The robustness is one of the structural properties of a complex system which measures its ability of continuing perform well, subject to failures or attacks. It is needed to quantify the measure of robustness in order to decide whether or not, a given system is robust. The most common measure of robustness of a network to random failures of components is all-terminal reliability polynomial, the probability that there is an operating communications link between any two components in the system. The reliability of a system of interacting agents, can be determined by analyzing the reliability of the underlying graph. A graph is a pair of two setswhere is a set of nodes/vertices and is a set of edges/links that connect two elements of . The elements of a real system and the interactions patterns between them are represented by nodes and edges in a graph, respectability. Suppose we have a graph for which the nodes are always operational but for which each edge is independently operational with probability . The (all terminal) reliability of , denoted by is therefore defined to be the probability that the graph is connected when each edge is (independently of the others) present with probability . In other words the probability that the operational edges form a spanning connected subgraph of . Let set of operational edges be the state of network, i.e., the network is in state when all edges of are operational and all edges are failing. Let be the set of all operational states. Then is equal to
By knowing all states, the reliability is easily but not efficiently computed. The most amenable case is that in which all of the edge probabilities are identical, say . Therefore under the condition that for all we have simpler version of formula(1.1), which is called F-form of the reliability polynomial, as follows
where is the number of connected spanning subgraphs of size . There are different forms of presentation for the reliability polynomial . Also there are methods to examine the possible states of the network, for example the factorization method . However, computing the reliability polynomial, is not easy in general. Even the time required to calculate for an arbitrary connected graph grows exponentially with the size of the network, ,.
Because of this difficulty, network designers have often relied upon upper and lower bounds on , ,,,,,,. It would be of interest if one can obtain exact solutions for the reliability polynomial for some families of graphs . In , the authors built a framework based on percolation theory to calculate the network reliability. They considered homogeneous random graph models with Poisson degree distribution. But in the current work, the method which we used to assess the reliability polynomial can be apply for more general random graph with arbitrary degree distribution. More precisely, in this paper, we use the percolation concept as “measuring” tool to assess the reliability polynomial of systems which can be modeled as a general inhomogeneous random graph and scale-free random graph.
The rest of the paper is organized as follows. In Section 2 we describe the inverse percolation process. In Section 3 we explain how to obtain the threshold at which the network loses its connectivity. Section 4 presents the assessment of network reliability. Section 5 and 6, will illustrate our method for two special case of real network which can be modeled as inhomogeneous and scale free random graph, respectively.
2. Inverse percolation process
Random graphs have been used extensively as models for various types of real world networks. It provides techniques to analyze structure in a system of interacting agents, represented as a network. In mathematics, networks are often referred to as graphs. A random graph is a graph that is sampled according to some probability distribution over a collection of graphs.Random graphs are used to prove deterministic properties of the graphs. The most basic property of a graph is that of being connected. It has been shown that, the limiting probability that a random graph possesses connectedness, jumps from to (or vice versa) very rapidly. There exists a threshold when a transition occurs from not being connected to being connected. For example for the binomial random graph , where is the number of vertices of the graph and is the probability that the edge
is present, we say that a phase transition occurs if there exists a functionsuch that for , almost surely does not have the property of being connected, but for , almost surely has the property of being connected. Here is the threshold. The most important transitions is the emergence of a giant component, a connected component of size where is the size of , (, if there exist constants and , such that, , for ). The macroscopic behavior of networks, when faced with random removal of nodes or edges, is characterized in terms of an inverse percolation process in a random graph. Percolation theory characterizes this property of random graphs, especially infinitely large ones. It studies the behavior of the operational giant component, ensuring the global well functioning of the network. Once the random removal of nodes or edges is done (random failure process), the size of giant component, is expected to decrease. Using percolation theory, one can predicts the presence of a threshold, above which the network percolates, i.e., it has a giant component and below it the giant component disappears, i.e. the network is fragmented in many disconnected components of very small sizes. For a highly-connected graph sequence, the giant component is unique when the limit superior of the size of second-largest component by is for large , .
3. The percolation threshold
Let be the probability that edge is operational and be the probability that the node is operational. By being operational we mean that node/edge hasn’t failed or been removed from the graph. In particular cases, when in turn, or are uniform, we will specify them as node () or edge () operational probability. Let the degree distribution of our graph be , i.e., a randomly chosen node has degree with probability . We choose an edge at random and follow it to one of the nodes at its ends, then the number of edges incident on that node, other than the one we arrived along, follows the following distribution,
where , is the average degree for the entire graph.
Let be the edge between node and node , and be the probability that edge does not lead to a vertex connected via the remaining edges to the giant component. Then we have the following recursive expression,
where product is taking over all which are not equal to and are a neighbor of . Now, the average overall probability that a randomly chosen edge does not belong to the giant component is,
The equation has the trivial solution . Moreover if then has a unique root in . This condition implies that
The existence of a giant component has a threshold at , (the bond percolation threshold of the graph). Note that the lower bound for in (3.4) is meaningful, if , (the Molloy-Reed criterion for existence of giant connected components,).
4. The network reliability
Now we map the random failures process of a network into an inverse percolation problem. The inverse percolation problem consists of finding the critical fraction of the edges for which the giant component disappears. Suppose denotes the fraction of randomly removed edges.
The deletion of a fraction of edges corresponds to a random graph in which the
edges are occupied with probability . For small , the infinite cluster which is identified as the giant component is present. The threshold for the destruction of the giant component, ,
can be thus computed from the percolation threshold at which the infinite cluster
first emerges. In this case the phase transition corresponds to the separation of a
region of damages which still allow a connected network of appreciable size from
a region in which the system is totally fragmented, .
Let be the probability that edge is operational at time , (the reliability of edge at time ). Also we define , to be the probability that at time the node is operational. According to the previous section, if is less than a critical point , the giant cluster does not exist anymore. Define the instant at which this occurs as the lifetime of the network.
It is known that , the binomial (uniform) random graph loses its connectivity when the number of failed edges reaches , where is the number of remaining edges at critical point. Since is a threshold for a monotone property of being connected, we have ,.
Now using threshold , an assessment for the network reliability at time , denoted by is given by the following equation:
where and stand for the family of all -element subsets of . In case when, all of ’s are equal to , we have
In , authors have defined the probability that a node/edge is functional at a given time by . Then by the voting system based on the cite percolation critical value, they presented the network reliability at time , by the following equation:
where is the reliability of the generic node. In fact in their model, they assumed the same reliability for all nodes. They also gave an example in which different edges have different failure probability and then they compared the numerical results with the result from equation (4.3). It is clear that when reliability of nodes (i.e., ) are not same, the network reliability cannot be obtained from equation (4.3).
In the following example we will see is an -approximation for the reliability polynomial . In fact difference between the exact and approximate values for the reliability polynomial is zero.
The reliability of complete graph where for all edges in , at given time is
as any sub-graph on at least edges is operational and sub-graphs with edges that are operational are those that are a spanning tree of . There are no operational states with or fewer edges. On the other hand, for complete graph , we have . Therefore . Hence we get same value for the reliability of from equations (4.2) and (4.1).
The lifetime distribution of the network based on is
and the network lifetime, , is given by
Still, the computing of in (4.1) by enumerating all elements in for large , is not practicable. Thus we need to use approximation methods for computing
. One can think of Poisson approximation method, which is used for Poisson binomial distribution( the distribution of the sum of independent and non-identical random indicators). Fix. Let . Then by Poisson approximation method,
By Le Cam’s theorem , the approximation error for the Poisson approximation method is
In the following section, we look at an application of these results to some specific examples. We consider the system which can be modeled as
Inhomogeneous binomial random graph, and
Scale-free random graph.
5. Inhomogeneous binomial random graph
If edges in the binomial random graph, have different probability of being occupied, the resulting graph is called the inhomogeneous binomial random graph and denoted by , where , . The binomial random graph is retrieved when taking for all . For this random graph, the degree distribution will be asymptotically Poisson. Hence,
Suppose we have a graph on
nodes and the vector of edge probabilities at a given timeas follows:
In Figure 1, graph is presented.
For this graph the value of threshold is . The plot of verses is shown in Figure 2.
Note that drops abruptly at . The corresponding for this threshold present the life time of network (between 4 and 5). In Figure 3, the evolution of the network is studied.
6. Scale-free random graph
In the scale-free networks majority of the nodes have few neighbors but a small portion of them have a large number of neighbors. Many of them possess a power-law degree distribution, so that the probability that a node has neighbors scales with as , where is the power-law exponent . Topologies of many real-world networks are thought to be scale-free, such as World Wide Web, email networks, networks of Internet routers, and protein-protein interaction networks. Here we consider the degree distribution as follows:
where and are constant and is the polylogarithm function:
Figure 4 shows a power-law graph with exponent .
For power-law degree distribution, the giant component threshold is,
The divergence of the second momentis generally a sufficient condition to ensure the absence of a percolation threshold on a scale-free graph.
It is known that for a phase transition exists at a finite . Whereas for the transition takes place only at , .
Suppose we have a graph in which its degree distribution is as follows:
where is the Riemann zeta function. Therefore from (6.2) we have
Note that For many scale-free networks the degree exponent is between and , . The behavior of as a function of is depicted in Figure 5. As the figure shows, for and hence the transition never takes place (unless the network is finite). Also is between zero and one, only in the small range of the exponent .
As an example of real world network, we consider the internet. The degree distribution of the internet nodes follows a power law :
where is normalization constant and and are the minimal and maximal degrees, respectively, . For this model we have
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