Network Models: Growth, Dynamics, and
Failure
Sandip Roy
Chalee Asavathiratham
Bernard C. Lesieutre
Massachusetts Institute of Technology
George C. Verghese
Abstract— This paper reports on preliminary explorations,
both empirical and analytical, of probabilistic models of
large-scale networks. We first examine the structure of networks that grow by the addition of nodes and lines, using a
class of connection rules motivated by considerations of distance and prior connectivity. Second, we examine the dynamic behavior of the binary influence model — a particular
form of a more general model of networks in which each node
has a status (for instance: normal, or failed) that behaves
as a Markov chain, but with transitions that are influenced
by the present status of each neighboring node. Some interesting influence model examples are analyzed, including
one displaying a power-law relation between the frequency
of a failure event and its extensiveness.
I. Introduction
Extensive failures in large complex systems often result
from cascading events, in which an initial failure of one network component impacts other components and induces
their failure, eventually impacting a large portion of the
system. The tendency for cascading failures in designed
networks is not completely understood, but is necessarily
influenced by the underlying structure of these networks.
For example, a system comprising a collection of small disconnected regions will not exhibit large-scale failures. The
interconnection of a wide area introduces the possibility
for large-scale failures. In addition, the dynamic behavior
of network components is important in explaining the occurrence of cascading failures. The dynamics of designed
networks are varied and are often complex, so that the possibility for cascading failures in these networks is difficult
to predict in general. Understanding the nature of interconnections and dynamics that will make extensive failures
probable is one of the goals of this research.
Several approaches for understanding failures in networks have been discussed in the literature. From real
data, researchers have tabulated statistics of specific failure
events in certain types of networks. One such representation, for the electric power grid, is shown in Figure 1 (the
data are from [1]). The statistics represented in this plot
provide compelling evidence that large-scale outages on the
power grid are not anomalies, but should be expected as
infrequent, but regular, occurrences. While such a highlevel statistical representation of data is useful, it does not
explain why such behavior is observed, nor does it help
identify which features of the network structure and dynamics most influence it, or suggest how to apply resources
to mitigate failures.
A complementary approach is the study of failures using appropriate models of networks. The models can be
used in at least two different ways. First, one may perform
Fig. 1. A plot of cumulative frequency versus number of customers
affected by power system disturbances, 1984-1997.
numerous studies to obtain statistical data similar to that
compiled in the real world observations. (And if the models adequately represent real-world systems, the resulting
statistics should be similar.) Correlating the statistics with
certain quantities represented in the model may yield more
information about the system. Second, one may attempt to
analytically relate features of the network and dynamics to
resulting system behavior. While very difficult to obtain,
such relations would be the most valuable results.
In this article we look at growth models of networks using simple connection rules. We study the connectivity
properties of the resulting networks, some of which display
power-law characteristics. Later in the paper we compare
the connectivity statistics to failure statistics on the same
networks.
For dynamic studies we introduce the Influence Model,
a general probabilistic model to describe both the occurrence of initial failure events in networks and the influence
of these failures on other parts of the network. In the influence model each node has a status (for instance: normal,
or failed) that behaves as a Markov chain, but the transitions of each chain are influenced by the present status
of each neighboring node. The overall network can still be
described as a Markov chain, but with order equal to the
product of the orders of the individual node chains. We
have established that analysis of a greatly reduced model,
II. The Network Growth Model
In order to gain insight into the important features and
dynamics that affect cascading failure statistics, it is necessary to consider network models. If one is only interested in
a single, isolated network, one may model it and study it in
detail to learn much of its expected behavior. However, to
gain a more general understanding of typical behavior in a
class of systems, one must be able to generate many different models that will differ in detail but all of which display
similar characteristics. In some statistical sense, the generated study systems will all be similar to each other and
to the real class of systems of ultimate interest.
In this section we present a network growth model in
which new nodes and links between nodes are added sequentially using some connection criterion. By randomly
placing nodes in a space and applying a connection rule,
one is able to generate numerous random networks which
do differ in detail but are statistically similar. The statistic we find most useful for comparison is the vertex degrees
of nodes: the number of branches connected to individual
nodes.
Nodes and Branches in a Model Network
1
0.9
0.8
0.7
0.6
Y
of dimension equal to the sum of the individual node orders, suffices to characterize many dynamic and steadystate properties of interest. In this paper, we primarily focus on one special case of the influence model, known as the
Binary Influence Model. Each node has only two statuses
in the binary influence model, and further the evolution of
these statuses is solely determined by influences among the
nodes.
We assign influences to the branches in our structural
model and analyze the resulting cascading failure model.
We find that the underlying structure combined with certain influence characteristics between components lead to
a high frequency of large failure events. To highlight the
graph characteristics that lead to cascading failures, we
vary the structural characteristics of the model as well as
the influence model parameters and determine the resulting
failure sizes.
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
X
Fig. 2. An example of a network generated from the formation model
is shown. The nodes are uniformly and independently distributed
in the unit square, and a weighting function f (v) = v0.5 is used.
j that satisfies
j = arg
d(i, k)
,
i,1≤i≤k−1 f (Vik )
min
(1)
where Vik is the degree of node i in the network composed of
the first k − 1 nodes, and f (·) is a function that weights the
connection cost based on the degree of the node. We denote
as the cost function for connecting a
the quantity fd(i,k)
(Vik )
new node to an existing one and minimize this cost over
the set of existing nodes. Typical choices for this function
include f (v) = 1 and f (v) = v α with α positive. A typical
network generated with this model is shown (Figure 2). In
this example, the 1000 nodes are placed in the unit square
at random locations with uniform probability. Distance is
measured according to the 2-norm, and the connection cost
weighting function is f (v) = v 0.5 .
A. Description of the Model
B. Characteristics of the Model Networks
In our network structure model, the nodes in the network
form sequentially, and each newly formed node connects to
exactly one of the existing nodes based on a cost criterion.
Specifically, we consider the development of a network with
N nodes in total. Each node is given a location, defined
by an M dimensional vector. The location of node k is
denoted as xk . Component i of vector xk , labeled xki , is
assumed to be an independent random variable determined
from a probability density function pi (x). The distance between nodes k and l is labeled as d(k, l). In our simulations,
we assume that the distance between two nodes is given by
d(k, l) = xk − xl 2 , though other distance measures can
also be considered in this framework. To place the lines
in the network, nodes 1 and 2 are first connected with a
line. Next, the remaining N − 2 nodes are connected sequentially. Specifically, node k is connected with the node
In this section we highlight structural properties of our
model networks. For analytical simplicity in this preliminary work, the form of the model in which the location of
the nodes are given by a one-dimensional vector (M = 1)
is considered in our simulations.
The first observation about the model is that the generated networks are trees. Most real networks do contain
loops. However, some of these networks are nearly trees
with only a few additional lines, suggesting that tree models can perhaps be used to represent their behavior.
The distribution of vertex degrees in the generated network is a graph-theoretic measure for the structure of a
network. The distribution of vertex degrees is determined
by calculating the fraction of nodes in a network with a
certain degree and plotting these fractions as a function of
their degrees. We plot the distribution of vertex degrees
ture that the distribution of vertex degrees, or fraction of
nodes with certain vertex degrees, approaches a steadystate distribution for a sufficiently high number of nodes.
We attempt to describe this distribution.
First, for a graph with a sufficient number of nodes, the
fraction of nodes with a certain degree v is assumed to
approach a steady-state value λv . This assumption is reasonable and borne out by many trials. Second, to estimate these steady-state vertex degree densities, the effect
of adding a new node to a graph with N nodes is considered. When a new node is added, a node with degree 1 is
formed with certainty. In order to maintain the expected
fraction λ1 of nodes with degree 1, the new node must
connect to an existing node of degree 1 with probability
1 − λ1 . Similarly, a node with degree v will connect to the
new node with probability
Cumulative Distribution of Vertex Degrees
log10(Fraction of Vertices with Degree>=q)
0
−0.5
−1
−1.5
−2
−2.5
−3
−3.5
−4
−4.5
−5
0
5
10
15
Vertex Degree=q
Cv = 1 −
Cumulative Distribution of Vertex Degrees
−1
−1.5
−2
−2.5
−3
−3.5
−4
10
log (Fraction of Vertices with Degree>=q)
−0.5
−4.5
0
0.5
1
1.5
2
log (Vertex Degree=q)
2.5
λi
(2)
i=1
0
−5
v
3
10
Fig. 3. The distribution of vertex degrees is shown for f (v) = 1
and f (v) = v. In this case, the node locations are defined in one
dimension. The distributions are experimentally determined by
generating graphs with 50000 nodes.
for our model for f (v) = 1 and f (v) = v (Figure 3). These
distributions are determined by generating 50000-node networks with nodes uniformly distributed on a unit line. If
f (v) = 1 is used, the resulting distribution of vertex degrees is well-approximated by an exponential function. If
f (v) = v is used, the distribution of vertex degrees becomes
a power law with an exponent of approximately 2.5.
This connection criterion has a intuitive interpretation.
A new node is connected to a nearby node, and it benefits
from a connection to a node with high vertex degree. This
serves to lessen the effective distance to other nodes within
the network.
C. Analytical Estimation of the Distribution of Vertex Degrees
The simplicity of the weighted distance tree growth
model allows the possibility of some analysis. We conjec-
Third and finally, an arrival process model is used to
describe the connection of a new node to the existing tree,
and the two calculations for the effect of adding a new
node are combined to deduce a recursion for the λv . We
consider a new node placed uniformly on a unit line. If
the line is traversed in either direction from this new node
in search of nodes of a certain degree v, the distance to
these nodes can be modeled as an arrival process. Further,
the costs (ratio of distance to f (v)) of connecting to these
nodes also constitute an arrival process, labeled as the cost
process. The cost of connecting to each of the existing
nodes is the superposition of the arrival processes for the
different vertex degrees in the existing graph.
We claim that the location of the first arrival in the cost
process for a sufficiently large vertex degree approaches an
exponential distribution, and further that the location of
the first arrival in cost processes with degrees v and v + 1
are independent. To justify this claim, we first note that
the locations of the nodes for any given vertex degree are
uniform over the unit interval due to the symmetry of the
problem; however, the locations will be correlated. The
correlation occurs because the degree of a certain node is
affected by the distance of the node to other nearby nodes.
For example, two nodes of degree one are not likely to occur
within a very short distance because these two nodes would
then be connected. If nodes of high degree are sufficiently
sparse, however, the correlation between their locations will
be weak. Furthermore, high vertex degree implies that the
node is attached to many other nodes of varying degree,
so that correlation with a specific node of the same degree
should not be strong. Therefore, for a wide range of connection criteria, nodes with sufficiently high vertex degrees can
be treated as uniformly and independently distributed. Using similar reasoning, we claim that the locations of nodes
with degrees v and v + 1 are also approximately independent. Since the locations in each cost process are uniformly
and independently distributed for sufficiently high vertex
degrees, the distance to the first arrival in this process is
an exponential distribution.
Since the cost measure for the first arrival of a vertex of
degree v and a vertex of degree v + 1 are independent and
exponentially distributed, the ratio of the probabilities of
connecting to a vertex of degree v and a vertex of degree
v + 1 can be calculated. The probability ratio is the ratio
of the expected cost measure to the first vertex of degree
v + 1 and the expected cost measure to the first vertex of
degree v, or
1
λv+1 f (v+1)
1
λv f (v)
=
However, this ratio is equal to
λv f (v)
λv+1 f (v + 1)
Cv
Cv+1 ,
leading to the equation
(4)
The equation can then be solved for λv+1 in order to generate a recursion for the steady-state probabilities:
v
f (v)
(1 − i=1 λi )
λv f (v+1)
=
f (v)
1 − vi=1 λi + λv f (v+1)
(5)
For specific functions f (·), the limiting behavior of the
steady-state vertex degree fractions can be more exactly
analyzed. For example, for f (v) = 1, the limiting distribution of node degrees is an exponential function, and for
f (v) = v, the limiting distribution is a power law. To deduce the limiting behavior of the steady-state vertex degree
fractions, we first introduce a new variable, θv , defined by
θv =
1−
λ
vv
i=1
λi
.
(6)
Then (4) can be rewritten as
θv+1
θv
=
f (v)
.
f (v + 1)
(7)
In this paper we limit f (v) to take the following form:
f (v) = v α . With this specific representation, (7) becomes
α
v
θv+1
=
.
(8)
θv
v+1
From (8) it is immediately evident that the variable θv
follows a power law and can be described by
θv
=
kv −α
(9)
where k is a constant. Substituting this expression for θv ,
we find
λn
=
α
k Πv−1
r=1 r
.
v
Πr=1 (k + rα )
(10)
To consider the limiting behavior we examine the ratio
λv+1
vα
=
.
λv
k + (v + 1)α
vγ
(v + 1)γ
(11)
vα
k + (v + 1)α
≤
<
vα
.
(v + 1)α
(12)
After some algebra, the following relationship can be derived:
(3)
v
1 − i=1 λi
λv f (v)
=
v+1
λv+1 f (v + 1)
1 − i=1 λi
λv+1
1
= k+1
, which is a constant, so λv is an
For α = 0, λλv+1
v
exponential function with respect to v. For α > 0, we
:
consider the following upper and lower bounds on λλv+1
v
γ
≥ α + αk
v
vα
(13)
for large v. From this equation, we conclude
For α < 1, the required γ does not exist and the original
distribution is not bounded by a power law.
• For α = 1, γ = 1+k is the limiting exponent as v becomes
large.
• For α > 1, γ = α.
The distribution is necessarily
bounded asymptotically by a power law depending on α.
It is not surprising that power laws are observed. In
other growth models in which the probability of a connection to an existing node is weighted by the vertex degree,
power laws are observed. For example, it has been argued
that such weights are reasonable in models of the growth
of world wide web through the addition of new pages and
links; and empirical data shows power law relations [2],
[3], [4]. In [2], a model is developed in which a new node
is linked to several existing nodes based on a probability
distribution that favors nodes with higher vertex degrees.
Their model does not incorporate the distance criterion
imposed here, and consequently their limiting power law
distribution differs from ours. (Conceptually, their analysis could be altered to include a “distance” measure to
account for the increased likelihood that a new web page
will link to an existing page with similar content.) It is also
worth noting that the distribution we observe in (10) for
α = 1 reduces to the Beta distribution observed in many
disciplines. For example, see [5], for a distribution of this
form arising from a growth model used to describe the frequency of the occurrence of words in a text.
•
III. The Binary Influence Model
We now introduce the binary influence model, which
generates a Markov process that models propagation on
a graph. The binary influence model is a special case of
the general influence model introduced in [9]. A full analysis of the influence model and its applications can also be
found therein. Here we only cover the portions that are
necessary for our later sections.
A. Introduction
In the binary influence model, each node has a status
value that varies over time as it is ‘influenced’ by its neighbors. The status of each node at any given time step is
assumed to be 0 or 1, which may represent any two different statuses such as ‘on’ vs. ‘off’, ‘healthy’ vs. ‘sick’, or
‘normal’ vs. ‘failed’, for example. Interaction only occurs
between connected nodes in the model because each node
copies the status of a connected node or retains its own
status.
Because of the general way in which it is defined, the
binary influence model can potentially capture the qualitative behavior of a number of systems. In power systems, this model can be used as a simplified paradigm for
cascading blackouts. Here the graph would represent the
power grid, and each node would be a substation whose
status value roughly characterizes the level of its operations. To simulate cascading failure, we can start with a
graph in which every node is in ‘normal’ state and then initiate a node failure by turning the status at some node to
‘failed’. Cascading failure occurs when a failed node causes
its neighbors to fail, and those neighbors induce more failures and so on.
This same model could also be used to model an election
process with two candidates. Each node is now a person,
and the graph represents the social connections that influence a person’s opinion. The candidate whom he or she
favors constitutes the status value. At each time step, the
voter re-evaluates his or her choice, taking into account
the current opinion of the neighbors. Our model will be
general enough to take into account the various degrees to
which a voter believes in his or her own previous decision
and the degree to which a voter is influenced by his or her
neighbors.
all the edges are reversed. Thus, instead of having the sum
of outgoing edges equal to 1, the sum of edges pointing
into a site is 1. As we will see, this feature allows us to
treat each edge weight as the relative amount of influence
from the source node to the destination. An example of a
network matrix and its graph is shown in Example 1.
Example 1: This is an example of a 3-site network
graph. Notice that the sum of the edge weights into any
site is 1.


.2 .4 0
D = .7 0 1
.1 .6 0
B. Previous Results
At each time index k, site i has a status, denoted by si [k],
which can be either 0 or 1. The vector
The binary influence model that we are about to introduce can indeed be classified as a variation of a certain
particle system known in the mathematical literature as
the “voter model” [6], [7], [8]. Using the terminology of
existing subjects, our model can be called a discrete-time
voter model on general finite graphs. The only significant
difference between this model and the voter model cited
above is that our model takes place in discrete rather than
continuous time. Although this might seem like an unimportant difference, the discrete-time raises a question that
would not have been an issue otherwise, namely the issue
of periodicity. In a continuous-time model, the process is
guaranteed to reach a consensus, whereas in discrete-time
convergence is only guaranteed when the graph is aperiodic.
We emphasize that the influence model presented here is
only a simplified version of the general influence model in
[9], which would contain a more significant generalization
of the traditional voter model.
C. Model Description
Assume that we are given an n × n stochastic matrix (a
non-negative matrix with rows that sum to one) D = [dij ],
called the network matrix. Notice that if D were being used
to describe a Markov chain, it would have been called the
state-transition matrix, but we assign it a different name
because we are using it for a different purpose. Define the
network graph, denoted Γ(D ), as a graph on n nodes whose
edge from i to j exists if and only if dji > 0. Each node is
referred to as a site. Since it is defined from D as opposed
to D, the network graph is just like a Markov chain, except
0.2
0.7
1
2
0.4
1
0.6
0.1
3
Fig. 4. Example of a network graph Γ(D ).
s[k] = [ s1 [k] . . . sn [k] ]
is called the state vector or simply state of the graph at
time k.
The binary influence model refers to the following two
evolution equations. These are the calculation of the vector
r[k + 1] of probabilities, and the realization of s[k + 1]:
r[k + 1] = D s[k]
s[k + 1] = Bernoulli(r[k + 1])
(14)
(15)
In (14), r[k + 1] is the length-n vector whose ith entry
represents the probability that si [k + 1] = 1. In (15),
each status si [k] is randomly realized. The operation
Bernoulli(r[k + 1]) can be thought of as flipping n independent coins to realize the entries of s[k + 1], where the
probability of the ith coin turning up heads (status 1) is
ri [k + 1]. The initial state s[0] is assumed to be independently realized from some given distribution. Each ri [k +1]
is a valid probability because it ranges between 0 and 1. To
see this, we notice that ri [k + 1] ≥ 0 because both D and
s[k] are nonnegative, and ri [k + 1] ≤ 1 because
dij sj [k] ≤
dij = 1.
ri [k + 1] =
j
j
since each sj [k] ≤ 1.
The one-step dynamics of the binary influence model can
be easily understood once viewed on the network graph
Γ(D ). On this graph, an edge from i to j has a weight
of dji . This weight can be interpreted as the amount of
influence that i exerts on j relative to the total amount
of influence that j receives. The total amount of influence
received by any site is thus equal to the sum of incoming
edge weights, which is 1 as described earlier. Eq. (14)
shows that the probability of a site having status 1 at the
next stage is the sum of the influences from the neighbors
whose statuses are currently 1. This sum, therefore, always has a value ranging between 0 and 1. The following
example shows a particular run of the model.
Example 2: Figure 5 gives an example showing the first
few steps of a particular run based on the graph in Example
1. The top row of graphs shows the realized states s[k]. The
status of each site, which is either a 0 or a 1, is written in
the site. The bottom row of graphs shows the probabilities
r[k]. Inside each site is ri [k], the probability that that site
would have status 1 at time k.
Network states
s[0]
s[2]
s[1]
0.2
0.2
0.2
0.7
0
0.7
1
0.4
1
1
0
1
0.6
0.1
0
0.0
r[1]
0
0.3
1.0
0.6
0.1
1
0.7
1
0.4
1
0.6
0.1
0.7
0
0.4
1.0
0.0
Probabilities
r[2]
Fig. 5. Example of a particular path of the binary influence Process
in its first few steps.
If all of a site’s influencing neighbors have the same status – whether all 0’s or all 1’s – then that site will copy its
neighbors’ status with certainty in the next time step. If
a site has a self-loop then it is one of its own influencers.
This feature allows us to model situations where statuses
tend to persist, because the self-loop would influence the
site to repeat its previous status. In the power systems
context, the self-loop can be used to model failures that
are difficult to repair, for instance.
D. A Generalization of the Binary Influence Model
Unlike the binary influence model, the general influence
model allows the internal status of each node to change
spontaneously based on a Markov chain that is internal to
that node. Although we do not present this model, we
generalize in a different way the binary influence model to
allow some internal node dynamics.
The need for internal node dynamics is motivated by the
possibility that nodes can be healed or repaired internally
in reality. In a failure state, the binary influence model
only allows the healing of nodes through the influence of
other nodes. Here, we assume the the failed node i remains
failed between each binary influence model time step with
probability pi , independently of the other nodes and of the
binary influence model. We label pi as the failure retention probability. Again, the probability that each node fails
given the previous status of the nodes can be calculated. In
this generalization, the probability that a node fails is not
simply the sum of influence model weights of neighboring
failed nodes; instead, the influence from each neighboring
failed node (possibly including the node itself) must be
scaled by pi . Consequently, the probability vector for the
next step is given by
r[k + 1] = DP s[k],
(16)
where P is a diagonal matrix with entry i given by pi . The
new state s[k + 1] is again realized as
s[k + 1] = Bernoulli(r[k + 1])
(17)
One significant distinction between the binary influence
model and this model is the final status of the nodes after
a failure event. The nodes in the binary influence model
may converge to a state of all zeros or all ones, while the
node in this model will necessarily converge to a state of
all zeros if all pi are less than one.
IV. Simulation and Analysis of Failure Events
So far, we have formulated a plausible model for describing network structure and developed the influence model,
a probabilistic method for quantifying failures dynamics
in networks. Next, we simulate network failures using the
influence model, with the influences among nodes determined by the structural model defined in this article. Our
goal is the analysis of failure characteristics in large network models. Through the simulations and analysis, we
determine how structural and dynamic network characteristics affect network failure characteristics. Our analysis of
failure events in these networks is still incomplete, but preliminary simulation results and analysis of failure characteristics in these networks show several intriguing features.
A. Simulation of Failures in Network Models
To simulate the complete failure propagation model, we
first generate networks based on the developed structural
network model. We consider nodes located on a unit line
segment and use the functions f (v) = 1 and f (v) = v to
generate the graphs. Next, influences are assigned according to the branches in this network, and the influence model
is simulated. Specifically, the network matrix is chosen so
that the status of a node is equally influenced by its graph
neighbors and by itself. Mathematically, if node j has vertex degree v, row j of the network matrix D has entries
1
in the columns corresponding to all of the node’s
of v+1
neighbors and the node itself, and has entries of zero otherwise. All of the nodes in the graph are assumed to have a
failure retention probability p. We have simulated graphs
with p = 1 (the binary influence model), p = 0.75, p = 0.5,
and p = 0.25.
We measure two characteristics of failure events in our
simulations, the failure duration and the integrated failure
p=1
p=0.75
p=0.5
p=0.25
−0.5
−1
−1.5
−2
−2.5
−3
−3.5
We plot the cumulative distributions of failure durations
for each underlying structural graph model and self-healing
probability (Figure 6). For each structural graph model,
the distribution of failure durations is approximately exponential for p = 0.25, 0.5, and 0.75, while the distribution of
failure durations is approximately a power law for p = 1,
the binary influence model. Surprisingly, the distribution
of failure durations seems unrelated to the vertex degree
distribution of the underlying graph.
The cumulative failure duration distribution can be
bounded by an exponential function for p < 1. First, the
probability that each node is in a failed status at time k is
given by
(18)
−4
−4.5
0
50
100
150
Duration of Failure=q
Cumulative Distributions of Failure Durations, f(v)=v
0
log10(Fraction of Failures with Duration>=q)
B. Observations and Analysis of Failure Durations
E(s[k]) = (p)k Dk E(s[0])
Cumulative Distributions of Failure Durations, f(v)=1
0
log10(Fraction of Failures with Duration>=q)
size. The failure duration is the number of time steps in
the influence model after the initiation of the failure event
for which at least one node is failed. The integrated failure
size is the sum of the number of failed nodes in each time
step over the course of a failure event.
Failure duration and integrated size distributions were
developed for the two underlying structural graphs and the
four self-healing probabilities. For each simulation, 500
200-node graphs were generated, and between 8 and 32
failure events were simulated on each graph to determine
the duration and failure characteristics.
p=1
p=0.75
p=0.5
p=0.25
−0.5
−1
−1.5
−2
−2.5
−3
−3.5
Since each element in Dk E(s[0]) is less than one, the probability that each node is failed at time k is less than pk .
Although we have no information about the joint density
of the node statuses, the probability that at least one node
is failed is upper bounded by N pk , where N is the number
of nodes in the network. Therefore, the probability that
the failure duration is greater than or equal to k is upper
bounded by N pk . In fact, this bound is valid regardless of
the initial failure state of the system.
Our simulations suggest that this exponential bound becomes the dominant factor in determining the failure duration distribution if p is sufficiently small. If p equals 1 or is
near 1, the dynamics of the network matrix D and perhaps
of its underlying structural graph are relevant in determining the distribution of failure sizes. We have not analyzed
the failure duration distribution for p = 1, but our simulations suggest that this distribution will be heavy-tailed for a
variety of different graph structures. A heavy-tailed failure
duration distribution is plausible for the following reason:
if p = 1, a failed node is more likely to be repaired if it
has more neighbors. However, a node with more neighbors
is also more likely to induce failures in other nodes. Consequently, an isolated failure will persist for several time
steps, while a failure of a well-connected node will spread,
so that more time steps are needed to repair the failure.
Large failure durations will be frequent regardless of the
structure of the underlying graph.
−4
−4.5
0
50
100
150
Duration of Failure=q
Fig. 6. The normalized cumulative distribution of failure durations
is shown for f (v) = 1 and f (v) = v, for four different values of
p. The cumulative distribution corresponding to p = 1 has been
truncated in order to better display the other distributions.
C. Observations and Analysis of Failure Sizes
We also plot the cumulative distributions of integrated
failure size for each underlying structural graph and selfhealing probability (Figure 7). Unlike the failure duration distributions, the failure size distributions differ significantly for the two different network structure models
(corresponding to f (v)=1 and f (v)=v). For p = 1, the
integrated failure size distribution is heavy-tailed for both
structural models. However, for any other choice of p, the
distribution of integrated failure sizes is much more heavytailed for the network structure model with f (v) = v.
These simulations also suggest that the distribution of failures sizes with f (v) = v is well-approximated by a power
law; for f (v) = 1, the distribution of integrated failure sizes
is much less heavy-tailed than a power law, as shown in the
log-log plot of failure sizes. (For f (v) = v in Figure 7, it is
Cumulative Distributions of Integrated Failure Sizes, f(v)=1
p=1
p=0.75
p=0.5
p=0.25
−0.5
−1
−1.5
−2
−2.5
−3
−3.5
log
10
(Fraction of Failures with Size>=q)
0
−4
−4.5
0
0.5
1
1.5
log10(Size of Failure=q)
2
2.5
Cumulative Distributions of Integrated Failure Sizes, f(v)=v
p=1
p=0.75
p=0.5
p=0.25
−0.5
−1
−1.5
−2
−2.5
−3
−3.5
log
10
(Fraction of Failures with Size>=q)
0
−4
−4.5
0
0.5
1
1.5
log10(Size of Failure=q)
2
2.5
Fig. 7. The normalized cumulative distribution of integrated failure
sizes is shown for f (v) = 1 and f (v) = v, for four different values
of p. The cumulative distribution corresponding to p = 1 has
been truncated in order to better display the other distributions.
failure sizes will be heavy-tailed when the distribution of
vertex degrees is heavy-tailed. We assume that the distribution of vertex degrees is given by the probability mass
function pV (v) and corresponding cumulative distribution
FV (v), where FV (v) describes the probability that V is
greater than or equal to v. We also assume that the degree
of a vertex provides no information about the degrees of the
connected vertices. Our previous analysis of the distribution of vertex degrees for the developed model suggests that
the assumption of vertex degree independence is approximately true for vertices of sufficiently large degree. We additionally assume that failure events may be initiated with
equal probability at each node in the graph. Once the failure event has been initiated, the connections in the network
graph of the influence model determine the progression of
the failure. An analysis of the number of failed nodes after one step of the influence model is a simple (and rather
weak) lower bound on the cumulative size of failures. However, the one-step failure size is sufficient to demonstrate
that the failure size distribution will be heavy-tailed if the
vertex degree distribution is heavy-tailed. We consider an
initiating event at some node with vertex degree Vs . Given
that the original node has remained failed based on its internal behavior, each of the branches leaving the initiating
node independently causes the failure of another node in
∞ V (v)
. Therefore, the
one step with probability pf = v=1 pv+1
total number of newly failed nodes F1 after one iteration
of the influence model is represented by a binomial distribution with parameters Vs and pf , and mean and variance
given by E[F1 |Vs ] = pf Vs and var(F1 |Vs ) = Vs pf (1 − pf ).
Using Chebyshev’s inequality, the following inequality on
the probability mass function of F1 given Vs can be derived:
P (F1 ≥ Cpf Vs |Vs ) ≥ 1 −
(19)
where C is a positive constant less than one. For large
enough Vs , therefore, P (F1 ≥ Cpf Vs |Vs ) ≥ β, where β is
also between zero and one. Next, we can write
P (F1 ≥ f ) =
believed that the frequency of failure sizes eventually drops
off due to the finite size of the network.)
For p = 1, the heavy-tailed distribution of failure sizes
in both structural models is reasonable because the failure
duration distribution is heavy-tailed, so that large integrated failures will occur regardless of the number of failed
nodes at each time step. The difference in the failure size
distribution between the two structural models for p < 1
suggests that the structure of the underlying network becomes important when self-healing of the nodes is allowed.
In this case, we have shown that the duration of the failure
is bounded by an exponential function and so is unlikely to
be very large. However, if the distribution of vertex degrees
is heavy-tailed, then a significant number of nodes can be
affected in a few time steps before the failure event ends.
We have determined a lower bound on the failure size
distribution which demonstrates that the distribution of
1 − pf
,
(1 − C)2 pf Vs
∞
P (F1 ≥ f |Vs = v)pV (v).
(20)
v=1
For v such that Cpf v ≥ f , P (F1 ≥ f |Vs = v) ≥ β in
equation (20) leads to the inequality
pV (v).
(21)
P (F1 ≥ f ) ≥ β
f
v> Cp
Since Cpf is a constant,
f
v> Cp
f
f
pV (v) is a heavy-tailed
function whenever pV (v) is a heavy-tailed function, and
so P (F1 ≥ f ) must be a heavy-tailed distribution. We
have assumed so far that the original node remains failed
after the the first time-step, which actually occurs with
probability p in the generalization of the binary influence
model. By conditioning on the internal status of the node
after one time step, we can find an expression for P (F1 > f )
in the general case where nodes can internally be repaired.
In this case,
P (F1 ≥ f ) ≥ pβ
active Networks and Systems (EPRI project WO-8333-06,
ARP project DAAG55-98-1-3-001).
pV (v),
(22)
f
v> Cp
f
which also is heavy-tailed if pV (v) is heavy-tailed. Finally,
we note that P (F1 ≥ f ) is a lower bound for the cumulative
distribution of the total integrated failure size during an
influence model failure event.
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[1]
[2]
[3]
[4]
V. Conclusions
We have proposed a class of models for growing trees that
incorporates distance and prior connectivity in the design
process. The resulting distribution of vertex degrees for
nodes was examined analytically and empirically. Next, we
introduced the binary influence model, a general stochastic model that can be used to represent failure propagation
in networks. Finally, we have simulated failure propagation on the developed structural graphs using the influence
model and have explored characteristics of the resulting
failure events.
In our studies we found that the distribution of failure
duration depends on the dynamic characteristics of the influence model and is largely unaffected by the structure
of the underlying graph. Furthermore, with any effect of
self healing present in the model, the distribution of failure
duration is limited by an exponential function.
Conversely, the distribution of integrated failure sizes depends strongly on the structure of the network and the dynamics of the influence model. For networks characterized
by a heavy-tailed distribution of vertex degrees, a heavytailed distribution of integrated failure sizes is observed.
The details of the distribution depend upon the self-healing
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We are encouraged that in this exploration we have been
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ways; most obviously in the structure of the resulting networks. Physically-based networks are not typically trees.
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unrelated to the virtual network.) More work needs to be
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practice, and then the type of study presented here may
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outages on such systems.
VI. Acknowledgments
The authors gratefully acknowledge EPRI and DoD for
support of this work under an initiative for Complex Inter-
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