Network theory is applied to an array of streamflow gauges located in the Coast Mountains of British Columbia (BC) and Yukon, Canada. The goal of the analysis is to assess whether insights from this branch of mathematical graph theory can be meaningfully applied to hydrometric data, and, more specifically, whether it may help guide decisions concerning stream gauge placement so that the full complexity of the regional hydrology is efficiently captured. The streamflow data, when represented as a complex network, have a global clustering coefficient and average shortest path length consistent with small-world networks, which are a class of stable and efficient networks common in nature, but the observed degree distribution did not clearly indicate a scale-free network. Stability helps ensure that the network is robust to the loss of nodes; in the context of a streamflow network, stability is interpreted as insensitivity to station removal at random. Community structure is also evident in the streamflow network. A network theoretic community detection algorithm identified separate communities, each of which appears to be defined by the combination of its median seasonal flow regime (pluvial, nival, hybrid, or glacial, which in this region in turn mainly reflects basin elevation) and geographic proximity to other communities (reflecting shared or different daily meteorological forcing). Furthermore, betweenness analyses suggest a handful of key stations which serve as bridges between communities and might be highly valued. We propose that an idealized sampling network should sample high-betweenness stations, small-membership communities which are by definition rare or undersampled relative to other communities, and index stations having large numbers of intracommunity links, while retaining some degree of redundancy to maintain network robustness.

Network theory is the practical application of graph theory, which is itself
the study of the structures formed by a system of pairwise relationships

There are many diagnostics used to characterize the topology and behaviour of
networks, but we will primarily be concerned with three major and widely used
properties: the degree distribution,

The application of these three fundamental graph theoretical measures to real
networks has revealed the existence of a diverse range of network topologies

The simplest network is a regular network, where, by definition, each node has the same number of degrees. A simple example is a 3-D Cartesian grid. In the special case where each node is connected to every other node, the network is said to be fully connected. Regular networks display a wide range of properties because there are many ways to construct them while keeping the degree uniform across all nodes. In general, however, regular networks are highly clustered, and therefore said to be stable, but have long average path lengths, implying inefficiency. In the context of complex networks, stability means that the removal of any randomly chosen node will have little effect on the network as a whole, while efficiency means that information may easily be propagated across the network because the average path length is small. Another fundamental type is the random network. Random networks are networks whereby pairs of nodes are connected randomly. Random networks have a small clustering coefficient and a small average path length, which means that they tend to be unstable but efficient.

While regular and random networks serve as useful idealizations, they are not
often observed in real-world phenomena. Instead, the so-called
“small-world” network has been found to describe a number of networks found
in nature and engineering. Small-world networks are regarded as a hybrid of
random and regular networks because they are highly clustered (like regular
graphs) and have short path lengths (like random graphs)

One subset of small-world networks, known as scale-free, has been
particularly successful in describing real systems. The degree distribution
for these networks asymptotes to a power law relationship for large

Here, we apply the analytical and interpretive framework of complex network theory to streamflow data, with two goals in mind. The first is simply to broach an interesting and fundamental scientific question: might regional streamflow data be quantitatively represented as a formal network, and, if so, what are the corresponding network theoretic properties, and, in particular, into what fundamental class of network architecture do streamflow data fall? That is, we explore the use of network theory and historical streamflow observations to characterize a regional system of stream gauges. Indeed, the very fact that a collection of stream gauges is typically referred to as a “network” begs for the application of network analysis. We accomplish this task by applying generally accepted approaches of network analysis to daily flow data and then assessing how our outcomes relate to established network topologies. In doing so, minor analytical or interpretive adjustments from prior applications of network theory need to be considered, as discussed in due course below. The overall notion, however, is straightforward in principle: we test the idea that stream gauges constitute nodes in a formal graph theoretic construct as described generically above, and the relationships between the flow time series measured at each such station form the links.

Our second goal is to assess whether these network theoretic results might inform the optimal design of hydrometric monitoring systems. As network theory describes the complex relationships between a system of measurement points – in our case, hydrometric stations – it seems reasonable to conjecture that certain outcomes from this theory might contain insight that could be useful in hydrometric monitoring system design. Because our implementation of network theory is based on historically observed hydrologic time series, this information would take the form of guidance on deciding which existing stations are most important, least important, or important in various different respects. More specifically, the results might be used to guide decisions about the placement or removal of gauges within the region while retaining the maximum amount of information. In other words, our analysis helps address questions such as the following: what is the degree of redundancy in the current network? Are there under-sampled regions? Is the network, in its current state, stable and efficient?

The study is conducted within the geographic context of the Coast Mountains of British Columbia and Yukon. As discussed in more detail below, this region, which spans almost 2000 km along the Pacific coast of Canada and adjacent interior regions, exhibits a distinctive range of streamflow regimes. It receives high annually averaged precipitation, and the extreme vertical relief, exceeding 4000 m over short distances, lends itself to microclimates and complicated hydrologic dynamics which are strongly varied in both space and time. Both the forest and glacial hydrology of the region, for example, are highly complex and remain incompletely understood. Furthermore, using stream gauges to capture such complexity over a large swath of difficult terrain is challenging, especially under the constraint of a finite operating budget and logistical challenges associated with establishing and maintaining gauging stations, so that any additional guiding information regarding sampling system design may be useful.

The work presented here has some practical limitations which should be
recognized. As a first-of-its-kind investigation, we elect to maintain
simplicity in certain aspects of the analysis. Earth science applications of
network theory are growing rapidly, but remain in their relative infancy. The
preponderance of these applications appears to focus on global climate
dynamics

Similarly, practical hydrometric sampling system design is a function of many
considerations, and some of the most powerful of these are in some sense
non-scientific. Factors influencing real-world gauge placement include
capital and maintenance costs, remoteness, legal authorization for land
access, occupational health and safety considerations, availability of
hydrodynamically and geomorphologically suitable sites for gauge installation
and stable rating curve development, and specific engineering or
socioeconomic drivers for station placement. Examples of the latter include
the need to monitor a particular river at a particular location to constrain
the design of a bridge or highway, set instream flow requirements for a river
with special ecological significance, monitor high-flow conditions for a
downstream inhabited flood plain, estimate water availability for a
particular water supply utility, provide key input information to an
environmental assessment process around a proposed natural resource
development project, and so forth. That said, there is a long history of
using quantitative analysis of environmental data to provide information that
might enable improved sampling system design, including correlation, cluster,
principal component, information theoretic (entropic), geostatistical, and
other types of analysis

With this in mind, our results confirm that network theory can indeed be successfully used to describe inter-gauge hydrologic relationships, and to guide sampling system design in a novel way which seems fruitful and warrants further investigation by the hydrologic community. The results additionally add to the broader literature in network theory by quantitatively identifying the network properties and, in particular, the fundamental network topology associated with the terrestrial hydrologic cycle.

In general, streamflow is determined by the interaction of weather and
climate with the terrestrial environment. The specific factors which
determine the nature of observed daily streamflows (i.e., the hydrograph) in
the Coast Mountains are numerous. The region consists primarily of temperate
rain forest, but also includes extensive glaciated alpine areas and some
drier inland locations. The broad meteorological context involves the
progression of a series of North Pacific frontal
storms propagating roughly eastward across the
region over the November-to-March storm season, occasionally with warmer
tropical or sub-tropical moisture feeds associated with atmospheric rivers.
Generally drier conditions prevail during the summer. The first-order
controls on local terrestrial hydrologic responses to this meteorological
forcing are drainage elevation and drainage area, which can be viewed as
gross descriptors incorporating or parameterizing a number of complex
characteristics and processes (precipitation type, ice cover, forest cover,
groundwater, soil moisture, storage, and so forth). Drainages in the Coast
Mountains exhibit a wide range in mean basin elevation and drainage area,
which in turn creates a variety of hydrograph types. Broadly speaking,
however, streamflow hydrographs in the Coast Mountains can be classified by
their dominant freshwater source: rainfall, snowmelt, and glacier melt

Systems dominated by rain are typically found on the windward (western) side
of the Coast Mountains, and they tend to have small, low-elevation drainage
areas which receive precipitation mostly in the form of rain. Peak flows are
often observed during autumn and winter, concurrent with peak rainfall, while
low flows occur in late summer when rainfall is at an annual minimum
(Fig.

Selected examples illustrating the four main types of annual
hydrographs found in the Coast Mountains of British Columbia and Yukon as
described by

Daily discharge data for all of Canada are maintained and archived by the
Water Survey of Canada. In this study, only stations with continuous daily
discharge records were selected, and geographic range was constrained to
stations on rivers originating in the Coast Mountains (Fig.

Map of the Canadian west coast showing the 127 Water Survey of Canada (WSC) streamflow gauging stations used in this study. The stations are coloured according to the first three characters in the WSC naming convention (example – 08M), which defines the stations according to subdivisions of the major drainage basins. The size of each circle scales with the logarithm of the drainage area. The streamflow database was subsetted for stations draining the Coast Mountains.

A total of 127 stations met the selection criteria. The distribution of
stations primarily reflects the population distribution, meaning that the
greatest density of stations is found near the dense urban centres of
southwestern British Columbia. Drainage elevation statistics were computed by
constructing a digital elevation model (DEM) for each gauged basin. Gridded
tiles from three DEM products were used: the 25 m British Columbia Terrain
Resource Information Management (TRIM), the 30 m USGS National Elevation
Database, and the 30 m Yukon DEM. Mean elevation was calculated as the
average of all cells for each gauge basin using the ESRI ArcGIS Arc/Info and
Spatial Analyst/GRID software. Mean drainage elevation ranges from 127 to
2252

In some applications of network theory, the decision of whether to assign a
link to a pair of nodes is straightforward. For example, in a social network,
friendships define the links between people. In the case of the Internet,
websites can be unambiguously connected by hyperlinks. In other applications,
there might not be a straightforward binary relationship between nodes,
meaning it becomes necessary to consider empirical relationships. A simple
and common method is to assign links to node pairs which share a linear
(Pearson) correlation coefficient,

If links are defined by a threshold correlation coefficient, then the
question of which threshold to choose naturally arises. A few specific
methods have been explored in prior studies. Here, we use

When calculating the correlation matrix, a pairwise-complete method was
chosen to avoid the errors that could otherwise be introduced by
interpolating over missing data. The correlation matrix is then thresholded
at

The network formed by the 127 streamflow records distributed across the Coast
Mountains has a total of 1247 pairwise links between the stations. The
average number of degrees per node is 19.6, the minimum is 0 (station numbers
08AA009, 08EE0025, 08FF006, and 08MH029), and the maximum is 43 (08EE020).
The connections are illustrated in Fig.

Georeferenced representation of the streamflow network. A line is
drawn between each pair of stations if their linear correlation coefficient
exceeds 0.7. The station colours are based on the WSC designated subregion as
in Fig.

Discrete representation of the degree distribution for the
streamflow network (grey bars). Also shown are ensemble means of the
equivalent degree distributions for a random network (solid line) and for
a scale-free network with

As discussed in the introduction, we can place the streamflow network in
context with the known network topologies by computing three network
properties, the degree distribution (

Therefore two possibilities remain – small-world (but not scale-free) or
random. The difference between these cases lies in the clustering coefficient
and average path length. A network is considered small-world if

As noted in the introduction, small-world networks are characterized by stability and efficiency. A stable network is one that retains its integrity even if nodes are removed because of the high degree of clustering. In other words, the removal of a node at random will likely not fragment the network. In the context of the streamflow network, this means that if a randomly selected station is removed then it should be possible to recover most of its information through the interdependence of the stations. Network efficiency is sometimes thought of as the ease with which information propagates across the network. A network with a small average path length is highly efficient because two arbitrary nodes are likely to be separated by only a few links.

While assigning links to stations sharing a correlation coefficient in excess
of 0.7 assures that the links are statistically and intuitively meaningful,
one might question whether the specific threshold value has any impact on the
structure of the network. An excessively low threshold, below perhaps 0.4 or
so, causes identification of links where, in general, none exists in any
statistically or (potentially) physically meaningful way. In the limit of

However, there is still a range of reasonable threshold values which deserve
some attention. To assess whether global network properties of the streamflow
network are sensitive to the choice of threshold, we evaluated the network
for two additional values of the selected threshold,

The streamflow network degree distribution undergoes a few obvious changes
when

Degree distribution,

Network clustering coefficient,

A change in global network properties as a function of correlation threshold
was observed by

Additionally, we explored the impacts of using Spearman rank correlation in
place of Pearson linear correlation, and of deseasonalized anomaly time
series in place of the observed hydrographs. Both affected certain details –
for example, the network contains fewer links at a given threshold
correlation coefficient when the seasonal cycle is removed from the data
because much of the variance in streamflow is associated with seasonality.
Use of Spearman correlation has a tendency to increase the number of links
between stations because rank correlation allows for more complex (yet
monotonic) relationships. However, these choices do not affect the global
network structure as diagnosed by the clustering coefficient or average path
length. Note also that when making the decision to use absolute or anomalous
values, we may additionally refer back to one of the major impetuses for this
paper, which is to use network theory to assess how well the current array of
streamflow gauges samples the hydrology of the Coast Mountains and to explore
how network theoretic insights might help guide future decisions on
streamflow monitoring system design. That is, the emphasis lies on actual
river flows, as might be required for water supply, ecology, civil
engineering, or other potential applications. These actual discharge values
are influenced to a considerable degree by seasonal forcing, and therefore
require direct sampling by a hydrometric monitoring system. Additionally,
sharing a common seasonal flow regime, especially within our study region
(where seasonal regimes exhibit great basin-to-basin heterogeneity as
discussed in detail above), is a fundamentally meaningful and operationally
important physical link between two stations. That is, we would in general
wish the network analysis, and a streamflow monitoring system, to directly
capture such connections. Further discussion on the use of anomalous values
of geophysical data and network analysis can be found in

Many networks consist of distinct groups of highly interconnected nodes,
which are often referred to as communities. This is particularly true of
small-world networks observed in nature

Consider Fig.

In this section we will formally analyze the streamflow network for community structure and show that the delineation made above is an oversimplification, but still accurate in the most general sense. We then explore what causes community structure in the streamflow network, and also what the community structure can tell us. It is important to note that the following does not require assumptions regarding network topology. The corresponding results are, therefore, in some sense independent of the foregoing conclusions.

Graph representation of the streamflow network. The vertices were
arranged by the algorithm of

Many algorithms have been developed to find community structures in graphs

Given the rather imprecise definition of a community, we cannot expect that
there will be a single correct algorithm which can find the one true answer.
Thus the task of choosing an algorithm comes down to practical
considerations. For example, run times can vary considerably between the
algorithms because the computational costs of some scale linearly with the
number of nodes or edges, while others scale exponentially

Although we cannot assess whether an algorithm can find the single true
answer (if such a thing exists), we can compare the algorithms to see if they
find the same answer. We therefore applied eight such algorithms to the
hydrologic data: walk trap, fast greedy, leading eigenvector, edge
betweenness, multi-level, label propagation, info map, and optimal. A review
of these various algorithms is beyond the scope of our article. Interested
readers may refer to

In addition to finding similar community structures, the algorithms return a similar, but not identical, number of communities (between 8 and 10). In general, all of the algorithms find three large communities, and five to seven smaller ones. The three largest communities contain between 84 and 94 % of the total number of stations. All of the algorithms find a handful of communities which contain only one member (station nos. 08AA009, 08EE025, 08FF006, and 08MH029). This is a trivial result (in a strictly graph theoretic sense) because these particular stations have no links to the network. The edge betweenness algorithm (discussed below) also identified a community composed of a single station which, unlike the cases just mentioned, had links to other stations (08AA008, two links).

If we consider the reasonable consistency in the number of communities found by each algorithm, the tendency for most stations to fall within three large communities, and the high NMI scores, it is apparent that choice of algorithm is not of critical importance. We therefore proceed by using the edge betweenness algorithm to isolate the communities because it is well documented, and because its NMI index ranges from 0.86 to 0.94, indicating a good agreement with the other algorithms.

The edge betweenness algorithm works as follows. The algorithm identifies
communities by finding bottlenecks (or bridges) between highly clustered
regions of the graph. These bridges are found by exploiting a property known
as edge betweenness

Comparison of community detection algorithms with the normalized
mutual information (NMI) index

WT: walk trap; FG: fast greedy; LE: leading eigenvector; EB: edge betweenness; ML: multi-level; LP: label propagation; IM: info map; and O: optimal.

More specifically, the algorithm works by first calculating edge betweenness
scores for every edge in the network. The edge with the highest score is
removed, which in some sense splits the network, and the edge betweenness for
the resulting network is calculated again. The algorithm is reminiscent of
hierarchical divisive (top-down) clustering methods in statistical analysis
and data mining, partitioning larger-scale communities into progressively
smaller ones in a dendritic fashion. At each step, a measure of the optimal
community structure called modularity is calculated

Application of the edge betweenness algorithm to the streamflow network sorts
the stations into 10 communities. Communities 3, 4, and 8 are the largest,
and together they contain 90 % of the stations. Five communities consist
of a single station. A summary of the community membership, along with a
basic description of a typical station in each community is given in
Table

The geographic distribution of the communities is mapped in
Fig.

Summary of the community analysis. The communities were found using
the edge betweenness algorithm

If the streamflow communities are not solely defined by the geographic distribution of their members, then what forms them? The answer must lie in the hydrographs, since the network was defined by their covariance. To investigate this, a representative hydrograph was computed for each community by first forming a median annual hydroclimatology for each station using the same 10-year time series that defined the network. The climatological median discharge for each station was then normalized by drainage area to form the unit area discharge. Finally, the median unit area hydrographs were averaged by community to form a representative annual hydrograph.

Streamflow station map coloured according to community membership.
The communities were identified with the edge betweenness algorithm

Representative unit area hydrographs for each of the 10 communities.
The hydrographs were created by averaging the 10-year median climatology for
all stations within the community. The line colours are consistent with the
map in Fig.

The representative annual hydrographs are shown in
Fig.

How can two stations of the same hydrologic type be poorly correlated? The
average annual cycle and its overall physical controls are only one aspect of
a river's dynamical properties. As an example, consider two small pluvial
basins, one on an island of Haida Gwaii on the northern BC coast, and the
other 800 km away on Vancouver Island on the southern BC coast. Although
peak flow for both stations occurs in winter, when rainfall is highest, the
rainfall is episodic because it is caused by frontal systems embedded in low
pressure cyclones. Even if the same weather system impacts both stations, the
travel time between stations will create a phase lag which is large enough
compared to the falling limb to create a weak zero-lag correlation. More
importantly, in many cases a specific storm will affect one region but not
another 800 km away. Indeed, precipitation teleconnections to El
Niño–Southern Oscillation and the Pacific Decadal Oscillation differ
fundamentally between the southern and northern BC coasts

A similar argument can be made for nival stations, although the mechanisms might be different. Day-to-day, basin-to-basin variability in the snowpack and/or melt rates (set by temperature or rain-on-snow events) can affect peak flow timing or the length of the falling limb, and therefore impact the correlation between two stations. Although the dominant forcing causing snowmelt is seasonal, the spatial scale of specific forcing anomalies (i.e., weather) could easily create spatial variability on scales smaller than the distance separating two different nival basins.

It is also interesting to explore how these network theoretic communities
might reflect different catchment properties. For example, both the
day-to-day streamflow dynamics and the overall seasonal hydrologic regime
exhibited by data from a particular hydrometric station are determined to a
significant extent by the elevation of the upstream basin area since in the
Coast Mountains elevation determines in large part whether the basin receives
daily precipitation as rain, snow, or some mixture of the two, and also what
time of year the corresponding runoff occurs. Thus it might be possible to
understand the community structure, at least in part, in terms of basin
elevation. Consider Fig.

We can also test whether the communities are influenced by the drainage area
upstream of the stream gauge. Drainage area impacts hydrological time series
because it might indicate the potential for storage mechanisms (lakes,
groundwater, etc.), which would in turn dampen impulsive precipitation events
and “redden” the spectrum of a theoretical hydrograph. This means all large
basins might have similar hydrographs (all else being held equal) and
therefore fall within the same community. The drainage areas, sorted by
community, are shown in Fig.

Boxplots of mean basin elevation grouped by community. The colours
are consistent with the map in Fig.

Boxplots of upstream basin drainage area grouped by community. The
colours are consistent with the map in Fig.

Alternatively, the division between communities 3 and 4 might also be driven
by the increased likelihood for stations in community 3, which extends
further north than community 4, of having more permanent ice coverage or a
thicker snowpack. Unfortunately this cannot be tested quantitatively because
ice cover data were not readily available for about half of the stations in
this analysis. However, mid-to-late summer differences in median hydrograph
form are consistent with this interpretation, with community 3 exhibiting a
more seasonally extensive melt freshet than community 4
(Fig.

The edge betweenness community detection algorithm placed 90 % of the stations into three communities, while the remaining 10 % fell within single-member and small-membership communities. Small-membership communities have daily streamflow dynamics that are uncommon because they represent undersampled and/or rare hydrometeorological regimes, which we will argue makes them important if the goal of a hydrometric network is to sample the inherent hydrometeorological diversity of the Coast Mountains. As we will show here, there are also several additional important stations which were not directly identified by the community analysis.

A closer inspection of the streamflow network representation in
Fig.

The local network property that sets them apart is called betweenness, a concept we broached briefly in our discussion of community detection algorithms. Formally, the betweenness of a node is the number of geodesic paths passing through it, where a geodesic path is the shortest path between a node pair. In fact, the concept of edge betweenness, which was used to identify the community structure, is an extension of the concept of node betweenness. A high betweenness node would host a great amount of geodesics in the same way that a bridge hosts a great amount of traffic in a transportation network. As for the community-finding process, no assumptions are required regarding network topology.

The bar plot in Fig.

The seven stations together connect communities 3, 4, and 8, the three
largest communities in the streamflow network. Community 3 occupies a large
part of the Coast Mountains but, interestingly, the high-betweenness stations
within it are all located in southern BC. These particular stations contain
links to stations in communities 4 (southern–central BC) and 8 (Vancouver
Island and southwestern BC). Intuitively, we expect the hydrograph of a
high-betweenness station to bear some resemblance to the multiple communities
it joins. This appears to be borne out
in practice: the climatological hydrographs for each of these seven stations
resembles the mixed rain–snow regime (e.g., Fig.

In terms of network theory, high-betweenness stations are important to network stability given their role as bridges between communities. For this reason we argue that they are essential members of the network, but not in the same way as the stations forming the small-membership communities. The loss of just a few high-betweenness stations would fragment the network into isolated communities. Information flow, or in our context, transferability of discharge measurements across locations, would be restricted in their absence.

Bar plot of the betweenness scores for every station, with several
high-betweenness stations highlighted. The station colours are based on the
WSC designated subregion shown in Fig.

The various network diagnostics and tools have provided micro-level (i.e., individual stations) and macro-level (community structure and network architecture) descriptions of the streamflow network. The question now becomes: how can we use these results to inform and guide streamflow network design? We begin by first summarizing what the network analysis told us about the data from the current monitoring system. As discussed above, the architecture of the streamflow network is consistent with the small-world class of networks. Small-world networks are considered stable, meaning that the removal of a node at random is unlikely to fragment the network. In terms of the streamflow monitoring system, this implies there may be a sufficient amount of redundant information, or a relatively large number of station pairs with high correlation coefficients. A randomly selected station will likely have 19.6 connections (the network-wide node degree average). As such, the loss of any one station selected at random will probably not result in the loss of a significant amount of information or a fragmented network. However, if a high-betweenness station is lost, then the likelihood of fragmenting the network is increased. Moreover, the loss of a station which belongs to a single-membership community is essentially the loss of unique and therefore unrecoverable information because there is no means to reconstruct its streamflow.

The edge betweenness community detection algorithm identified 10 communities
within the streamflow network, but 90 % of the stations fell within just
3 communities. A community, defined on the basis of network theoretic
analysis, shares specific elements which can be tied back to two general
physical hydrologic characteristics: mean annual hydrograph form reflecting
similar precipitation phasing in this transitional rain–snow region, in turn
largely a function of basin elevation or secondarily latitude and
continentality, and geographic proximity reflecting shared day-to-day
local-to-synoptic scale meteorological forcing. Therefore, the number of
communities reflects the hydrometeorological diversity of the Coast
Mountains, and the number of stations per community sets the extent to which
each distinct hydrologic “family” is sampled. The stations within each
community having the highest number of intracommunity links can be thought of
as index or reference stations (explicitly summarized for the three largest
communities in Table

The community-detection and node-betweenness algorithms identified two types of “outlier” stations. The first type consists of those stations belonging to small-membership communities. These stations represent rare or undersampled hydrometeorological regimes. Such communities may exhibit a median annual hydrograph similar to other communities, but they appear to be sufficiently distant in space that they do not, in general, share the same meteorological forcing with those other communities. Thus, the streamflow time series from one such community cannot be accurately indexed by, or easily reconstructed from, the streamflow time series from another.

The most highly connected (in an intracommunity sense) stations in
each of the three largest communities and the number of intracommunity links
(

The second type of outlier station consists of those with high betweenness scores. These stations contain intercommunity links, which serve to bridge disparate communities. The hydrographs of these stations can be regarded as hybrids of the communities they connect. These might be viewed as do-it-all stations, which provide information about several communities of hydrometeorological variation, though incompletely. The loss of such stations would fragment the network, in principle making it more difficult to recover information.

There is a substantial amount of redundancy (many pairs of stations with a high correlation coefficient) within the three large communities identified in this paper. Stations having a low betweenness score, a high number of degrees, and membership to a large community might be regarded as redundant and thus, perhaps, candidates for decommissioning under, for example, budgetary pressure. However, the network theoretic perspective suggests that this type of redundancy could alternatively be considered a strength of the hydrometric monitoring system, insofar as it implies that the stream gauges, in their present arrangement, form a stable network which is resilient to the unintended loss of a node (as might occur operationally due to equipment failure, for example). Much of the high interconnectedness within each of the three large communities may simply be driven by seasonal snow and ice melt from mid- to high-elevation basins, or, in the case of the pluvial drainages of Vancouver Island and the low-elevation regions of southwestern BC, a dense array of gauges sampling a sufficiently small region.

Given the insights gained by analyzing the current network, what might the optimal sampling network look like? As discussed in the introduction, this depends on many practical considerations which are far beyond the scope of this study and, perhaps, any statistical data analysis-based method for hydrometric monitoring system design. Some of these considerations include budget constraints, station accessibility, or special applications (such as fisheries studies, climate variability and change detection, or the need to monitor a particular river for a particular purpose, such as an assessment for microhydropower generation potential or the design of bridge crossings, for example). In the absence of these considerations, or in addition to them, a sampling program would ideally capture all of the possible types of streamflow dynamics in the region. In the context of network theory, this amounts to maximizing the number of communities sampled because the number of communities reflects hydrometeorological diversity. The number of members in each community should be large enough to provide some redundancy as a safeguard to ensure minimal information is lost if a station fails or is decommissioned; that said, redundancy might also be viewed as an argument in favour of station closure, as noted above. In any event, the small number of stations having high betweenness, and the stations which are members of a small community, constitute two types of particularly high-value stations which should not be removed from the streamflow monitoring system under cost-cutting, for example. Additionally, stations with a high number of intracommunity links might be identified as index or reference stations for their respective communities, and should be viewed as high-value stations.

In this paper, we have analyzed the hydrology of the Coast Mountains by applying network analysis tools to a collection of streamflow gauges. Our motivation was to characterize the existing network and place it in context with idealized and observed networks, with an eye to informing streamflow network design.

Daily streamflow data in this region proved amenable to network theoretic analysis. In particular, it was found to display properties consistent with the small-world class of networks, a common type observed in many disciplines. A small-world network implies stability, and that its structure is resilient to the loss of nodes. Interestingly, the results also suggest that the streamflow network in this region is not of the scale-free type. There is precedent for small-world, non-scale-free networks, but they appear uncommon.

Community-detection algorithms separated the network into three main groups, each containing dozens of stations, plus a handful of smaller groups. We then show that these 10 individual communities appear to be defined by both (i) their typical annual hydrograph forms, which in turn correspond to various considerations such as basin elevation, and (ii) their geographical proximity, which in turn corresponds to shared or different meteorological forcing. That is, (i) and (ii) together form distinct classes of daily-to-seasonal hydrological dynamics which are identified by the community-finding algorithm. The number of communities reflects the diversity of such hydrologic dynamical classes, and the number of stations per community sets the extent to which each regime is sampled.

The network theoretic outcomes provide a different way of viewing spatiotemporal hydrologic patterns and, in particular, a novel perspective on the old question of optimal hydrometric monitoring system design. We argue that the idealized sampling strategy should span the full range of dynamical classes described above, and additionally that it should retain some redundancy in the event of station failure, which may be facilitated by the small-world topology identified for this network. Furthermore, we identified a number of stations which warrant special attention because they characterize rare, undersampled, or information-rich hydrometeorological dynamics. Specifically, we propose that from a monitoring system design perspective, the most important stations are (1) those which have a large number of intracommunity links and thus serve as indices for their respective communities, (2) those with high betweenness values, and which thus serve as do-it-all stations embedding information about multiple communities, and (3) those which are members of single-membership or small-membership communities, as their hydrometeorological dynamics are poorly sampled by the existing monitoring system and cannot be readily reconstructed from other hydrometric stations.

The network analysis as applied in this paper required us to choose a number of parameters. For example, it was necessary to fix the threshold correlation coefficient to define the pairwise relationships between streamflow gauges. We reiterate that our analysis showed that the network architecture, a global property, is not sensitive to the threshold coefficient within a realistic range of values. However, we do expect that changing the coefficient will likely impact the details of community membership and the individual high-value stations identified by community detection and betweenness. This is obviously due to the fact that some pairwise relationships will simply change as the threshold correlation coefficient is varied. Care should be taken to understand which stations share correlation coefficients near the threshold before using a community or betweenness analysis to guide practical decisions on whether to alter the streamflow monitoring system.

In addition to hydrometric monitoring system design, this work will hopefully
inspire further applications of network theory to regional hydrology. As
such, and given the relative newness of network theoretic applications within
water resources science as discussed in the introduction, one could envision
any number of (potentially) useful extensions or refinements. A few are
listed as follows. Repeating the analysis with deseasonalized discharge time
series might be interesting because it would remove the seasonally driven
component of serial correlation, and therefore more clearly reveal regional
climate or weather effects, but might be less useful for hydrometric network
design as it would not speak directly to actual streamflow values. The
analysis could also be repeated with time periods of different lengths, or
with climate-conditioned networks formed by selecting data from particular
seasons or years (e.g., winter only, or El Niño years). Application of the
methods in different regions could prove interesting, as the results were
found to reflect (in part) hydrologic regime types which, generally speaking,
would be different elsewhere. Another option is to apply these methods to
derived streamflow metrics, such as annual time series of peak flow, freshet
start date, or minimum 7-day mean discharge, though it remains to be seen
whether the attendant reduction in the number of samples (by a factor of 365,
essentially) might be debilitating to the network analysis algorithms. Our
application of network theoretic community detection algorithms to streamflow
data could be seen as a new approach to watershed typing, and the success of
this procedure provides some confirmation of the possibility, raised by

The edge betweenness community finding algorithm identified 10 communities
within the streamflow network. In Table

Membership table of the communities in the streamflow network as determined by the edge betweenness algorithm.

The authors would like to thank Judy Kwan at Environment Canada for her GIS expertise in drainage elevation statistics, and the referees Mishra Ashok and Bellie Sivakumar for their valuable comments. Edited by: J. Vrugt