## 1 Introduction
### A Resilience-Oriented Approach Using Graph Invariants for C-RAN Backhaul Design  
The design of communication networks has been discussed for decades. There is a consensus in the literature on treating this problem as a multi-objective optimization problem [12; 13; 14; 15; 16; 17], which maps aspects such as cost, performance, and survivability into constraints. Whereas this consensus suggests a common methodology for network design, it reduces the design problem to a general cost optimization formulation, where the topology is a mere input of this problem, neglecting the role of the topology properties as a relevant component of the solution. Nevertheless, several works have shown how the topology has great impact on network capabilities, such as fault tolerance [18; 19], resource consumption (i.e., number of wavelengths) [20; 21], and throughput [22]. C-RAN backhaul topologies can also be planned considering capacity [23] and energy consumption [24]. Moreover, in scenarios where CAPEX and OPEX can be decreased, the backhaul is planned considering the possibility of sharing links through the usage of already installed metropolitan optical networks and other underused fiber in a city [25]. In this case, our main novelty is to improve reliability and survivability of the network by optimizing link selection according to intrinsic topology properties, instead of optimizing capacity or energy consumption. In this paper, we propose a method for systematic refinement of existing optical networks toward service of C-RAN backhaul, grounded in graph theoretic optimization. Our approach leverages graph invariants to quantify the requirements of wireless networks and constraints of backhauling infrastructure. A graph invariant is a measure that characterizes the topological structure of a graph, and does not depend on the labels of its nodes. One of the most studied graph invariants is the Wiener index, proposed in 1947 [26] with the purpose of estimating the boiling point of alkane molecules. This graph invariant gave rise to hundreds of topological indices in the field of mathematical chemistry [27; 28]. Interestingly, the same invariant was independently proposed 40 years later in [29], motivated by communication network problems. Graph invariants have been used since 1993 for designing novel chemical compounds [30; 31]. However, to the best of our knowledge, this is the first time a work applies graph invariants to the design of backhaul communication networks. Graph invariants provide the means to capture several important features of a backhaul topology, including the number of nodes or edges as well as the maximum degree of nodes, the topology diameter, and the average distance (or average hop count). Note that these invariants do not depend on the specific technology adopted by the backhauling network and instead focus on the structure of the topology. Importantly, this aspect of invariants enables their use in determining the fundamental ability of a backhaul topology to meet latency and resilience requirements. Moreover, a grounding in graph theory enables the use of mature tools. Therefore, a basis in graph theory enables our approach to provide fundamental insight into the ability of a backhaul design to meet the requirements of wireless network and systematically refine this ability. Through refining existing optical network topologies in service of C-RAN backhaul, we jointly consider the needs of wireless networks and the associated optical operation. More specifically, we investigate how C-RAN backhaul resilience requirements can be expressed in terms of the graph invariants described in the literature. Furthermore, we propose a new distance-based graph invariant, called Wiener impact, which is related to resilience aspects of the backhaul topology design.