State transitions in geomorphic responses to environmental change

Geomorphology 204 (2014) 208–216

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State transitions in geomorphic responses to environmental change Jonathan D. Phillips ⁎ Tobacco Road Research Team, Department of Geography, University of Kentucky, Lexington, KY 40506-0027, USA

a r t i c l e

i n f o

Article history: Received 4 April 2013 Received in revised form 28 July 2013 Accepted 10 August 2013 Available online 17 August 2013 Keywords: State transition State transition model Geomorphic response Environmental change Synchronization Complex response

a b s t r a c t The fundamental geomorphic responses to environmental change are qualitative changes in system states. This study is concerned with the complexity of state transition models (STM), and synchronization. The latter includes literal and inferential synchronization, the extent to which observations or relationships at one time period can be applied to others. Complexity concerns the extent to which STM structure may tend to amplify effects of change. Three metrics—spectral radius, Laplacian spectral radius, and algebraic connectivity—were applied to several generic geomorphic STMs, and to three real-world examples: the San Antonio River delta, soil transitions in a coastal plain agricultural landscape, and high-latitude thermokarst systems. While the Laplacian spectral radius was of limited use, spectral radius and algebraic complexity provide significant, independent information. The former is more sensitive to the intensity of cycles within the transition graph structure, and to the overall complexity of the STM. Spectral radius is an effective general index of graph complexity, and especially the likelihood of amplification and intensification of changes in environmental boundary conditions, or of the propagation of local disturbances within the system. The spectral radius analyses here illustrate that more information does not necessarily decrease uncertainty, as increased information often results in the expansion of state transition networks from simpler linear sequential and cyclic to more complex structures. Algebraic connectivity applied to landscape-scale STMs provides a measure of the likelihood of complex response, with synchronization inversely related to complex response. © 2013 Elsevier B.V. All rights reserved.

1. Introduction The most fundamental landscape responses to environmental change are not quantitative changes in rates of, e.g., erosion, deposition, or shoreline retreat. Rather, the most important changes are qualitative changes in system states. In a coastal environment, for instance, changes to marsh surface accretion rates are less important than transitions of tidal to marsh to, say, open water or to supratidal marsh. This study is concerned with models of such state transitions—specifically, the extent to which networks of geomorphic state transitions may be prone to complex responses to environmental change. Specifically, this study addresses the complexity of networks of state transitions, stability of patterns of geomorphic change, and synchronization of geomorphic responses to change. Complexity in this case concerns the extent to which the structure of networks of (potential) geomorphic transitions may tend to amplify effects of change. The concern with synchronization is chiefly with respect to the extent to which state transitions are contemporaneous or lagged at the landscape scale, an issue related directly to the geomorphic concept of complex response (Schumm, 1973, 1977). A state transition is a change that results in a qualitatively different landform, geomorphic environment, or landscape unit. Thus, for instance,

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an increase or decrease in shoreline erosion rates would not constitute a state transition. However, a change from an eroding to a stable or accreting condition would be a state transition, as would the changes among nearshore, beach, dune, backbarrier, and marsh environments. There exist a number of conceptual, analytical, and interpretive models of state transitions in landscape response to environmental changes and disturbances. These may take some standard or canonical forms, including linear sequential, cyclical, radiation, cross, mesh, and random patterns. This study will examine properties of these standard forms and compare them to three field examples with respect to network complexity, stability, and synchronization. Rather than analysis or prediction of state transitions at fixed locations, the concern here is with landscape-scale responses to environmental change reflected in the spatial pattern of geomorphic system states and the nature of change propagation. The use of ecological state transition models (STMs) has recently been extended into this type of application (Perry and Enright, 2002; Bestelmeyer et al., 2009, 2011), and Phillips (2011a) explored the use of graph theory for this type of analysis in geomorphic and pedologic systems. Here complexity is concerned with the extent to which changes (whether due to internal development or interactions, or external disturbances or changes in boundary conditions) tend to be amplified by the pattern of transitions. This reflects, for example, the likelihood that a localized disturbance may produce responses elsewhere in the landscape, and the uncertainty with respect to predicting those responses. Network complexity also reflects whether a landscape-scale environmental change is

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likely to produce homogeneous or heterogeneous responses within the landscape. The index of stability analyzed in this study reflects the spatial resolution or time step necessary to observe stability. The more unstable a network is, the smaller or finer the time step must be to observe stability. In a geomorphic context, the relative stability indicates the likelihood that localized changes or disturbances will persist and grow over time. Synchronization may be literal—i.e., are state transitions in response to environmental change contemporaneous throughout the landscape? Phillips (2013) also discussed inferential synchronization in geomorphic systems: the extent to which observations or inferences at a given time can be applied to historical reconstructions or future predictions. 2. State transitions 2.1. Standard models of state transitions Patterns of state transitions, whether temporal sequences or spatial gradients, may be analyzed as networks, using graph theory. Thus the standard, canonical, or benchmark models are discussed here in terms of the structure of a graph depiction. Single-path, linear sequential succession-type models describe a pattern whereby after a change or disturbance the environmental system follows a regular progression of stages or states, before reaching some final stable condition. This is the case, for example, in many channel evolution models for incised or channelized alluvial streams (c.f. Schumm et al., 1984; Simon, 1989; Bledsoe and Watson, 2002; Doyle et al., 2002; Watson et al., 2002). The classic model of the stages of karst landscape evolution originating with Cvijic (1918) is another example; classical Clementsian-type ecological succession models are also of this type. The chrono-, topo, bio-, hydro-, and lithosequence models in soil geomorphology are also linear sequential forms, though they are based on the notion of holding all but a single soil-forming factor constant (Birkeland, 1999; Schaetzl and Anderson, 2005). Cyclical state transition models are characterized by a repeated singlepath cycle through a sequence of states. W.M. Davis's (1922) cycle of erosion is one example, whereby uplifted peneplains evolve through stages of youthful, mature, old age, and peneplain topography, with uplift renewing the cycle. Sequence stratigraphy models of sedimentary system tracts in response to sea-level change are linear sequential in the context of a single episode sea-level rise or fall, but cyclical in the context of both transgressive and regressive episodes (e.g., Christie-Blick and Driscoll, 1995; Miall and Miall, 2001; Catuneanu, 2006). Radiation-type models involve landscape divergence from a single state or condition into multiple discrete states. One example is the degradation of semiarid grasslands, where the uneroded grasslands may be transformed into a mosaic of uneroded, vegetated, nutrient-rich patches; minimally-vegetated nutrient-poor patches; and unvegetated, highly eroded sectors (e.g., Parsons and Abrahams, 1996; Bergkamp, 1998; Puigdefabregas et al., 1999). Another is the fragmentation of salt marsh surfaces into a mixture of marsh, mudflats, salt pannes, and open water in response to relative sea level rise (e.g. Reed, 1990; Reed and Cahoon, 1992; Day et al., 2011). Convergence models are the conceptual opposite of radiation—in this case different states all transition to the same end or attractor state. For instance, if various straight, convex, and concave slope profiles all eventually evolve toward a convexo-concave (convex upper, concave lower) profile, as sometimes occurs in humid climates, this is a convergent pattern of state transitions. This type of pattern is implied in several conceptual models of landform and landscape evolution, including the traditional dynamic equilibrium model and hypotheses of convergence toward a critical state (c.f. Phillips, 1999a). Likewise, some narratives of desertification postulate expansion of ergs or other desert environments at the expense of other landforms and ecosystems (for a summary and critique, see Thomas and Middleton, 1994). The

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network structure of convergence and divergence/radiation models are mathematically identical, so they will be treated together in that respect. A cross type pattern has been identified in spatial patterns of connections among habitat types in landscape ecology (Cantwell and Forman, 1993). The cross graph has a single key component connected to all other nodes, but the other nodes are connected only to the central key node, and to two other adjacent nodes. No obvious geomorphic STM examples are known, but the cross represents one standard type of graph structure that is much more strongly connected than the linear sequential, cyclical, radiation, or convergence types. The cross model is potentially applicable to patterns of geomorphic spatial interactions or mass fluxes with a critical central transfer point or corridor. The most strongly connected type is the maximum connectivity graph, where any state can potentially transition into any other. Some vegetation community STMs identified for U.S. rangelands have this structure (NRCS, 2013). A geomorphic example is the bedform state of sandbed streams, where rapid changes in flow conditions can result in changes among plane bed, ripple, dune, and antidune states without necessarily passing through intervening bedform states. Notwithstanding these examples, the maximum connectivity graph is independently useful as a benchmark or reference structures. Other models of geomorphic state changes are variations on the types above, or have a mesh structure (Cantwell and Forman, 1993; Phillips, 2011b). The mesh is a relatively well-connected network (for graphs with N 5 nodes, lying between the cross and maximum connectivity types above in this regard), with a relatively uniform number of edges associated with each node. More complex channel evolution models incorporating multiple pathways may have this structure (e.g., Leyland and Darby, 2008; Phillips, 2012a), along with some ecological STMs. These standard graph types are summarized in Table 1. In this paper key properties of graphs representing patterns of geomorphic state changes will be derived for the standard structures described above, and for three case studies. Though the case studies below all involve decadal-scale state transitions, the examples of standard graph structures given above illustrate the fact that state transitions occur at, and can be analyzed as such at, a broad range of spatial and temporal scales.

Table 1 Standard graph structures for models of geomorphic and other environmental state changes. Graph structure

Geomorphic examples

Comments

Linear sequential

•Simple channel evolution models

Traditional vegetation succession models also linear sequential

Cyclical

•Karst landscape evolution model •Soil state factor sequences •Davisian cycle of erosion •Sequence stratigraphy

•Fragmentation of degraded grasslands •Marsh fragmentation Convergence •Steady-state attractors (“dynamic equilibrium”) •Desertification by dune encroachment Cross •Spatial interaction or flux models with a critical central intersection or bottleneck point Mesh •Complex channel evolution models •Some ecological state-andtransition models Fully •Some ecological state-andconnected transition models •Sand-bed fluvial bedforms Radiation

Incomplete cycles are linear sequential; linear paths reset by disturbance are cyclical Represents divergent evolution

Represents convergent evolution

Involves a single key state connected to all others For N N 5, intermediate in connectivity between cross and fully-connected structures Commonly a reference condition representing a random system where any state can transition to any other

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2.2. Geomorphic state transitions Three examples are presented here, related to geomorphic changes in (1) the San Antonio River delta, Texas, USA associated with fluvial dynamics and sea-level rise; (2) a coastal plain agricultural landscape in North Carolina, USA; and (3) thermokarst systems in northern North America. The first two are related to previous field-based research by the author; the thermokarst example was chosen to represent an example driven directly by climate change.

2.2.1. San Antonio River delta An STM of soil-geomorphic changes in the San Antonio River delta (SARD; Fig. 1) was developed in earlier work (Phillips, 2011c), and graph theory methods were applied to determine the potential complexity of responses to changes in sea-level, fluvial water and sediment inputs, and internal interactions within the delta. Phillips (2012b) examined the history of and controls on avulsions in the SARD. These studies provide the basis for the state transition model shown in Fig. 2. More complete background on the environmental setting, key processes, and interactions are available in those references and in a technical report (Phillips, 2011c). The study area is along the Gulf of Mexico coastal plain in south Texas. The San Antonio River is about 386 km long, with a drainage area of 2419 km2. The San Antonio River joins the Guadalupe River in a tidally-influenced delta area near Tivoli, Texas, about 11 km upstream of the Guadalupe/San Antonio Bay estuary. Climate in the study area is humid subtropical. The geologic framework is entirely Quaternary. Alluvial, deltaic, and coastal deposits within the valley consist primarily of Holocene fills, but some Pleistocene terrace remnants occur within the valley. In the past 80 years at least two major avulsions have occurred,

Fig. 2. Geomorphic state transition model for the San Antonio River delta, Texas.

one of which is still active, and there is field evidence of at least seven others within the past 3 ka (Phillips, 2012b). The SARD encompasses five major geomorphic environments or landscape units, in addition to active distributary and tributary channels. In the lower delta low marsh units are flat except for occasional ponds, frequently flooded (daily), and characterized by saline soils with high organic matter content. High marsh/swamp environments are slightly higher or further upstream. These are occasionally flooded (multiple times per year), with saline soils/sediments. They include a

Fig. 1. Shaded relief model derived from a 3-m resolution digital elevation model, San Antonio River delta, Texas. Inset shows approximate location in Texas.

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i) Direct observation (possible due to ongoing channel shifts occurring during fieldwork; see Phillips, 2011c, 2012b). ii) Spatial adjacency of units. iii) Geomorphic indicators of channel and floodplain sedimentation and erosion (see Phillips, 2011c). iv) Comparison of aerial photographs and images over the 1930–2012 period, and of maps from 1835 to the present. v) Sedimentological and pedological evidence of change (see Phillips, 2011a,c). vi) Previously published sedimentological, paleoecological, and archeological evidence (Donaldson et al., 1970; McGowen et al., 1976; Morton and Donaldson, 1978; Ricklis and Blum, 1997; Weinstein and Black, 2009).

2.2.2. Littlefield Littlefield is a 71 ha agricultural site in Pitt County, North Carolina, on the Atlantic Coastal Plain of the U.S.A. Soils were mapped in detail in support of studies of sediment fluxes and soil redistribution by a combination of fluvial, aeolian, and tillage processes (mass wasting is not significant in this low-relief landscape). Results of those studies are reported elsewhere (Phillips et al., 1999a, 2000; Gares et al., 2006; Lecce et al., 2006, 2008). The site (Fig. 3) is about half row crop agriculture, and half mixed pine and hardwood forest. Maximum relief is 8 m, and slope gradients are entirely less than 0.03 and mostly less than 0.02. This site also has artificial drainage (field ditches), draining to a partially channelized natural stream. Climate is humid subtropical, and the site is on the Pleistocene Wicomico marine terrace. Soils are mainly acidic Paleudults or Kandiudults and Paleaquults (U.S. Soil Taxonomy). Three of the previous studies (Phillips, 2000, 2011a,b) were explicitly concerned with soil transformations associated with surface sediment redistribution, and water table changes caused by artificial drainage. This provides the basis for Fig. 4, which generalizes the soil series at the site according to depth to a seasonal high water table (well vs. poorly-drained soils), the relative thickness of sand and loamy sand surface horizons, and field-edge dunes or sandy deposits with evident podzolization. The transitions are driven by surface deposition or erosion, local water table fall or rise, and either translocation or leaching of iron and humus. As above, Fig. 4 was converted to a graph and adjacency matrix.

A graph is shown in Fig. 2, based on the transitions among geomorphic units and the observed or apparent driving processes. These include processes increasing local surface elevations—sedimentation and accretion by alluvial, deltaic, and coastal processes. Also included are processes reducing elevations—surface erosion or stripping, and subsidence or autocompaction (primarily in the low and high marsh environments). Processes associated with crevasses, some of which result in avulsions, are critical, as are relative water level increases, associated with sea-level rise and channel aggradation. The transformations shown in Fig. 2 were converted to a simple, undirected graph (as all transitions are two-way) and associated adjacency matrix for analysis of the parameters described below.

2.2.3. Thermokarst Thawing and degradation of permafrost at high northern latitudes lead to, under various circumstances, transitions to thermokarst depressions, lakes, and peat accumulations, with reverse transitions possible during climatic cooling. A state transition model for thermokarst transitions is shown in Fig. 5, which is the basis for a graph and adjacency matrix. Thawing drives transitions from the permafrost/incipient thermokarst state to form thermokarst depressions and then thermokarst lakes. Subsequent infilling of the latter can lead to peat-filled depressions. Cooling and freezing can drive transitions from the depression states back to permafrost, and lake drainage from lakes back to thermokarst

combination of alluvial and deltaic deposits and marsh and swamp vegetation. Alluvial floodplain environments include backswamps and flats. These are typically flooded several times per year, and are non-saline. Natural levees are treated as a separate unit, and occur along channels, mainly in the usually freshwater, non-tidally-influenced portion of the delta. The frequency of avulsions in the SARD, and the common landforms associated with avulsions, justifies the definition of a crevasse landform unit. This includes crevasse splays, and infilled or infilling paleochannels abandoned by avulsions. Transitions among the geomorphic units were determined based on a combination of the following:

Fig. 3. Littlefield study area, North Carolina. Slightly modified from Phillips et al. (1999b).

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Fig. 4. Geomorphic state transition model for the Littlefield, North Carolina soil landscape.

depressions. Fire or oxidation in peat-filled depressions can convert these to lake status. The landscape units and transitions shown here are derived from work in the U.S.A. (Alaska), Canada, and Siberia (e.g., Osterkamp et al., 2000, 2009; Jorgenson et al., 2001; Pokrovsky et al., 2011; Sannel and Kuhry, 2011; Karlsson et al., 2012). 3. Graph theory, complexity, stability, and synchronization 3.1. Geomorphic state transitions as graphs A geomorphological model based on transitions among qualitatively different stages or system states can be considered as a network or graph. The system states are the nodes or vertices of the graph, and the transitions between them are the links or edges of the graph. Many geomorphic models of this type are directional in concept (that is, they are concerned with development in a particular sequence among states, such as recovery pathways after disturbance or the sequence of stages in a cycle). However, in this study state transitions are assumed to be bidirectional—that is, if two states are linked, transitions may occur in either direction. This recognizes the potential role of disturbances, environmental changes, and human interventions in resetting or reversing geomorphic changes. For example, the post-

incision development trend of an incised alluvial channel may be reversed by a new episode of incision, or a general downwasting trend of topographic development might be reversed by renewed uplift. Thus, the geomorphic transition graphs are simple and undirected. The graphs are also assumed to be connected (i.e., there are no isolated nodes with no connections to other nodes). Each such graph is characterized by a symmetric adjacency matrix A, where the entries are 0,1, depending on whether there is a link or edge connecting the row and column nodes (with zeros on the diagonal). Application of graph theory methods to models of this type is described by Phillips (2011b,c). A is an N × N matrix, where N is the number of system states or graph nodes, and therefore has N (possibly complex) eigenvalues λι, i = 1, 2,..., N. For a simple undirected graph these are all nonnegative, and λ1 N λ2 N ... λN. The largest eigenvalue λ is called the spectral radius, and governs or indicates a number of properties. The spectral radius is a general index of the complexity of the graph, and is related to the number of cycles within the system, the extent to which externally imposed changes are likely to be amplified by state transitions, and transitions to coherence or chaos (Tinkler, 1972; Biggs, 1994; Restrepo et al., 2006, 2007; Yuan et al., 2008). The Laplacian matrix L of A is given by L = D – A, where D is the degree matrix of A (the degree of a node is the number of links or edges incident to that node). The N eigenvalues of the Laplacian λ(L)i include at least one zero (λ(L)N = 0), and the largest λ(L)1 is simple. Two of these, the largest λ(L)1 and the second-smallest λ(L)N − 1, have special significance with respect to system dynamics. The largest Laplacian eigenvalue is called the Laplacian spectral radius, and is hereafter denoted as μ(G), where G represents the graph under study. The second-smallest is called algebraic connectivity (Fiedler, 1973), denoted α(G). Gong et al. (2005) showed that the dynamical stability of G is linearly related to μ(G). Algebraic connectivity is a widely used indicator of the synchronization properties of graphs, and of the rate of convergence (e.g., Biggs, 1994; Lafferrierre et al., 2004). The spectral radius, Laplacian spectral radius, and algebraic complexity, and their relevance to geomorphic state transitions are discussed in more detail below. These address, respectively, complexity, stability, and synchronization. 3.2. Spectral radius The maximum spectral radius for a graph of given number of nodes or states and edges or transitions is λ1;max ¼ ½2mðN−1Þ=N

Fig. 5. Geomorphic state transition model for thermokarst systems.

0:5

ð1Þ

where m is the number of links. Based on this, maximum λ1 for the archetypal transition model structures can be determined based on m associated with a given number of N. Eq. (1) overestimates maximum λ for the linear sequential, cyclical, and radiation structures; these were calculated directly. The linear sequential pattern has m = N − 1, and for a cyclical graph m = N. The radiation and convergence patterns also have m = N − 1. For the cross pattern m = 2(N − 1). Maximum connectivity graphs have m = (N2 − N)/2. For any graph m = Nd/2, where d is the mean degree of the nodes. The concept of the mesh pattern suggests that this can be constrained based on 2 ≤ d ≤ N − 2. For this reason the spectral radius (and other metrics discussed below) for mesh patterns lies between the values for the cross and the maximum connectivity structures. In geomorphological and ecological applications, λ1 has been interpreted in terms of amplification of effects of environmental change, complexity of food webs and population structures, and sensitivity of geomorphic systems and ecosystems (Schreiber and Hastings, 1995; Fath, 2007; Phillips, 2011b,c, 2012c, 2013; Logofet, 2013).

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3.3. Laplacian spectral radius Gupta et al. (2003) considered the stability of formations of dynamically varying agents represented as graphs. They showed that stability is directly related to the Laplacian spectral radius of the adjacency matrix. For analyses based on discrete iterations of the system, they derived the stability condition for graphs of the type considered in this paper: hb2=μ ðGÞ

ð2Þ

where h is the step size or time increment. The time step is of explicit relevance in a numerical modeling context. However, the relative time step required for stability is potentially a useful index of the relative stability of the system, and at least partially circumvents the scale-dependence of other indicators of stability of geomorphic systems (Phillips, 1999b). Amrikitar and Hu (2006) also showed that the Laplacian spectral radius reflects stability of timevarying networks. In addition, μ(G) is also related to synchronization of networks (e.g., Atay et al., 2006; Duan et al., 2009), considered in more detail below via the algebraic connectivity. Theoretical bounds for μ(G) can be established for various graph types (Liu et al., 2004; Shi, 2007; Wang et al., 2011). However, for the relatively simple graphs and N ≤ 5 graphs considered here, direct calculation is straightforward. In geomorphology, μ(G) has not been widely applied, though Phillips (2013) employed it as one measure of historical contingency in network representations of historical change in Earth surface systems.

Table 2 Spectral radius (largest eigenvalue) for standard graph structures, and calculated values for the delta, soil landscape, and thermokarst geomorphic systems, for N = 4 and 5. Graph structure or geomorphic system

Linear sequential Cyclical Radiation or convergence Cross Mesh Maximum connectivity Delta Soil landscape Thermokarst

4=ND ≤λN −1 ≤κ ðAÞ

ð3Þ

Vertex connectivity is the minimum number of vertices or nodes that could be removed to disconnect the graph, while diameter is the maximum shortest path (number of links) between any two vertices. Vertex connectivity is bounded by edge connectivity (minimum number of edges that could be removed to disconnect the graph) of A such that κ(A) ≤ edge connectivity ≤ minimum degree. Vertex connectivity is N − 1 for the maximum connectivity form, 2 for cross and mesh patterns, and 1 for all others. D = N − 1 for the linear sequential and D = N − 2 for the cyclical sequential cases. For the radiation types, D = 2, while D = 1 for maximum connectivity. For most conceivable mesh patterns, 2 ≤ D ≤ N/2. Synchronization, algebraic connectivity, and the Laplacian spectral radius are sensitive to local variations in network structure, even if gross graph properties remain constant, and to the difference between the largest and smallest node degrees (i.e., most and least connected states; Atay et al., 2006). λ(L)N − 1 reflects the rate of convergence, so that larger algebraic connectivity is associated with greater synchronizability (e.g. Lafferrierre et al., 2004). 4. Results 4.1. Spectral radius The spectral radius values for the standard transition graph structures (maxima in the case of cross, mesh, and maximum connectivity), and calculated values for the geomorphic systems considered here are shown in Table 2. The geomorphic transition pattern of the coastal plain soil landscape is the most complex, and the thermokarst transitions the least, as

Spectral radius N=4

N=5

1.618 2.000 1.732 NA b3.000 3.000 NA NA 2.000

1.732 2.000 2.000 3.578 3.578 b λ1,max b 4.000 4.000 2.686 2.856 NA

it represents a cyclical pattern. The SARD and Littlefield λ1 values are substantially higher than those of the simplest graph structures, but well below those of the more highly connected standard structures. Not surprisingly, the complexity of state transition graphs indicated by the spectral radius is lowest for the linear sequential structure and highest for the maximum connectivity type. In general, the potential variation in spectral radius varies with the size of the graph with respect to N. As N increases, the absolute potential range between the smallest (linear sequential structure) and largest (maximum connectivity) possible values increases due to larger maximum eigenvalue values, but the ratio between the maximum spectral radius for a linear sequential (LS) graph and that for maximum connectivity varies as the −0.5 power of N:

3.4. Algebraic connectivity Algebraic connectivity is the second-smallest eigenvalue λLN − 1, also the largest non-zero eigenvalue, of the Laplacian matrix L(A) for an undirected, connected graph. Algebraic connectivity measures the synchronizability of the system (Biggs, 1994; Duan et al., 2009). It is bounded by the vertex connectivity κ(A) and graph diameter D:

213

λ1;LS =λ1;max ¼ 1:414N

−0:5

ð4Þ

The cyclical pattern, of which the thermokarst system is an example, is unique in that λ1 = 2, regardless of N. 4.2. Laplacian spectral radius The Laplacian spectral radius values, reflecting stability properties, are shown in Table 3. The common values for the radiation/convergence, cross, and maximum connectivity structures, and for the SARD and soil landscape examples for N = 5 suggest that this parameter is not sensitive to variations in graph structure at low N. This is also indicated by the same values for cyclical and radiation and convergence structures (and for the thermokarst example) for N = 4. The μ(G) metric may be more useful for higher-N systems; see, e.g., Amrikitar and Hu (2006), Atay et al. (2006), and Phillips (2013). 4.3. Algebraic connectivity Graphs with larger differences between the highest and lowest degree nodes are more difficult to synchronize, but equality of degree

Table 3 Laplacian spectral radius μ(G) for standard graph structures, and calculated values for the delta, soil landscape, and thermokarst geomorphic systems, for N = 4 and 5. Graph

μ(G)

μ(G)

N=4

N=5

Linear sequential Cyclical Radiation or convergence Cross Mesh Maximum connectivity Geomorphic system Delta (N = 5) Soil landscape (N = 5) Thermokarst (N = 4)

3.414 4.000 4.000 NA NA 4.000 μ(G) 5.000 5.000 4.000

3.618 3.802 5.000 5.000 NA 5.000

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Table 4 Algebraic connectivity α(G) for standard graph structures, and calculated values for the delta, soil landscape, and thermokarst geomorphic systems, for N = 4 and 5. Graph

Linear sequential Cyclical Radiation or convergence Cross Mesh Maximum connectivity Geomorphic system Delta (N = 5) Soil landscape (N = 5) Thermokarst (N = 4)

α(G)

α(G)

N=4

N=5

0.586 2.000 1.000 NA b4 4.000 α(G) 1.000 2.000 2.000

0.382 0.753 1.000 3.000 3 b α(G) b 5 5.000

nodes does not necessarily imply ease of synchronization (Atay et al., 2006). This is illustrated by the low algebraic connectivity values for cyclical graph (Table 4). The more highly connected structures are less prone to synchronize. 5. Discussion Complexity of state transition graphs indicated by the spectral radius is lowest for the linear sequential structure and highest for the maximum connectivity type. For a given N, the more connected standard structures are more complex; and for a given structure, higher-N graphs are more complex. However, there is a nonlinear relationship between the ratio between maximum and minimum possible λ1 values and the number of nodes, increasing as N2. One important implication is that increasing information about geomorphic state transitions may actually increase predictive uncertainty. This will occur when new knowledge results in the expansion of simple patterns of potential transitions such as linear sequential, cyclical, and radiation or convergence to more highly connected relationships, and when new nodes (geomorphic states) are discovered. Phillips (2012c) showed how the relative contributions to the spectral radius of the number of links or connections and the specific arrangement or wiring of the graph can be assessed. Relatively small graphs such as those treated here can sometimes be assessed visually. Fig. 5, for instance, can readily be identified as a cyclical pattern. However, results also show that simple visual assessments are potentially misleading. Figs. 2 and 4, representing the delta and soil landscape systems, look fairly complicated and cannot be readily identified as one of the canonical types. They indeed have spectral radii higher than those of the simpler low-connectivity graph structures. However, in both cases λ1 is also well short of the values for cross, mesh, or maximum-connectivity structures of the same N. The Laplacian spectral radius, while a useful stability indicator for larger systems, is not very sensitive to variations in graph structure for the N ≤ 5 models considered here. μ(G) also influences synchronization, with smaller values implying better synchronizability as opposed to larger values of algebraic connectivity, α(G). However, comparing Tables 3 and 4 shows that α(G) is more sensitive to graph structure. In any case, Laplacian spectral radius is most important with respect to synchronization for graphs where all λ(L) b 1 (Atay et al., 2006), which is not the case for the graphs considered here. Unless there is a direct concern with modeling time steps, a Routh–Hurwitz stability analysis of the directed form of models such as those in Figs. 2, 4, and 5 (with self-effects included) is a better indicator of dynamical stability (Phillips, 1999a). Algebraic connectivity is the best metric of synchronizability, which may reflect either the contemporaneity of responses at the landscape scale, and/or the extent to which observations or relationships at a particular time are applicable to past or future situations. This is directly relevant to the geomorphic concept of complex response, articulated first by Schumm (1973, 1977). Complex response refers to the fact that

different locations or components of a geomorphic system often do not respond in the same way, at the same time, or at the same rate to a given environmental change. Algebraic connectivity and graph synchronization are negatively related to complex response—lower synchronizability implies a higher likelihood of complex response sensu Schumm. Thus, with respect to transitions among geomorphic states, α(G) is a quantitative index of the propensity for complex response. The San Antonio River delta STM has a spectral radius indicating an intermediate level of complexity, and algebraic connectivity suggesting low synchronization and a high potential for complex response. This suggests that changes affecting the entire system, such as climate, sea level, or freshwater inflow regimes are unlikely to be manifested similarly or simultaneously within the delta. Localized changes such as an avulsion, while eventually triggering state transitions beyond the immediate site, would not be expected to trigger widespread transitions in the short term. Soil state transitions in the agricultural soil landscape at Littlefield also show medium complexity as indicated by the spectral radius, but algebraic connectivity double that of the SARD example. The latter indicates greater synchronization and less complex response. This is mainly because in the delta the drivers of state transitions, such as crevasses and erosion or accretion, are likely to be localized within the system. At Littlefield, water table rise and fall are likely to influence the whole area, and erosion and deposition will be linked in that removal is connected to deposition elsewhere within the site. These differences are reflected in the specific network of transitions, and are reflected in the algebraic connectivity. The cyclical graph structure characterizing the thermokarst state transition network has spectral radius values higher than the linear or radiation type models (though for N = 5, λ1 is equal for the cyclical and radiation structures), and less than for the more highly connected patterns. Synchronization is relatively high for this system, in part because the climate-driven changes that would presumably affect all parts of the landscape are involved in all but one of the state transitions. These case studies suggest that the extent to which factors that drive state transitions is localized, as opposed to affecting the entire landscape, is an important determinant of the structural properties of state transition networks. This deserves further investigation, via application to state transitions in other geomorphic systems. A broader implication is that the algebraic graph theory perspective supports the intuitive notion that more knowledge or information often increases rather than decreases uncertainty. Expanding the number of potential state transitions from those represented by simple linearsequential and cyclical patterns produces STMs with higher complexity. This is consistent with several recent overviews of models in Earth and environmental sciences more generally, where increasingly realistic models or conceptual frameworks open up new possibilities and reduce the (apparent) level certainty in less realistic representations (c.f. Oreskes, 2003; Phillips, 2007). The approach used here can be used to quantitatively examine qualitative models, which (as stated in the introduction) often portray the most fundamental geomorphic interactions and relationships. The analyses are not limited to simple, undirected graphs. Transition probability matrices (e.g., Markov matrices) or contingency tables may also in some cases be treated as weighted graph adjacency matrices. The interpretation of the largest eigenvalue of directed graphs in the context of dynamical stability in geomorphic systems is well established (Phillips, 1999a), and Wu (2005) extended algebraic connectivity to measure synchronization in directed graphs. There is therefore a strong potential to expand use of these methods in analyses of geomorphic systems. 6. Conclusions Surface processes and landforms may respond to climate and other environmental changes not just via accelerations or decelerations in

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process rates, but also by state changes. The latter, constituting fundamental transitions among geomorphic environments and/or developmental pathways, are directly addressed via state transition models. Two metrics, spectral radius and algebraic connectivity, are useful in analyzing the complexity and synchronization properties of landscapescale responses to environmental change. The Laplacian spectral radius was found to be of limited utility to the systems analyzed here. These were applied to several generic graph/network structures for geomorphic STMs, and to three real-world examples, dealing with a lower coastal plain delta, soil transitions in a coastal plain agricultural landscape, and high-latitude thermokarst systems. While there is some redundancy between spectral radius and algebraic complexity, these metrics provide independent information. The spectral radius is more sensitive to the intensity of cycles within the transition graph structure, and to the overall complexity of the STM. Spectral radius is a general index of graph complexity; and especially the likelihood of amplification and intensification of changes in environmental boundary condition, or of the propagation of local disturbances within the system. Among other things, the spectral radius applications here illustrate that more information does not necessarily decrease uncertainty with respect to predicting responses to environmental change. Commonly, increased knowledge and information result in the expansion of state transition networks from simpler linear sequential and cyclic to more complex structures. The analysis also suggests that state transitions driven by processes likely to affect an entire landscape, such as climate change, have fundamentally different impacts on system behavior than transitions triggered by local disturbances within the landscape, such as river avulsions. Algebraic complexity, a standard measure of synchronization in algebraic graph theory, applies both to synchronization in a literal sense, and to inferential synchronization (the extent to which observations or relationships at one time period can be applied to others). Algebraic connectivity applied to landscape-scale STMs provides a measure of the likelihood of complex response, in the sense of Schumm (1973, 1977), with synchronization inversely related to complex response.

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