Defining the fundamental unit
Within all the disciplines explored the fundamental unit employed in any connectivity analysis depends on the spatio-temporal context of the study and the specific research question – this applies even where a clear fundamental unit might be self-evident (e.g., the individual in social network science, or the individual neuron or cortical area in neuroscience). The spatial and temporal scale of the fundamental unit may span orders of magnitude within a single discipline, and may thus have to be redefined for each particular study. For example, whilst for some applications in Neuroscience it is appropriate to adopt the neuron as the fundamental unit, for others the cortical area (many orders of magnitude larger in size) may be more appropriate – notably in cases where it becomes challenging to address adequately the connectivity of neurons due to computational limitations. This issue is also present in Geomorphology where adopting individual sediment particles as the fundamental unit would become too computationally demanding. In this sense, there are parallels between connectivity and the field of numerical taxonomy (Sneath and Sokal,
1973) where, despite the obvious taxonomic unit being the individual organism, an arbitrary taxonomic unit (termed an operational taxonomic unit) was employed. The exception to this general statement is the field of Ecology, where the ecosystem provides a conceptual unit that can be applied at any spatial scale. The concept of the ecosystem was introduced by Tansley (
1935) and has been subject to much debate since. Despite the shortcomings of the ecosystem concept, within connectivity studies it is nonetheless useful to have an overarching concept that can be employed at any scale. The ecosystem concept is particularly useful when the interactions (connectivity) between different organizational levels are of interest, with an ecosystem at a lower hierarchical level forming a sub-unit of an ecosystem at a higher hierarchical level. Many systems are hierarchically organised, and therefore a key question for other disciplines is whether identifying something theoretically similar to the ecosystem concept may be useful. For many applications in connectivity studies, appropriate conceptualisation and operationalisation of the fundamental unit will depend on the purpose of investigations. For example where interventions within a system have the goal of managing or repairing a property of that system, the scale of the fundamental unit may be specified, to work within the certain system boundaries for a particular purpose. But as noted in the case of Ecology (section
Defining the Fundamental Unit) it is critical that whilst defining the FU, relationships that cross scales are also defined clearly.
Although in most disciplines the fundamental unit corresponds to some physical entity, in Social Network Science for example, it may be more abstract, i.e. a unit of interaction. More abstract conceptualisations of the fundamental unit may be fruitful in other disciplines where the definition of a fundamental unit as a physical entity has proved difficult (e.g. Geomorphology), or in modelling approaches to examining connectivity (e.g. random graph models). Furthermore, the notion (in Systems Biology) that the fundamental unit is a concept dependent upon the current state of knowledge of the system under study is a valuable point that merits wider consideration.
Separating SC and FC
There is general consensus that SC is derived from network topology whilst FC is concerned with how processes operate over the network. In all the disciplines considered, the separation of SC from FC is commonplace, due to the ease with which they can be studied separately – especially in terms of measuring and quantifying connectivity. The success separating SC and FC in Systems Biology has been attributed to the fact that the structural properties and snapshots of biological function are typically measured in independent ways, whereas elsewhere it is common for FC to be inferred from measurements of SC. Whilst structural-functional feedbacks are widely recognised, the extent to which these feedbacks are explored/accounted for varies considerably, in accordance with factors such as how advanced the discipline is, and the sophistication of available tools within that discipline. Separating SC and FC, of course, imposes severe limitations on one’s ability to address these feedbacks. For example, in Geomorphology it is well established that structural-functional feedbacks drive system evolution and emergent behaviour, and whilst it is common in some applications to explore these feedbacks (e.g., landscape evolution models), there are still many applications where it remains common to look at relations between SC and FC in one direction only (i.e., the effect of structure on function). There is a similar tendency in Neuroscience to focus on structural-functional interactions rather than the full suite of reciprocal feedbacks between structure and function. However the increasing recognition within Geomorphology and Neuroscience of reciprocal feedbacks is heightening the need for additional tools that will allow the evolution of SC and FC and the development of emergent behaviour to be understood more fully. The importance of such feedbacks is highlighted in Computational Neuroscience, in the case where frequently used networks persist, whilst rarely used links are degraded leading to the development of network topology over time. Nevertheless, separating SC and FC does permit insights into the behaviour of systems insofar as it permits predictive models of function from structure that are amenable to experimental testing.
The ease and meaningfulness with which SC and FC can be separated will also depend on the timescale over which feedbacks occur within a system. Structural connectivity can only be usefully studied independently of FC if the timescale of the feedbacks is large compared to the timescale of the observation of SC. Any description of SC is merely a snapshot of the system. For that snapshot to be useful it needs to have a relatively long-term validity. Thus, for meaningful separations of SC and FC to be made, it is paramount to know how feedbacks work, the timescales over which they operate, and how connectivity helps us to understand these feedbacks. There are striking examples from several of the disciplines explored here of the ways in which feedbacks between SC and FC can lead to the co-evolution of systems towards a phase transition point – this is seen in Computational Neuroscience, and in Ecology and Geomorphology where system-intrinsic SC-FC feedbacks shift a system to an alternate stable state.
Linked to SC-FC relations and the validity of separating the two is the concept of memory. Memory is about the coexistence of fast and slow timescales. Qualitatively speaking, the length of distribution cycles in a graph can be viewed as (being related to) a distribution of time scales. Changes to SC in response to functional relationships imprint memory within a system. Thus, key questions are: How far back does the memory of a system go? Is memory cumulative? In systems subject to perturbations (possibly true for all discipline studied here) which perturbations control memory and its erasure? What are the timescales of learning in response to memory? In Neuroscience, the discipline which gives us the term ‘memory’, it is argued that the final imprint of memory is diffuse across the brain, and consequently difficult to assess. Other disciplines have similarly struggled to comprehend the instantaneous non-linear behaviour of their systems in terms of memory. In Ecology, ‘legacy effects’ make empirical approaches to the study of ecological interactions across space and time challenging (van der Putten
et al.,
2009). In Social Network Science it is possible to speak of culture, which raises the notion of a hierarchy of memory effects on connectivity: one that has not yet been explored. In Geomorphology memory is related to feedbacks and/or thresholds – thus, exploring the coexistence of fast and slow timescales of processes and mechanisms is a potential avenue for future research. Of all the key challenges facing the use of connectivity, memory appears to be one which no discipline has yet resolved.
Understanding emergent behaviour
Emergence is a characteristic of complex systems, and is intimately tied to the relationship between SC and FC. The choice of fundamental unit will have implications for one’s ability to understand emergent behaviour, since connectivity at larger spatial scales emerges from connectivity at smaller spatial scales, and thus, microscopic units produce macroscopic behaviour through emergent properties. In this sense, a fundamental unit is an emergent property of microscopic descriptions.
An important question is how far does the analysis of connectivity help understand emergence? As noted in (section
Understanding emergent behaviour), the co-evolution of SC and FC offers an interesting possibility for the overarching perspective of self-organization and emergent behaviours, as the system now can, in principle, tune itself towards phase transition points. Thus, by separating SC and FC in our analyses of connectivity, we remove the opportunity to understand and to quantify emergence – to understand how a system tunes itself towards phase transition points (and the role of external drivers). Without tools that can deal with SC and FC simultaneously, it is challenging to see how connectivity can be used to improve understanding of emergent behaviour. However, some suitable tools do exist. For example, adaptive networks that allow for a coevolution of dynamics on the network in addition to dynamical changes of the network (Gross and Blasius,
2008) provide a powerful tool that have potential to drive forward our understanding of how connectivity shapes the evolution of complex systems. Approaches are used in Computational Neuroscience that look at the propagation of excitation through a graph showing waves of self-organization around hubs, thus allowing exploration of conditions that lead to self-organised behaviour. However, even in this example, there is still great demand for new ideas that will more easily accommodate the study of memory effects (in all its various guises) and emergent properties. In Geomorphology and Ecology, key studies demonstrate how incorporating SC and FC into studies of system dynamics allows for the development of emergent behaviour (e.g. Stewart et al.,
2014). However, such examples are relatively rare, which highlights the scope for trans-disciplinary learning which may help to drive forward our understanding of emergent behaviour.
Link prediction is a potentially useful tool that has been applied for example in Systems Biology and Social Network analysis. It can be used to test our understanding of how connectivity drives network structure and function (Wang et al.
2012; Zhang et al.
2013), and our understanding of emergent behaviour. If a comprehensive understanding of a system has been derived of the SC and FC of a network and their interactions, then we should be able to predict missing links (Lu et al.,
2015). Lu et al. (
2015) hypothesize that missing links are difficult to predict if their addition causes huge structural changes, and therefore the network is highly predictable if the removal or addition of a set of randomly selected links does not significantly change the networks structural features. Thus, prediction and network inference – even though blurring the distinction between SC and FC (see section
Neuroscience) – can be used to identify the most important links in a network – i.e. where SC, FC and their interactions are most important.
Measuring connectivity
In view of the widespread adoption of the concept of connectivity it may seem surprising that actually measuring connectivity remains a key challenge. However, such is the case. Because connectivity is an abstract concept, operationalizing models into something measurable is not straightforward. The imperative here is to consider SC and FC separately. For the former, some disciplines (e.g. Geomorphology, Ecology) have developed indices of connectivity (e.g., Bender et al.,
2003; Borselli et al.,
2008; Cavalli et al.,
2013). Such indices measure only SC, and their usefulness is a function of the timescales of the interaction of SC and FC. Furthermore, there is a concern as to what is the usefulness of such indices, other than as descriptions of SC: as might equally be said of clustering coefficients and centrality measures. To what extent can/do they enhance our understanding? Systems Biology, on the other hand, does not attempt to measure SC
per se, but infers SC based on knowledge accumulation of the system. In that sense, connectivity may be seen as a means of describing current understanding. Neuroscience, in contrast again, measures connectivity directly through experimentation. How far such an approach could be applied in other disciplines raises the issue of ethics, as discussed in 3.6.4. Only in the case of Computational Neuroscience, which deals with analysed entities (the properties of which are defined
a priori) is measuring SC straightforward.
Of the two, FC poses the greater measuring problem. In network terminology, FC may be thought of as the links in the network that are active, and is thus easier to derive if a network description of the system’s SC exists. Without such a description, FC can be derived from fluxes (e.g. movement of animals), but the measurement of fluxes may present its own difficulties (e.g., in Geomorphology).
Link prediction is also a potentially useful tool in deriving a network-based abstraction of a system where it is infeasible to collect data on SC and FC required to parameterise all links, or where links, by their very nature, are not detectable (Cannistraci et al.,
2013). This problem of observability is inherent in Systems Biology where link types can be very diverse and it has already been noted that databases will drift in time. Therefore, the topological prediction of novel interactions in Systems Biology is particularly useful (Cannistraci et al.,
2013). The use of link prediction also raises the possibility that data can be collected to represent the subset of a network (therefore reducing data collection requirements), and link prediction be used to estimate the rest of the network (Lu et al.,
2015).
Separate, but directly linked to measuring connectivity, is analysis of the measurements. The most commonly applied approach is the use of graph theory. This powerful mathematical tool has yielded significant insights in fields as diverse as Social Network Science, Systems Biology, Neuroscience, Ecology, and Geomorphology. However, in many applications of network-based approaches simply knowing if a link is present or absent (i.e. a binary approach) is too basic or artificial, and characterising the capacity of a link or the relative significance of a link within a network is important. This issue can be dealt with by providing a more detailed representation of the network using weighted or directional links. The use of weighted links is common within network science (see for example Barratt et al.,
2004), and is likely to be of particular importance when, for example, applying network-based approaches to hydrology where the capacity of flow pathway (channel) is an important structural element of links within the hydrological network (e.g. Masselink et al.,
2016b). Using a weighted network can provide an additional layer of information to the characterisation of a network that carries with it advantages for specific applications, and to ignore such information is to throw out data that could potentially help us to understand these systems better (Newman,
2004a); hence, the importance of using measures that incorporate the weights of links (Opsahl and Panzarasa,
2009). More recently, further advances have been made in network-based abstractions of systems, for example, in Ecology, multi-layer networks are being increasingly used, which overcome the limitations of mono-layer networks, to allow the study of connections between different types (or layers) of networks, or interactions across different time steps. Similarly, bipartite networks have been used to provide a more detailed representation of different types of nodes in a network. These more complex network-based approaches carry with them advantages that a more detailed assessment of connectivity within and between different entities can be assessed. However, whilst there are many advantages in using more complex network-based abstractions of a system (weighted, bipartite and multi-layer networks), there are also inherent limitations as many of the standard tools of statistical network analysis applicable to binary networks are no longer available.
In the case of weighted networks, even the possibility of defining and categorizing a degree distribution on a weighted network is lost. In some cases there are ways to modify these tools for application to weighted networks, but one loses the comparability to the vast inventory of analysed natural and technical networks available. A further problem of assigning weights to network links is that it requires greatly increased parameterisation of network properties, which may in turn start to drive the outcome of using the network to help characterise SC and FC and may influence any emergence we might have otherwise seen. However, in recognition of not throwing away important information associated with the weights of links, there are increasingly tools available to deal with weighted links, including: the revised clustering coefficient (Opsahl and Panzarasa,
2009); node strength (the sum of weights attached to links belonging to a node) (Barratt et al.,
2004); average node strength (the average strength,
s, of nodes of degree
k, i.e.
s = s(k), which describes how weights are distributed in the network) (Menichetti et al.,
2014); and the inverse participation ratio (the average inverse participation ratio of the weights of the links incident upon nodes of degree
k, i.e. Y = Y(k), which describes how weights are distributed across the links incident upon nodes of degree
k) (Menichetti et al.,
2014).
As already discussed in the case of Ecology, a limitation of bipartite networks is that to analyze these networks, a one-mode (monopartite) projection of the network is required, as many of the tools available for monopartite networks are not so well developed for bipartite networks. An important issue when analyzing bipartite networks is therefore devising a way to obtain a projection of the layer of interest without generating a dense network whose topological structure is almost trivial (Saracco et al.,
2017). Potential solutions to this issue include projecting a bipartite network into a weighted monopartite network (Neal,
2014) and only retaining links in the monopartite projection by only linking nodes belonging to the same layer that are significantly similar (Saracco et al.,
2017). A further issue is that it is often not possible to recover the bipartite graph from which the classical form has been derived (Guillaume and Latapy,
2006). Developments are being made in our ability to analyze bipartite networks directly; for example, progress has been made in developing link-prediction algorithms applicable to bipartite networks (e.g. Cannistracti et al.,
2013).
Similarly, to apply standard network techniques to multi-layer networks requires aggregating data from different layers of a multi-layer network to a mono-layer network (De Dominico et al.,
2013) which can result in a lot of valuable information being discarded (Kivela et al.,
2014), although approaches are being developed to reduce the amount of information loss (De Dominico et al.,
2014).
Careful consideration of the most appropriate tools is thus required when measuring connectivity using a network-based abstraction. Key questions are: Is it practical/doable to collect the data required to parameterise the network-based model of the system? Can a sensible projection of a bipartite network be derived, to facilitate analysis of the network? Is it possible to derive a monoplex abstraction of a multiplex network without losing too much information?