explain why such behavior is observed, nor does it help identify which features of the network structure and dy- namics most influence it, or suggest how to apply ...
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
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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.
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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
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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)
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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 =
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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
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To consider the limiting behavior we examine the ratio λv+1 vα = . λv k + (v + 1)α
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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
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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]
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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
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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)
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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
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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.
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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.
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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
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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.
References [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 property of the influence model. We are encouraged that in this exploration we have been able to correlate some observed statistics to properties of the underlying network structure and the model dynamics. This preliminary study is, of course, lacking in many ways; most obviously in the structure of the resulting networks. Physically-based networks are not typically trees. Furthermore, it is not clear that a connection criterion that rewards connection to a highly connected node is reasonable. In many physical applications, the opposite is true, for example, to prevent congestion. In the case of the World Wide Web, in which such connections are plausible, it is questionable to consider cascading outages on that virtual network. (The underlying physical network of the Internet is a different matter; however, its structure is largely unrelated to the virtual network.) More work needs to be done to accurately model growing networks that occur in practice, and then the type of study presented here may help identify characteristics that contribute to cascading outages on such systems. VI. Acknowledgments The authors gratefully acknowledge EPRI and DoD for support of this work under an initiative for Complex Inter-
[5] [6] [7] [8] [9]
North American Reliability Council, Disturbance Analysis Working group, www.nerc.com/dawg. Barabasi, A.-L., R. Albert, H. Jeong, “Mean-field theory for scale-free random networks,” Physics A, 272:173-187, 1999. M. Faloutsos, P. Faloutsos, and C. Faloutsos. On Power-Law Relationships of the Internet Topology. SIGCOMM 1999 B. A. Huberman and L. A. Adamic. Evolutionary Dynamics of the World Wide Web. http://www.parc.xerox.com/istl/groups/iea/www/growth.html, unpublished. Simon, H.A., Models of Man: Social and Rational, Chapter 9, On a Class of Skew Distribution Functions, John Wiley and Sons, New York, 1957. R. A. Holley and T. M. Liggett. Ergodic theorems for weakly interacting infinite systems and the voter models. Annals of Probability, 3:643–663, 1975. R. Durrett. An introduction to infinite particle systems. Stochastic Processes and Their Applications, 109–150, 1981. T. M. Liggett. Interacting Particle Systems. Springer-Verlag, 1985. C. Asavathiratham. The Influence Model: A Tractable Representation for the Dynamics of Networked Markov Chains Ph.D. thesis, Elec. Eng. and Comp. Sci. Dept., MIT, 2000.
