We present a simple model of network dynamics that can be solved analytically
for uniform networks. We obtain the dynamics of response of the system to
perturbations. The analytical solution is an excellent approximation for random
networks. A comparison with the scale-free network, though qualitatively
similar, shows the effect of distinct topology.Comment: 4 pages, 1 figur

Bar-Yam, Yaneer

de Aguiar, M. A. M.

Epstein, Irving R.

Physical Review E

English

arXiv

We present a simple model of network dynamics that can be solved analytically for fully connected networks. We obtain the dynamics of response of the system to perturbations. The analytical solution is an excellent approximation for random networks. A comparison with the scale-free network, though qualitatively similar, shows the effect of distinct topology.726

de Aguiar, MAM

Epstein, IR

Bar-Yam, Y

Repositorio da Producao Cientifica e Intelectual da Unicamp

Analytically solvable model of probabilistic network dynamics
M. A. M. de Aguiar,1,2 Irving R. Epstein,1,3 and Yaneer Bar-Yam1
1New England Complex Systems Institute, Cambridge, Massachusetts 02138, USA
2Instituto de Física Gleb Wataghin, Universidade Estadual de Campinas, 13083-970 Campinas, São Paulo, Brazil
3Department of Chemistry, MS015, Brandeis University, Waltham, Massachusetts 02454, USA
Received 16 March 2005; revised manuscript received 29 August 2005; published 7 December 2005
We present a simple model of network dynamics that can be solved analytically for fully connected net-
works. We obtain the dynamics of response of the system to perturbations. The analytical solution is an
excellent approximation for random networks. A comparison with the scale-free network, though qualitatively
similar, shows the effect of distinct topology.
DOI: 10.1103/PhysRevE.72.067102 PACS numbers: 89.75.Hc, 02.50.Ey, 84.35.i
Recent advances in the understanding of complex social
1, biological 2, and technological 3 systems have re-
vealed widespread if not universal properties of the topology
of networks of association, interaction, and communication.
These properties include small-world global connectivity 4,
scale-free local connectivity distribution 5, and characteris-
tic local motif structures 6. Central to our understanding of
complex systems 7 is characterizing their response to envi-
ronmental stimuli. While much of the focus has been on
robustness to random perturbation or directed attack 8, the
effectiveness of response requires satisfying a wider range of
conditions including, for example, sensitivity to particular
stimuli 9. Indeed, one of the main functions of biological
and social systems is the detection of specific stimuli that
require collective large scale response in seeking desirable
resources foraging or responding to dangers “fight or
flight”. Understanding system function from network topol-
ogy requires mapping the topology onto system response
10.
Insofar as the network characterizes the internal interac-
tions, it must provide key information about response to ex-
ternal perturbations. In this paper we propose to use a model
of influence similar to that studied in physics and the social
sciences as the “voter model” and in biology as a model for
propagation of mutations through a populations. We study
the probabilistic discrete dynamics of this model and solve it
analytically for fully connected networks, and for uniform
node degree networks after aggregating states into state
classes. We show that the model can reveal the differences in
dynamic response of distinct topologies. The exact solution
we obtain can serve as a basis for perturbative studies of
diverse topologies.
We consider a general network with N nodes. To each
node i is assigned an internal state i that can take the values
0 or 1. At each time step the state of a node is updated
according to the following rule: either the state does not
change, which happens with probability p, or, with probabil-
ity 1− p, it copies the state of one of its neighbors. This
process describes many systems in which mutual influence
occurs, for example, high school students changing their
dress style based upon that of others, or the propagation of a
mutation through a species 11. For a fully connected net-
work and p=1/N the model is equivalent to the Moran pro-
cess 12, and for p=0 to the voter model 13. Since the
states of a node are abstract labels, the change of one node to
adopt the state of another can be considered a general model
of influence propagation, with each node state a label for its
own relevant physical, biological, or social property. We will
study the impact of a perturbation that changes the state of a
subset of nodes as it propagates through the system.
The 2N states of the network can be labeled by a string of
zeroes and ones describing the internal state of each node in
sequence N−1 ,N−2… ,1 ,0. Alternatively, the states can
be labeled by integers via x= j=0
N−1 j2
j, with x varying be-
tween 0 and 2N−1.
Let Ptx be the probability of finding the network in the
state x at time t and let the network evolve through asynchro-
nous updates, where a single node is allowed to change at
each time step. To find how this probability changes with
time we define the auxiliary state x̃k which is equal to x at all
nodes except at node k, which has the opposite internal state.
The probability of finding the network in the state x at time
t+1 can now be written as a sum of three terms: a the
probability that the network was in state x at time t and that
the selected node did not change plus b the probability that
it was in the state x and the selected node copied the state of
an identical neighbor plus c the probability that the network
was in the state x̃k at time t and that the node k was selected
and its state ̃k=1−k changed to k:
Pt+1x = pPtx +
1 − p
N

k
PtxProbk → k
+ Ptx̃kProb̃k → k .
The probability Probk→k is just the number of
neighbors of node k in the state k divided by the total num-
ber of neighbors the degree dk=i=0
N−1Cik, where Cik is the
connectivity or adjacency matrix. This can be written as
1
dk

i=0
N−1
Cik1 − i − k .
The probability Prob̃k→k is also given by this formula,
since ̃k=1−k and ̃i=i for ik. Using these relations,
we obtain the following master equation for the network dy-
namics:
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Pt+1x = pPtx +
1 − p
N

k=0
N−1
1
dk

i=0
N−1
Cik
1 − i − kPtx + Ptx̃k . 1
Finding Ptx for networks with arbitrary topologies can
be very difficult. However, the problem can be completely
solved for fully connected networks, where dk=N−1. In this
case, by symmetry, the states of the network can be labeled
simply by counting the number of nodes in the internal state
1, given by nx=ii. The probability of finding the net-
work in the state labeled by n is related to Px by
P„nx… = PxBN,n , 2
where BN ,n=N ! / n ! N−n !  is a binomial coefficient.
The last two terms on the right-hand side of Eq. 1 can be
simplified if we separate the sum over k into the cases
k=1 and k=0 and multiply both sides of this equation by
BN ,n. After some algebra we obtain
Pt+1n = pPtn +
1 − p
NN − 1
nn − 1
+ N − nN − n − 1Ptn
+ N − n + 1n − 1Ptn − 1
+ N − n − 1n + 1Ptn + 1 . 3
For uniform networks where dk=d0N−1 is the same for
all nodes, we can use this equation if we classify together all
the network states with the same nx. In this case the factor
dk in the denominator is replaced by d0 in Eq. 1. Averaging
with respect to the set of states labeled by n, the counting of
the number of neighbors must be multiplied by d0 / N−1, so
that d0 cancels and N−1 remains in the denominator.
Therefore Eq. 3 holds in this case as well. For random
networks the degree of each node is nearly constant; tests
reported below show that Eq. 3 is indeed a reasonable ap-
proximation for the dynamics in this case.
We now calculate the transition probabilities Ptn, defin-
ing a vector Pt of N+1 components. The master equation 3
can be written in matrix form as Pt+1=UPt where the evolu-
tion matrix U is tridiagonal. The propagation of an initial
probability vector requires the calculation of powers of U.
Alternatively, we can diagonalize U and use its eigenvectors
as a basis. In what follows we shall calculate explicitly all
the eigenvalues and eigenvectors of U, obtaining the com-
plete solution of the dynamical problem.
The eigenvalues of U can be calculated for small matrices
and extrapolated to matrices of arbitrary size. They are given
by
r = 1 −
1 − p
NN − 1
rr − 1
with r=0,1 ,… ,N. The only degeneracy occurs for 0=1
=1. The other eigenvalues are all smaller than 1 and decrease
towards N= p.
Since U is not symmetric, its eigenvectors do not form an
orthogonal set. Let ar	 and 
br be the right and left eigen-
vectors of U, with components arm and brm. Then

r=0
N
1
r
ar	
br = 1,
where =
br ar	=rrr and r=marmbrm.
An initial vector v0	 containing the information about
the probability of the different states at time zero can be
projected using this resolution of unity and easily evolved:
vt	 = Utv0	 = 
r=0
N
1
r

brv0	r
t ar	 .
The transition probability between two network states
with n=M and n=L after a time t can be calculated by taking
the components of the initial vector as vm0=M,m and pro-
jecting the evolved state onto the state with components L,m:
PL,t;M,0 = 
r=0
N
1
r
brMarLr
t .
The coefficients arm follow a recursion relation that can be
derived directly from the eigenvalue equation for U. For
r=0 and r=1 the eigenvectors can be found immediately:
a0	 =
1
0
0

0
1
 , a1	 =
1
0
0

0
− 1
 4
and
b0	 = 1 1 1 ¯ 1 1  ,
b1	 = N N − 2 N − 4 ¯ − N + 2 − N  . 5
In order to calculate the remaining eigenvectors, we de-
fine the auxiliary eigenvalues r by
r = 1 − r
NN − 1
1 − p
= rr − 1 6
and the auxiliary coefficients
Arm = mN − marm, m = 1,2,…,N − 1. 7
The recursion relation for the Arm can be written explicitly as
Arm+1 − 2Arm + Arm−1 = −
r
N
Arm
m
+
Arm
N − m
 . 8
A generating function is now defined as
frx = 
m=1
N−1
Armx
m
note that Ar0=ArN=0. Multiplying Eq. 8 by xm and sum-
ming over m we get, on the left-hand side,
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fr
x
1 − x2 − Ar1 + ArN−1xN.
In order to write down the right-hand side of Eq. 8 we
define the auxiliary functions
grx = 
m=1
N−1
Arm
m
xm, hrx = 
m=1
N−1
Arm
N − m
xm. 9
It is easy to check that dgr /dx= fr /x and dhr /dx
=Nhr /x− fr /x. After multiplying Eq. 8 by xm and summing
over m, we differentiate both sides with respect to x to obtain
d
dx
 1 − x2
x
frx − ArN−1xN = − rx hrx . 10
The solution of the differential equation for hr can be
obtained in terms of its Green function, satisfying
dG /dx−NG /x=x−y. In this case G is given by x /yN if
x	y and zero otherwise. Therefore
hrx = xN
 − 
−
x fry
yN+1
dy . 11
Substituting Eq. 11 into Eq. 10, re-arranging the terms
and differentiating once again with respect to x, we obtain
d
dx
 1
xN−1
d
dx
 1 − x2
x
frx = r frxxN+1 .
Finally, defining
Frx =
1 − x2
x
frx 12
we obtain the differential equation
Fr −
N − 1
x
Fr −
r
x
Fr
1 − x2
= 0. 13
Now let rx=armxm= gr+hr /N. Differentiating with
respect to x, using Eq. 11, dividing by xN−1 and differenti-
ating again, we find
r −
N − 1
x
r −
1
x
Fr
1 − x2
= 0.
Comparing with Eq. 13 we see that r=Fr /r. Therefore,
except for normalization, the generating function for the co-
efficients arm , rx, is equal to Frx.
For r=0 or r=1, r=0 and the two independent solutions
of Eq. 13 are F0x=1+xN and F1x=1−xN, which corre-
spond to the two degenerate eigenvectors a0	 and a1	. For
r=2 and r=3 the solution can also be found explicitly; the
general formula can then be extrapolated from these simple
cases. We find
Frx = 1 − x1−r1 + 
p=1
r−1
drpx
p . 14
with
drp = − 1p
Br − 1,pBN + r − 1,p
BN − 1,p
.
Finally, the coefficients of the rth eigenvector are given
by
arm =
1
m!
dmFrx
dxm

x=0
.
Working out the derivatives, we find the explicit formula
valid for r2:
arm = 
p=0
r−1
Bm − p + r − 2,r − 2drp 15
for m=1,2 ,… ,N−1, with ar0=1 and arN= −1r.
From the recursion relations we find that the coefficients
of the left eigenvectors are given by
brm = armmN − m/N 16
for m=1,2 ,… ,N−1, with br0=brN=0. Finally, the normal-
ization factors r can also be obtained explicitly:
r =
r ! BN + r − 1,r
2r − 1BN,r
17
for r=2,… ,N and 0=2 , 1=2N.
We apply our calculations to consider a perturbation
propagating through the system. Specifically, we seek the
probability that a perturbation that initially affects M nodes
will lead to a response by L nodes at a time t later. Starting
with n=M sites with the value 1 and all other sites 0 we look
at the transition probability PL , t ;M ,0 of finding the sys-
tem at a later time t with n=L. Except for the three special
cases L=0, M , N, this probability must start from zero, go
through a maximum and then asymptotically go to zero. The
case L=M starts from one and decreases monotonically to
zero, and the cases L=0, N start from zero and increase
FIG. 1. Transition probabilities for a network with N=101 and
p=0.1. The lines correspond to our theoretical calculation thick
and to simulations for random dotted and scale-free thin net-
works, both with an average of six connections per node. The nu-
merical probabilities were computed running the simulations
2105 times. The dotted line is nearly indistinguishable from the
thick line.
BRIEF REPORTS PHYSICAL REVIEW E 72, 067102 2005
067102-3
monotonically to their nonzero equilibrium values, as they
are the absorbing steady states of the system. The initial and
final state conditions are independent of topology; however,
the intermediate dynamics can be affected by the topology
and reveal key differences in system response to perturba-
tions. PL , t ;M ,0 can be computed using Eqs. 4, 5, 15,
16, and 17:
PL,t;M,0 = N − M
N
−
3MN − M
N + 1N
2
t L0
+ M
N
−
3MN − M
N + 1N
2
t LN
+
6MN − M
NN2 − 1
1 − L01 − LN2
t
+ r=3
N 1
r
brMarLr
t .
All eigenvalues except for 0 and 1 are smaller than 1,
so the transition probability at long times is dominated by
0=1=1 and by the second largest eigenvalue 2, whose
contribution we obtain explicitly. For large networks
2
t exp− 21− pt /N2, so that the characteristic duration
of the transition process is =N2 /21− p, which increases
with the square of the network size. The time to the maxi-
mum can be estimated as tmax=NL−M /21− p. The
asymptotic probability of finding the network in states n=0
or n=N, having started from n=M, is N−M /N and M /N,
respectively.
Figure 1 shows a comparison between our analytical re-
sult and numerical calculations for a random and a scale-free
network with N=101. The estimates 6000 and
tmax1000 for Fig. 1a are in agreement with the numeri-
cal calculation. The fluctuations not shown of the transition
probabilities for random and scale-free networks are very
similar, which is presumably due to self-averaging. The
probability of initially selecting a highly connected node in a
scale-free network is low, but as the perturbation spreads,
such nodes will be encountered with increasing frequency,
because their influence scales with their connectivity.
Summarizing, we have completely solved a simple dy-
namical process on fully connected networks. Our analytical
result is an excellent approximation for the average behavior
of sparsely connected random networks. For scale-free net-
works the qualitative behavior is the same, but there are de-
viations reflecting the significant topological differences,
showing the utility of the dynamic model in describing the
distinct responses of different topologies.
M.A.M.A. acknowledges financial support from CNPq
and FAPESP. I.R.E. was supported in part by the David and
Lucile Packard Foundation.
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Analytically solvable model of probabilistic network dynamics

An analytically solvable model of probabilistic network dynamics

An analytically solvable model of probabilistic network dynamics

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