Network reliability analysis based on percolation theory
- a School of Reliability and Systems Engineering, Beihang University, Beijing, China
- b Chair on Systems Science and the Energetic Challenge, European Foundation for New Energy-Electricite׳ de France, Ecole Centrale Paris and Supelec, France
- c Politecnico di Milano, Milano, Italy
- d Department of Physics, Bar Ilan University, Ramat-Gan, Israel
- Received 20 February 2013, Revised 31 March 2015, Accepted 25 May 2015, Available online 17 June 2015
Highlights
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Based on percolation theory, we address questions of practical interest such as “how many failed nodes/edges will break down the whole network?”
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The percolation threshold naturally gives a network failure criterion.
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The approach based on percolation theory is suited for calculations of large-scale networks.
Abstract
In this paper, we propose a new way of looking at the reliability of a network using percolation theory. In this new view, a network failure can be regarded as a percolation process and the critical threshold of percolation can be used as network failure criterion linked to the operational settings under control. To demonstrate our approach, we consider both random network models and real networks with different nodes and/or edges lifetime distributions. We study numerically and theoretically the network reliability and find that the network reliability can be solved as a voting system with threshold given by percolation theory. Then we find that the average lifetime of random network increases linearly with the average lifetime of its nodes with uniform life distributions. Furthermore, the average lifetime of the network becomes saturated when system size is increased. Finally, we demonstrate our method on the transmission network system of IEEE 14 bus.
Keywords
- Network reliability;
- Percolation theory;
- Phase transition;
- Criticality;
- Random network
Notation
- V
a set of vertices
- E
a set of arcs
- Γ (V, E)
a network defined as an undirected graph with V, E
- N
the total number of nodes in a network

the binomial coefficient
- <a>
the average value of the random variable a
- p
the probability that a node/edge is functional
- pc
the percolation threshold
- Ts
the average lifetime of the network
- a⁎b
product of a and b
- [a]
the largest integer less than or equal to a
1. Introduction
In modern society, technological networks are pervasive as they provide essential services including materials [1] and [2], energy [3] and [4], information [5] and even social communication [6]. It is not surprising, then, that network reliability is receiving particular attention, on one side as a value requested by the users and on the other side as a challenge for the service providers and network operators. One way to address the problem is to consider the structure connectivity of the network as a graph Γ (V, E) consisting of a vertex set V={v1, v2,…, vn} and an arc set E={e1, e2,…, em}. Within this abstraction, terminal reliability can be defined as the probability of achieving connectivity from the source nodes to the terminal nodes [7]. The terminal reliability of networks can be characterized by assessment methodologies [8] such as Reliability Block Diagram (RBD), Fault Tree Analysis (FTA) [9] and so on. Typical algorithms for computing terminal reliability include the state enumeration method [10], sum of disjoint products method [11], factorization method [12], minimal cuts method [13] and cellular automata [14] and [15].
However, in the consideration of terminal connectivity, the identification of the operational limits of a network is missing [8], where a critical fraction of functional components to sustain the network is considered instead of studying paths in the terminal reliability. Percolation theory [16] and [17] provides us with an opportunity to overcome this gap, by referring network failure to the situation whereby a critical fraction of network components have failed [18], [19] and [20]. In the percolation theory, the failure of a node/edge of network is modeled by removal. As the removal of nodes/edges increases, the network undergoes a transition from the phase of connectivity (functional network) to the phase of dis-connectivity (nonfunctional network). The probability threshold signifying this phase transition can be found theoretically or computed numerically by percolation theory. The probability threshold can be used as a statistical indicator for the operational limits of the network, which is not considered in traditional terminal reliability analysis. Thus, percolation theory, based on statistical physics, can help to understand the macroscopic failure behavior of networks in relation to the microscopic states of the network components. It can address questions of practical interest such as “how many failed nodes/edges will break down the whole network?”
In this paper, we define “network reliability” by using concepts of percolation theory and exploit the related statistical physics techniques to calculate it. We analyze the network failure process and network reliability properties by percolation theory, providing a new framework for network reliability analysis. In Section 2, we further explain the operational limits of a network. In Section 3, we relate the network reliability problem to percolation theory. In Section 4, we analyze theoretically the network reliability and lifetime distribution, referring to random networks. In Section 5, we present simulation results, which are extended to real networks in Section 6.
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