Cyber security is a tremendous challenge facing not only individual corporations but also nations around the world. The social and economic impact from a systematic cyber attack is potentially devastating for an organization. Understanding and overcoming the vulnerabilities of an enterprise cyber system is therefore critical to continued operation of an organization. However, cyber systems are complex in nature and hard to model in a uniform setting. A graph
is an ordered pair whereis a set of vertices (or nodes) and is a set of edges (or links). An edge represents a binary relationship between two vertices. Graph is a simple but powerful tool for modeling complex systems. Unique entities of a system form the vertex set , and binary relationships between vertices are represented in the edge set . Driven by their simplicity, graph-theoretic modeling and analysis has been used extensively for cyber security [1, 2, 3, 4, 5]. Although powerful, graph-based modeling of cyber systems has limited applicability. A fundamental limitation of this approach is that vertices and edges of a graph need to be homogenous in order to make graph-theoretic analysis meaningful. However, cyber systems are fundamentally heterogeneous in nature. Graph-theoretic models are also limited in their ability to capture the dynamics and physics of a given cyber system accurately.
To further illustrate these limitations consider the following situation. An enterprise has access to several different sources of information such as electronic-mail traffic within its domain, network flow within and across its perimeter, events generated by anti-virus software running on individual computers, and vulnerabilities of individual computers quantified as a function of the operating system and services that run on them. Individually, each source provides a limited perspective on the collective actions of human and software agents within an enterprise. By focusing on the information related to a particular user group or a specific software service, strong correlation or dependencies can be observed across multiple data sources. Based on an aggregated information, addressing questions such as: “How likely would host get compromised?” or “Given that host has been compromised, how can we ensure continued operation of a service running on ?” will become possible. It is not only necessary to discover independent local events, but also to follow the trail of dependencies and discover new events and vulnerabilities at a global scale. Therefore, we argue that it is imperative for a cyber system model to encompass different entities (hosts, users, software processes, and events) and their diverse interactions (user-host, host-host, host-process, etc.).
Further, the benefits of modeling a complex cyber system using a graph-theoretic approach are tremendous. However, the above definition of a graph can only capture binary relations between homogeneous entities. It does not provide a mechanism to distinguish between heterogeneous entities and different types of entity relationships. Semantic networks  and conceptual graphs  support the notion of heterogeneous entities and multiple types of edges. However, this expressivity comes at a price. For example, consider a simple breadth-first search on a semantic network to detect paths between hosts. A multi-hop path not only includes hosts but may also include other vertex types with specific rules. Thus, to discover which hosts are reachable in hops we need to extend traditional graph-theoretic tools to incorporate the notion of heterogeneity. In summary, there is a critical need to extend traditional graph-theoretic models to successfully model complex cyber systems.
Motivated by the interdependencies between different critical infrastructure networks, the networks-of-networks (NoN) model was developed to address some of the limitations of graph-theoretic modeling of single networks . For instance, infrastructure networks such as electric power grids, communication and control networks, oil and natural gas pipelines, and transportation networks are all interconnected and interdependent on each other. While each of these networks can be efficiently modeled and studied individually using graph-theoretic concepts and tools, the interdependencies are hard to model in a uniform setting. However, the inclusion of these dependencies fundamentally alters our understanding of a complex network, and is therefore being labeled as the next frontier of network science . Using a two-layered model, Kurant and Thiran  made a distinction between the physical and the logical aspects of a network. A mapping of the logical network onto the physical network was then added. This approach enabled a better modeling of the load on a (railroad) network and the vulnerabilities caused due to failures in different layers. Further, the notion of NoN naturally emerges in the context of cyber-physical systems. For example, the interactions between a supervisory control and data acquisition (SCADA) network and the underlying power transmission network that SCADA controls. The connectivity and dynamics of these two networks are fundamentally different, but the two networks are interdependent on each other. By modeling this system as a NoN critical properties and combined vulnerabilities can be expressed mathematically [11, 8].
Contributions: In this position paper, we propose a novel three-layered NoN model for an enterprise cyber system (Section !II). By explicitly modeling the different facets of a cyber system, our goal is to bring together different concepts from traditional graph theory and network science to bear on the problems in cyber security (Section III). We aim to develop an efficient tool for modeling complex cyber systems and provide graph-theoretic metrics to express different functionalities and vulnerabilities of such a system. While NoN models have been proposed for several cyber-physical systems, our approach is to apply this model exclusively for cyber systems, and is thus the novelty of our approach.
Ii A NoN-Based Model for Cyber Systems
Mathematically, we represent a network of networks as a graph where the vertex-set , and the edge-sets and , such that , and for . The graph can have weights associated with vertices and/or edges , and can be directed or undirected depending on the network that is being modeled. Pair-wise interconnections between two networks can be represented as a bipartite graph , such that and an edge always connects a vertex in to a vertex in . Further, we define an edge between two vertices with its end-points as where appropriate. We define a walk in a graph as a finite sequence of edges (or vertices) such that any two consecutive edges in are adjacent to each other in the graph. A walk that does not traverse any edge twice and visits each vertex only once is called a path. We now briefly present a three-layered NoN model to represent an enterprise cyber system.
Ii-B Three Layered Model
For the purposes of representing an enterprise cyber system, we propose a three-layered NoN model, where the layers are:
Layer-1: Physical (Hardware) layer,
Layer-2: Logical (Software; Functional) layer, and
Layer-3: Social (User; Computer) layer.
This simple model, illustrated in Figure 1, can be further extended to include other aspects of an enterprise as needed. For example, modern telecommunication systems critically depend on the underlying communication networks for operation – this can be modeled as a new layer. Another example would be the network of power cables supporting the communication network, which forms a new layer. We now provide details on how to model each layer and to build their interdependences.
The physical layer (Layer-1) is built from the network topology that is readily available to network engineers and administrators. This network consists of computers that are interconnected via switches and routers to the enterprise backbone. From a graph-theoretic perspective the connectivity of this network is fundamentally different from other layers. For example, let different workstations (computers) be represented as vertices, and two computers that can connected by a certain physical link be represented by edges. In this representation, all the computers connected to a switch would form a clique or a strongly connected component (a path exists from each node to every other node in the component). If we treat virtual machines as independent servers (vertices), then all the virtual machines residing on a physical server would also form a clique.
The logical layer (Layer-2) can be built in several different ways. One simple approach would be to model the different software applications (services) running in an enterprise system as a graph. Here the nodes would correspond to different services and edges would represent pair-wise relationships between them. Other approaches such as attack graphs  that describe vulnerability dependencies and system exploits would also form a network for the logical layer. Similar to the approach taken by Karagiannis et al. , information collected from analyzing NetFlow  data can lead to the functional relationships between different services. When a logical network is built to represent different services, the relationship with the physical layer can be straightforward – services run on specific hardware. On the other hand, if relationships are not obvious simplifying assumptions can be made. From a graph-theoretic perspective the connectivity structure of networks at this layer is different from those at the physical layer. For example, attack graphs are typically a collection of directed paths from an attack source to an attack destination (target computer).
The social layer (Layer-3) can be built in several different ways. Our goal at this abstraction layer is to capture the social aspects of a network: How do we find users with similar activity-profile? How do different users within the enterprise interact? The interactions among different users can be collated from several sources such as electronic mail exchanges, instant messenger communications and possibly through participation in social networks. Similarly, social interaction of computers via their Internet Protocol (IP) addresses can be collated from network monitoring tools such as NetFlow. Graphs describing behavioral similarity can be built by clustering the user activities. An edge is added between two users/hosts if both of them participate in the same cluster by satisfying a threshold on membership. From a graph-theoretic perspective such networks have been studied extensively .
In general, social interaction networks tend to have the following structural properties: small-world nature, where any two nodes in the graph are connected via a short geodesic path; large clustering coefficient, where the neighbors of a node are also interconnected; and scale-free nature, where a large number of nodes have a small degree (number of neighbors) and a small number of nodes have large degrees. As a consequence, social networks tend to be vulnerable to targeted attacks but resilient to random failures.
Ii-C Interactions between layers
Following the work of Buldyrev et al. we use special edges (dependency links) to connect different networks between layers [8, 9]. Dependency edges are directed edges. Further, we make a distinction between dependent and independent edges. Independent edges represent internetwork relationships where the node in one network does not depend on the node in another network for operation. Independent edges are modeled as undirected (bidirectional) edges. This distinction will become clear for a given context and the graph algorithms that are being used, which will be discussed next.
Iii Graph-Theoretic Analysis for Mission Assurance
Our goal is to study and quantify different aspects of an enterprise cyber system. Globally at an enterprise level, we are interested in understanding and quantifying the collective robustness (or vulnerability) of a cyber system due to targeted or random failures at different layers (networks). And, locally at a subsystem level, we are interested in developing metrics that can be computed from information collected locally but provide enough information to guide continuous monitoring and maintenance. The rest of this section is dedicated to answering the above questions using the NoN model. We present a number of graph-theoretic approaches that measure and describe different facets of a cyber system.
Iii-a Critical nodes and their relationships
Identifying critical components (represented as nodes and/or edges of a graph) in a network is an important activity. Many graph-theoretic approaches can be used to identify such components. For example, betweenness centrality  measures the importance of a node (or an edge) as the ratio of the number of paths (for example, shortest paths as described in Section II-A) that flow through a particular node to the number of all possible paths in the network. If the ratio is high for a given node then it implies that this node is critical to the connectivity of the graph. While betweenness centrality is one of the popular measures, there are several other variants of centrality that quantify different aspects of connectivity in a graph . Metrics based on network flow, nodes along a maximum-flow minimum-cut in a graph, have also been used in the context of cyber security to identify critical nodes . In the context of networks-of-networks, computation of these metrics needs to be adapted carefully. As an illustration, consider the bottom two layers (logical-physical) of the three-layer NoN model illustrated in Figure 1. In the logical network, is an important node (since it controls the flow from the left part of the network to the right). Similarly, in the physical network, is an important node. If node is dependent on any node in , then the failure of node in will be detrimental to the connectivity of the logical network. Therefore, identifying critical nodes and their interrelationships is an important activity towards making a cyber system robust.
Quantifying the relationships between critical nodes is an important next step. Let us assume that we have identified critical nodes in different layers based on their degree (number of neighbors) distributions. In the context of a single network, Newman defined the concept of assortativity as the association of nodes that are similar in some fashion, degree for example . Similarly, a network is called a disassortative when dissimilar nodes get associated with each other. Assortativity is measured as Pearson correlation coefficient of the degrees of nodes at the two ends of edges. The metric is positive for assortative networks, negative for disassortative networks, and zero for networks with no specific associations. The notion of assortativity can be extended to networks-of-networks and provides a critical metric to gain insight to the interdependencies. Parshani et al.
define this as the joint probability of a dependency edge connecting a node in the first network with degree j to a node in the second network with degree k, and call it inter degree-degree correlation (IDDC). Using simulations and empirical data modeling a two-lawyer port-airport system, they show that coupled systems with higher assortativity between layers are more robust to random failures. While vertex degrees are important, we need the ability to use other metrics to define assortativity in cyber systems. The notion of assortativity needs to be further modified for NoNs where we might get a zero assortativity when non-critical nodes in one network are connected to critical nodes in another. Another closely related metric of interest for us is the notion of clustering coefficient which can be extended to networks-of-networks. Local clustering coefficient of a vertex in a graph is defined as the ratio of the number of actual edges to the total number of potential edges between all neighbors of this vertex. Parshani et al. extend this to NoN as the ratio of the number of neighbors of a node in the first network that are also dependent on the corresponding neighbors of its dependent node in the second network. They call this metric as the inter-cluster coefficient (ICC) . Similar to IDDC, this metric provides insight on the collective robustness (vulnerability) of NoN.
Iii-B Reachbility Analysis
Reachability analysis can be a useful tool in determining the vulnerability of a network. If it is important that a chosen vertex in (in Figure 1) be reachable quickly from another vertex also in , then we can count the number of shortest paths (as defined in Section II-A) between and in . However, it might be possible that the shortest path between two vertices in a NoN model may go through other layers (consider vertices and in layer from Figure 1). Disturbances in one layer would result in the loss of efficient connectivity in the other (the dependent layer). On the other hand, the fact that there are paths between and that go through , and paths that go through can be positive in the sense that the network as a whole becomes more resilient. Thus, by suitably expressing reachability across multiple layers, we can measure robustness as a function of reachability. Using a single-network approach we used reachability as a metric to express vulnerability of a cyber system , and plan to extend this approach to multi-layer networks in our future work.
Iii-C Cascading Failures
A topic that originated in the context of electric power grids but has found applications in a wide range of networks is the concept of cascading failures, popularly known as blackouts in power grids. The dynamics of network failures can be modeled in several ways. Cascading failures have been studied using different models under different names. We will extend a few models that are relevant to NoN based models of an enterprise cyber system. A commonly studied topic under this theme is of graph perturbation where changes are made to a graph and the impact of these changes is studied . Different metrics can be used to measure the impact of node (or edge) failures: connectivity of the graph expressed using the number of disconnected components, or the size of the largest connected component, or the fraction of nodes remaining in the largest connected component; flow in the network; or the overall modifications to the size of the network expressed via the number of nodes and/or edges.
Another analysis that will be useful for addressing aspects of cascading failures is optimal security hardening . In order to protect a system from cyber breaches, all known weaknesses in a system needs to be hardened. However, there is an associated cost to this and a fixed budget dictates which security controls can be implemented and which weaknesses need to be left unhardened. Thus, first and foremost, an optimal strategy needs to be determined that minimizes residual damage (damages due to having weaknesses in a system) to the NoN that is not perfect. In order to perform such analysis, we can model the NoN as a dynamic graph and focus on the degree distribution of the nodes in the NoN as it changes over time. Of most importance will be nodes that have high centrality – the higher the centrality the more valuable a node is. Starting with nodes with high centrality, further node reachability questions can be asked that will allow us to identify critical nodes that potentially contribute to cascading failures and design efficient defense strategies to overcome them. This information can then be used in a multi-objective optimization problem  that minimizes the total security control cost as well as minimizes the residual damage. We can constrain this optimization problem by allowing a maximum degree of perturbation in the NoN model.
Iii-D Subgraph Pattern Mining
Evidence of a cyber attack can be extracted from specific signatures expressed as subgraphs or graphlets. Therefore, the ability to detect and count a large number of signatures from cyber data is desirable. Detection of emerging patterns can be used to counter active attacks. Kargiannis et al. use this approach for classification at the application level [5, 22]. However, we can use the graphlet degree distribution (GDD) proposed by Pržulj  as a general metric to study the existence of patterns not only in the static data but also to express similarity between two networks.
While the identification of specific signature subgraphs and graphlets can provide insight into the behavior represented by the NoN, mining for more general subgraph patterns and anomalies has shown promise for cyber-security applications [24, 25]. Incorporating multiple views of the domain (physical, logical, social) into one graph potentially allows for the discovery of rich patterns involving structure from multiple views; however, such approaches would not scale to the size and rate of change over the entire NoN. Mining within layers will be more efficient, because an individual layer is smaller and more homogeneous. But more importantly, subgraph patterns and anomalies in one layer serve to focus attention in other layers. For example, a frequent subgraph pattern in the social layer can be expanded to include the induced subgraph from the logical layer to improve understanding of the services supporting the normative social behavior. Likewise, an anomaly in the social layer gives us direct links to the related structures in the logical and physical layers, thus providing an explanation of lower-level anomalies. And the reverse forensics can link patterns and anomalies in the physical or logical layers to the responsible entities in the social layer. The NoN also improves our ability to mine for patterns in the presence of dynamics, because change in the structure of one layer can be processed independently from the change in another. This allows us to improve graph mining efficiency while still retaining the connections between layers in order to perform a more focused analysis of the dynamics in other layers.
Iv Summary and Future Work
We proposed a novel networks-of-networks (NoN) approach for modeling an enterprise cyber system. We claim that such an approach enables efficient modeling of interactions between heterogeneous entities and yet allows us to leverage on the traditional graph-theoretic tools. We presented a novel three-layered model to demonstrate how the NoN approach can be used for continuous system monitoring and mission assurance. Specifically, we discussed in depth the following graph-theoretic methods for cyber security: detection of critical nodes and their dependencies; reachability analysis; models for cascading failures; and subgraph pattern mining. We described how each of these applications can be built on top of our proposed NoN model, thus extending the state-of-the-art.
Building a framework to extract each of the unique networks from Netflow, electronic mail and event logs is the next logical step for our work. Discovering dependencies across these networks and developing theories to translate these dependencies into answering questions on risk management and mission assurance remain candidates for our future work. NoN is an emerging novel concept and we look forward to exciting research in the near future.
This work is funded by the Asymmetric Resilient Cyber (ARC) Initiative at the U.S. Department of Energy’s Pacific Northwest National Laboratory (PNNL). PNNL is operated by Battelle Memorial Institute under Contract DE-ACO6-76RL01830. We thank John Feo, Pradeep Ramuhalli and George Muller for stimulating discussions on this topic.
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