12] L. L¨u et. al., Phys. Rep. 519, 1 (2012).
[13] E. W. Montroll and G. H. Weiss, J. Math. Phys. 6, 167
(1965); J. W. Haus and K. W. Kehr, Phys. Rep. 150,
263 (1987).
[14] N. Masuda, M. A. Porter, R. Lambiotte, Phys. Rep 716,
1 (2017).
[15] R. Burioni and D.Cassi, J. Phys. A : Math. Gen. 38, R45
(2005).
[16] Moez Draief and Laurent Massouli, Epidemics and Ru-
mours in Complex Networks. (Cambridge University
Press, New York, 2010).
[17] A. Dvoretzky and P. Erdos, in Proceedings of the Sec-
ond Berkeley Symposium on Mathematical Statistics and
Probability, (University of California, Berkeley, 1951).
[18] H. Larralde, P. Trunfio, S. Havlin, H. E. Stanley, and
G. H. Weiss, Phys. Rev. A 45, 7128 (1992); ibid, Nature
355, 423 (1992).
[19] G. H. Weiss et. al., Physica A 191, 479 (1992).
[20] S. B. Yuste and L. Acedo, Phys. Rev. E 61, 2340 (2000).
[21] A. Kundu, S. N. Ma jumdar, and G. Schehr, Phys. Rev.
Lett. 110, 220602 (2013).
[22] B. D. Hughes and M. Sahimi, J. Stat. Phys. 29, 781
(1982).
[23] C. De Bacco, S. N. Majumdar, and P. Sollich, J. Phys.
A48, 205004 (2015).
[24] S. Lee, Soon-Hyung Yook and Y. Kim, Physica A 387,
3033 (2008).
[25] E. Almaas, R. V. Kulkarni, and D. Stroud, Phys. Rev. E
68, 056105 (2003)
[26] R. Albert and A. L. Barabasi, Rev. Mod. Phys. 74, 47

6
(2002).
[27] M. E. J. Newman, in Handbook of Graphs and Networks:
From the Genome to the Internet, (Wiley-VCH, Berlin,
2003)
[28] N. Alon et. al., arXiv:0705.0467
[29] B. F. Maier and D. Brockmann, Phys. Rev. E 96, 042307
(2017).
[30] Jure Leskovec and Andrej Krevl, SNAP Datasets at
http://snap.stanford.edu/data (2014).
[31] V. Batagelj and A. Mrvar, Pajek datasets at
http://vlado.fmf.uni-lj.si/pub/networks/data/ (2006).
[32] This value provided in http://konect.uni-
koblenz.de/test/networks/
[33] R. Albert, J. Cell Sci. 118, 4947 (2005).