Aging is a major risk factor for chronic diseases, which in turn can provide information about the aging of a biological system. This publication serves as an introduction to systems biology and its application to biological aging. Key pathways and processes that impinge on aging are reviewed, and how they contribute to health and disease during aging is discussed. The evolution of this situation is analyzed, and the consequences for the study of genetic effects on aging are presented. Epigenetic programming of aging, as a continuation of development, creates an interface between the genome and the environment. New research into the gut microbiome describes how this interface may operate in practice with marked consequences for a variety of disorders. This analysis is bolstered by a view of the aging organism as a whole, with conclusions about the mechanisms underlying resilience of the organism to change, and is expanded with a discussion of circadian rhythms in aging. Finally, the book presents an outlook for the development of interventions to delay or to reverse the features of aging.
The publication is recommended to students, researchers as well as professionals dealing with public health and public policy related to an aging society.
1 - 17: Introduction to the Theory of Aging Networks
This chapter will briefly address the history of systems biology and complexity theory and its use in understanding the dynamics of aging at the ‘omic' level of biological organization. Using the idea of treating a biological organism like a network, we will examine how network mathematics, particularly graph theory, can provide deeper insight and can even predict potential genes and proteins that are related to the control of organismal life span. We will begin with a review of the history of network analysis at the cellular level and follow that by an introduction to the various commonly used network analysis variables. We will then demonstrate how these variables can be used to predict potential targets for experimental analysis. Lastly, we will close with some of the challenges that network methods face.
In this chapter, we will briefly address the history of systems biology and complexity theory and their use in understanding the dynamics of aging at various levels of biological organization. Using the idea of treating a biological organism like a network, we will examine how network mathematics, focusing on graph-theoretic methods, can provide deeper insight and can even predict potential genes and proteins that are related to the control of organismal life span and perhaps even related to diseases associated with age-related changes within the organism or health span. We will begin with a review of the history of network analysis as related to the study of aging and follow that by an introduction to the various commonly used network analysis constructs. We will then demonstrate how these network variables can be used to further understand and possibly predict potential targets for experimental analysis. Lastly, we will close with some of the challenges that network methods face.
Aging - being old - is defined both biologically and psychosocially , ‘The geriatric or elderly patient is defined as an individual whose biological age is advanced. By definition, such an individual has one or more diseases, one or more silent lesions in various organ systems.' In addition, physiological changes affect the response to or handling of various medications. Social aspects of aging are also complex, and they include adapting to lessened physical capabilities and often to reduced income and to reduced social network support. For example, many older persons find themselves living alone after decades of marriage, partnership, and/or child rearing. Aging is an intricate spatial and temporal hierarchy of dynamic behaviors that are coupled together in a complex dance across the life span. Thus, aging is a complex, multidimensional, hierarchical process not easily dissected into disjoint subprocesses. How then do we grapple with the problem of understanding such systems?
In the Beginning: Reductionism
Historically, the pursuit of science has taken place by breaking objects apart and subsequently trying to understand how the pieces work at increasingly smaller and smaller levels of organization, the reductionist methodology. It was tacitly assumed that one could just glue the pieces back together and understand the behavior of the unbroken original system. Reductionist methods have been and continue to be widely used to understand biological systems and their dynamics. For example, the early genomic studies of aging identified numerous single genes related to survival . If survival is related to ‘aging' and the connections between genes/proteins are known, then perhaps networks of genes/proteins can be constructed that should predict other genes/proteins related to aging. If we understand how these genes and proteins function within an organism, then perhaps we can find ways to extend health span , control mortality and morbidity and better treat diseases associated more commonly found in elders of a population. Reductionist science has certainly yielded numerous insights into mechanisms underlying the processes of aging, the control of life span and the dynamics of age-related disease/decline in vitality. We now know many more genes and related proteins that appear to control or to be connected with these processes and we have even identified network pathways of importance . Thus, reductionist approaches have led us part of the way down the path to understanding the processes of life span control. However, as we shall soon see, understanding these systems is not as straightforward as simply gluing genes together to form networks and subsequently gluing networks together to form the whole organism .
As we will be making use of a large number of terms, any number of which may be unfamiliar to the readers of this text. We begin by defining terms so that we may all begin with a uniform understanding of the chapter vocabulary and how these concepts apply to the study of biological systems as a whole and ‘aging' in particular. We begin by defining the words ‘complex' and ‘complicated'.
Is a System Complicated or Is It Complex?
The terms ‘complicated' and ‘complex' are frequently used interchangeably in much the same way that the words ‘sex' and ‘gender' are now assumed to be linguistically equivalent, though they refer to significantly different conceptual constructs. The same can be said about the words complicated and complex. Given that a system has many parts, a system is said to be complicated if infinite knowledge of the behaviors of the system's components allows an experimenter to predict all possible behaviors of the system. For example, a pocket watch would satisfy the complicated but not complex criteria. We can understand the behavior of all of the cogs, wheels and springs in the system and, with some effort, we can arrive at what would be considered reasonable inferences concerning what the watch does and how it works. Breaking apart an organism costs information about how the ‘whole' organism functions. This begs the question of whether or not aging can even be reduced to discrete causes, or whether it involves a ‘complexity effect' that no single part or collection of parts can fully explain. Systems that lose information in breaking them apart are called ‘complex' systems. But what does a complex system actually look like? What might its properties be?
Properties of Complex Systems
If we were to examine a large collection of different complex systems, we would find that complex systems have certain common or unifying characteristics:
• They demonstrate emergent behavior; behavior that cannot be inferred from a linear analysis of the behavior of the components.
• They contain many components that are dynamically interacting (feedback, controllers, detectors, effectors and rules). There is no master controller. The parts interact extensively at their local level with nearest neighbors.
• The components are diverse, thereby leading to a significant diversity of information in the system.
• The components have surrendered some of their uniqueness or identity to serve as elements of the complex system. This is called dissolvence.
• All interactions of the components within the system and the system acting as a component in a higher hierarchy occur locally. There is no action at a distance.
• These interactions take place across a number of scale levels, and they are arranged in a hierarchical structure where fine structure (scale) influences large-scale behavior.
• They are able to self-organize, to adapt and to evolve.
As we can see, complex systems have properties that we do not expect to see in a pocket watch. Complex systems possess additional properties (e.g. control features, feedback loops and branches) that add order, robustness and stability to the system. Complex systems also exhibit an ability to adapt (i.e. evolve) to changing conditions. For example, changes in one free radical-scavenging pathway can up- or downregulate other pathways. Another way to think of complex systems is that they are systems in which the whole is greater than the sum of the parts . Why is this distinction important?
One of the most important properties that distinguish complex systems from complicated systems is the property of emergence. Consider the following examples. Infinite knowledge of a single bird or fish would not allow an experimenter to predict the phenomena of swarming or schooling or the synchronization of firefly lights . Infinite knowledge of a single female's menstrual cycle would not predict cycle locking in a college dorm room. These systems are termed complex . They have ‘emergent properties', meaning that a behavior that was not predicted from infinite knowledge of the parts emerges as part of the system's behaviors . Living systems, whether they are cells or ecosystems, do not function like pieces of a jigsaw puzzle. Instead, they are often fuzzy or stochastic, with backup systems and redundancies that belie their true structure. An examination of these systems requires a different conceptual framework. From a Positive Psychology perspective, Maddi  makes the argument to ‘…consider creativity as behavior that is innovative…'. We could easily argue that innovative behavior is emergent behavior, and therefore creativity is an emergent and unpredictable process. Thus, in order to understand complex systems, we must understand them through a reverse engineering perspective rather than a reductionist perspective.
Nonlinear Dynamics and Aging
By the early 1800s, studies of biological systems, ecosystems in particular, were observed to demonstrate a variety of nonlinear behaviors; particularly oscillations, apparently chaotic time series and radical behavioral changes that could not be explained by traditional reductionist constructs . From the early work of von Bertalanffy  and many others emerged the concepts of systems dynamics and systems theory as applied to a variety of living systems. Very early on, ecologists saw the value of systems theoretic approaches in understanding the complex ecological systems with which they worked. However, it was not until the work of Rosen  on MR systems and the subsequent work of May  and others who began to write about simple nonlinear models with complex dynamics (these are classic papers) that we began to see the emergence of previously described nonlinear phenomena such as chaos.
Nonlinear systems theory and multifractal analysis have already been used to understand fall safety in elders, frailty in the elderly, wandering in community-dwelling older adults, understanding interactions of geriatric syndromes and disease and in understanding the brain structures of Alzheimer patients. Network analytic methods have been used to construct longevity gene-protein networks and to predict potential gene targets of importance to longevity and perhaps to pharmacological intervention. Consequently, systems biology is now emerging as a powerful paradigm for understanding networks of longevity genes and proteins. With the sequencing of the human genome, massive amounts of data have been generated by the ‘omics' disciplines over the past twenty years; including genomics, proteomics, metabolomics, transcriptomics, and interactomics. An excellent discussion of complex systems dynamics and nonlinear dynamics may be found in Strogatz .
The application of the pantheon of mathematical and computational tools of systems biology has the potential to help transform the massive amounts of data into useful information that can be used to understand the biomedical processes associated with human disease and potentially how they relate to the dynamics of aging. By integrating omic data with the identification of critical networks and pathways associated with specific diseases of age and with vitality and longevity, greater understanding of these biological processes can be achieved. This enhanced understanding can help biomedical researchers design new and better approaches to treat or to manage the diseases of age and to help develop strategies to promote enhanced vitality and longevity, what is more currently known as health span. As the ‘baby boomers' move into their 60s and 70s, increased demand for care for the diseases of age and for approaches to enhance vitality and promote longevity means that new and improved remedies and interventions will be required. Consequently, a systems approach to the study of aging and its processes offers promise as a means of attaining potentially significant gains in the management and treatment of age-related diseases.
On the one end of the spectrum, we have reductionist methods that have allowed us to see into the organism and determine genes associated with life span. At the other end of the spectrum, we have holistic or complexity theoretic methods that allow us to probe an organism with minimal perturbation. Where does Systems Biology fit in?
The Emergence of Systems Biology
Systems science takes a middle ground approach, neither reductionist nor holistic . It attempts to look at the parts and it tries to glue them back together under the assumption that whatever complexity-related information is lost does not profoundly affect understanding the behavior of the system. Like a jigsaw puzzle, pieces are linked into chains that are then used to form small networks from which a picture of the process begins to emerge. While it was often possible to gain insights into the system behavior by gluing parts back together, for many systems it just did not work. This was particularly true for living systems in all of their forms and beauty. Life, it seems, was far more ‘complex' than had been thought . However, given the early lack of data on the pieces of biological systems and the minimal knowledge on how they were connected, it seemed that the only obvious approach was to try to glue pieces into potential networks, then glue the networks into hierarchies and finally see what results were obtained. The initial developments, particularly as applied to studies in gerontology and geriatrics, evolved from the idea of building reliable biological organisms.
An Example: Reliability Theory
One of the earliest aging-related uses of systems biological approaches was the use of what is now called reliability theory. The constructs of reliability theory emerged from the 1950s gedankt experiments of the computer scientist John Von Neumann . Von Neumann's interest [see ] was in how one would go about building a reliable biological organism out of unreliable parts. This question led to the development of the field of reliability theory and the subsequent adaptation of the field of reliability theory to become what is now known as the field of survival theory. Until the thought experiments of von Neumann, the concept of reliability had not been well defined.
Von Neumann's argument proceeded as follows. He began by defining the concept of the conditional instantaneous failure rate, denoted by λ(t). We interpret this as follows. The condition is that the failure has not occurred at time t given that the organism has survived until time t. With this in mind, we may then define the reliability R(t) of an organism as the probability of no failure of the organism before time t. If we let f(t) be the time to (first) failure (this is the same as the failure density function), then the reliability R(t) is given by R(t) = 1−F(t), where
How do we actually obtain an equation for the reliability R(t)? We do this as follows. Suppose we ask what is the reliability R(t+ Δt) where Δt is a small time increment. In other words, suppose that we know the reliability of the organism at time t and we want to know the organism's reliability at a small time increment Δt later than time t. In order for the organism to be operational at time t+ Δt, the organism must have been operational until at least time t and then not have failed in the time interval (t,t+ Δt). We can express this mathematically as follows. The reliability R(t+ Δt) is given by
R(t+ Δt) = R(t) − λ(t)R(t)Δt (1)
Reading equation 1, we see that to be functional (operational) at time t+ Δt, the organisms had to be functional at time t [denoted by the reliability term R(t) on the right hand side of the equation]. Next, we have to subtract out all of the items that failed in the time interval (t,t+ Δt); given by the second term on the right hand side of equation 1. What remains after this subtraction is all of the organisms or items that remain functional at time t+ Δt. A bit of algebraic rearrangement and we have
It follows that letting Δt → 0 (remembering our calculus), equation 1 becomes the simple differential equation given by
Thus, if we can specify the form of the function λ(t), we can solve for R(t). The literature in these fields often uses the term ‘failure rate function' interchangeably with the term ‘hazard' function. For those readers who have dabbled in demography or survival analysis, these constructs should seem quite familiar. Most people are familiar with either the Gompertz mortality rate (hazard rate/failure rate) λ(a) = h₀eγa where h₀ and γ are parameters typically estimated from population data for a given organism. Given the large literature on different mortality rate functions and their applicability to population modeling, we direct the interested reader to the relevant literature in that field. An excellent starting place may be found in Carnes et al. .
Systems biology, as applied to the biology of aging was simultaneously and independently originated by Doubal , Gavrilov and Gavrilova , Koltover , Witten and Bonchev , in the mid-to-late 1980s. Additional application of network theory to aging may be found in Kirkwood  and more recently in Qin  and Wieser et al. [53.] These papers focused on two application areas, genetic and general network theoretic applications. The thinking was that biological systems, particularly cellular systems, could be thought of in the same way as networks with interconnected parts that had certain failure rates. The death of the organism, and hence its life span, could be thought of as a network failure. The discipline of reliability theory, coupled with network analysis/graph theory, allowed these researchers to hypothesize certain network structures and to subsequently calculate failure curves for those network structures. In a number of cases, the shapes of the network survival curves mimicked the population survival curves seen in real biological populations, suggesting that reliability theoretic approaches, coupled with network assumptions, might have something to offer in understanding aging at a demographic level. This is because concepts of reliability have direct analogs to the longevity and lifespan of an organism. The most obvious one is that life span can be thought of as ‘the time to failure' of an organism. If death can be viewed as a failure, then there is a natural linkage between survival and reliability. Thus, the ideas of reliability mutated and the mutation became what we now know as the field of survival theory. Reliability theory allows researchers to predict the age-related failure kinetics for a system of given architecture (reliability structure, network, graph) and given reliabilities of its components.
During the past decade, with the increase in pathway ‘omic' information, there has been an increased use of complexity theoretic and systems biological tools and techniques to address putting the pieces of cellular networks back together so that their network properties can be better understood . These methods have also been applied to understanding the dynamics of cellular and molecular aging networks [28,29,30]. The systems biology approach has begun to allow researchers to understand the effects of multiple complex interactions in these aging networks, thereby further advancing our understanding of how longevity, vitality, and aging-related diseases may be managed. While reductionist approaches are still important, systems biology methods and complex systems theory constructs such as dynamical systems theory, network analysis, fractal dynamics, multi-level computational modeling and swarm theory can extract real information out of terabytes of data, and the role of systems biology and complex systems theory is now emerging as the front-running paradigm for understanding molecular and cellular networks of longevity genes and proteins. How then do we begin to understand networks?
Networks and Graphs
Much of the early work in graph theoretic applications to aging was based upon assumptions about how the genes were connected, as large databases of genes and networks simply did not exist. As available biological data increased, theoretical approaches, though more rigorously tied to experimental data, still struggled with questions around accuracy and reliability of the ‘omic' data being used. As the data became cleaner, it became possible to connect single life span-related genes and proteins into component networks. Other networks that controlled heat shock and other biological processes began to be identified. And now, with GWAS methods, we can begin to tie multiple cellular networks to the longevity gene networks. These networks could then be represented as mathematical structures called graphs [31,32]. These graphs could then be analyzed using the techniques of mathematical graph theory, particularly in light of the recent developments in network topology  and its implication for small-world theory , scale-free theory , redundancy , robustness , frailty , evolvability  and resilience  of the original biological network.
From Data to Graphs
A graph G is simply a set of nodes or verticesn1, n2, ... nG and edgesEij that connect some or all of the nodes to each other. From a biological perspective, we can consider the nodes to be genes or proteins and the edges as paths between them. We represent the overall network connectivities in a matrix format called the adjacency matrix which we denote with the symbol A. The elements of A are denoted aij and are simple; if node ni is connected to node nj, we enter the number one in the (i,j)th element of the matrix A, otherwise we enter a zero. Observe that if ni is connected to nj, then it follows that nj is connected to ni so that the matrix A is a symmetric matrix. In the case where there are multiple edges connecting the same nodes, we enter the number of edges. Thus, if two different edges connect ni to node nj, we would enter the number two. Nodes that are not connected to anything in the graph G are called islands. The edges can have weights, denoted wij, assigned to them where, for example, the weight value may correspond to a rate of reaction. An edge Eij can also have a direction assigned to it. For example, if E12 represents the edge between nodes n1 and n2, we might denote the fact that n1 is upstream of n2 by E1→ 2. An edge that does not have any direction assigned to it is said to be undirected, whereas edges that have a direction assigned to them are called directed edges. Note that we can have other types of edges in a network. For example, multi-edges are multiple edges between nodes, and self-edges occur when a node is connected to itself. With this simple set of definitions, we have some powerful tools with which to investigate the structure of a network and how it might inform us about the biological dynamics of the overall network. We begin with the concept of connectivity.
Network Structures and Connectivity
It is natural to conclude that the more edges going in and out of a node, the more likely that the given node is going to be of importance to the network. Hubs or nodes with large numbers of connections are known to play central roles in keeping complex networks connected. This is important when we consider, in an upcoming section, the concepts of robustness, resilience and frailty of a network. The number of connections ki going in and out of a node ni is called the connectivity or degreeki of the ith node. Sometimes you will see the degree of a node expressed using d(ni). In mathematical terms
where N is the number of nodes in the network and aij is the (i,j)th element of the adjacency matrix A. Computing the connectivity of a large set of nodes leaves us with nothing more than a frequency table, and it is hard to interpret this string of numbers ki, particularly if the number of nodes in the network is large. In order to assist us in understanding the connectivity structure of the network, we create a connectivity plot. To do this, we first count the number of nodes with a given connectivity k, where the connectivity varies from zero to the maximum connectivity value. The number of nodes with a given connectivity k is called the frequency of that connectivity and is denoted f(k). Next, plot the frequency f(k) versus the connectivity k.
Studies of the statistical behavior of various network structures [41,42] have shown that networks can have a small variety of overall topologies : random, regular, small world and scale free. Moreover, each of these network topologies has a classic pattern form for its degree distribution plot. Random networks are just what you would imagine them to be; nodes are randomly connected to each other. Regular networks can be thought of as lattices where there is a repetitive pattern of connections such as a grid. Small-world and scale-free networks are of greater interest because they have some fascinating underlying properties. Moreover, many real-world networks can be shown to be small world or scale free . A small-world network can be described as a network in which most nodes are not neighbors of one another, but most nodes can be reached from every other node by a small number of hops or steps . A scale-free network may appear to be a random network; however, in a scale-free network the links between the nodes are preferentially attached to the most highly connected nodes, thereby creating a greater frequency of links connected to a smaller number of nodes . Because scale-free networks are ubiquitous and highly relevant to our discussion, let us look at them a bit more closely. In examining figure 1, we see that not all nodes in the network have the same number of edges. If we divide the y-axis in figure 1 by the total number of nodes in the network, call that N, then f(k)/Nrepresents the probability P(k) that a randomly selected node has exactly kedges. In a randomly connected graph, the edges are placed at random, and one can show that the majority of the nodes will have approximately the same connectivity which is close to the average connectivity <k>. In fact, it has been shown that the connectivities k in a random network follow a Poisson distribution with a peak at <k>.
What became interesting is that, for larger networks like gene, protein and metabolic networks, these networks did not follow the traditional Poisson probability distribution. Rather, they followed a probability distribution where the connectivity probability P(k) was a power law of the form
P(k) = Bk-γ (4)
Observe that since P(k) is a probability, when we sum over all of the values of k, the result had better add up to one. Thus, the parameter value of B is chosen so that this is true. We will not get into all of the varied aspects of scale-free networks [31,33]. However, how can we determine if we have a scale-free distribution?
Power Plots and Scale-Free Networks
We observe that if we take the log of both sides of equation 4 the better the fit, the more linear the plot should be. Thus, networks whose connectivity structure follows a power law of the form f(k) = Bk-γ, where B and γ are parameters to be estimated and should look like negative slope lines if they are scale free. The simplest way to estimate the parameters is to perform a linear regression on the log-log transformed f(k) versus k data, dropping the k = 0 data point because there are no connectivities. The more linear this curve, the more the connectivity behaves like a power law.
Why should we even care about degree distributions and power law networks in the first place? As we noted earlier, scale-free networks are ubiquitous in living and even nonliving systems. Scale-free networks are also special in that they are built in a unique way. To build a scale-free network, you start off with a set of N nodes in which each node in the network is connected to all of the other nodes. Next, to add a new node, you make k connections to existing nodes in the network. However, whether a new node m is connected to an already existent network node ni is determined by the degree of the given node ni; the greater the degree of ni the more likely m is going to be connected to ni. In other words, the probability that node m will be connected to node ni is given by
Notice that this connectivity algorithm means that if you are already very tightly connected in the network, then you are more likely to get even more connected in the network. Many scale-free networks have an exponent γ ≈ 3. However, the exponent value very much depends upon the rule used for the probability of new node connection. Equation 5 is a very simple example. Given the large number of biological networks, particularly at the cellular level, that have been shown to be small-world formations, this suggests that tendency to create small-small world networks is a natural evolutionary pathway.
Categorizing Small-World Networks
Because of the unique nature of scale-free networks, a log-log connectivity plot is enough to let you know if you are dealing with a scale-free network or not. However, this does not work for other network forms. Because many biological systems also demonstrate small-world network behavior , we briefly examine how to determine whether or not a network is a small-world network. Remember, a working definition of a small-world network is a network in which most of the neighbors of a node are neighbors themselves (think regular network here, lattice structure for example). However, in addition to this property, the average number of connections between two chosen random nodes in the network ni and nj is small (similar to the properties of a randomly connected network). To help characterize small-world networks, we introduce a few new network descriptors. The first is the average path length of a network. Path length is the distance or number of edges between two nodes in the network. So, choose two random nodes in the network, figure out all of the different paths between them and count the number of edges in each of the paths. Then compute the average number of edges and you have the average path length. We can use the idea of path length to construct the minimum path length between node ni and node nj and denoted ℓij and the average minimum length of a network as <ℓ> using the same ideas as the average path length.
Another common network term is the centrality of a node. Centrality is a measure of the ‘position' or relative importance of a node in a network. In the literature, there are four main measures of centrality of a node: degree centrality, betweenness centrality, closeness centrality and eigenvector centrality. From an aging-related perspective, understanding node centrality of the nodes in a network could lead to potential targets for pharmaceuticals that might help hinder disease progression or extend life span. The simplest of the centrality measures is degree centrality. Degree centrality of a node is defined by CD(ni) = d(ni). In other words, the degree centrality of a node ni is simply the number of edges that are connected to the given node ni. Obviously, this measures the chance that a given node ni in the network will receive something flowing along the network. In the case where the graph is directed, or we know the flow along the edges (upstream, downstream), we can define two new concepts CinD and CoutD as the number of edges going in and out of ni. These are called indegree and outdegree, respectively. Closeness centrality, denoted CC(ni) is the idea that the more central that a node is in a network, the lower its total distance is to all of the other nodes in the network. In other words, if a node ni is very close to all of the nodes, it should take a small number of edges to get to every other node in the network. From a biological perspective, closeness can be thought of as a measure of how long it will take to send a chemical or other biological signal out from ni to all of the other nodes in the network. Betweenness centrality, denoted CB(ni)looks at how often, in a network, a given node ni acts as a bridge along the shortest path between two other nodes. From a biological perspective, knocking out a node with high betweenness centrality would force a signal to reroute itself along a path that was not the shortest path. Lastly, eigenvector centrality, denoted CE(ni)is a measure of the ‘influence' of a node in a network. Here, the idea is that not all connections between nodes are equal. That is, if a node is influential and it is connected to another node, it is likely that it will have more influence on that node than a node that is not that influential. An excellent discussion of the various concepts of centrality can be found in Opsahl et al. .
Clusters and Hierarchies
Earlier on, we mentioned the concepts of clustering. Many biological networks, metabolic networks and protein interaction networks demonstrate both clustering and scale-free properties . When examining network structures of this class of networks, we find that they are often modular and hierarchical in nature. That is, networks that exhibit the combination of small-worldness and clustering appear to be built out of modules that are themselves networks. One measure of the intrinsic hierarchical nature of a network is to make use of the mathematical result that deterministic scale-free networks that are hierarchical tend to have a clustering coefficient that goes as C(k) ≈ k-1. That is, if a node ni has k connections, then its clustering coefficient is approximately k-1. Thus, the higher a node's degree, the smaller its clustering coefficient. Moreover, the larger k gets, the more likely the clustering coefficient of the given node behaves as k-1 studies of many biological systems have, indeed, shown that the networks demonstrate modularity .
Robustness, Resilience and Frailty
We all have an intuitive idea of what robustness, resilience and frailty mean. From an intuitive perspective, resilience can be defined as the ability of a system, when perturbed, to return to its original state of operation . Some people loosen the definition to allow the system to return to a state of operation that is close to the original state of operation, where closeness is defined in such a way that the system is still functional as if it were still in its original state. Like most of the terms that we have been using, resilience is a complexity-related concept. For example, a system can take a short time or a long time to return to its operational zone. Are both of these the same degree of resilience? Surely not! A system can be perturbed for a fixed length of time and then the perturbation stops. What if the return to normalcy time depends upon the length of the perturbation? Are systems that return faster more resilient than ones that take longer to return? Can resilience be used up or built up? Thus, the term resilience encompasses a number of facets, most of which are ignored or tacitly assumed when talking about the subject of resilience. What we need to understand is that resilience is a system response property that allows the system to compensate after it has been perturbed. Since it is a global system property, complexity theory teaches us that it can have unpredictable outcomes due to its inner complexity. Bonanno et al.  point out that there are many ‘independent predictors of resilient outcomes'. This suggests two things. First, it suggests that resiliency analysis requires nonlinear methods in order to more effectively represent it. Second, it suggests that the human ‘resilience system' may be built with some form of redundancy/back-up system, some form of alternative and/or compensatory pathways in case some portion of the resilience system fails. Notice that the constructs of backup and redundancy tie back to our discussion on reliability of network systems . This type of organizational structure suggests that the human ‘resilience system' may have a fractal dimension that lies in what is often called the ‘robust to attack' domain. That is, the resilience system may have evolved in such a way that it is not frail; not easily vulnerable to attack and/or perturbation. If the system is fractal in nature, then this also suggests the various paths to prototypical outcome trajectories .
In the previous discussion, we noted that resilience is a measure of the system's ability to return to an operational space upon perturbation. The fact that the system was able to be perturbed indicates that it was not able to resist the forces of perturbation. This brings us to the concept of robustness. There are many definitions for robustness, and they are all context dependent. In one sense, robustness and resilience are opposite concepts. Robustness can be viewed as resistance to perturbation, and resilience is the ability to return to one's original state, or - in the case of positive psychology - to subsequently ‘[achieve a] positive adaptation despite major assaults on the developmental process' after a perturbation. Thus, robustness to stress implies that it is hard for stress to move a person off their current trajectory (life course), while resilience says that if the individual is moved away from the life course trajectory then how long, if you will, does it take to get back to the original trajectory or to a trajectory that will serve as an acceptable surrogate for the original trajectory.
Systems that are robustly designed are often more difficult to study because they have built in ability to resist perturbation and, if they blow a circuit or an operational unit, they often have backup systems that can keep the living system (or nonliving system) operating in a functional way . Robustness, or ‘resistance to perturbation' can be enhanced by developing the client's strengths. Thus, the construct of robustness may be seen as an emergent consequence of a system design that has built in buffers against assaultative pressures. Thus, like resilience, robustness is a global catch-all term designed to describe a system's ability to defend itself against perturbation. Because robustness represents a system's ability to resist perturbation, robustness could also represent a threat to evolvability and adaptability.
Evolvability is the system's ability to alter itself in response to changes in external forces in such a way as to allow the system to continue to function - adaptive evolution. A living system is said to be evolvable if it can acquire novel functions through genetic change, functions that help the organism survive and reproduce. Because the concept of evolvability originally arose in the field of organismal evolution, it carries with it the ideas associated with genetic evolution in the face of exogenous pressures. However, it is not unreasonable to think of living systems such as organizations as having constructs that are equivalent to genes and to ask how an organization might evolve in the face of external pressures such as economic stress. Thus the important nodes in the hierarchy can change as can the connections . These changes can lead to new emergent dynamics that could be considered adaptations to the exogenous stressors or forces.
In this chapter, we reviewed the literature on systems biology in aging from a historic as well as current perspective. We discussed how systems biology of aging emerged from the dynamic interplay of reductionist methodological approaches and the need to address concepts of complexity theory that began to develop in the early 1970s. We showed how these considerations and the perspective of considering a biological system as a network allowed scientists to use graph-theoretic methods to begin to understand the impact of an organism's network structure on its behavior. We then examined some basic concepts of network theory and how they apply to studying cellular aging systems. We discovered that many longevity-related cellular networks have small-world and/or scale-free network properties. Further, it appears that these networks are also very modular in nature. It has been pointed out that this suggests that modularity is important, not because modules exist as somewhat independent subnetworks, but rather because they are combined so that they are tightly connected to each other. These modules are then used to build higher-level network components and networks themselves. We raised questions around what happens when networks ‘age'. We saw that this raises questions around network robustness, frailty and susceptibility to various forms of attack, and we examined how network structure may or may not make a network more vulnerable to perturbation, thereby potentially reducing its resilience/robustness and potentially making it more frail. In this chapter, we have focused solely upon graph-theoretic approaches in systems biology. There are many other systems methodological approaches to understanding systems biology of aging. For example, Kriete et al.  use a rule-based systems approach to modeling aging systems. Albert et al.  use a Boolean network simulation approach. And Huang et al.  use a machine learning approach.
The bulk of network aging studies are based upon a snapshot of a single time point in the organism's life span, a cross-sectional picture of the network. In order to really understand how an organism ages (how its networks age), we must better understand how networks evolve over the life span of the organism. There is no reason to believe that network nodes remain active throughout the whole life span, nor should we assume that network edges remain present throughout the life span of an organism. Longitudinal network analyses are now needed. Determining the weights and directions of network connections in aging-related network structures is now important so that accurate simulations of the network dynamics can be developed .
In the following chapter, we apply some of these concepts to actual aging networks and illustrate for the reader how to make some of the calculations we discussed in this chapter. Due to chapter restrictions, many relevant references were left out of both this chapter and the following chapter. For this, the author can only apologize for the citation choices made. To compensate for this, I am providing an exhaustive reference list for both chapters. This reference bibliography may be found at http://www.people.vcu.edu/∼tmwitten.
The author would like to thank many individuals for their respective support and collaborative kindnesses during her career path. In alphabetical order I would like to acknowledge my colleagues and friends: Danail Bonchev, Bruce Carnes, Caleb Finch, Ari Goldberger, Leonard Hayflick, S. Michal Jazwinski, Tom Johnson, Vitaly Koltover, Andres Kriete, George Martin, Ed Masoro, Robert May, Robert Rosen, Bernie Strehler, F. Eugene Yates and B.P. Yu.