# How reliable are Finite-Size Lyapunov Exponents for the assessment of ocean
dynamics?

Ismael Hernández-Carrasco  Cristóbal López  Emilio Hernández-García  IFISC,
Instituto de Física Interdisciplinar y Sistemas Complejos (CSIC-UIB), 07122
Palma de Mallorca, Spain  Antonio Turiel  Institut de Ciències del Mar, CSIC,
Passeig Marítim de la Barceloneta 37-49, 08003 Barcelona, Spain

###### Abstract

Much of atmospheric and oceanic transport is associated with coherent
structures. Lagrangian methods are emerging as optimal tools for their
identification and analysis. An important Lagrangian technique which is
starting to be widely used in oceanography is that of Finite-Size Lyapunov
Exponents (FSLEs). Despite this growing relevance there are still many open
questions concerning the reliability of the FSLEs in order to analyse the
ocean dynamics. In particular, it is still unclear how robust they are when
confronted with real data. In this paper we analyze the effect on this
Lagrangian technique of the two most important effects when facing real data,
namely noise and dynamics of unsolved scales. Our results, using as a
benchmarch data from a primitive numerical model of the Mediterranean Sea,
show that even when some dynamics is missed the FSLEs results still give an
accurate picture of the oceanic transport properties.

Lagrangian viewpoint Griffa.1996 ; Buffoni.1997 ; Iudicone.2002 ; Mancho.2006
; Molcard.2006 . Lagrangian diagnostics exploit the spatio-temporal
variability of the velocity field by following fluid particle trajectories, in
contrast with Eulerian diagnostics, which analyze only frozen snapshots of
data. Among Lagrangian techniques the most used ones involve the computation
of local Lyapunov exponents (LLE) which measure the relative dispersion of
transported particles Artale.1997 ; Aurell.1997 ; Haller.1998 ; Haller.2000 ;
Boffetta.2001 ; Iudicone.2002 . LLEs give information on dispersion processes
but also, and even more importantly, can be used to detect and visualize
Lagrangian Coherent Structures (LCSs) in the flow like vortices, barriers to
transport, fronts, etc.Haller.2000b ; Joseph.2002 ; Koh.2002 ; Lapeyre.2002 ;
Beron-Vera.2008 ; Chaos.2010 .

The standard definition of Lyapunov exponents Ott.1993  involves a double
limit, in which infinitely-long times and infinitesimal initial separations
are taken. These limits can not be practically implemented when dealing with
realistic flows of geophysical origin. Over real data, LLEs are defined by
relaxing some of the limit procedures. In finite-time Lyapunov exponents
(FTLE) Ott.1993 ; Lapeyre.2002  trajectory separations are computed starting
still at infinitesimal distance, but only for a finite time. In the case of
finite-size Lyapunov exponents (FSLE), the key tool used in our work
Aurell.1997 ; Boffetta.2001 , one computes the time which is taken for two
trajectories, initially separated by a finite distance, to reach a larger
final finite distance.

FSLEs are attracting the attention of the oceanographic community Artale.1997
; Iudicone.2002 ; dOvidio.2004 ; Molcard.2006 ; Garcia-Olivares.2007 ;
Haza.2008 ; dOvidio.2009 ; TewKai.2009 ; Poje.2010 . The main reasons for this
interest, within the framework of their ability for studying LCSs and
dispersion processes, are the following: a) they identify and display the
dynamical structures in the flow that strongly organizes fluid motion (the
above mentioned LCSs, which are defined as ridges of the FSLEs fields); b)
they are relatively easy to compute; c) they provide extra information on
characteristics time-scales for the dynamics; and d) they are able to reveal
oceanic structures below the nominal resolution of the velocity field being
analyzed. In addition, FSLEs are specially suited to analyze transport in
closed areas Artale.1997 .

Despite the growing number of applications of FSLEs, a rigorous analysis of
many of their properties is still lacking. There are two main concerns before
applying FSLE to real data, namely the effect of noise and the role of
observation scale. Concerning noise, real data are discretized and noisy, and
this can affect the reliability of FSLE-based diagnostics (recent related
studies for FTLE can be consulted in Chaos.2010 ). Concerning scale
properties, FSLEs can be obtained over a grid finer than that of the data.
This enables to study submesoscale processes under the typical mesoscales
(below $10$ kilometers) that nowadays provide, for example, altimetry data
dOvidio.2009 ; TewKai.2009 . But then the question is if finer-grid LCSs are
meaningful or just an artifact. On the other side, we can have access to a
limited-resolution velocity field, and then ask ourselves if any refinement in
the velocity grid (by improved data acquisition, for instance) is going to
modify our previous assessment of LCS at the rougher scale. The main objective
of this paper is to address these questions, in particular with reference to
their potential applications into ocean dynamics. A related study of the
sensitivity of relative dispersion of particles statistics, when the spatial
resolution of the velocity field changes, has appeared recently Poje.2010 .

The benchmark for the study of FSLEs properties used in this work is a the
two-dimensional velocity field of the marine surface obtained from a numerical
model of the Mediterranean Sea. The first half of the paper is devoted to
study what we have just called scale properties of the FSLE field. By changing
gradually the resolution of the grid on which they are computed, we will show
that they have typical multifractal properties. This means, in particular,
that FSLEs obtained for a finer resolution than that provided by the data
provides non-artificial information. Subsequently, we will consider a somewhat
opposite case, i.e., what happens to the FSLEs if the velocity-data grid is
changed. Their robustness under data-resolution transformations will be
discussed. The second half of the work will analyze the effect of noise.
Again, two different scenarios are considered: a) uncertainties in the
velocity data, and b) noise in the particle trajectories. FSLEs will be shown
to be robust against these two sources of error, and the reasons for this will
be discussed.

##  I Description of the data

We analyze a velocity dataset generated with the DieCast (Dietrich for Center
Air Sea Technology) numerical ocean model adapted to the Mediterranean Sea
Fernandez.2005 . The dataset has been already used in previous Lagrangian
studies dOvidio.2004 ; Schneider.2005a ; Mancho.2008 . DieCast is a primitive-
equation, z-level, finite difference ocean model which uses the hydrostatic,
incompressible and rigid lid approximations. At each grid point, horizontal
resolution is the same in both the longitudinal, $\phi$, and latitudinal,
$\lambda$, directions, with resolutions $\Delta\phi=1/8^{o}$ and
$\Delta\lambda=\Delta\phi\cos\lambda$. Vertical resolution is variable with 30
layers. Annual climatologic forcing is used, so that it is enough to keep a
temporal resolution of one day. We will use velocity data corresponding to the
second layer, which has its center at a depth of 16 meters. This sub-surface
layer is representative of the marine surface circulation and is not directly
driven by wind. We have recorded daily velocities for five years, and
concentrate our work in the area of the Balearic Sea. In Figure 1 we show a
snapshot of the velocity field from the DieCAST model.

Figure 1: Snapshot of the surface velocity field of the Balearic Sea
corresponding to day $640$ in the DieCAST simulation.

##  II Definition and implementation of FSLEs

FSLEs provide a measure of dispersion as a function of the spatial resolution,
serving to isolate the different regimes corresponding to different length
scales of oceanic flows, as well as identifying the LCSs present in the data.
To calculate the FSLEs we have to know the trajectories of fluid particles,
which are computed by integrating the equations of motion for which we need
the velocity data $(u,v)$. FSLE are computed from $\tau$, the time required
for two particles of fluid (one of them placed at $(x,y)$) to separate from an
initial (at time $t$) distance of $\delta_{0}$ to a final distance of
$\delta_{f}$ , as:

$\Lambda(x,y,t,\delta_{0},\delta_{f})\;=\;\frac{1}{\tau}\log\frac{\delta_{f}}{\delta_{0}}.$
(1)

In principle obtaining $\Lambda(x,y,t,\delta_{0},\delta_{f})$ would imply to
consider all the trajectories starting from points at distance $\delta_{0}$
from our basis point; in practice, when confronted with regular, discretized
grids, only the four closest neighbors are considered. It is convenient to
choose $\delta_{0}$ to be the intergrid spacing among the points on which the
FSLEs will be computed, i.e., it is the resolution of the “FSLE grid”. The
details of the calculation of the FSLEs are in Appendix A.

##  III Effect of sampling scale on FSLEs

Notice that we distinguish two types of grids, namely the FSLE grid (in the
following “F-grid”) and that where the velocity field is given (called
velocity grid or “V-grid”). These two grids need not to coincide. We explore
first the effect of changing the resolution of FSLE grid.

In Figure 2 we show an example of the FSLEs derived using four different
F-grids with increasing resolutions. Visually, as the resolution of the F-grid
is increased the structures already observed in the coarser version are kept,
just increasing their detail level. In addition, when resolution is increased,
new less active structures are appearing in areas previously regarded as
almost inactive. Taking a F-grid finer than the associated V-grid would make
no sense if FSLE was an Eulerian measure obtained from single snapshots of the
velocity field. But FSLE is a Lagrangian measure, i.e. they are computed using
trajectories which integrate information on the history of the velocity field.
This allows capturing the effects of the large scales on scales smaller than
the V-grid. This does not mean that we reconstruct all the effects taking
place at the smaller scales, but only the ones that have been originated by
the relatively large velocity features which are resolved on the V-grid.

Figure 2: Snapshots of FSLEs backward in time starting from day $640$ at
different F-grid resolutions: a) $\delta_{0}=1/8^{\circ}$, b)
$\delta_{0}=1/16^{\circ}$, c) $\delta_{0}=1/32^{\circ}$, d)
$\delta_{0}=1/64^{\circ}$. In all of them we take $\delta_{f}=1^{\circ}$. The
values in the color bar have units of $day^{-1}$.

The main structures of the flow, which are essentially filaments, become finer
as resolution is increased, behaving much like the geometrical persistence of
a fractal interface Falconer.1990 . The question naturally rises about the
possible multifractal character of FSLEs. Multifractality is a property
characteristic to turbulent flows, and it is associated to the development of
a complex hierarchy accross which energy is transmitted and utterly dissipated
Frisch.1995 ; Turiel.2008 . The study of the scaling properties of the
distribution of FSLEs at the different resolution scales reveal that they are
multifractal (see Appendix B). This implies that changes in scale are
accounted for by a well-defined transformation, namely a cascade
multiplicative process Frisch.1995 ; Turiel.2008 . It also implies that
information is hierarchized Turiel.2002  and so what is obtained first, at the
coarsest scales, is the most relevant information. Due to multifractality,
small-scale structures, as unveiled by the FSLEs, with typical sizes smaller
than that of the velocity resolution, are determined by the larger ones and
the multi-scale invariance properties. Thus, no artificiality is induced by
this calculation and the robustness of FSLE analysis under changes in scale is
confirmed.

A different question concerns the robustness of FSLEs when the V-grid is
changed. For instance, when a diagnosis is obtained with a low-resolution
velocity field, is this diagnosis compatible with a later improved observation
of the velocity? The answer is yes. In Figure 3 we show the FSLEs obtained at
a fixed F-grid resolution of $1/8^{\circ}$ for varying velocity resolutions
(namely, $1/8^{\circ}$, $1/4^{\circ}$ and $1/2^{\circ}$). The change of
resolution of the velocity grid is performed as indicated in Appendix C. We
observe that the global features observed with the coarser resolution V-grid
are kept when this is refined. Obviously, as the velocity field is described
with enhanced resolution new details (with short-range effect on the flow
structure and so no contradicting the large-scale picture) become apparent in
the FSLEs. However, the effect of refining the V-grid is not only introducing
new small-scale structures: there is a consistent increase in the values of
the FSLEs as the V-grid is refined. The histograms of the FSLEs,
$\Lambda_{f}$, for a given velocity resolution conditioned by the FSLEs,
$\Lambda_{c}$, obtained from a coarser velocity field are shown in Figure 4.
The modal line (the line of maximum conditioned probability) is close to a
straight line. The best linear regression fits are
$\Lambda_{1/8^{\circ}}=1.08\Lambda_{1/4^{\circ}}+0.05$ (correlation
coefficient, $\rho=0.69$) and
$\Lambda_{1/4^{\circ}}=0.99\Lambda_{1/2^{\circ}}+0.04$ ($\rho=0.69$).
According to these results, we can approximate the finer FSLEs $\Lambda_{f}$
in terms of the coarser FSLEs $\Lambda_{c}$ as:

$\Lambda_{f}(\vec{x})\;=\;\Lambda_{c}(\vec{x})\>+\>\Delta\Lambda_{fc},$ (2)

where the quantity $\Delta\Lambda_{fc}$ determines the contribution to the
FSLE by the small-scale variations in velocity not accounted for by the lower
resolution version of this field. It is hence independent of $\Lambda_{c}$, so
the intercept of the vertical axis with the linear regression equals the mean
of this quantity.

Figure 3: Snapshots of FSLEs backward in time starting from day $640$ at
different initial V-grid resolutions: a)$\Delta_{0}=1/8^{\circ}$, b)
$\Delta_{0}=1/4^{\circ}$, c) $\Delta_{0}=1/2^{\circ}$. In all of them we take
the same F-grid resolution of $\delta_{0}=1/8^{\circ}$, and
$\delta_{f}=1^{\circ}$. The color bar has units of day-1.

A linear dependence of $\Lambda_{f}$ with $\Lambda_{c}$ when scale is changed
implies that FSLEs follow a multiplicative cascade Turiel.2006 ; Turiel.2008 ,
an essential ingredient in multifractal systems which gives further
confirmation to our previous results. We have hence shown that a) the
dependence of FSLEs on both types of scale parameters reveals a multifractal
structure; b) what is diagnosed at the coarser scales is still valid when
scale is refined (although as V-grid resolution is increased the reference
level of FSLEs is increased by a constant).

Figure 4: Probability distributions (coded as grey levels) of FSLEs derived at
a coarse velocity resolution ($\Lambda_{c}$, vertical axis) conditioned by a
finer velocity grid ($\Lambda_{f}$, horizontal axis). The range of values of
both axes run linearly from $0$ to $0.5$ day-1. To obtain a statistics large
enough, we have considered $30$ FSLEs snapshots, starting form $t=640$ up to
$t=1075$ days, in steps of $15$ days. The brightest color (pure white)
corresponds to the maximum probability at each column; the darkest color (pure
black) corresponds to zero. Left panel: FLSEs derived from $1/4^{\circ}$
velocities conditioned by FSLEs at original $1/8^{\circ}$ velocities. Right
panel: FSLEs from $1/2^{\circ}$ velocities conditioned by FSLEs from
$1/4^{\circ}$ velocities.

##  IV Effect of noise on FSLEs

We compute the FSLEs after applying a random perturbation to all components of
the velocity field. The velocity is changed from $(u,v)$ to
$(u^{\prime},v^{\prime})$, with
$u^{\prime}(x,t)=u(\textbf{x},t)(1+\alpha\eta_{x}(\textbf{x},t))$ and
$v^{\prime}(\textbf{x},t)=v(\textbf{x},t)(1+\alpha\eta_{y}(\textbf{x},t))$.
$\\{\eta_{x}(\textbf{x},t),\eta_{y}(\textbf{x},t)\\}$ are sets of Gaussian
random numbers of zero mean and unit variance. $\alpha$ measures the relative
size of the perturbation (it gives the ratio of the mean amplitude of noise
with respect to mean amplitude of the velocity). We introduce three different
kinds of error: uncorrelated noise, i.e. different and uncorrelated values of
$\\{\eta_{x}(\textbf{x},t),\eta_{y}(\textbf{x},t)\\}$ for each x and $t$;
correlated in time and uncorrelated in space (uncorrelated for different x but
the same values at given x for different $t$); and correlated in space and
uncorrelated in time (uncorrelated values for different $t$, but the same
values for different x at fixed $t$). Note that the perturbation is
proportional to the original velocity. Figure 5 shows a snapshot of FSLEs for
the velocity field perturbed by uncorrelated noise of a relative size
$\alpha=10$, i.e., noise is $10$ times larger than the amplitude of the
initial velocity field. The computed Lagrangian structures look rather the
same, despite the large size of the perturbation introduced.

Figure 5: Snapshots of FSLEs calculated backwards in time starting from day
$600$ at fixed spatial resolution ($\delta_{0}=1/64^{\circ}$), and at
different $\alpha$: a)$\alpha=0$, b) $\alpha=10$. In both of them we take
$\delta_{f}=1^{\circ}$. The color bar has units of $day^{-1}$. Initial
conditions for which the separation $\delta_{f}$ has not been reached after
$600$ days are assigned a value $\Lambda=0$.

In order to quantify the influence of the velocity perturbation in the FSLE
calculation we compute the relative error (RE) of perturbed FSLEs with respect
to unperturbed FSLEs, $<\epsilon(t)>$, at a given instant of time, and then
averaging in time (we have $M=100$ snapshots $:t=t_{1},...,t_{M}$) as:

$\epsilon(t_{i})=\sqrt{\frac{1}{N}\sum_{\textbf{x}}\dfrac{|\Lambda^{\alpha}(\textbf{x},t_{i})-\Lambda(\textbf{x},t_{i})|^{2}}{|\Lambda(\textbf{x},t_{i})|^{2}}},\
\ \ \hfill<\epsilon(t)>\equiv\frac{1}{M}\sum_{i=1}^{M}\epsilon(t_{i})\ .$ (3)

$\Lambda(\textbf{x},t_{i})$ and $\Lambda^{\alpha}(\textbf{x},t_{i})$ are the
FSLEs fields without and with inclusion of the perturbation in the velocity
data, respectively. The sum over x runs over the $N=2679$ spatial points.
Figure 6 shows the average RE as a function of $\alpha$. It must be remarked
that the RE has always small values: even for $\alpha=10$ the RE remains
smaller than $0.23$ for the three kinds of noise. To get an idea of how
relevant these quantities are, we have computed the RE of shuffled FSLEs
(permuting locations at random) with respect to the original ones, and
obtained a value of $1.143$. FSLEs are thus robust against uncorrelated noise;
the reason is the averaging effect produced when computing them by integrating
over trajectories which extend in time and space, that tends to cancellate
random, uncorrelated errors.

Figure 6: Relative error $<\epsilon(t)>$ of the FSLE fields for different
perturbation intensity $\alpha$ in the velocity data. Solid line is for
uncorrelated noise in space and time, dashed-dotted line is for uncorrelated
noise in time and correlated in space, and dotted line is for uncorrelated
noise in space and correlated in time. $<\epsilon(t)>$ is obtained by
averaging the RE in $100$ snapshots (see Eq. (3)). The error bar is the
statistical error of the temporal average $<\epsilon(t)>$. Left: spatial
resolution $\delta_{0}=1/8^{\circ}$. Right: spatial resolution
$\delta_{0}=1/64^{\circ}$. In all calculations we take $\delta_{f}=1^{\circ}$

Now we proceed by adding noise to the particle trajectories. This is a
simplified way of including unresolved small scales in the Lagrangian
computations Griffa.1996 . To be precise we solve numerically the system of
Equations (13) and (14) (see Appendix D), where a stochastic term with a
Gaussian random number and an effective eddy-diffusion, $D$, has been added.
For the diffusivity we use Okubo’s empirical formula Okubo.1971 , which
relates the effective eddy-diffusion, $D$ in $m^{2}/s$, with the spatial
scale, $l$ in meters: $D(l)=2.055\ 10^{-4}\ l^{1.15}$. If we take $l=12\ km$,
which is the approximate length corresponding to the $1/8^{\circ}$ DieCAST
resolution at Mediterranean latitudes, we obtain $D\sim 10\ m^{2}s^{-1}\equiv
D_{0}$.

In Figure 7 we show particle trajectories without (top panel) and with (bottom
panel) eddy diffusion. As expected diffusion introduces small scale
irregularities on the trajectories, and also substantial dispersion at large
scales. In Figure 8, FSLEs with $\delta_{0}=1/64^{\circ}$ $D=0~{}m^{2}s^{-1}$
and $D=0.9~{}m^{2}s^{-1}$ (obtained for this scale by Okubo’s formula for
spatial resolution) are shown. We can see that the main mesoscale structures
are maintained, but small-scale filamental structures are lost since filaments
become blurred. This is somehow expected a priory because diffusion introduces
a new length scale $l_{D}$ proportional to $\sqrt{D}$. A pointwise comparison
of noiseless and noise-affected FSLEs makes no sense, since the noise-induced
blurring disperses FSLEs values specially at places with low values, see
Figure 8. But for Lagrangian diagnostics high FSLE values are much more
relevant, so we compute the error restricted to the places where $\Lambda>0.2$
for $D=0$. Left panel of Figure 9 shows the RE respect to the $D=0$ case
(applying Equation (3)) for different values of $D$. The RE monotonously
increases with $D$, but remains smaller than $0.6$ for the largest value of
$D$ considered. Again, to get an idea of how relevant these relative errors
are, we have to compare them with the RE of shuffled FSLEs, of value $1.143$.

Figure 7: Trajectories of five particles without diffusion (top) and with
diffusion (bottom). The difference in the initial positions of all five
particles is about $0.06^{\circ}$, and we use these initial conditions in both
computations. The trajectories were computed for $50$ days of integration. We
used the eddy-diffusion $D_{0}\sim 10m^{2}s^{-1}$ assigned by the Okubo
formula to the resolution of the DieCAST model at Mediterranean latitudes.

Figure 8: FSLEs computed backwards from day $500$ at the same spatial
resolution ($\delta_{0}=1/64^{\circ}$), and for different eddy-diffusion
values: a)$D=0~{}m^{2}s^{-1}$ b) $D=0.9~{}m^{2}s^{-1}$. We take
$\delta_{f}=1^{\circ}$. The color bar has units of $day^{-1}$. Initial
conditions for which the separation $\delta_{f}$ has not been reached after
$500$ days are assigned a value $\Lambda=0$.

As a matter of fact, diffusion introduces an effective observation scale, and
one should not go beyond that limit to obtain senseful results; this is
illustrated in the right panel of Figure 9. As shown in the figure, for fixed
eddy diffusivity (in the case of the figure, $D_{0}=10m^{2}s^{-1}$), when
$\delta_{0}$ becomes greater the error diminishes. Hence, a fixed diffusion
will eventually be negligible at a scale large enough, in this way determining
an observation scale. A completely different situation is given when the eddy-
diffusion depends on the scale according to Okubo’s formula (in our case, at
$\delta_{0}=1/16^{\circ}$ $D=4,5m^{2}s^{-1}$; at $\delta_{0}=1/32^{\circ}$
$D=2m^{2}s^{-1}$; and at $\delta_{0}=1/64^{\circ}$, $D=0.9m^{2}s^{-1}$). Now
$<\epsilon(t)>$ takes a constant value close to $0.45$, meaning that Okubo’s
diffusion behaves the same at all scales. This is expected since Okubo’s law
is based on the hypothesis that unsolved scales act as diffusers, like in
turbulence Frisch.1995 . Our result is consistent with the ideas behind
Okubo’s hypothesis.

Figure 9: Left: Relative error $<\epsilon(t)>$ of the FSLE at the different
values of $D$ in the particle trajectories, with respect to the $D=0$ case.
Spatial resolution is $\delta_{0}=1/8^{\circ}$, and $\delta_{f}=1^{\circ}$.
$<\epsilon(t)>$ is obtained by temporally averaging the relative errors in
$100$ snapshots. The (small) error bars indicate the statistical error in the
$<\epsilon(t)>$ average. Right: Dotted line is the RE $<\epsilon(t)>$ of the
FSLE at different spatial resolution $\delta_{0}$ and at one assigned eddy-
diffusion $D$ for every spatial resolution in the particle trajectories with
respect to the $D=0$ case. Solid line is the RE $<\epsilon(t)>$ of the FSLE at
different spatial resolution $\delta_{0}$, and at the same eddy-diffusion
$D_{0}=10m^{2}s^{-1}$ in the particle trajectories with respect to the $D=0$
case. Dashed-dotted line is the RE of shuffled FSLE with respect to the
original case ($D=0$) at different spatial resolution. $<\epsilon(t)>$ by
temporally averaging the RE in $100$ snapshots. The (small) error bar
indicates the statistical error in the $<\epsilon(t)>$ average. In all of them
we take $\delta_{f}=1^{\circ}$.

We can characterize the effective diffusion scale from the properties of the
histograms. In Figure 10 we present the histograms of FSLEs at different
F-grid resolutions, and including or not eddy diffusion which takes always the
same value $D_{0}=10m^{2}s^{-1}$. For $\delta_{0}=1/8^{\circ}$ the histograms
with and without diffusion are almost coincident. This is due to the fact that
the value of diffusion we are using is the one corresponding, by the Okubo’s
formula, to $1/8^{\circ}$. i.e., we are parameterizing turbulence below
$1/8^{\circ}$, and this has no effects on the FSLE computations if the minimum
scale considered is also $1/8$. However, this behavior is different for
smaller $\delta_{0}$ (always keeping $D_{0}=10m^{2}s^{-1}$). The histograms
for $\delta_{0}=1/16,1/64^{\circ}$, with and without diffusion, are clearly
different, those including diffusion becoming closer to the histogram for
$\delta_{0}=1/8^{\circ}$.

Figure 10: Comparison between probability density function for the FSLEs at
different resolutions with different values of eddy-diffusion, and without
diffusion. It is obtained from the temporal average ($30$ snapshots) of
histograms. Solid line for $\delta_{0}=1/8^{\circ}$ with diffusion, dotted
$\delta_{0}=1/8^{\circ}$ without diffusion, dashed $\delta_{0}=1/16^{\circ}$
without diffusion, dashed-dotted line for $\delta_{0}=1/64^{\circ}$ with
diffusion, and circle-line for $\delta_{0}=1/64^{\circ}$ without diffusion.

##  V Conclusions

In this paper, we have analyzed the sensibility of FSLE-based analyses for the
diagnostic of Lagrangian properties of the ocean (most notably, horizontal
mixing and dispersion). Our sensibility tests include the two most important
effects when facing real data, namely dynamics of unsolved scales and of
noise. Our results show that even if some dynamics are missed (because of lack
of sampling or inaccuracy of any kind in the measurements) FLSEs results would
still give an accurate picture of Lagrangian properties, valid for the solved
scales. This does not mean that scale and/or noise leave FSLEs unaffected, but
the way in which they modify this Lagrangian diagnostics can be properly
accounted.

Lagrangian methods provide answers to problems which have a deep impact on
risk management (e.g. control of pollutant dispersion) as well as on ecosystem
analysis (e.g. tracking nutrient mixing and transport, identifying the role of
horizontal mixing in primary productivity). They utterly will give hints about
energy exchanges in the upper ocean and will help in understanding processes
driving global change in the oceans. The use of Lagrangian techniques for the
assessment of the transport and mixing properties of the ocean has grown in
importance in the latest years, with increasing efforts devoted to the
implementation of appropriate techniques but few studies on the validity of
the results when real data, affected by realistic constraints, are used. Our
work will serve to unify and interpret the analyses provided by Lagrangian
methods when real data are processed.

###### Acknowledgements.

This work is a contribution to OCEANTECH project (CSIC PIF-2006). IH-C, CL and
EH-G acknowledge support from project FISICOS (FIS2007-60327) of MEC and
FEDER.

##  Appendix A Calculation of FSLE.

It is natural to choose the initial points $(x,y)$ on the nodes of a grid with
lattice spacing coincident with the initial separation of fluid particles
$\delta_{0}$. In this way one obtains maps of values of $\Lambda$ at a spatial
resolution that will coincide with $\delta_{0}$. To compute $\Lambda$ we need
to know the trajectories of the particles. The equations of motion that
describe the horizontal evolution of particle trajectories in the velocity
field are:

$\displaystyle\frac{d\phi}{dt}$ $\displaystyle=$
$\displaystyle\frac{u(\phi,\lambda,t)}{R\cos{\lambda}},$ (4)
$\displaystyle\frac{d\lambda}{dt}$ $\displaystyle=$
$\displaystyle\frac{v(\phi,\lambda,t)}{R},$ (5)

where $u$ and $v$ represent the zonal and meridional components of the surface
velocity field coming from the simulations described above, $R$ is the radius
of the Earth (6400 km in our computations). Numerically we proceed integrating
Eqs. (4) and (5) using a standard fourth-order Runge-Kutta scheme with an
integration time step $dt$ = 6 hours. Spatiotemporal interpolation of the
velocity data is achieved by bilinear interpolation. Since this technique
requires an equally spaced grid and this is not the case for the spherical
coordinates $(\phi,\lambda)$, for which the grid is not uniformly spaced in
the latitude coordinate, we first transform it to a new system $(\phi,\mu)$
where the grid turns out to be uniformly sampled Mancho.2008 . The latitude
$\lambda$ is related to the new coordinate $\mu$ according to:

$\mu\;=\;\log|\sec\lambda\>+\>\tan\lambda|$ (6)

Using the new coordinate variables the equations of motion become:

$\displaystyle\frac{d\phi}{dt}$ $\displaystyle=$
$\displaystyle\frac{u(\phi,\mu,t)}{R\cos{\lambda(\mu)}}$ (7)
$\displaystyle\frac{d\mu}{dt}$ $\displaystyle=$
$\displaystyle\frac{v(\phi,\mu,t)}{R\cos{\lambda(\mu)}},$ (8)

and one can convert the $\mu$ values back to latitude $\lambda$ by inverting
Eq. (6): $\lambda=\pi/2-2\arctan e^{-\mu}$. Once we integrate the equations of
motion, Eqs. (7) and (8), we compute the FSLEs applying Eq. (1) for the points
of a lattice with spacing $\delta_{0}$. We will only compute FSLEs integrating
backwards-in-time the particle trajectories, since LCSs associated to this has
a direct physical interpretation, but all our results are similar for forward-
in-time dynamics.

##  Appendix B Multifractal character of FSLE.

To study if FSLEs behave like multifractal systems we have computed their
probability density distribution, $P(\delta_{0},\Lambda)$, for different
resolutions $\delta_{0}$. It must follow that for any resolution scale
$\delta_{0}$ of FSLE grid we should observe:

$P(\delta_{0},\Lambda)\propto\delta_{0}^{d-D(\Lambda)},$ (9)

Notice that as $d-D(\Lambda)$ is a positive quantity, $P(\delta_{0},\Lambda)$
becomes smaller as $\delta_{0}$ is reduced. In fact, a characteristic
signature of multifractal scaling is having a scale-dependent histogram which
becomes more strongly peaked as the resolution scale becomes smaller. We know
that there is a FSLE for each domain point, so we can normalize the FSLE
distribution by its maximum (that will be attained for a value $\Lambda_{c}$)
and then retrieve the associated singularity spectrum according to the
following expression:

$D(\Lambda)\;=\;d-\frac{\log{\frac{P(\delta_{0},\Lambda)}{P(\delta_{0},\Lambda_{c})}}}{\log{\delta_{0}}}.$
(10)

In Figure 11, top, we show the histograms (averaged over $30$ snapshots
distributed among $15$ months) normalized to have the same unitary are. One
can see that when the resolution gets finer the $P(\delta_{0},\Lambda)$
narrows, and the peak height increases, in agreement with Equation (9).
According to the definition of Microcanonical Multifractals Isern.2007 ;
Turiel.2008 , the system will be multifractal if the curves $D(\Lambda)$
estimated at different resolutions $\delta_{0}$ using Equation (10) are all
equal. This can be observed in Figure 11, bottom. The collapse of the four
curves is not perfect due to the lost of translational invariance produced by
the small size of the domain -the Balearic basin- that we analyze. Thus, the
interfaces of constant $\Lambda$ values build an approximate multifractal
hierarchy and generalized scale invariance is present in the FSLE field.

Figure 11: Top: Comparison of the probability density functions
$P(\delta_{0},\Lambda)$ for the FSLEs at different resolutions. It is obtained
from the temporal average ($30$ snapshots) of histograms. Dotted line is for
$\delta_{0}=1/8^{\circ}$, dashed-dotted line $\delta_{0}=1/16^{\circ}$, dashed
$\delta_{0}=1/32^{\circ}$, and solid line for $\delta_{0}=1/64^{\circ}$.
Bottom: $D(\Lambda)$ for different values of $\delta_{0}$. Dotted for
$\delta_{0}=1/8^{\circ}$, dashed-dotted line $\delta_{0}=1/16^{\circ}$, dashed
$\delta_{0}=1/32^{\circ}$, and solid line line for $\delta_{0}=1/64^{\circ}$.

##  Appendix C Change of resolution of the velocity grid.

In order to reduce the resolution of a given velocity grid, the easiest way
would be by subsampling the existing grid points or by block-averaging the
values of the velocity and assign the result to the central grid point.
However, in the case of a complex boundary such as the Mediterranean coast
such a strategy is strongly inconvenient, as the coarsening of the grid would
imply to change land circulation barriers (islands, straits). The
disappearance of a land barrier or the creation of a new one as a consequence
of the coarsening would imply a dramatic change in the value of FSLEs at all
points affected by the modified circulation; if the circulation patterns are
rather complex almost every point could be affected. We have thus preferred to
smooth the velocity with a convolution kernel weighted with a local
normalization factor, and keeping the original resolution for the data so land
barriers are equally well described than in the original data.

We define the coarsening kernel of scale factor $s$, $\kappa_{s}$, as:

$\kappa_{s}(x,y)\;=\;e^{-\frac{x^{2}+y^{2}}{2s^{2}}}$ (11)

We disregard the introduction of a normalization factor at this point since we
will need to normalize locally later. The coarsened version of the velocity
vector would hence be given by the convolution of this kernel with the
velocity, denoted by $\kappa_{s}\star\vec{v}$. A coarsening convolution kernel
turns out to be convenient with almost horizontal fields, as the derivatives
commute with the convolution operator so if $\nabla\vec{v}\approx 0$ hence
$\kappa_{s}\star\nabla\vec{v}\approx 0$. However, this coarsening scheme needs
to be improved. By convention, we take the velocity $\vec{v}$ as zero over
land points. For that reason, a simple convolution does not produce a correct
coarsened version of the velocity because points close to land would
experience a loss of energy. The easiest way to correct this is to normalize
by the weight of the sea points. Let us first define the sea mask $M(x,y)$ as
$1$ over the sea and $0$ over the land. The normalization weight is given by
$\kappa_{s}\star M$. For points very far from the land, this weight is just
the normalization of $\kappa_{s}$. For points surrounded by land points the
weight takes the contributions from sea points only. We thus define the
coarsened version by a scale factor $s$ of the velocity, $\vec{v}_{s}$, as:

$\vec{v}_{s}\;=\;\frac{\kappa_{s}\star\vec{v}}{\kappa_{s}\star M}$ (12)

In Figure 12 we show two examples of coarsened velocities. We can see that
typical circulation patterns are coarsened as $s$ increases, while land
obstacles are preserved. In fact, if $\Delta_{0}$ is the velocity resolution
scale, the effective resolution scale of $\vec{v}_{s}$ is $s\Delta_{0}$ (the
nominal resolution, on the contrary, is the original one, $\Delta_{0}$).

Figure 12: Top: Velocity field coarsened by a scale factor $s=2$, for a
equivalent resolution $\delta_{0}=1/4^{\circ}$. Bottom: Velocity field
coarsened by a scale factor $s=4$, for a equivalent resolution
$\delta_{0}=1/2^{\circ}$.

##  Appendix D Introducing noise in the particle’s trajectories.

In order to introduce noise in the particle’s trajectories we resolve the
following system of stochastic equations:

$\displaystyle\frac{d\phi}{dt}=\frac{u(\phi,\lambda,t)}{R\cos(\lambda)}+\frac{\sqrt{2D}}{R\cos(\lambda)}\xi_{1}(t),$
(13)
$\displaystyle\frac{d\lambda}{dt}=\frac{v(\phi,\lambda,t)}{R}+\frac{\sqrt{2D}\xi_{2}(t)}{R}.$
(14)

$\xi_{i}(t)\ \ i=1,2$ are the components of a two-dimensional Gaussian white
noise with zero mean and correlations
$<\xi_{i}(t)\xi_{j}(t^{\prime})>=\delta_{ij}\delta(t-t^{\prime})$.Eqs. (13 and
14) use a simple white noise added to the trajectories. A more realistic
representation of small-scale Lagrangian dispersion in turbulent fields
requires using other kinds of correlated noises Griffa.1996  but, as we are
interested in examining influences of the missing scales, it is convenient to
use white noise, since this would represent the extreme case of very irregular
trajectories which gives an upper bound to the effects of more realistic
smoother small scales. Thus the tests presented here are similar to the ones
considered before (perturbation of the velocity) when adding uncorrelated
perturbations to the velocity, but here the perturbation acts at arbitrarily
small scales, as appropriate for a turbulent field, instead of being smooth
below a cutoff scale, as appropriate for modelling observational errors.

## References

  * (1) V. Artale, G. Boffetta, A. Celani, M. Cencini, and A. Vulpiani.  Dispersion of passive tracers in closed basins: Beyond the diffusion coefficient.  Physics of Fluids, 9:3162–3171, 1997.
  * (2) E. Aurell, G. Boffetta, A. Crisanti, G. Palading, and A. Vulpiani.  Predictability in the large: an extension of the concept of Lyapunov exponent.  Journal of Physics A, 30:1–26, 1997.
  * (3) F.J. Beron-Vera, M.J. Olascoaga, and G.J. Goni.  Oceanic mesoscale eddies as revealed by Lagrangian coherent structures.  Geophysical Research Letters, 35:L12603, 2008.
  * (4) G. Boffetta, G. Lacorata., G. Redaelli, and A. Vulpiani.  Detecting barriers to transport: a review of different techniques.  Physica D, 159:58–70, 2001.
  * (5) G. Buffoni, P. Falco, A. Griffa, and E. Zambianchi.  Dispersion processes and residence times in a semi-enclosed basin with recirculating gyres: An application to the Tyrrhenian sea.  Journal of Geophysical Research - Oceans, 102:18699–18713, 1997\.
  * (6) F. d’Ovidio, V. Fernández, E. Hernández-García, and C. López.  Mixing structures in the mediterranean sea from finite-size lyapunov exponents.  Geophysical Research Letters, 31:L17203, 2004.
  * (7) F. d’Ovidio, J. Isern-Fontanet, C. López, E. Hernández-García, and E. García-Ladona.  Comparison between Eulerian diagnostics and Finite-Size Lyapunov Exponents computed from altimetry in the Algerian basin.  Deep-Sea Research I, 56:15–31, 2009.
  * (8) K. Falconer.  Fractal Geometry: Mathematical Foundations and Applications.  John Wiley and sons, Chichester, 1990.
  * (9) V. Fernández, D.E. Dietrich, R.L. Haney, and J. Tintoré.  Mesoscale, seasonal and interannual variability in the Mediterranean sea using a numerical ocean model.  Progress in Oceanography, 66:321–340, 2005.
  * (10) U. Frisch.  Turbulence: The legacy of A.N. Kolmogorov.  Cambridge Univ. Press, Cambridge MA, 1995.
  * (11) A. Garcia-Olivares, J. Isern-Fontanet, and E. Garcia-Ladona.  Dispersion of passive tracers and finite-scale Lyapunov exponents in the Western Mediterranean Sea.  Deep Sea Research I, 54:253–268, 2007.
  * (12) A. Griffa.  Applications of stochastic particle models to oceanographic problems.  In R. Adler, P. Muller, and B. Rozovskii, editors, Stochastic modelling in physical oceanography. Birkhauser, 1996.
  * (13) G. Haller.  Finding finite-time invariant manifolds in two-dimensional velocity fields.  Chaos, 10(1):99–108, 2000.
  * (14) G. Haller and A. Poje.  Finite time transport in aperiodic flows.  Physica D, 119:352–380, 1998.
  * (15) G. Haller and G. Yuan.  Lagrangian coherent structures and mixing in two-dimensional turbulence.  Physica D, 147:352–370, 2000.
  * (16) A. C. Haza, A. C. Poje, T. M. Özgökmena, and P. Martin.  Relative dispersion from a high-resolution coastal model of the Adriatic Sea.  Ocean Modelling, 22:48–65, 2008.
  * (17) J. Isern-Fontanet, A. Turiel, E. Garcia-Ladona, and J. Font.  Microcanonical multifractal formalism: application to the estimation of ocean surface velocities.  Journal of Geophysical Research, 112:C05024, 2007.
  * (18) D. Iudicone, G. Lacorata, V. Rupolo, R. Santoleri, and A. Vulpiani  Sensitivity of numerical tracer trajectories to uncertainties in OGCM velocity fields.  Ocean Modelling, 4:313-325, 2002.
  * (19) B. Joseph and B. Legras.  Relation between kinematic boundaries, stirring, and barriers for the antartic polar vortex.  Journal of Atmospheric Sciences, 59:1198–1212, 2002.
  * (20) T.-Y. Koh and B. Legras.  Hyperbolic lines and the stratospheric polar vortex.  Chaos, 12(2):382–394, 2002.
  * (21) G. Lapeyre.  Characterization of finite-time Lyapunov exponents and vectors in two-dimensional turbulence.  Chaos, 12(2):688–698, 2002.
  * (22) A.M. Mancho, D. Small, and S. Wiggins.  A tutorial on dynamical systems concepts applied to lagrangian transport in oceanic flows defined as finite time data sets: theoretical and computational studies.  Physics Reports, 437:55–124, 2006.
  * (23) A.M. Mancho, E. Hernández-García, D. Small, S. Wiggins, and V. Fernández.  Lagrangian transport through an ocean front in the North-Western Mediterranean Sea.  Journal of Physical Oceanography, 38:1222–1237, 2008.
  * (24) A. Molcard, A.C. Poje, and T. M. Ozgokmen.  Directed drifter launch strategies for Lagrangian data assimilation using hyperbolic trajectories  Ocean Modelling, 12: 268-289, 2006.
  * (25) A. Okubo.  Ocean diffusion diagramas.  Deep Sea Research, 18:789–802, 1971.
  * (26) E. Ott.  Chaos in Dynamical Systems.  Cambridge Univ. Press, Cambridge (UK), 1993.
  * (27) T. Peacock and J. Dabiri.  Introduction to focus issue: Lagrangian coherent structures.  Chaos, 20:017501, 2010.
  * (28) A.C. Poje, A. C. Haza, T. M. Ozgokme, M. G. Magaldi, and Z. D. Garraffo.  Resolution dependent relative dispersion statistics in a hierarchy of ocean models.  Ocean Modelling, 31:36–50, 2010.
  * (29) J. Schneider, V. Fernández, and E. Hernández-García.  Leaking method approach to surface transport in the Mediterranean Sea from a numerical ocean model.  Journal of Marine Systems, 57:111–126, 2005.
  * (30) E. Tew Kai, V. Rossi, J. Sudre, H Weimerskirch, C. López, E. Hernández-García, F. Marsac, and V. Garcon.  Top marine predators track lagrangian coherent structures.  Proceedings of the National Academy of Sciencies of the USA, 106:8245–8250, 2009.
  * (31) A. Turiel and A. del Pozo.  Reconstructing images from their most singular fractal manifold.  IEEE Trans. Im. Proc., 11:345–350, 2002.
  * (32) A. Turiel, C. Pérez-Vicente, and J. Grazzini.  Numerical methods for the estimation of multifractal singularity spectra on sampled data: a comparative study.  Journal of Computational Physics, 216(1):362–390, July 2006.
  * (33) A. Turiel, H. Yahia, and C. Pérez-Vicente.  Microcanonical multifractal formalism: a geometrical approach to multifractal systems. Part I: Singularity analysis.  Journal of Physics A, 41:015501, 2008.


Competition Dynamics in the Meme Ecosystem
Trenton Ford, Rachel Krohn, and Tim Weninger
Department of Computer Science and Engineering
University of Notre Dame

{tford5, rkrohn<EMAIL_ADDRESS>
The creation and sharing of memes is a common modality of online social interactions. The goal of the present work is to better understand the collective dynamics of memes in this accelerating and competitive environment. By taking an ecological perspective and tracking the meme-text from 352 popular memes over the entirety of Reddit, we are able to show that the frequency of memes has scaled almost exactly with the total amount of content created over the past decade. This means that as more data is posted, an equal proportion of memes are posted. One consequence of limited human attention in the face of a growing number of memes is that the diversity of these memes has decreased at the community level, albeit slightly, in the same period. Another consequence is that the average lifespan of a meme has decreased dramatically, which is further evidence of an increase in competition and a decreasing collective attention span.

§ INTRODUCTION

With the rise of social media platforms, the cost historically associated with producing and consuming information has decreased to unprecedented levels; users and organizations can easily share their thoughts, stories, and others' content with diverse and widespread audiences with very little effort. Due to the ease of production, the volume of content produced has increased to the point that any individual user can only see a small portion of what is available. The reduction in content production cost and increase in availability has induced a change in scarcity dynamics: from content scarcity to consumer scarcity <cit.>. This shift in scarcity has birthed new research areas to help users see relevant content – such as recommender systems – as part of the broader attention economy <cit.>.

Meme image with text meant to be condescending to the subject. The text of this meme and others like it are frequently used without the image in humorous or sarcastic contexts.

The attention economy seeks to explain the allocation of cognitive resources in the creation and consumption of information. Though this concept existed long before the advent of social media <cit.>, recent work has focused on how this model governs the dynamics of content consumers and curators in the socio-digital space <cit.>. The main focus has been on the consumption of information <cit.>, but others focus on the production and curation of information <cit.>. One of the primary questions at the center of online social media is this: how does the limited attention of users shape the information landscape?

One particularly compelling subset of the information landscape is the production and resharing of compelling memes, which are short phrases and images. For example, the image in Figure <ref> is a meme with text that is meant to be condescending to the subject; oftentimes, the text of the meme is independent of its image and is written in plaintext in comments and tweets. The dynamics of these viral messages are not well understood despite widespread attempts to predict and simulate their spread or popularity <cit.>. Yet this narrow subset of the information landscape is an increasingly visible and influential communication mode with exciting properties. A meme's popularity can be quantified by how many times it is reproduced or shared, how long it stays relevant, and how many times it is mutated – in the case of meme images.

Intuition about how memes are created, transmitted, consumed, or mutated are often derived from their association with genes and the process of gene evolution <cit.> and more recently, memes have been considered through the lens of epidemiology and disease transmission <cit.>. Indeed, its etymology is a portmanteau of mind+gene, which begs the question: rather than continuing the economic analogy, are memes better situated in the realm of ecology? And if so, what kind of understanding can be gleaned from this perspective?

In the present work, we derive findings about competition and diffusion of information from the ecological perspective. There are many compelling examples that motivate this perspective. Foremost is the concept of competition, which is a driving force in genealogical, economic, epidemiological, and ecology fields <cit.>. In analogical terms, the goal of competing memes is their continued existence within the minds and communication patterns of people. Survival, therefore, follows as a natural extension of competition, which presumes that memes are designed with survival in mind <cit.>.

The differences between the ecological perspective and others are nuanced. Fundamentally, each perspective offers a unique interpretation of information dynamics. For example, within the economics perspective, human behavior (, attention) is the primary focus, and memes just one of many possible factors. From an epidemiological perspective, memes are treated as a contagion (, a virus), but epidemiological models typically do not consider landscapes with multiple viruses and their interactions. The genealogical perspective treats memes as genes and explores gene-gene interactions, but the gene perspective does not natively consider gene-environment interactions.

In taking the ecological perspective, we consider a meme to be a single species existing within the same environment or habitat. The ecological perspective shifts the focus away from the human users and back to the memes and the environments they exist within – wherein memes seek both longevity and a large population, competing for limited environmental resources – human attention.

Within the perspective of the meme ecology, we ask the following research questions:

RQ1: How does the collective user attention scale? Do more users permit a larger or smaller number of memes?

RQ2: How do memes compete for attention? How does the introduction of a new meme impact the ecosystem of existing memes?

RQ3: How have the dynamics of collective attention changed over time?

In summary, by using well-known metrics and concepts from ecology, we perform an ecological analysis of the dynamics of text-memes on Reddit. The results of this analysis and the behavior they suggest are compelling and strongly support the case for the ecology of memes. We find that memes comprise a relatively constant fraction of all activity on the platform, even as social media increases in popularity. This suggests that as more memes are created their lifecycle duration becomes shorter, which further suggests that the collective human attention span on social media is decreasing.

Although the current work focuses on short, frequently repeated texts, , memes, we further hypothesize that our findings are likely to apply to a number of other communication modalities like image-memes and hashtags.

Tokens     Count     Examples

1     69      thicc; yeet; wat; mfw; impossibru

2     80      moms spaghetti; zerg rush; y tho

3     69      winter is coming; u wot m8

4     48      do you even lift; kill it with fire

5     25      hello darkness my old friend

6     25      shrek is love shrek is life

7     18      still a better love story than twilight

8     18      this is why we cant have nice things

Meme dataset consists of 352 text memes, ranging in length from 1 to 8 tokens. Some memes reference current pop-culture events, while others seem unconnected to trends of the time.

(A) Stacked line plot representation of Reddit contributions between 2010 and 2020. The lower (orange) region shows the number of posts per month; the upper (blue) region shows the number of comments per month. (B) The unigram background model is used to compute normalized meme-frequencies. This background behavior closely mirrors the growth of Reddit, but is one order of magnitude larger.

§ DATA COLLECTION

Using a comprehensive dataset of the 352 most popular memes from KnowYourMeme.com, we identified their individual occurrences on Reddit. The memes were selected from the Confirmed category on KnowYourMeme, and include text-based memes that ranged from 250 thousand to 13 million page views each. Note that the tracking of rapidly-evolving image templates is outside the scope of the present work; therefore, image-memes are not included in this analysis. Extended meme text (, copypasta) is truncated to include only the 8-token prefix. The final set contains meme-phrases that range in length from 1 to 8 word tokens as shown in Table <ref>.  Additionally, we collected all posts and comments from Jan. 2010 to Jan. 2020; the number of monthly posts and comments is plotted in Figure <ref>.

The questions raised in the present work are considered human subjects research, and relevant ethical considerations are present. We sought and received research approval from the Institution Review Board of redacted.

§ COLLECTIVE ATTENTION TO MEMES IS STATIONARY

Previous work has shown that innovation and technological development is accelerating. Moore's Law is one example of this phenomenon where a compounding increase in circuit density has led to remarkable increases in computational power <cit.>; similar effects have been shown in genome sequencing <cit.> and telecommunications bandwidth <cit.>. In online social systems, the early empirical evidence suggests that a similar pattern exists: that social innovations are accelerating <cit.>.

This is the basis for RQ1: How does collective user attention of memes scale? Does the presence of larger groups result in super-scaling effects like those found in population densities <cit.> and software development <cit.> where collections of individuals produce more than the sum of their parts?

At first glance, Figure <ref> appears to show that our data supports these claims: more posts, comments, and memes are being made at an accelerating pace year over year. But how much attention is paid to individual memes? To answer this question, we first need to measure collective attention.

§.§ Measuring Collective Attention

Because we cannot collect the number of users who viewed or thought about a particular meme, instead, we estimate the collective attention for each meme by the number of times it appears, , its frequency. So our first task is to extract the daily frequency $F_m(t)$ of each meme, such that $F_m(t)$ is the frequency of meme $m$ on day $t$.

Given the scale variance of social media sites, simply counting tokens would be insufficient to fully determine whether memes are truly increasing in prevalence or if their frequency increases mirror the increase in content that Reddit has experienced overall. In order to produce as accurate assessment of the meme ecology, we need to carefully normalize the frequency of meme occurrences.

To do this, we constructed a set of 5000 randomly selected words from Reddit to serve as a background language model.
Then, for each day, we count the number of occurrences of each token in the background set, such that $B(t)$ gives the background sum of all 5000 words on day $t$. As the number of posts and comments on Reddit grows over time, so too do the occurrences of the background set. As seen in Figure <ref>, the background growth closely mirrors the growth of Reddit.

Likewise, we expect the number of meme occurrences to increase at a similar rate. In order to control for the growth of Reddit over time we compute the normalized meme-frequency for each meme $\hat{F}_m(t) = F_m(t) / B(t)$.

Average normalized meme-frequency from 2010–2020 and 95% confidence interval (shaded region). Light grey lines show the individual normalized meme-frequency for a random 10% sample of individual memes. Overall, meme occurrence has remained consistent over the past decade (Pearson $R=+0.03$, $p$-value$<0.01$).

Figure <ref> illustrates the mean average normalized meme-frequency along with its 95% confidence interval from 2010 to 2020. A selection of individual memes are also plotted in light grey. We find that the occurrence of memes remains remarkably consistent when controlled for Reddit's overall activity, even as the occurrence of individual memes varies widely. Correlation analysis finds almost no association between time and the normalized meme-frequency (Pearson $R=+0.03$, $p$-value$<$0.01).

Competition for limited attention is not a new concept in socio-digital media studies. However, much of the previous analysis has focused on network-level simulation or mathematical modeling as an attempt to predict the behavior of large social systems <cit.>. User-level attention analysis seeks to understand the impact of competition and content volume on individual actors <cit.>. Other work attempts to determine which features will make a meme successful within a competitive environment <cit.>. These works are valuable extensions of the classic attention economy, yet fail to demonstrate the system-wide effect of competition. By showing that meme activity accounts for a stable fraction of all Reddit content, we have demonstrated the ability for competition to act as a kind-of global bandwidth cap. Even with a rapidly expanding user base, and a seemingly insatiable hunger for fresh content, Reddit appears only to sustain a relatively limited population of memes.

§ COMPETITION AMONG MEMES

The prevalence of individual memes rises and falls over time. Popularity is fleeting, but as a meme dies out, another always seems to rise to take its place. The ebb and flow of what is or is no longer popular has long been studied as the diffusion of innovations  <cit.>. The study of the production and evolution of innovations (, new ideas, behaviors, memes, and other information patterns) has recently become particularly compelling because of the availability of vast amounts of human behavior happening through online social platforms.

The dynamic behavior of memes, in particular, suggests that these ideas exist in constant competition with one another. The competition among memes is not unlike competition observed in markets, where innovations eventually replace outdated products, or ecological systems, where competition among individuals in a group exerts a selective pressure that rewards certain genetic innovations.

Average Simpson's Diversity Index across subreddits, and number of subreddits, over time. Meme diversity on Reddit is decreasing (Pearson $R=-0.63$, $p$-value$<0.01$) with a small slope (0.04% per month) despite an increase in the number of subreddits containing memes.

Bump chart representation of the yearly subreddit innovation ranking, for the top meme beachheads on Reddit. This version is limited to the top 20 global/all-time beachheads.

Bump chart representation of the yearly subreddit innovation ranking, for the top meme beachheads on Reddit. This version has no restriction on the subreddits - so we can see the turnover as subreddits suddenly become beachheads, or fade into obscurity.

Rank     2011     2013     2015     2017     2019

1     /r/pics     /r/funny     /r/4chan     /r/me_irl     /r/aww

2     /r/AskReddit     /r/AskReddit     /r/WTF     /r/aww     /r/ComedyCemetery

3     /r/funny     /r/WTF     /r/gaming     /r/pics     /r/NBA2K

4     /r/gaming     /r/pics     /r/TumblrInAction     /r/stevenuniverse     /r/Right_Wing_Politics

5     /r/reddit.com     /r/videos     /r/Smite     /r/AskReddit     /r/gaming

6     /r/politics     /r/AdviceAnimals     /r/pics     /r/worldnews     /r/funny

7     /r/WTF     /r/trees     /r/AskReddit     /r/woahdude     /r/madlads

8     /r/comics     /r/gaming     /r/funny     /r/nba     /r/TheNewsFeed

9     /r/IAmA     /r/4chan     /r/dogecoin     /r/Drama     /r/wow

10     /r/fffffffuuuuuuuuuuuu     /r/IAmA     /r/sex     /r/TwoBestFriendsPlay     /r/OutOfTheLoop

Top 10 most innovative subreddits by year. Colored subreddit names show the top 10 most innovative subreddits from 2010 to 2020 in aggregate.

Continuing the ecological analogy, one way to assess the health of an ecological system is to examine the diversity of the species living within the system. From this perspective, the ecology of a social media environment ought to have several different memes simultaneously appearing in abundance. This leads us to RQ2: How do memes compete for attention? Specifically, as Reddit grows and new memes are introduced, does the overall diversity increase or decrease?

One way to measure the diversity of an ecological system is with Simpson's Diversity Index ($D$) <cit.>. This index takes into account both the number of species present, as well as the population of each species. It is formally defined as:

\begin{equation}
D = 1 - \frac{\sum_{i=1}^{R} n_i(n_i-1)}{N(N-1)}
\end{equation}

Where $R$ is the total number of species in a community (numbered $1$ through $R$), $N$ is the total number of organisms, and $n_i$ is the number of individuals belonging to species $i$. For our purposes, we consider a subreddit to be a community, each meme as its own species, and each meme occurrence as a species entity. $D$ ranges from 0 to 1, where 0 indicates no diversity and 1 indicates infinite diversity.

We compute the Simpson's Diversity Index for each subreddit in each month. We then compute the average monthly Simpson Diversity across all subreddits, resulting in a monthly average diversity of Reddit at the community level. Figure <ref> shows the average diversity and 95% confidence intervals over time. Overall, the diversity of Reddit communities appears to be decreasing at a small (0.48% per year) but steady rate (Pearson $R=-0.63$, $p$-value$<$0.001).

This indicates that subreddit communities, on average, are using slightly less diverse meme subsets. This finding, coupled with the growth in the number of subreddits using memes, suggests a slow Balkanization of  Reddit communities. That is to say, as more subreddit communities come into existence, human attention – and the memes which that attention supports – moves towards more niche subreddits, which aligns with findings from other subreddit analysis <cit.>.

§.§ Innovative Communities

Ecological research focuses on the environments in which species exist; the communities and biomes that spawn new species are often the focus of study because they can host exotic species along with high levels of species diversity <cit.>. The same is true in social media, where the study of the creation and diffusion of innovations within social environments is a central topic of research. The detection and analysis of highly innovative communities are of great interest in this line of work <cit.>. As a direct consequence of our findings in RQ2 we further ask: Are there certain communities that consistently introduce new memes? These meme-nurseries are the places where many new linguistic and cultural innovations are born and cultivated. We further ask: do these innovative communities persist over time, or is their high-innovation only temporary?

To answer these questions, we first define a meme entry event as the first time that a meme is used within the Reddit ecosystem. The subreddit in which it first appears is defined as the landing space or beachhead for the new meme. From an ecological perspective, a new meme is analogous to new species, and subreddits are the host habitats. Because Reddit itself exists within the even larger environment of the social internet, tracing the true spread of a meme between specific communities is intractable. For this reason, it is necessary to consider that a single meme may have multiple beachheads within Reddit.

Under these circumstances, we elect to rank subreddits based on the order in which a meme first appears; this methodology reasonably controls for the uncertainty about which community was first, second, etc., to articulate the innovation. We constrict our analysis to the first 1000 subreddits to use each meme. Based on this ordering, we compute the mean reciprocal rank (MRR) of each community with respect to each of our 352 memes. These MRR values consider the rank for each subreddit for each year in the dataset. The top 10 most innovative subreddits in the odd-numbered years (due to space constraints) are given in Table <ref>

It is clear that certain subreddits contribute a disproportionate number of new memes to Reddit. The colors in Table <ref> represent the top 10 subreddits from all 10 years combined. It is unclear why these subreddits are more innovative than others, but there may be conditions within these particular communities that increase their ability to produce viral content more regularly than others. No matter the cause, optimal meme-nursery conditions appear to be transient, as subreddits can be highly innovative one year, and not the next.

There are a few conclusions to be drawn here. Early in Reddit's history, massive and highly contributive subreddits – like /r/reddit.com and /r/AskReddit – were the primary beachheads for new memes. However, as years progressed the set of top contributing subreddits became less consistent. Each new year comes with new meme beachheads.

We quantify the degree of change in subreddit ranks by computing Kendall's ($\tau$) coefficient between consecutive years. Larger $\tau$ values indicate more similarity, smaller $\tau$ indicates less similarity, and a negative $\tau$ represents dissimilarity. Figure <ref> illustrates $\tau$ for each pair of years where solid bars represent statistical significance $p<0.01$ and hollow bars vice versa $p>0.05$; there were no $p$-values between 0.01 and 0.05. Until the 2017/2018 pairing the rank correlation trended downward, indicating increased turnover in the topmost innovative subreddits.

Kendall's rank correlation coefficient ($\tau$) of subreddit innovation rankings for pairs of consecutive years. Solid bars represent $p<0.01$ and hollow bars represent $p>0.05$. Subreddits that consistently use new memes before other communities are ranked higher, but rankings change each year. A higher $\tau$ means more correlation between year-pairs, while lower indicates more change in their relative rankings. In general, subreddit beachhead rankings have become less stable, indicating greater turnover in the top subreddits.

The 2017 to 2018 evaluation showed a return to high-rank correlation, indicating less change in the top-ranked subreddits. There are a few potential explanations for this behavior. First, this may be due to a major Reddit policy change: beginning in June 2017, Reddit removed the default subreddits, which included many /r/pics, /r/funny, and many of the other most innovative subreddits, and instead introduced /r/popular, which was a mix of posts from various subreddits as a means to expose new users to a wider variety of communities[<https://www.reddit.com/r/announcements/comments/6eh6ga/reddits_new_signup_experience/>]. This change essentially means that users “subscribe” to a wider variety of subreddits by default, providing a greater opportunity for innovations from niche subreddits to become more easily accessed. Another potential explanation for this trend reversal is due to the fact that the meme set used for our analysis is biased towards more popular, and therefore older, memes. Memes created during the last years of our analysis window are less likely to have become popular enough to appear in our top-memes dataset. This may result in fewer meme entries in more recent years.

§ CHANGING DYNAMICS OF THE MEME ECOSYSTEM

Now that we have established some of the consequences of the competition of memes in a social media ecosystem, we turn our attention towards the dynamics of collective attention. Existing recent work suggests that these dynamics are accelerating <cit.>, that is, new concepts are becoming viral faster and stay viral for a shorter duration. Instead of focusing on specific cultural artifacts like memes, the previous work on general collective behavior focused on hashtags on Twitter, comments on Reddit, and n-grams in books, etc. Does this acceleration hold true for memes?  This is the basis for RQ3: How have the dynamics of collective attention on memes changed over time? Are we cycling through memes faster than we were a few years ago?

While we find this to be true in some ways, it is not true when the growth of the community is accounted for.
In other words, relative to the number of words produced on Reddit, the number of memes is not changing. But what about the dynamics of individual memes? How have they changed over time?

§.§ Investigating Meme Dynamics

(A) Average relative meme-frequency, time-shifted so that the maximum frequency occurs on day 0. Shaded areas indicate 95% confidence intervals. The width of the primary peak has not changed, suggesting that memes have not experienced significant acceleration. (B) Probability distribution of relative meme velocities divided into gains (right) and losses (left). Points give true distribution values, lines are fitted log-normal distributions. While small magnitude gains and losses have shifted slightly, larger velocities do not change. These stable velocities again indicate that memes have not accelerated.

Lorenz-Spreen et al. <cit.> used a variety of methods to analyze collective dynamics in the online social sphere. Here, we apply their methodology to our meme dataset.

Results of lifespan analysis on meme set for $\alpha = 0.005, 0.01$, and $0.02$. Lifespan starts on the first day with a frequency $\geq \alpha \hat{F}_m(t_{\textrm{peak}})$, and ends on the last day with frequency $\geq \alpha \hat{F}_m(t_{\textrm{peak}})$, such that all days in the lifespan have a frequency $\geq \alpha$ times the maximum normalized meme-frequency. (A) Number of active memes per month, where a meme is active only during its defined lifespan. Overall, the raw number of active memes has increased. The dip in later years is likely the result of bias toward older memes in our dataset. (B) Number of active memes per month, normalized by total Reddit contributions. Reddit growth is outpacing the number of active text memes. (C) Average meme lifespan (in days) over time and corresponding best fit lines. Shaded area indicates 95% confidence interval. Over the course of 10 years, the average lifespan has decreased ($\alpha = 0.005$: Pearson $R = -0.85$, $p$-value $<$0.01; $\alpha = 0.01$: Pearson $R = -0.77$, $p$-value $<$0.01; $\alpha = 0.02$: Pearson $R = -0.59$, $p$-value $<$0.01), indicating that memes are cycling faster.

First, we focus on the peaks of memes on Reddit to assess the pace of collective attention. For each meme, we compute its frequency across all of Reddit each day, such that $F_m(t)$ gives the frequency of meme $m$ on day $t$. We also identify the peak frequency for each meme $F_i(t_{\textrm{peak}})$ and when this peak occurred. To ensure all memes are on the same scale, the frequencies of each meme are then normalized by that meme's peak to get a relative meme-frequency: $R_m(t) = F_m(t) / F_m(t_{\textrm{peak}})$. In Figure <ref>(A), we illustrate the average relative peak frequency $R_m(t)$ for all memes in our dataset and group by the year of each meme's peak. Overall, there appears to be no change in peak dynamics over time. The difference between the peak and the baseline frequency (, frequency before and after the peak) remains relatively stable, nor does it change a statistically significant amount from year to year. Furthermore, the changes seen do not show a trend over time. This suggests that memes have not exhibited a significant acceleration over the past decade.

Next, we look closer at the velocities of memes. For each meme we compute relative gains $[\Delta F_m^{(g)} / F_m](t) = (F_m(t) - F_m(t-1) / F_m(t-1)$ and relative losses $[\Delta F_m^{(l)} / F_m](t) = (F_m(t) - F_m(t+1) / F_m(t+1)$, where gains are $>0$ and losses are $<0$. We analyze the distributions of losses and gains of all memes at all times, grouped by year. Both distributions fit well to a log-normal distribution. Gains and losses are shown in Figure <ref>(B). While we observe some shift in gains and losses with small magnitudes, the larger velocities do not change significantly or regularly across years.

Taken together, both of these analyses indicate that once the growth of Reddit is controlled for, the collective dynamics of Reddit memes have not accelerated. Rather, meme dynamics have remained remarkably consistent, even surrounding its peak.

§.§ Meme Lifespans are Shrinking

The previous analysis raises another interesting question about the collective dynamics of memes: Are meme lifespans growing or shrinking?

To answer this question we define the lifespan of a meme as follows. For a meme with a peak frequency of $F_m(t_{\textrm{peak}})$, the lifespan begins on the first day $u$ where $\hat{F}_m(u) \geq \alpha \hat{F}_m(t_{\textrm{peak}})$. Recall that $\hat{F}_m(t)$ is the normalized meme-frequency and is computed as $\hat{F}_m(t) = F_m(t) / B(t)$. The lifespan ends on the last day $v$ where the meme experiences a normalized frequency $\hat{F}_m(v) \geq \alpha \hat{F}_m(t_{\textrm{peak}})$, such that all days between the beginning, peak, and the end are continuous and above $\alpha \hat{F}_m(t_{\textrm{peak}})$. The threshold value $\alpha$ is very small; here we present results for $\alpha = 0.005, 0.01$, and $0.02$. Other values produced similar results.

By defining the lifespan this way, each meme's lifespan captures the majority of its occurrence, but does not include very early, late, or anomalous uses. The lifespan is also determined using normalized meme-frequencies to control for Reddit growth. We compute the lifespan length in number of days for each meme. For a threshold of $\alpha = 0.005$, these lifespans range from a high of 4140 days to a low of 1 day. (For completeness, we include the full history of Reddit beginning in 2005 when defining lifespans.)

Based on this definition, we consider a meme as active on all days contained within its lifespan. On any single day within our analysis window, there may be many active memes. Figure <ref>(A) illustrates the number of memes active within each month. As expected, the number of active memes has increased over time.

To more accurately assess the number of active memes, it is important to control for Reddit's growth. Figure <ref>(B) shows the normalized count of active memes, , the number of active memes per month divided by $B(t)$. With this normalization applied, it is apparent that over time, the relative number of active memes is decreasing. This decrease in the normalized number of active memes occurs despite our finding that normalized meme-frequency has remained stable (Figure <ref>). The most likely explanation for these contrasting findings is that the lifespan of memes is decreasing.

We can test this hypothesis directly by computing how lifespans have changed over time.

Figure <ref>(C) plots the average meme lifespan and 95% confidence interval in days for the active memes within each month. For example, if there were 40 memes active in June 2012 and the mean-average lifespan of these 40 memes was 2,400 days, then we would plot 2,400 for June 2012 (and the confidence interval). The lifespan is heavily dependent on the $\alpha$ threshold. So, we repeat these calculations for $\alpha = 0.005, 0.01$, and $0.02$. In all cases, we found that the average lifespan decreased over time ($\alpha = 0.005$: Pearson $R = -0.85$, $p$-value $<$0.01; $\alpha = 0.01$: Pearson $R = -0.77$, $p$-value $<$0.01; $\alpha = 0.02$: Pearson $R = -0.59$, $p$-value $<$0.01). Simply put, memes are rising and falling more quickly now than in the past.

To recap: the relative fraction of Reddit focused on memes has not changed over time (Figure <ref>), the meme diversity of subreddits has decreased (Figure <ref>), and the relative number of active memes has declined (Figure <ref>(B)). These results indicate that the production and consumption of new memes are accelerating. There is simply not enough attention to go around. As fast as new content is created, old content must be forgotten. Given the increasing volume of content on Reddit every day, the necessary consequence is that the collective attention span of Reddit is decreasing.

An exception to this rule can be found in long-lasting memes, many of which live beyond the end of our data analysis window. For example, the meme `cool story bro' is still active on the last day of 2019, following an extended lifespan beginning in 2009; this is true for all $\alpha$ values tested. These long-lasting memes have transcended their origins and evolved from memes into slang, maintaining a consistent presence even as Reddit grows, and other memes come and go. However, despite their seemingly `undying' status, these durable memes must still compete for attention within the broader ecosystem.

In a content-consumption system driven by user attention, memes are forced into competition with one another for scarce attention. As collective attention decreases, memes appear to rise and fall at an accelerated rate. In a system that favors the newest, freshest content, no meme is immortal.

A reasonable question then becomes: if active memes these are not decreasing though and meme lifespans are both decreasing, but the overall fraction of meme activity remains constant, where has all the meme activity gone? To reconcile this seemingly contradictory result, we next investigate how the maximum frequency of memes has changed.

^^this paragraph needs to conclude and then I think we end this section.
not before that paragraph?
perhaps... I'd like to put a slightly bigger bow on it though.

I'm not sure how maximum meme frequency answers the contradiction above.
I'm actually not convinced that there is a contradiction above.

i'm not confident that this following paragraph is particularly interesting or informative
rachel's lower comment may be pre-asking the same question.

For each meme, we find its maximum frequency $F_i(t_{peak}$, and normalize it by the background set to get the normalized maximum meme frequency. For each month in our analysis window, we compute the average normalized maximum meme frequency for all memes whose maximum frequency occurred in that month. This gives us the average maximum frequency for each month - an aggregate measure of the maximum attention given to any one meme at that time. We visualize this quantity in Figure <ref>. As with the total attention paid to memes, the maximum attention is also stable when normalized by Reddit's growth. This further supports the notion of a limited attention economy: even in an ever-expanding social network, memes can only command so much user interaction.

having just written that - this max freq/peak analysis (last 3 paragraphs) bit may fit better in section 1 than here - since it actually has nothing to do with lifespans, and very little to do with meme dynamics. thoughts?

Average maximum normalized meme-frequency occurring in each month. Shaded regions indicate 95% confidence intervals. Maximum frequency of memes is consistent when growth of Reddit is controlled for, because limited attention bounds each meme.candidate for deletion I can't get the y-label to line break, somebody HELP! WARNING!!!!!!! don't resurrect this plot without updating the data first!!!!!

§ CONCLUSIONS

The three research questions in the present work coalesce into an emerging theory of meme dynamics in online social bulletin boards like Reddit. Taking an ecological perspective, we have shown that although Reddit as a whole continues to grow, the fraction of Reddit devoted to text memes is consistent. This reinforces the notion of a meme ecology, which, like the economic and epidemiological perspectives of information diffusion, has consequences that derive from limited user attention.

We have also shown that the diversity of memes across communities is decreasing slightly, even as the number of communities continues to grow. This represents yet another consequence of the ecological perspective: as social media communities continue to Balkanize, , split into narrow and self-referential communities, the number of shared references to memes appears to be decreasing.

The competition among memes is also evident in the increasing turnover of beachhead communities. As memes fight for user and subreddit attention, cycling in and out of collective attention faster, large and general communities are unable to consistently integrate new memes before other smaller communities. Instead, subreddits may rise to the top if they happen to introduce the meme-of-the-moment, only to quickly fade into obscurity.

Finally, we have shown that meme lifespans have decreased significantly. Yet, unlike similar work on general n-grams, hashtags, citations, etc., we do not find that all aspects of meme dynamics are accelerating, and even the accelerations are bounded by the effect of increasing competition for limited user attention. The patterns of relative meme growth, even surrounding a meme's peak, remain relatively unchanged by the expansion of Reddit. And although the absolute number of memes active on Reddit has increased over time, this growth has been outpaced by the growth of Reddit itself.

It is important to recognize the limitations of this study. Our evaluation is principally limited by the KnowYourMeme dataset. Because the popularity metric is accumulative on KnowYourMeme, it is likely that newer memes, which have not yet accumulated their full/final popularity, are less likely to appear in our dataset. Our focus on the textual artifacts of memes may also limit the generalizability of our findings. Much of the online meme culture, especially recently, is conveyed through visual depictions of memes <cit.>. We necessarily assume that textual representations of visual memes follow proportionately, but we are unable to validate this claim in the current work. Future work should endeavor to confirm these findings on visual imagery. Finally, the present work used a strict text-matching algorithm to identify memes. Additional follow-up work is needed to better understand and trace how these memes evolve and how the shape of a meme's popularity and evolution affects its lifespan and reach.

§ ACKNOWLEDGEMENTS

We would like to thank Satyaki Sikdar for his help preparing this manuscript. This work is funded by the US Army Research Office (W911NF-17-1-0448) and the US Defense Advanced Research Projects Agency (DARPA W911NF-17-C-0094).