I Introduction
Network topologies in wireless environments are generally dynamic in nature, as the connectivity between nodes is determined according to their timevarying link signaltonoise ratio (SNR) value. Channel impairments such as fading and path loss make it essential to monitor the quality of each link.
Based on the corresponding SNR value, the outage status can be determined and used as a performance indicator for each link. In case of an outage (where the SNR value falls below a certain threshold), two nodes are deemed to be disconnected; otherwise, they remain connected. Outage probability is a convenient measure of communication system performance[1]. Here we investigate network outage, i.e., the outage probability of communication between a source node and a terminal node over a network of relay nodes. Assuming links are in outage independently, network outage can be measured by using individual link outage probabilities. The behavior of the outage probability in the high SNR regime also gives an intuitive understanding of performance limits of the network [2]. In [3], high SNR error performance of any communication network (coded or uncoded) is represented by diversity and coding gains. Diversity gain is a measure of the number of independent copies of the transmitted signal captured by the receiver [4], while coding gain represents the difference in the outage probability curve relative to a benchmark performance in the high SNR region [3].
In this paper, we show that the diversity gain and the coding gain between a source node and a terminal node can be determined through the network outage polynomial. This approach dates back at least to Shannon and Moore [5], who provide a reliability analysis of relayaided systems by considering the unreliability probabilities of relay nodes. It is proven that the endtoend reliability of a given network can be increased through these unreliable relay nodes. When a sufficient number of relay nodes is used, the probability of network unreliability approaches zero [5]. However, transforming a complex network into equivalent seriesparallel projection may not always be possible. When the seriesparallel representation of a given network is not available, the reliability analysis of generalized networks becomes more difficult. There are various methods proposed to calculate network reliability, such as state enumeration, factorizing, path enumeration, and cutset enumeration [6, 7, 8, 9].
Although network reliability is a wellstudied subject, its extension to wireless networks it is still relatively unexplored [10, 11, 12, 13, 14]. As the popularity of wireless communication systems increases when compared to their wired counterparts in many different areas, the reliability analysis of wireless communications becomes more important, yet challenging, as wireless links are more prone to errors and erasures. Firstly, an unrealistic deterministic channel model is used when investigating the interference effect of the wireless channels [10]. The reliability analysis of wireless multihop networks is conducted regarding shadowing effect of the wireless channel in [11, 12]. Both [11, 12] do not consider the correlation effect of shadowing and this gap is filled by [13]. The reliability analysis of wireless multihop networks, which proposes a mathematical model to represent the network reliability of correlated shadowing wireless channel, is given in [14].
In [8], pathenumeration and cutset enumeration methods are used to calculate network reliability of generalized schemes. An algorithm based on a pathenumeration method is presented in [15] to determine the reliability of telecommunication networks from the capacity of the networks by considering different link capacities. In [16], a pathbased algorithm with a reduced computational time is modeled to obtain network reliability of wired communications. Instead of considering all cutsets of the network, some cutsets that have as many elements as the size of the size of the minimum cut are used to obtain an approximate network reliability expression with reduced complexity [17]. Hence, a lower bound for network reliability is attained by providing a more practical algorithm.
The network outage polynomial gives the probability that the network has zero instantaneous capacity. The investigation of network capacity is an attractive problem since the maximum capacity of any network is restricted by the size of the minimum cut of the graph. Hence, the ergodic capacity of any network can be calculated by using zeroto capacity polynomials, where the th polynomial gives the probability that the network has instantaneous capacity . In the literature, there are some works about the calculation of the capacity polynomials that determine the value of the maximum flow of arbitrary networks with random capacity edges by utilizing subset decomposition method [18, 19]. In [18], a subspace decomposition principle is used to determine the value of the maximum flow of arbitrary networks with random capacity edges. The value of maximum flow analysis of arbitrary networks with random edge capacities is conducted in [19], based upon Bernoulli statistics.
The aforementioned works have focused on obtaining only network reliability expressions. On the other hand, these works do not introduce any fundamental performance analysis. In this work, the essential goal is to obtain performance limits of an arbitrary network topology comprised of links that are prone to errors and erasures. The main contributions of this work can be listed as follows:

We establish a framework to calculate the network outage polynomial, as a tool to obtain network outage performance of communication networks.

We determine the network outage polynomial of some simple directed networks, in both correlated channels and uncorrelated channels. Three methods, namely the pathenumeration method, the cutset enumeration method, and the terminal reliability based method are proposed.

We extract the diversity order and the coding gain of a wireless network for arbitrary topology based on its graph properties.

We establish a relationship between the maxflow mincut theorem of graph theory and the diversity gain definition and show that the diversity order corresponds to the size of the minimum cut of the wireless network graph. We also prove that the coding gain is equal to the number of cutsets which have the size of the minimum cut, and also be easily determined from the network graph.

We provide the ergodic capacity analysis of networks in terms of individual link outage probability. Hence, an upper bound for the achievable transmission rate is determined.
Using this analysis, optimization of resource utilization can be realized thanks to the information about the diversity order and the ergodic capacity of any topology in wireless networks. For example, efficient multiple access schemes can be obtained by considering user demands and network limitations (the diversity order and the ergodic capacity).
The rest of the paper is organized as follows. Firstly, in Section II, methods for the calculation of outage polynomials of wireless networks are given. In Section III, diversity order analysis and ergodic capacity derivations are presented. In Section IV, to demonstrate the validity of theoretical results, numerical results are presented. Finally, the paper concludes with a summary of the findings and suggestions for future work in Section V.
Ii Outage Polynomials of Wireless Networks
Graph representations of communication systems are frequently used to analyze system performance; hence, key graph theory concepts can often be matched with the elements of communication systems. In literature pertaining to wired networks, the link outage probability is generally ignored since links are generally highly reliable. Thus, for wired networks, the connections between nodes can be represented by deterministic edges. The links in wireless channels, on the other hand, are subject to random SNR values, and so the connections between nodes must be modeled probabilistically.
We model a communications network as a directed acyclic network compromising of a finite vertex set of communication nodes, a multiset of directed edges representing communication links between nodes, a designated source vertex and a designated terminal vertex where . An edge from vertex to vertex is denoted as .
A directed path in from to is a sequence of edges with and . We suppose that there are distinct paths in from to . Nodes and are said to be connected if .
A subset of edges whose removal from the network disconnects and is called an separating cut, or simply a cutset. We suppose that there are distinct cutsets ; the collection of all cutsets is denoted as .
A cutset is called minimal if no proper subset of is itself a cutset. The collection of all minimal cutsets is denoted as . A cutset is called a minimum cutset if it is a cutset of minimum possible size, i.e., having the least number of edges among all cutsets. The collection of all minimum cutsets is denoted as , and the size of any minimum cutset is denoted as . Although each minimum cutset is certainly a minimal cutset, the converse is not true in general, thus .
Network outage is a convenient measure of a communication system’s performance, as the overall system performance can be obtained using individual outage probabilities of the links in the system. To enable communication between a source node and a terminal node , there must be at least one path from to . Hence, we can obtain an overall performance result by considering individual link outages. The network outage polynomial concept, which has been proposed for switching networks [5, 20], is also suitable as performance observation tool for wireless communication. Network outage is random due to individual link outages. In order to obtain the network outage polynomial for an arbitrary topology, we use three different methods: path enumeration, cutset enumeration, and reliability polynomial calculation. The required method can be selected to realize the target aim, as detailed below.
In the following, we consider the network at a given time instant, and denote by the probability that link is in outage at that instant. For example, if the wireless channel gain has a Rayleigh distribution (a frequent assumption in the wireless communication literature), then the outage probability of is equal to
where represents the average SNR of the link [4].
Link outages induce a random subgraph of , called the residual network, with edges that are in outage removed. In the residual network, it may happen that and are not connected. The network outage polynomial, which gives the probability that no path exists between and in the residual network, is then formally a polynomial function of , denoted as .
Throughout this paper, for any positive integer , we will denote the set as .
Iia Network Outage Polynomial Calculation Based on Path Enumeration
Firstly, we investigate the path enumeration method to obtain the network outage polynomial. We suppose that the edges comprising a path in from to are indexed by the set , i.e., , .
Let denote the event that path is available, i.e., that none of its links are in outage. The outage probability of the network is then given by
By the principle of inclusionexclusion [21], we have
(1) 
Assuming that individual links are in outage (or not) independently, we have
(2) 
IiB Network Outage Polynomial Calculation Based on CutSet Enumeration
The network outage polynomial of an arbitrary network can also be calculated by enumerating cutsets of the network, which is dual to the process of path enumeration. If the edges of any cutset are all in outage, the network is in outage.
We suppose that the edges comprising a cutset are indexed by the set , i.e., , .
Let denote the event that cutset is active, i.e., that all of its links are in outage. The outage probability of the network is then given by
Again by the principle of inclusionexclusion we have
(3) 
where
(4) 
IiC Network Outage Polynomial Calculation Based on TwoTerminal Polynomial
Finally, we derive the network outage polynomial expressions of a network based on the reliability polynomial concept [20], which is a useful function to reflect the performance of a network.
Consider, for any cutset , , the event that all the edges of are in outage while all other edges of the network are not in outage. Since is disjoint from when , we have
Again assuming that individual links are in outage (or not) independently, we have
In the special case where for all , we have
Writing for the outage polynomial in this case, we get
(5) 
where
(6) 
and where the coefficient of enumerates the number of cutsets of size .
It can be deduced from the minimum cutset definition that is equal to the number of distinct minimum cutsets and . In addition, is equal to 1. The outage polynomial can be also expressed in terms of the reliability polynomial associated with the  connectedness problem, [20, Sec. 1.2].
IiD Bounds on the Outage Polynomial
We may write some simple bounds on the outage polynomial as follows.
Firstly, if we use the inequality of in (5), we get
(7) 
To derive another upper bound expression, we can use the fact that every cutset must contain a minimal cutset. Since the probability that edges of a cutset are in outage is , we get that
(8) 
We also have the lower bound
(9) 
which is obtained by retaining just the first term in the expansion .
IiE Presence of Correlated Channels
In the previous subsections, we have assume that the state of each link is independent of the others. This assumption may be unrealistic in many situations (e.g., multiantenna systems) because of spatial correlation. The correlated channel case needs to be considered to determine the limitations of the wireless networks.
We adopt a simple correlation model, as follows. Firstly, the set of links is partitioned into disjoint nonempty subsets, , , …, , so that
To subset is associated a Bernoulli (
valued) random variable
, with . If , the link states (in outage or not) for all links in are chosen to be equal, while if , the link states for the links in are chosen independently at random. Suppose that has size , and let be any subset of of size , where . Then the probability that the links of are in outage while the links of are not in outage is given as(10) 
We assume that the random variables are mutually independent. Note that the previously considered case (of independent linkstates) is obtained by considering , or, equivalently, by partitioning into singleton sets where for all .
Now, given any subset of edges (e.g., a cutset), the probability that all edges of are in outage while all edges in are not in outage is given by
Thus the network outage polynomial is obtained as
(11) 
Iii Diversity Order and Ergodic Capacity Analyses for Arbitrary Network Topologies
In this section, performance limitations of an arbitrary network are determined via the outage polynomial. Firstly, expressions for diversity gain and coding gain are derived. Secondly, the ergodic capacity is considered.
Iiia Diversity Order Analysis
In order to provide further insight into the obtained outage probability expression, an asymptotic expression of outage probability is derived. The network is in outage if there is no defined path between a source and terminal nodes. Coding and diversity gains can represent the network outage probability in the limit as , referred to as the high SNR regime. The high SNR performance of any system determines the performance limits of a wireless network. In the high SNR regime, the outage probability expression of an arbitrary given network is given as
where , the diversity gain, measures the number of independent copies of the transmitted signal that are received at the terminal node, and where , the coding gain (usually expressed on a decibel scale), is a measure of the performance difference between the given system and a baseline system having [24].
For the purposes of the following theorem, we say that two functions and are asymptotically equal, written , if
Theorem 1.
In a network with outage polynomial ,
Thus the diversity order of such a network is equal to the size of a minimum cutset, i.e., , and the coding gain is equal to the number of distinct minimum cutsets, i.e., .
Proof.
We have
(12) 
∎
The value of maximum flow (the size of the minimum cut) can be calculated by enumerating the number of cutsets in a dual manner for unit capacity graphs. For dense network graphs, the FordFulkerson algorithm can be used to determine the size of the minimum cut value [25].
It is obvious that adding new edges to a network cannot reduce the size of any cutsets. If newly added edges (e.g., a lineofsight edge) provide a new edgedisjoint path from to , then the cardinality of all cutsets, and hence the diversity order of the network, increases by one.
IiiB Ergodic Network Capacity
Suppose now that each network link (when not in outage) provides unit transmission capacity. It is well known, e.g., [26], that the instantaneous  unicast capacity is equal to the size of the minimum separating cut in the network subgraph induced by the links that are not in outage; this transmission rate can be achieved by routing information along edgedisjoint paths between and (which, by Menger’s Theorem, exist in sufficient number). As the linkstate is random, the instantaneous capacity is a random variable. Indeed, the outage polynomial gives the probability that . It is also clear that is bounded by , the minimum cutset size. As takes integer values in a bounded set, it has a welldefined expected value, called the ergodic network capacity.
For , the event arises when all minimal cutsets contain at least links not in outage, and at least one of these cutsets contains exactly links not in outage. In other words, arises when the minimum number of nonoutage links among minimal cutsets is equal to . More precisely, let denote the set of edges not in outage at a given time instant. For any minimal cut , let
(13) 
be the function that indicates whether contains at least edges not in outage. The event then arises if
(14)  
(15) 
For every , the probability that is given by some polynomial . The ergodic capacity can then be obtained, in terms of , as
(16) 
When the minimal cut sets are disjoint, the th capacity polynomial can be calculated as follows. For any minimal cut of size , let denote the probability that contains at least links not in outage; thus
The probability that every minimal cut contains or more links not in outage is then given as . The probability that is then given as the probability that every minimal cut contains or more links in nonoutage but not every minimal cut contains more links in nonoutage, namely
The computational complexity of the ergodic network capacity is made up of the enumeration of all minimal cutsets, (13), (14), and (15). The enumeration of all minimal cutsets has a complexity of as given in [27]. The total complexity of the functions defined in (13), (14), and (15) is equal to , where . Hence the total computational complexity of the ergodic network capacity expression is equal to . Note that and increase exponentially with the size of [22].
Iv Numerical Results
In this section, we present numerical results to clarify the theoretical expressions on the network performance. We provide two instructive examples to clarify theoretical expressions derived in the previous sections.
Firstly, consider the example network presented in Fig. 1(fig:N1) with edges as labeled. In this network, there are edges with the size of the minimum cut . The cutsets, minimal cutsets, and minimum cutsets are
We have , thus, the outage polynomial for is calculated as
The bound expressions are also given by
Fig. 2 shows that the given upper bounds become tight when . As , all bounds have the same outage performance with the exact expression. In addition, the first order approximation of given as has a close performance with along with .
Based on the given capacity assurance sets, capacity polynomials of the network can be calculated as:
By using (16), the ergodic capacity of can be found as:
The obtained capacity polynomials of are presented in Fig. 3 (fig:capacity_3edges_a). While , is highly probable when compared to for . On the other hand, in the case of . It can be deduced from Fig. 3 (fig:capacity_3edges_b), the average capacity of the network increases while is decreasing. In addition, the maximum value of the average capacity of the network is equal to for .
We give another example to illustrate the correlated case results. The depicted extended graph of Fig. 1 (fig:N1) with 4 edges labeled as is given in Fig. 1 (fig:N2).
The network has the following sets:
In the uncorrelated case, the outage polynomial can be calculated as:
where and . If the correlated edge assumption given in (10) is used, the disjoint edge sets are given as
where . By using (11), the outage polynomial of the correlated case is derived as:
(17) 
The numerical results of (17) are presented in Fig. 4. The uncorrelated case () has the best outage performance as expected. When , the outage performance is worse than the uncorrelated case. As increases, the outage performance gets worse. When , 4edges network has the close performance of 3edges system handled in Example 1. On the other hand, if is equal to it means that highly correlated links are available, the outage performance of 4edges networks approaches to 2edges system model labeled as is given in Fig. 1 (fig:N3).
The capacity polynomials of is given in Fig. 4 can be calculated as:
Hence, the ergodic capacity of is given by
The numerical results of the given polynomials are shown in Fig. 5 (fig:capacity_4edges_a). While , is high than . On the other hand, in the case of . Hence, the maximum value of the ergodic capacity of the network which is shown in Fig. 5 (fig:capacity_4edges_b) is equal to for .
In order to obtain further insight about the derivations, the outage polynomial and the ergodic capacity results of , , and depicted in Fig.s 1 (fig:N4), (fig:N5) and (fig:N6), respectively, are investigated. By using (5), the outage polynomial expressions of , , and can be respectively calculated as:
Here, the three graphs have the same coding gain with . On the other hand, and have the same diversity order equal to 2 and the diversity order of is equal to 3. The ergodic capacity results of the three networks can be respectively given as:
The outage polynomial and the ergodic capacity results of all the networks shown in Fig. 1 are presented in Fig. 6. It can be deduced from Fig. 1(fig:outage_all_graphs) that has the best outage performance with the highest diversity order . The two worst outage performance with belongs to and , as expected. , , and have close outage performance results with . The ergodic capacity results are in accordance with the outage polynomial results. Hence, the best performance belongs to .
V Conclusion
In this paper, we have obtained the performance limits of generalized wireless communication networks by using the concepts of graph theory. We have evaluated the network outage polynomial by utilizing individual link outages, through the use of path enumeration, cutset enumeration and terminalreliability approaches. For highSNR region, diversity order and coding gain have been extracted from the graph model of wireless networks. We have proven that the diversity order of any wireless communication network is minimum cutset size of the network graph and the coding gain is the number of distinct minimum cutsets. We have also presented the ergodic capacity analysis of arbitrary networks to obtain the ergodic capacity polynomials. The theoretical expressions have been illustrated by numerical examples. Hence, we have provided a comprehensive tool can be used to determine asymptotic performance of unstructured wireless networks and to specify their performance limitations under various implementation schemes.
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