PHYSICAL REVIEW E 75, 056103 共2007兲
Seed size strongly affects cascades on random networks
1
James P. Gleeson1 and Diarmuid J. Cahalane2
Applied Mathematics, University College Cork, Cork, Ireland
2
MACSI, University College Cork, Cork, Ireland
共Received 16 February 2007; published 3 May 2007兲
The average avalanche size in the Watts model of threshold dynamics on random networks of arbitrary
degree distribution is determined analytically. Existence criteria for global cascades are shown to depend
sensitively on the size of the initial seed disturbance. The dependence of cascade size upon the mean degree z
of the network is known to exhibit several transitions—these are typically continuous at low z and discontinuous at high z; here it is demonstrated that the low-z transition may in fact be discontinuous in certain parameter
regimes. Connections between these results and the zero-temperature random-field Ising model on random
graphs are discussed.
DOI: 10.1103/PhysRevE.75.056103
PACS number共s兲: 89.75.Hc
Dynamical models on networks have attracted a great deal
of recent research interest; see the reviews 关1–3兴 and references therein. Of particular interest are the effects of the
topological structure of the network upon the time evolution
of dynamical quantities. Under certain circumstances a
change of state at a small number of network nodes may
cause a cascade of changes over the whole network—such
cascades may occur as, for example, overload failures 关4–9兴,
avalanches in sandpile models 关10,11兴, evolution of species
关12兴, or the spread of rumors and fads in social networks
关13–15兴.
In this paper, we analyze a model of dynamics on random
networks from the class of problems known as binary decisions with externalities 关16兴. Each node of an undirected network represents an agent, and may be in one of two states,
called active and inactive. The network is chosen from an
ensemble of graphs with a specified degree distribution pk,
i.e., the probability that a node has degree k is pk, with
兺pk = 1 关1,17兴. Each agent also has a 共frozen兲 random threshold r, chosen from a distribution with F共r兲 denoting the
probability that an agent has threshold less than r. Initially
the network is seeded by activating a randomly chosen fraction ␳0 of the N nodes. When updated, an inactive agent
calculates the fraction of its neighbors who are currently active 共m / k, where k is the degree of the node and m is the
number of active neighbors兲, and it becomes active if this
fraction exceeds his threshold r. Once active, an agent cannot
become inactive; this will be referred to as the permanently
active property 共PAP兲. Isolated nodes 共degree k = 0兲 are never
updated. It can be shown that 共unlike some other sequential
spin-flip models 关18兴兲 the steady state of the system does not
depend on whether the updating of all agents is performed
synchronously, or in a random asynchronous fashion.
Watts introduced the above model 关13兴 to demonstrate the
interaction between network topology and individual thresholds in the spreading of rumors and fads. He used percolation
methods to determine whether a single activated initial agent
can induce network-spanning cascades of activation. We focus here on calculating ␳, the average fraction of active
nodes in the steady state, where the ensemble average is over
realizations of networks and threshold values, and in the
limit of infinite network size: N → ⬁. Watts’ criterion for glo1539-3755/2007/75共5兲/056103共4兲
bal cascades is recovered in the ␳0 → 0 limit 共since ␳0 = 1 / N
for single-node initialization兲 but important differences are
found when the seed size is finite. The quantity ␳ measures
the average avalanche size, and has recently been studied for
dynamics on directed random graphs, with applications to
random Boolean networks 关19兴, and on small-world topologies 关20兴. The magnetization in the zero-temperature
random-field Ising model 共RFIM兲 关21,22兴 is also closely
analogous to ␳; the agents’ threshold values correspond to
the local quenched random fields of the RFIM. There are,
however, two important differences: 共i兲 the RFIM does not
exhibit the PAP, since spins may flip either up or down depending on their local field; 共ii兲 the RFIM has not been studied on random graphs of arbitrary degree distribution. Analytical solutions for the RFIM have been developed in the
mean-field case 关21兴 and on Bethe lattices 关22,23兴. Our approach reduces to the latter in the case of regular random
graphs, where each node has exactly z neighbors 共pk = ␦kz for
integer z兲, provided that ␳0 is zero.
Our main result is that the average final fraction ␳ of
active nodes is given by
⬁
k
␳ = ␳0 + 共1 − ␳0兲 兺 pk 兺
k=1
m=0
冉冊
k
m
q⬁m共1 − q⬁兲k−mF
冉冊
m
, 共1兲
k
where q⬁ is the fixed point of the recursion relation
qn+1 = ␳0 + 共1 − ␳0兲G共qn兲
for n = 0,1,2 . . . ,
共2兲
with q0 = ␳0, and the nonlinear function G is defined by
⬁
k−1
冉 冊
冉冊
k−1 m
k
m
G共q兲 = 兺 pk 兺
q 共1 − q兲k−1−mF
.
k
m
k=1 z m=0
共3兲
Here z is the mean degree of the network, z = 兺kpk. We postpone the derivation of 共1兲 to the end of the paper, and first
examine some of its consequences.
Figure 1 shows ␳ in the case of uniform thresholds 共all
nodes have identical threshold r = R兲 on Poisson 共ErdősRényi兲 random graphs with degree distribution pk = e−zzk / k!.
In Fig. 1共a兲 ␳ is color coded on the R , z parameter plane for
seed fraction ␳0 = 10−2; Fig. 1共b兲 compares ␳ at R = 0.18 with
numerical simulations for several ␳0 values, as a function of
056103-1
©2007 The American Physical Society
PHYSICAL REVIEW E 75, 056103 共2007兲
JAMES P. GLEESON AND DIARMUID J. CAHALANE
1
15 (a)
10
10
0.5
z
z
0.5
5
5
0
0
0.1
0.2
R
0
0.3
0
0
1
0.1
0.2
R
2
4
0.3
0
0.4
1
(b)
(b)
ρ
ρ
1
15 (a)
0.5
0.5
0
0
2
4
z
6
8
0
0
10
FIG. 1. 共Color online兲 Average density ␳ of active nodes in a
Poisson random graph of mean degree z and uniform threshold
value R. 共a兲 Color-coded values of ␳ from Eq. 共1兲 on the R , z plane
with seed fraction ␳0 = 10−2. Lines show approximations to the global cascade boundaries: the solid line encloses the region where 共5兲
is satisfied; the dashed line represents the extended cascade boundary from Eq. 共6兲. 共b兲 Values of ␳ at R = 0.18 from Eq. 共1兲 共lines兲 and
numerical simulations 共symbols兲, averaged over 100 realizations
with N = 105. Seed fractions are ␳0 = 10−3 共solid兲, 5 ⫻ 10−3 共dashed兲,
and 10−2 共dot-dashed兲. The arrow marks the cascade boundary
given by Eq. 共5兲 in the limit of zero seed fraction.
z. The theory shows excellent agreement with simulations.
Note the smooth transition from ␳ ⬇ 0 to ␳ ⬇ 1 near z = 1, and
the discontinuous transition back to ␳ ⬇ 0 at larger z values.
The z location of the upper transition clearly depends sensitively on the initial activated fraction ␳0.
A global cascade occurs with high probability when a
small seed ␳0 results in a large value of ␳. Writing G共q兲 as
⬁
兺ᐉ=0
Cᐉqᐉ with coefficients
⬁
Cᐉ =
ᐉ
兺 兺
k=ᐉ+1 n=0
冉 冊冉 冊
k−1
ᐉ
冉冊
ᐉ
k
n
共− 1兲ᐉ+n pkF
,
z
n
k
共4兲
and linearizing Eq. 共2兲 near q = 0 gives a 共first-order兲 condition for global cascades to occur: 共1 − ␳0兲C1 ⬎ 1, since this
guarantees that qn increases with n, at least initially. This
cascade condition may also be written as
⬁
兺
k=1
冋冉冊 册
k共k − 1兲
1
1
pk F
− F共0兲 ⬎
.
z
k
1 − ␳0
6
8
10
FIG. 2. 共Color online兲 As Fig. 1, but threshold distribution is
Gaussian with mean R and standard deviation 0.2, and seed fraction
␳0 = 0. 共a兲 Cascade boundaries as in Fig. 1; the arrow marks the
critical point 共Rc , zc兲 described in the text; 共b兲 theory and numerical
simulation results 共N = 105, average over 100 realizations兲 at R
= 0.2 共solid兲, 0.362 共dashed兲, and 0.38 共dash-dotted兲.
␳0 = 10−3 and 10−2 共close to the z value marked by the arrow兲,
but the simulations and full theory show that the respective
transitions are actually near z ⬇ 6.4 and 9.2. Condition 共5兲
clearly does not accurately represent the effects of finite seed
size 关nor of nonzero F共0兲; see below兴, because the function
G is not well approximated by a straight line near the critical
parameters for cascade transitions.
We seek an improved cascade condition by extending the
series for G共q兲 to order q2. This approximation results in a
quadratic equation for the fixed point q⬁ which we represent
as aq⬁2 + bq⬁ + c = 0. Under the approximation of small q values, we interpret the existence of a positive root of this quadratic equation to imply q⬁ Ⰶ 1, and hence the impossibility
of global cascades. Note that the first-order approximation
共5兲 requires b ⬎ 0 for global cascades. However, when the
nonlinear behavior of G is important, it is still possible for
global cascades to occur when b is negative, provided that
the full solution of the quadratic equation precludes positive
real roots. We therefore extend the cascade boundary to include regions where either b ⬎ 0 共as in 共5兲兲 or b2 − 4ac ⬍ 0,
the latter giving 共to first order in ␳0兲
共5兲
In the limit ␳0 → 0 and with F共0兲 = 0, this reduces to the
condition derived by Watts using percolation arguments 关13兴.
On the R , z plane of Fig. 1共a兲 the condition 共5兲 is satisfied
inside the solid line; for R = 0.18 in Fig. 1共b兲, 共5兲 predicts
cascade transitions at z values between 5.7 and 5.8 for both
z
共C1 − 1兲2 − 4C0C2 + 2␳0共C1 − C21 − 2C2 + 4C0C2兲 ⬍ 0.
共6兲
It is straightforward to plot this extended cascade condition:
the dashed lines in Figs. 1共a兲 and 2共a兲 clearly give much
improved approximations to the actual boundaries for global
cascades.
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PHYSICAL REVIEW E 75, 056103 共2007兲
SEED SIZE STRONGLY AFFECTS CASCADES ON RANDOM…
1
(a)
q
0.5
0
0
2
4
2
4
2
4
z
6
8
10
6
8
10
6
8
10
1
(b)
q
0.5
0
0
z
1
(c)
q
0.5
0
0
z
FIG. 3. 共Color online兲 Bifurcation diagrams as described in text
for dependence of q⬁ on z for ␴ = 0.2 and ␳0 = 0 at R = 共a兲 0.35, 共b兲
0.371, and 共c兲 0.375.
Figure 2共a兲 shows ␳ on Poisson random graphs with
thresholds drawn from a Gaussian distribution of mean R and
standard deviation ␴ = 0.2. Unlike the assumption in 关13兴 that
F共0兲 = 0, a Gaussian distribution necessarily implies the presence of negative-valued thresholds among the population, so
F共0兲 ⬎ 0. Negative-threshold agents act as a natural seed,
since they activate regardless of the states of their neighbors.
The presence of these innovators 关13兴 allows us to set ␳0
= 0 in this case. The extended cascade condition 共6兲 again
gives a good approximation to the discontinuous ␳ transition
at high z values. Figure 2共b兲 focuses on the low-z transition
and highlights the existence of a discontinuous transition in z
for certain threshold distributions. This is qualitatively different from the previously-studied case 关13兴 where only continuous low-z transitions were found.
Bifurcation analysis of Eq. 共2兲 elucidates this result. In
Fig. 3 we plot the roots of the fixed-point equations G共q兲
− q = 0 共recall that ␳0 = 0 here; extension to nonzero ␳0 is
straightforward兲 as functions of z, for different values of the
mean threshold R. Thick solid and dashed lines denote stable
and unstable fixed points respectively 关24兴. The PAP means
the value of q⬁ achieved at a given z is that of the lowest
stable branch above q = ␳0. The occurrence of triple roots as
R is increased causes the smooth low-z transition seen in Fig.
3共a兲 to become discontinuous 关as shown by the thin solid line
in Fig. 3共b兲兴, as previously seen in the numerical simulations
of Fig. 2共b兲. The discontinuous low-z transition occurs for
R ⬎ Rc, where the critical coordinates 共Rc , zc兲 and the value
q = qc where the triple root appears are found by numerical
root finding for the system of three equations q = ␳0
+ 共1 − ␳0兲G共q兲, 共1 − ␳0兲G⬘共q兲 = 1, and G⬙共q兲 = 0. For ␴ = 0.2
this yields 共Rc , zc兲 = 共0.3543, 3.136兲; this point is marked with
an arrow in Fig. 2共a兲. We remark that the discontinuous transition from q⬁ ⬇ 1 to q⬁ ⬇ 0 关which induces a similar transition in ␳ through Eq. 共1兲兴 occurs due to a saddle-node bifurcation 关24兴. This behavior is quite generic, occurring for a
wide variety of parameters, with the exception of the special
case studied by Watts. For ␳0 = 0 and F共0兲 = 0 as in 关13兴, the
coefficient C0 is zero and the fixed-point equation always has
a root at q = 0, with transcritical bifurcations on the q = 0 line
giving rise to the observed transitions. However, any nonzero
seed size replaces the transcritical bifurcations with saddlenode bifurcations as described above. We have confirmed the
accuracy of these results 关and Eq. 共1兲兴 against numerical
simulations on other configuration model network topologies
关1兴, including power-law degree distributions 共with exponential cutoff兲: pk ⬀ k−␶ exp共−k / ␬兲 关17兴.
We turn now to the derivation of Eqs. 共1兲–共3兲. Our analytical approach is based on methods introduced by Dhar et
al. to study the zero-temperature random-field Ising model
on Bethe lattices 关22兴. The RFIM is a spin-based model of
magnetic materials, and its zero-temperature limit has been
extensively studied as a model for systems exhibiting hysteresis and Barkhausen noise 关21兴. A Bethe lattice of coordination number z 共for integer z兲 is an infinite tree where every
node has exactly z neighbors. Dhar et al. derive analytical
results valid on Bethe lattices, but their numerical simulations show that the theory also applies very accurately to
random graphs where every node has exactly z neighbors,
provided that short-distance loops are rare. To analyze Watts’
model we extend the approach of 关22兴 in two ways. First, we
consider treelike random graphs with arbitrary degree distributions, rather than the Bethe lattices of 关22兴. Second, we
account for the PAP, which is the essential difference between Watts’ update rule and standard RFIM dynamics. This
difference between the update rules is crucial to our derivation of the ␳0 dependence of the activated fraction ␳.
We begin by replacing the given random graph 共with degree distribution pk兲 by a tree structure. The top level of the
tree is a single node with degree k, and this is connected to
its k neighbors at the next lower level of the tree. Each of
these nodes is in turn connected to ki − 1 neighbors at the next
lower level, where ki is the degree of node i. The degree
distribution of the nodes in the tree is given by p̃k = 共k / z兲pk,
which is the distribution for the number of nearest neighbors
in a connected graph 关1,25兴. To find the final density ␳ of
active nodes, we label the levels of the tree from n = 0 at the
bottom, with the top node at an infinitely high level 共n
→ ⬁兲. Define qn as the conditional probability that a node on
level n is active, conditioned on its parent 共on level n + 1兲
being inactive. Consider updating a node on level n + 1, assuming that the nodes on all lower levels have already been
updated. With probability p̃k the chosen node has k neighbors: one of these is its parent 共on level n + 2兲, and the remaining k − 1 are its children 共on level n兲. Since a fraction ␳0
of nodes were initially set to be active, there is a probability
␳0 that we have chosen one of these nodes. In this case the
state of the node remains unchanged. On the other hand, with
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PHYSICAL REVIEW E 75, 056103 共2007兲
JAMES P. GLEESON AND DIARMUID J. CAHALANE
probability 共1 − ␳0兲 the node in question is inactive. In this
case we must consider its neighbors. Each of the k − 1 children is active with probability qn. We also assume its parent
is inactive. Thus the node has m active children 共and therefore k − 1 − m inactive children兲 with probability 共 km−1 兲qm
n 共1
− qn兲k−1−m. The probability that its threshold ri is less than its
fraction of active neighbors m / k is given by F共m / k兲, and
combining the independent probabilities yields Eq. 共2兲 for
qn+1. The probability that the single node at the top of the
tree is active is given by adding the probabilities for two
independent cases: either it is already active as part of the
originally activated fraction 共with probability ␳0兲, or it was
initially inactive 共with probability 1 − ␳0兲. In the latter case, it
will become active if sufficiently many of its k neighbors 共all
on the next lower level兲 are active. Noting that the top node
has degree k with probability pk, we conclude that the total
probability of it being active is given by Eq. 共1兲.
Although the theory is defined in terms of level-by-level
updating on a tree, these results also apply to the randomgraph Watts model, provided that 共i兲 the network structure is
locally treelike, 共ii兲 the state of each node is altered at most
once, and 共iii兲 the steady state is independent of the order in
which nodes are updated 关23兴. We note that condition 共i兲 is
true in configuration model networks, where the clustering
coefficient scales as 1 / N as N → ⬁ 关1兴. It does not, however,
hold in small-world 关20,26兴 and other real-world networks
where clustering and loops play an important role. Condition
共ii兲 is guaranteed by the PAP, and 共iii兲 holds for this and
related unordered binary avalanche models 关19兴.
In summary, our main result is Eq. 共1兲 for determining the
expected fraction of active nodes 共mean avalanche size兲 ␳
due to the activation of a seed fraction ␳0 of random nodes.
The simple conditions for global cascades 共5兲 and 共6兲 follow;
these substantially extend Watts’ original result to include
nonzero initial seed sizes. Values of ␳0 as low as 0.1% have
dramatic effects on the location of cascade transition points
关see Fig. 1共b兲兴. Transitions in the dependence of ␳ upon the
mean network degree z are explained by the appearance of
saddle-node bifurcation points. Of particular note is the fact
that in certain parameter regimes the low-z transition may be
discontinuous.
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This work is funded by Science Foundation Ireland under
Investigator Grant No. 02/IN.1/IM062 and the Mathematics
Applications Consortium for Science and Industry in Ireland
共MACSI兲, and by the Irish Research Council for Science,
Engineering and Technology under the Embark Initiative,
Grant No. RS/2005/16.
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