11institutetext: Ben Gurion University of the Negev, Israel22institutetext:
HUB Security, Israel

# Anomaly Detection for Fraud in Cryptocurrency Time Series

Eran Kaufman 11 Andrey Iaremenko 33

###### Abstract

Since the inception of Bitcoin in 2009, the market of cryptocurrencies has
grown beyond initial expectations as daily trades exceed $10 billion. As
industries become automated, the need for an automated fraud detector becomes
very apparent. Detecting anomalies in real time prevents potential accidents
and economic losses. Anomaly detection in multivariate time series data poses
a particular challenge because it requires simultaneous consideration of
temporal dependencies and relationships between variables. Identifying an
anomaly in real time is not an easy task specifically because of the exact
anomalistic behavior they observe. Some points may present pointwise global or
local anomalistic behavior, while others may be anomalistic due to their
frequency or seasonal behavior or due to a change in the trend. In this paper
we suggested working on real time series of trades of Ethereum from specific
accounts and surveyed a large variety of different algorithms traditional and
new. We categorized them according to the strategy and the anomalistic
behavior which they search and showed that when bundling them together to
different groups, they can prove to be a good real-time detector with an alarm
time of no longer than a few seconds and with very high confidence.

###### Keywords:

Cryptocurrency,Anomaly detection,Machine Learning, Deep nets,Fraud detection

##  1 Introduction

Blockchain technology has become pervasive today in many industries, such as
the financial industry, cloud infrastructure, supply chain management and
others. Blockchain is a distributed decentralized ledger with guarantee of
immutability of the data stored in the ledger across multiple computational
nodes. This cryptography-based technology removes the central authority and
enables fast and trusted sharing of data and exchange of value directly
between two or more parties. Today there are many kinds of blockchains,
customized to specific needs and environments. In the monetary sector there
are multiple cryptocurrencies traded on the open markets, just like a regular
forex. Moreover, in the US, Japan and Europe cryptocurrency trading is
officially regulated. Blockchain based payments with cryptocurrencies (e.g.
Bitcoin, Ethereum) provide fast, cheap, anonymous and secure payments
worldwide. However, the immutability of the payment transaction presents a
problem when a malicious actor obtains control of the access to the Bitcoin
account and moves the tokens to another account on the blockchain. In
particular, it is practically impossible to rewind the transaction and
retrieve the money. Thus the largest security issue with cryptocurrencies is
the access to the private key (a random number of 32 bytes), which represents
the entire monetary holdings of a personal account on the blockchain. This
risk is exponentially higher in the case of institutions handling large
amounts of money and managing third party assets in an automated and round-
the-clock manner. Therefore automated malicious activity detection mechanisms
are required to keep pace with the organizational use of blockchain. For the
financial use of blockchain there are relatively few parameters and data
fields to inspect with automated anomaly detection algorithms, similar to
regular banking fraud detection. For Bitcoin transactions, the following
fields are most relevant for anomaly detection:

  1. 1.

Payment amount. Each account has relatively unique behavior patterns regarding
transaction amounts as stand-alone metric (like groups of values) and also
relative to the time scale (time of day, weekly and monthly scales).

  2. 2.

The destination address. Amount value behavior is coupled to the recepient
address field. There may be a small number of addresses that receive similar
payments over time, and one-time addresses with a single payment.

  3. 3.

The Fee value. Fee is a transactional surcharge for the miners (or
maintainers) of the blockchain infrastructure. Bitcoin fees are traditionally
predefined in the wallet application.

For Ethereum transactions, the following fields are most relevant to anomaly
detection:

  1. 1.

The payment amount value, similar to Bitcoin.

  2. 2.

The destination address, similar to Bitcoin.

  3. 3.

Gas limit and Gas price. These two parameters reflect the fee paid to the
maintainers of the blockchain infrastructure.

In ethereum the fee is calculated in real time for each transaction, based on
the current network load and on the required speed of transaction approval on
the blockchain. In the Ethereum network, a transactional request is an actual
computation in a dedicated virtual machine in the network. Gas is the unit of
work to execute specific computational operations on the blockchain. Each
transaction has a base fee to be included in the blockchain, based on the
complexity of the required calculations. The transaction initiator can pay an
additional fee to secure faster processing and approval time. Thus the fee
component could be an indicator of account fraud. All of the above parameters
are trackable per account and per unit of time for use in anomaly detection
processes. Each account has a different behavior over time and in between
temporal epochs. So the anomaly detection methods should combine training and
inference models. The algorithms seek anomalies in the spending behavior of
the account over a period of time, for example a comparably large transaction
amount or a group of many small transactions.

####  1.0.1 Our contribution.

We focus our attention on the spending behavior of an account over time.
Unlike previous work in this field which focused on documented Ponzi schemes,
our aim is to find discrepancies in an online working account. We conducted
comparative research using multiple anomaly detection algorithms from a vast
range of algorithms and models and compared their results in order to find the
best suited algorithms for both researchers and practitioners alike.

The algorithms must be both very reliable, and also run in real time, meaning
the detection alarm must be raised within a few seconds of its occurrence.
Since we work on unsupervised data, we found that ‘hand labeling’ anomalies
rarely works well. This is because the question often posed to an arbiter –
when is an anomaly really an anomaly? – is not a simple one to answer. For
this reason we ran a large scale of algorithms from different approaches and
categorized them into different categories. An anomaly was decided to be an
anomaly only when the majority out of multiple algorithms from the _same_
category made the same decision. (We found that deciding anomalies based on
different categories led to poor results.) Secondly, In regard to the feature
space, out of the many possible features, we identified three features as
being the most significant: These are payment amount, Gas price, and gas limit
(as explained above). Moreover, we did both a univariate and a multivariate
analysis of the data, i.e. we referred to the vector of features both as
individual time series as well a combined three-vector. By doing so we mark
the fact that while each feature may not be ”out of range” individually, the
vector as a whole can be anomalous as, for example, the gas price and gas
limit should be correlated.

We combined both pointwise anomalies based on representations of the data in
the sample space along with contextual and collective anomalies based on the
time space representation since for fraud detection not only the value but
also the frequency is an important factor. Whilst a certain price may be of
regular value in a certain context, it may be the case that a very frequent
request implies a D.o.S. (Denial of Service) attack, or conversely, close
prices may tend come together in bundles (as this is the current price of the
market) where as, a single request followed by long ’silent’ periods may be a
sign of a fraud.

##  2 Related Work

###  2.1 Anomalies in Time-series Data

An anomaly in time-series data is a data point (or points) at a certain time
step (or steps) that shows unexpected behaviors differing significantly from
the behavior of previous time steps. Anomaly detection can be categorized in
the follwing way [2]:

  1. 1.

Pointwise anomalies. Also known as global outliers, these points lay outside a
user defined sensitivity parameter over the entirety of a data set. This user
defined threshold is used to balance between type $1$ and type $2$ statistical
errors.

  2. 2.

Contextual anomalies. Also referred to as conditional outliers, these
anomalies have values that significantly deviate from other data points that
exist in the same context (usually a period) but are not significant in the
global sense. The value exists within global expectations but may appear
anomalous within certain seasonal data patterns.

  3. 3.

Collective anomalies, when a subset of _continuous points_ within a set is
anomalous to the entire dataset. In this category, individual values are not
anomalous globally or contextually but only the entire subset. Individual
behavior may not deviate from the normal range in a specific section, but when
combined with other sections these anomalies become apparent.

###  2.2 Anomaly Detection Approaches

Another way to categorize anomalies is based on the implementation method:

####  2.2.1 Statistical Models.

Statistical models generate statistical measures, such as mean, variance,
median, quantile, kurtosis, skewness, and many more. With the generated model,
a newly added time-series data can be inspected to determine whether it
belongs to the normal boundary [18].

####  2.2.2 Predictive Models.

Predictive models are among the common approaches to anomaly detection. These
methods forecast future states based on past and current states. We can deduce
the anomaly according to the severity of the discrepancy between the predicted
value and the real one. For example, autoregressive integrated moving average
(ARIMA) [4] is frequently employed to forecast time-series.

The ARIMA model is composed of three parts:

  1. 1.

An auto-regressive (AR) component which is composed of a weighted sum of
lagged values, and can model the value of a random variable $X$ at time step
$t$ as:

$AR(P):X_{t}=\phi_{1}X_{t-1}+\phi_{2}X_{t-2}+\ldots+\phi_{p}X_{t-p}+\epsilon_{t}$
(1)

where $\\{\phi_{i}\\}$ are autocorrelation coefficients and $\epsilon$ is
white noise. The parameter $p$ is the order of AR model.

  2. 2.

A moving-average (MA) component which computes the weighted sum of lagged
prediction errors and is formulated as:

$MA(q):X_{t}=\epsilon_{t}-\theta_{1}\epsilon_{t-1}-\theta_{2}\epsilon_{t-2}-\ldots--\theta_{q}\epsilon_{t-q}$
(2)

where $\\{\theta_{i}\\}$ are moving-average coefficients, $\epsilon_{t}$
denotes a model prediction error at time step $t$, and $q$ is the order of MA
model.

  3. 3.

An integrated component representing the time-series using differences, and
thus a data point at time step $t$ is $\hat{X}=X_{t}-X_{t-1}$ where $d$
denotes the order of differencing.

The differencing process makes the time-series stationary, resulting in ARIMA
being very effective for non-stationary time-series. If the time-series data
has a seasonal variation, we can use a variant called seasonal ARIMA (SARIMA)
[30]. In this case, we introduce the additional parameters: P, D, and Q, which
deal with the seasonality. These parameters are used in the same manner as p,
d, and q.

The parameters for the S/ARIMA can be learned in a supervised or non-
supervised manner. In the non-supervised manner the order of $p$ and $q$ can
be determined by using the sample autocorrelation function or via the Akaike
information criterion.

In the supervised manner the parameters are learned from a training set using
cross-validation. We tested both approaches in our experiments, and saw very
similar results.

Another commonly used algorithm is the season-trend decomposition better known
as STL. STL is a statistical method for breaking down time-series data into
three different uncorrelated components that contain seasonality, trend and
residues. The trend analysis of the data shows a general direction of the
overall data, while the seasonal analysis presents a pattern which is repeated
at a fixed time intervals. The residue (noise) is the random oscillation or
unexpected change. The frequency meta parameter for this algorithm can be
found either by using the autocorrelation function in the time domain or by
calculating the bandwidth of the Fourier transform in the frequency domain.

In the category of predictive models we also include the SVM regressor, the
nearest neighbor regressor and neuralnets based regressors. These methods
assume a ’natural’ trade line to the dataset behavior and exclude points which
are out side a certain ’strip’ of the predicted line. These are usually global
or contextual anomalies.

####  2.2.3 Clustering Models.

Clustering based methods are choices for grouping the data into disjoint
groups. Once a time-series is mapped into a multidimensional space, clustering
algorithms group them close to a cluster depending on their similarities. Here
we assume that there is some metric in which closely related points are close
to each other and the outliers are far away. They can vary from centroid based
clustering like k-means to hierarchical based clustering like BDSCAN and
others. These algorithms are usually collective anomalies and are represented
in the time (not sample) domain. Popular data clustering methods include the
k-means algorithm [16], one-class support vector machine (OCSVM) [17], or
Gaussian mixture model (GMM) [19].

OCSVM [17] is an unsupervised algorithm based on dividing the points into
disjoint spaces with the margin between spaces being as wide as possible.

Instead of working in feature space it is often better to work in the kernel
space instead. Working in kernel space provide two principal benefits. The
first being that it implicitly induces a non-linear feature map, which allows
for a richer space of classifiers. The second is that, when the kernel trick
is available, it effectively replaces the dimension of the feature space with
the size of the sample space which allows for faster computations. Among the
most popular kernels are linear kernel, polynomial kernel and radial basis
kernel (RBF).

####  2.2.4 Dimensionality Reduction Models.

Dimensionality reduction models assume that a large scale system can be
represented using a few significant factors. Thus, by extracting the main
features via dimensional reduction we can reconstruct the data and separate
the expected data from noise. Linear algebra based methods include principal
component analysis (PCA) and singular value decomposition (SVD) [28].

Tree based methods include isolation forest and neural network based method
include the neuralnet AutoEncoder (AE). Isolation forest is an algorithm that
detects anomalies by the depth of the tree needed for isolating each point. At
the basis of the algorithm there is an assumption that outliers are easier to
separate from the rest of the points, compared to ordinary points. In
principle, anomalies are further away in the feature space and fewer splits
are required than a normal point. Anomalous points will therefore be defined
as points where the number of partitions required until they are insulated is
low.

####  2.2.5 NeuralNet Based Models.

Using deepnets can come in some of aforementioned techniques i.e. either as a
predictive model or as a dimensionality reduction model. In time-series
applications, the temporal context should be considered when modeling the
series. For this reason two general kinds neural net are considered

  1. 1.

RNNS. RNNs have been extended with other variants, such as LSTM [10] and GRU
[7]. LSTM and GRU address the vanishing or exploding gradient problem, where
the gradient becomes too small or too large as the network goes deeper. There
are multiple gates in an LSTM and a GRU cell, and they can learn long-term
dependencies by determining the number of previous states to keep or forget at
every time step. Meanwhile, the dilated RNN is proposed to extract multiscale
features while modeling long-term dependencies by using a skip connection
between hidden states. Shen et al. [25] for instance, adopt a three layer
dilated RNN and extract features from each layer to jointly consider longterm
and short term dependencies. RNN-based approaches are generally used for
anomaly detection in two ways. One is to predict future values and compare
them to predefined thresholds or the observed values. This strategy is applied
in [25, 12, 21, 14]. The other is to construct an autoencoder in order to
restore the observed values and evaluate the discrepancy between the
reconstructed value and observed one. This strategy is used in [11, 15, 15,
26, 9].

  2. 2.

CNNs. Although the RNN is the primary option for modeling time-series data,
CNN sometimes shows better performance in several applications that work with
short term data. [8, 29, 31]

By stacking convolutional layers, each layer learns a higher level of
features. In addition, the pooling layers introduce non-linearity to the CNN,
allowing them to capture the complex features in the sequences. Instead of
explicitly capturing the temporal context, the CNN models learn patterns in
segmented time-series.

In the field of the cryptocurrency anomaly detection many previous works were
done on finding anomalies using machine learning techniques. Bartoletti et al.
[3], Chen et al. [5], [6] and Jung et al. [13] analyzed Ponzi schemes of
Ethereum trade developed as smart contracts.

Chen et al. [5] used a regression tree model (XGBoost) to detect Ponzi schemes
on Ethereum. A subsequent work of the same authors [6] improved the results by
using the Random Forest (RF) classifier. Jung et al. [13] analyzed both
behavior and code frequency of contracts to detect Ponzi schemes, and obtained
a precision of $99\%$ and recall of $97\%$.

Bartoletti et al. and Toyoda et al. used addresses clustering techniques
[22],[20] to extend their datasets of scam addresses, and both applied
_supervised_ machine learning techniques to detect Ponzi schemes. In
particular, Toyoda et al. used XGboost and Random Forest classifiers, and
Bartoletti et al. used a cost-sensitive Random Forest classifier,.

Schnaubelt [23], Shahbazi [24] and Tanwar[27] showed the use of deep
reinforcement learning to find the optimal placement of cryptocurrency limit
orders and Akba et al. [1] used predictive models for detection of a
manipulator in the cryptocurrency market.

##  3 Experiments

For our experiments we ran all the above mentioned algorithms and compared the
results for anomalies in the same category. In the category of predictive
models we used ARIMA, SARIMA, STL, LSTM based regressors, kernel ridge
regression, support vector regressor (SVR) RBF, SVR with a polynomial kernel
and nearest neighbor regressor. In the category of dimensionality reduction
models we used Isolation forest, LSTM AE and PCA based models. In the category
of cluster based models we used the OCSVM, k-means and DBSCAN based model.

The datasets for these experiments are in the public domain, collected from
etherscan.io and accessible through public APIs. The name of stock markets and
the active accounts are available on our GitHub.Also comprehensive results of
the different algorithms with their accuracy can be found on GitHub 111
https://github.com/erankfmn/anomaly_detection. Here we present some
illustrative results, while the conclusion section summarizes the result from
all datasets and algorithms. The algorithms are briefly explained in
subsection 2.2.

Figure 1: example of point summation for time representations

####  3.0.1 Processing the Data.

The data is given as the dates and amounts of the ethereum transaction for a
single account. Dates and time where no transaction took place are not given
in the dataset (since this is the common case). While some anomaly algorithms
work better in the sample space and even may be confused by a long period of
null transactions (essentially, marking every transaction as an anomaly).
Other algorithms which search for an anomaly of a point in temporal space need
their representation in time domain instead.

When treating the series based on time rather than on samples an issue arrises
as the data is typically given in the format of value and date, it maybe the
case that some exact dates cooccur (at the exact same second). While working
in the sample domain these points are regarded as different points however, in
the time domain it is unclear as to how they should be treated since there can
only be one data point for each sampling time. We decided in these cases to
merge these points into to a single point. Other possible approaches may
include taking the maximum or the average value, but after exhaustive trial
and error, we found that merging was more appropriate in the context at hand.
An example of such a manipulation is illustrated in figure 1.

Figure 2 shows the raw data in the sample space for the fields of payment
amount, gas limit and gas price. It is apparent that the different fields are
highly uncorrelated and demonstrate different anomaly behavior.

Figure 2: Raw Data of the payment amount gas price and gas limit

Figure 3 shows the concurrent result of ARIMA, SARIMA and STL algorithms for
the payment amount feature and at the bottom of the graph shows points which
more than two algorithms concur.

Figure 3: ARIMA SARIMA STL results for anomalies in the payment field

In order to estimate the parameters for S/ARIMA cross validation was used, the
data was split into train and test $70\%-30\%$ and validated over all possible
combinations of parameters over the range of $1-3$. Test values whose
difference between the prediction and actual value greater than $3\cdot RMS$
were detected as an outlier.

We can concluded that the three algorithms identified the same samples quite
similarly. For example, for SARIMA accuracy is $97\%$ and recall $96.6\%$ .
The ARIMA and SARIMA algorithms identified almost exactly the same anomalies,
probably due to the fact the that the seasonal component did not have
significant weight in the sampling classification.

The STL algorithm produced similar result to the two other algorithms.

The points that were identified were usually points at the beginning or end of
major trend changes.

The two following graphs in figure 4b are for the gas price and gas limit
yielding the same results.

Figure 4: ARIMA SARIMA STL results for anomalies in the gas fields

(a) Gas Price (b) Gas Limit

Figure 5 shows the results of LSTM and regression trees for the payment value.
Figure 6b shows the same result for gas price and gas limit. Since predictive
models are supervised models they are split to train and test sets. When
commerce start to rise the tendency of predictive models is to keep the
tendency line rising and when sudden and abrupt changes occur they are
classified as anomalies. The LSTM architecture was constructed by 3 layers of
LSTM consisting of $128,64,32$ units respectively, followed by a dropout layer
- this is the encoder part. Afterwards a repeated vector layer followed by the
decoder unit of 3 layers of LSTM $32,64,128$ units respectively.

Figure 5: Anomalies detected for payment by regression trees and LSTM Figure
6: Anomalies detected for the gas fields by regression trees and LSTM

(a) Gas Price (b) Gas Limit

Figure 7: Anomalies detected for payment by isolation forest and OCSVM Figure
8: Anomalies detected for gas fields by isolation forest and OCSVM

(a) Gas Price (b) Gas Limit

Figures 7 and 8b were taken from algorithms who look at collective anomalies.
For the isolation forest and the OCSVM collective points were prepared as a
continuous time-series, the sliding window was set to be $5$ continuous
minutes and each sampling point can belong to several vectors at the same time
in the time domain. These type of algorithms perform online learning, they
receive a set of samples of about five minutes and compares them to the
previous four day database. The database is also updated every four days as
detection needs to happen in real time.

We used OCSVM with RBF kernel. The regularization parameter $C$ was $5$-fold
cross-validated over the set $\\{2^{5},2^{3},\ldots,2^{15}\\}$, and for the
RBF kernel, the $\gamma$ parameter was five-fold cross-validated over the set
$\\{2^{5},2^{3},\ldots,2^{3}\\}$.222This is also true for the SVR where also
tried both polynomial and RBF kernels

##  4 Conclusion and Further Research

In this paper we evaluated the anomalistic behavior of Ethereum trade of
several accounts for a real time fraud detector. The detection searched for
pointwise as well as contextual and collective anomalies both in the sample
domain, the time domain and the frequency domain. We used many well known
algorithms traditional and new to try and compare their performances. We
categorized the algorithms according to the their strategy of detection. We
marked a point as an anomaly when there was a majority vote on that point. In
the category of predictive models we found that classical as well as neural
based methods have an accuracy of $97\%$ and a recall of $96.6\%$. They work
well together and tend to appear when trade line are changes abruptly.
Dimensionality reduction methods on the other hand are contextual and
collective anomalies and work well both in time or sample representations.
Here we found poor results since especially neuralnets AE detected more points
than other methods such as isolation forest and PCA and we believe them to be
more sensitive due to their complexity. Clustering methods have an accuracy of
$96.3\%$ and a recall of $93.2\%$. They are collective anomalies that work
well both in the time domain as well as in the sample domain. They also work
both as univariate and multivariate models detecting uncorrelated vectors. All
model give reliable real time results either pretrained or online learners.
The over all conclusion of our research is that for a real time anomaly
detector with unsupervised data the best approach is to have parallel
processing divided according to the anomalistic behavior: pointwise local or
global, contextual, collective,in the sample or time representation. And have
them work simustansouly while using a majority vote. Models from the same
category traditional and new tend to have a high rate of agreement and are
therefore reliable and relatively fast for real time fraud detection.

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# Interacting agegraphic quintessence dark energy in non-flat universe

A<EMAIL_ADDRESS>A. Bagheria and M.M.
<EMAIL_ADDRESS>aDepartment of Physics, Shahid Bahonar
University, P.O. Box 76175, Kerman, Iran
bResearch Institute for Astronomy and Astrophysics of Maragha (RIAAM),
Maragha, Iran

###### Abstract

We suggest a correspondence between interacting agegraphic dark energy models
and the quintessence scalar field in a non-flat universe. We demonstrate that
the agegraphic evolution of the universe can be described completely by a
single quintessence scalar field. Then, we reconstruct the potential of the
interacting agegraphic quintessence dark energy as well as the dynamics of the
scalar field according to the evolution of the agegraphic dark energy.

##  I Introduction

It is a general belief that our universe is currently experiencing a phase of
accelerated expansion Rie . Missing energy component with negative pressure
which is responsible for this expansion constitute a major puzzle of modern
cosmology. Despite the theoretical difficulties in understanding dark energy,
independent observational evidence for its existence is impressively robust.
Among the various candidates to explain the accelerated expansion, the
cosmological constant with $w=-1$ is located at a central position. Though, it
suffers the so-called fine-tuning and cosmic coincidence problems. In
quintessence QUINT  and Chaplygin gas KMP  $w$ always stays bigger than $-1$.
The phantom models of dark energy have $w<-1$ PHANT . However, following the
more accurate data analysis, a more dramatic result appears showing that the
time varying dark energy gives a better fit than a cosmological constant and
in particular, $w$ can cross $-1$ from above to below Alam .

An interesting attempt for probing the nature of dark energy within the
framework of quantum gravity is the holographic dark energy. This proposal,
that arose a lot of enthusiasm recently Coh ; Li ; Huang ; Hsu ; HDE ; Setare1
, is motivated from the holographic hypothesis Suss1  and has been tested and
constrained by various astronomical observations Xin . However there are some
difficulties in holographic dark energy model. Choosing the event horizon of
the universe as the length scale, the holographic dark energy gives the
observation value of dark energy in the universe and can drive the universe to
an accelerated expansion phase. But an obvious drawback concerning causality
appears in this proposal. Event horizon is a global concept of spacetime;
existence of event horizon of the universe depends on future evolution of the
universe; and event horizon exists only for universe with forever accelerated
expansion. In addition, more recently, it has been argued that this proposal
might be in contradiction to the age of some old high redshift objects, unless
a lower Hubble parameter is considered Wei0 . Another proposal to probe the
nature of dark energy within the framework of quantum gravity is a so-called
agegraphic dark energy (ADE). This model is based on the uncertainty relation
of quantum mechanics together with the gravitational effect in general
relativity. Following the line of quantum fluctuations of spacetime,
Karolyhazy et al. Kar1  argued that the distance $t$ in Minkowski spacetime
cannot be known to a better accuracy than $\delta{t}=\beta t_{p}^{2/3}t^{1/3}$
where $\beta$ is a dimensionless constant of order unity. Based on Karolyhazy
relation, Maziashvili discussed that the energy density of metric fluctuations
of the Minkowski spacetime is given by Maz

$\rho_{D}\sim\frac{1}{t_{p}^{2}t^{2}}\sim\frac{m^{2}_{p}}{t^{2}},$ (1)

where $t_{p}$ is the reduced Planck time. We use the units $c=\hbar=k_{b}=1$
throughout this work. Therefore one has $l_{p}=t_{p}=1/m_{p}$ with $l_{p}$ and
$m_{p}$ are the reduced Planck length and mass, respectively. The agegraphic
dark energy model assumes that the observed dark energy comes from the
spacetime and matter field fluctuations in the universe Cai1 ; Wei2 ; Wei1 .
Since in agegraphic dark energy model the age of the universe is chosen as the
length measure, instead of the horizon distance, the causality problem in the
holographic dark energy is avoided. The agegraphic models of dark energy have
been examined and constrained by various astronomical observations age ; shey1
; Setare2 . Although going along a fundamental theory such as quantum gravity
may provide a hopeful way towards understanding the nature of dark energy, it
is hard to believe that the physical foundation of agegraphic dark energy is
convincing enough. Indeed, it is fair to say that almost all dynamical dark
energy models are settled at the phenomenological level, neither holographic
dark energy model nor agegraphic dark energy model is exception. Though, under
such circumstances, the models of holographic and agegraphic dark energy, to
some extent, still have some advantage comparing to other dynamical dark
energy models because at least they originate from some fundamental principles
in quantum gravity. We thus may as well view that this class of models
possesses some features of an underlying theory of dark energy.

On the other hand, the scalar field model is an effective description of an
underlying theory of dark energy. Scalar fields naturally arise in particle
physics including supersymmetric field theories and string/M theory.
Therefore, scalar field is expected to reveal the dynamical mechanism and the
nature of dark energy. However, although fundamental theories such as string/M
theory do provide a number of possible candidates for scalar fields, they do
not uniquely predict its potential $V(\phi)$. Therefore it becomes meaningful
to reconstruct $V(\phi)$ from some dark energy models possessing some
significant features of the quantum gravity theory, such as holographic and
agegraphic dark energy model. The investigations on the reconstruction of the
potential $V(\phi)$ in the framework of holographic dark energy have been
carried out in Zhang . In the absence of the interaction between agegraphic
dark energy and dark matter, the quintessence reconstruction of the agegraphic
dark energy models have been established ageQ .

In this paper we intend to generalize the study to the case where both
components- the pressureless dark matter and the agegraphic dark energy- do
not conserve separately but interact with each other. Given the unknown nature
of both dark matter and dark energy there is nothing in principle against
their mutual interaction and it seems very special that these two major
components in the universe are entirely independent Setare3 ; wang1 ; shey2 .
We shall establish a correspondence between the interacting agegraphic dark
energy scenarios and the quintessence scalar field. We suggest the agegraphic
description of the quintessence dark energy in a universe with spacial
curvature and reconstruct the potential and the dynamics of the quintessence
scalar field which describe the quintessence cosmology.

This paper is outlined as follows. In the next section we demonstrate a
correspondence between the original agegraphic and quintessence dark energy
model. In section III, we establish the correspondence between the new model
of interacting agegraphic dark energy and the quintessence dark energy. The
last section is devoted to conclusions.

##  II Quintessence reconstruction of ORIGINAL ADE

We assume the agegraphic quintessence dark energy is accommodated in the
Friedmann-Robertson-Walker (FRW) universe which is described by the line
element

$\displaystyle
ds^{2}=dt^{2}-a^{2}(t)\left(\frac{dr^{2}}{1-kr^{2}}+r^{2}d\Omega^{2}\right),$
(2)

where $a(t)$ is the scale factor, and $k$ is the curvature parameter with
$k=-1,0,1$ corresponding to open, flat, and closed universes, respectively. A
closed universe with a small positive curvature ($\Omega_{k}\simeq 0.01$) is
compatible with observations spe . The corresponding Friedmann equation takes
the form

$\displaystyle
H^{2}+\frac{k}{a^{2}}=\frac{1}{3m_{p}^{2}}\left(\rho_{m}+\rho_{D}\right).$ (3)

We introduce, as usual, the fractional energy densities such as

$\displaystyle\Omega_{m}=\frac{\rho_{m}}{3m_{p}^{2}H^{2}},\hskip
14.22636pt\Omega_{D}=\frac{\rho_{D}}{3m_{p}^{2}H^{2}},\hskip
14.22636pt\Omega_{k}=\frac{k}{H^{2}a^{2}},$ (4)

thus, the Friedmann equation can be written

$\displaystyle\Omega_{m}+\Omega_{D}=1+\Omega_{k}.$ (5)

We adopt the viewpoint that the scalar field models of dark energy are
effective theories of an underlying theory of dark energy. The energy density
and pressure for the quintessence scalar field can be written as

$\displaystyle\rho_{\phi}=\frac{1}{2}\dot{\phi}^{2}+V(\phi),$ (6)
$\displaystyle p_{\phi}=\frac{1}{2}\dot{\phi}^{2}-V(\phi),$ (7)

Then, we can easily obtain the scalar potential and the kinetic energy term as

$\displaystyle V(\phi)=\frac{1-w_{D}}{2}\rho_{\phi},$ (8)
$\displaystyle\dot{\phi}^{2}=(1+w_{D})\rho_{\phi}.$ (9)

Now we are focussing on the reconstruction of the original agegraphic
quintessence model of dark energy. The original agegraphic dark energy density
has the form (1) where $t$ is chosen to be the age of the universe

$T=\int_{0}^{a}{\frac{da}{Ha}},$ (10)

Thus, the energy density of the original agegraphic dark energy is given by
Cai1

$\rho_{D}=\frac{3n^{2}m_{p}^{2}}{T^{2}},$ (11)

where the numerical factor $3n^{2}$ is introduced to parameterize some
uncertainties, such as the species of quantum fields in the universe, the
effect of curved space-time (since the energy density is derived for Minkowski
space-time), and so on. The dark energy density (11) has the same form as the
holographic dark energy, but the length measure is chosen to be the age of the
universe instead of the horizon radius of the universe. Thus the causality
problem in the holographic dark energy is avoided. Combining Eqs. (4) and
(11), we get

$\displaystyle\Omega_{D}=\frac{n^{2}}{H^{2}T^{2}}.$ (12)

The total energy density is $\rho=\rho_{m}+\rho_{D}$, where $\rho_{m}$ and
$\rho_{D}$ are the energy density of dark matter and dark energy,
respectively. The total energy density satisfies a conservation law

$\dot{\rho}+3H(\rho+p)=0.$ (13)

However, since we consider the interaction between dark matter and dark
energy, $\rho_{m}$ and $\rho_{D}$ do not conserve separately; they must rather
enter the energy balances

$\displaystyle\dot{\rho}_{m}+3H\rho_{m}=Q,$ (14)
$\displaystyle\dot{\rho}_{D}+3H\rho_{D}(1+w_{D})=-Q.$ (15)

Here $w_{D}$ is the equation of state parameter of agegraphic dark energy and
$Q$ denotes the interaction term and can be taken as $Q=3b^{2}H\rho$ with
$b^{2}$ being a coupling constant. This expression for the interaction term
was first introduced in the study of the suitable coupling between a
quintessence scalar field and a pressureless cold dark matter field Ame ; Zim
. In the context of holographic dark energy model, this form of interaction
was derived from the choice of Hubble scale as the IR cutoff Pav1 . Taking the
derivative with respect to the cosmic time of Eq. (11) and using Eq. (12) we
get

$\displaystyle\dot{\rho}_{D}=-2H\frac{\sqrt{\Omega_{D}}}{n}\rho_{D}.$ (16)

Inserting this relation into Eq. (15), we obtain the equation of state
parameter of the original agegraphic dark energy

$\displaystyle
w_{D}=-1+\frac{2}{3n}\sqrt{\Omega_{D}}-\frac{b^{2}}{\Omega_{D}}(1+\Omega_{k}).$
(17)

Figure 1: The evolution of $w_{D}$ for original ADE with different interacting
parameter $b^{2}$. Here we take $\Omega_{D0}=0.72$ and $\Omega_{k}=0.01.$
Figure 2: The evolution of $\Omega_{D}$ for original ADE with different
interacting parameter $b^{2}$. Here we take $\Omega_{D0}=0.72$ and
$\Omega_{k}=0.01$. Figure 3: The reconstruction of the potential $V(\phi)$ for
original ADE with different model parameter $n$, where $\phi$ is in unit of
$m_{p}$ and $V(\phi)$ in $\rho_{c0}$. We take here $\Omega_{m0}=0.28$ and
$\Omega_{k}=0.01$.  Figure 4: The reconstruction of the potential $V(\phi)$
for original ADE with different interacting parameter $b^{2}$, where $\phi$ is
in unit of $m_{p}$ and $V(\phi)$ in $\rho_{c0}$. We take here
$\Omega_{m0}=0.28$ and $\Omega_{k}=0.01$ Figure 5: The revolution of the
scalar-field $\phi(a)$ for original ADE with different interacting parameter
$b^{2}$, where $\phi$ is in unit of $m_{p}$ and we take here
$\Omega_{m0}=0.28.$ Figure 6: The revolution of the scalar-field $\phi(a)$ for
original ADE with different model parameter $n$, where $\phi$ is in unit of
$m_{p}$ and we take here $\Omega_{m0}=0.28$.

Differentiating Eq. (12) and using relation
${\dot{\Omega}_{D}}={\Omega^{\prime}_{D}}H$, we reach

$\displaystyle{\Omega^{\prime}_{D}}=\Omega_{D}\left(-2\frac{\dot{H}}{H^{2}}-\frac{2}{n}\sqrt{\Omega_{D}}\right),$
(18)

where the dot and the prime stand, respectively, for the derivative with
respect to the cosmic time and the derivative with respect to $x=\ln{a}$.
Taking the derivative of both side of the Friedman equation (3) with respect
to the cosmic time, and using Eqs. (5), (11), (12) and (14), it is easy to
show that

$\displaystyle\frac{\dot{H}}{H^{2}}=-\frac{3}{2}(1-\Omega_{D})-\frac{\Omega^{3/2}_{D}}{n}-\frac{\Omega_{k}}{2}+\frac{3}{2}b^{2}(1+\Omega_{k}).$
(19)

Substituting this relation into Eq. (18), we obtain the equation of motion of
agegraphic dark energy

$\displaystyle{\Omega^{\prime}_{D}}$ $\displaystyle=$
$\displaystyle\Omega_{D}\left[(1-\Omega_{D})\left(3-\frac{2}{n}\sqrt{\Omega_{D}}\right)-3b^{2}(1+\Omega_{k})+\Omega_{k}\right].$
(20)

We plot in Figs. 1 and 2 the evolutions of the $w_{D}$ and $\Omega_{D}$ of the
original ADE with different interacting parameter $b^{2}$. From Fig. 1 we see
that $w_{D}$ of the agegraphic dark energy can cross the phantom divide. It
was argued Wei2  that without interaction ($b^{2}=0$) $w_{D}$ is always larger
than $-1$ and cannot cross the phantom divide. In the presence of the
interaction the situation is changed. An interesting observation from Fig. 1
is that $w_{D}$ crosses the phantom divide from $w_{D}<-1$ to $w_{D}>-1$. This
makes it distinguishable from many other dark energy models whose $w_{D}$ can
cross the phantom divide. Fig. 2 shows that at the early time
$\Omega_{D}\rightarrow 0$ while at the late time $\Omega_{D}\rightarrow 1$,
that is the ADE dominates as expected.

Now we suggest a correspondence between the original agegraphic dark energy
and quintessence scalar field namely, we identify $\rho_{\phi}$ with
$\rho_{D}$. Using relation $\rho_{\phi}=\rho_{D}={3m_{p}^{2}H^{2}}\Omega_{D}$
and Eq. (17) we can rewrite the scalar potential and kinetic energy term as

$\displaystyle V(\phi)$ $\displaystyle=$ $\displaystyle
m^{2}_{p}H^{2}\Omega_{D}\left(3-\frac{\sqrt{\Omega_{D}}}{n}+\frac{3b^{2}}{2}\frac{(1+\Omega_{k})}{\Omega_{D}}\right),$
(21) $\displaystyle\dot{\phi}$ $\displaystyle=$ $\displaystyle
m_{p}H\left(\frac{2}{n}{\Omega^{3/2}_{D}}-3b^{2}(1+\Omega_{k})\right)^{1/2}.$
(22)

Using relation $\dot{\phi}=H{\phi^{\prime}}$, we get

$\displaystyle{\phi^{\prime}}$ $\displaystyle=$ $\displaystyle
m_{p}\left(\frac{2}{n}{\Omega^{3/2}_{D}}-3b^{2}(1+\Omega_{k})\right)^{1/2}.$
(23)

Consequently, we can easily obtain the evolutionary form of the field by
integrating the above equation

$\displaystyle\phi(a)-\phi(a_{0})=\int_{a_{0}}^{a}{\frac{m_{p}}{a}\sqrt{\frac{2}{n}{\Omega^{3/2}_{D}}-3b^{2}(1+\Omega_{k})}da},$
(24)

where $a_{0}$ is the present value of the scale factor, and $\Omega_{D}$ is
given by Eq. (20). Therefore, we have established an interacting agegraphic
quintessence dark energy model and reconstructed the potential of the
agegraphic quintessence as well as the dynamics of scalar field.

As one can see from the above equations, the analytical form of the potential
$V=V(\phi)$ is hard to be derived due to the complexity of the equations, but
we can plot the agegraphic quintessence potential versus $a$ numerically. For
simplicity we take $\Omega_{k}\simeq 0.01$ fixed in the numerical discussion.
Besides, $\rho_{c0}=3m_{p}^{2}H^{2}_{0}$ is the present value of the critical
energy density of the universe. The reconstructed quintessence potential
$V(\phi)$ and the evolutionary form of the field are plotted in Figs. 3-6,
where we have taken $\phi(a_{0}=1)=0$. Selected curves are plotted for the
different model parameter $n$ with fixed $b^{2}$ and different $b^{2}$ with
fixed $n$, and the present fractional matter density is chosen to be
$\Omega_{m0}=0.28$. From these figures we can see the dynamics of the
potential as well as the scalar field explicitly. They also show that the
reconstructed quintessence potential is steeper in the early epoch and becomes
very flat near today. Consequently, the scalar field $\phi$ rolls down the
potential with the kinetic energy $\dot{\phi}$ gradually decreasing.

##  III Quintessence reconstruction of NEW ADE

Soon after the original agegraphic dark energy model was introduced by Cai
Cai1 , a new model of agegraphic dark energy was proposed in Wei2 , while the
time scale is chosen to be the conformal time $\eta$ instead of the age of the
universe. It is worth noting that the Karolyhazy relation $\delta{t}=\beta
t_{p}^{2/3}t^{1/3}$ was derived for Minkowski spacetime
$ds^{2}=dt^{2}-d\mathrm{x^{2}}$ Kar1 ; Maz . In the case of the FRW universe,
we have $ds^{2}=dt^{2}-a^{2}d\mathrm{x^{2}}=a^{2}(d\eta^{2}-d\mathrm{x^{2}})$.
Thus, it might be more reasonable to choose the time scale in Eq. (11) to be
the conformal time $\eta$ since it is the causal time in the Penrose diagram
of the FRW universe. The new agegraphic dark energy contains some new features
different from the original agegraphic dark energy and overcome some
unsatisfactory points. For instance, the original agegraphic dark energy
suffers from the difficulty to describe the matter-dominated epoch while the
new agegraphic dark energy resolved this issue Wei2 . The energy density of
the new agegraphic dark energy can be written

$\rho_{D}=\frac{3n^{2}m_{p}^{2}}{\eta^{2}},$ (25)

where the conformal time $\eta$ is given by

$\eta=\int{\frac{dt}{a}}=\int_{0}^{a}{\frac{da}{Ha^{2}}}.$ (26)

Figure 7: The evolution of $w_{D}$ for new ADE with different interacting
parameter $b^{2}$. Here we take $\Omega_{D0}=0.72$ and $\Omega_{k}=0.01.$
Figure 8: The evolution of $\Omega_{D}$ for new ADE with different interacting
parameter $b^{2}$. Here we take $\Omega_{D0}=0.72$ and $\Omega_{k}=0.01$.
Figure 9: The reconstruction of the potential $V(\phi)$ for new ADE with
different model parameter $n$, where $\phi$ is in unit of $m_{p}$ and
$V(\phi)$ in $\rho_{c0}$. We take here $\Omega_{m0}=0.28.$ Figure 10: The
reconstruction of the potential $V(\phi)$ for new ADE with different
interacting parameter $b^{2}$, where $\phi$ is in unit of $m_{p}$ and
$V(\phi)$ in $\rho_{c0}$. We take here $\Omega_{m0}=0.28.$ Figure 11: The
revolution of the scalar-field $\phi(a)$ for new ADE with different
interacting parameter $b^{2}$, where $\phi$ is in unit of $m_{p}$ and we take
here $\Omega_{m0}=0.28.$ Figure 12: The revolution of the scalar-field
$\phi(a)$ for new ADE with different model parameter $n$, where $\phi$ is in
unit of $m_{p}$ and we take here $\Omega_{m0}=0.28$.

The fractional energy density of the new agegraphic dark energy is now given
by

$\displaystyle\Omega_{D}=\frac{n^{2}}{H^{2}\eta^{2}}.$ (27)

Taking the derivative with respect to the cosmic time of Eq. (25) and using
Eq. (27) we get

$\displaystyle\dot{\rho}_{D}=-2H\frac{\sqrt{\Omega_{D}}}{na}\rho_{D}.$ (28)

Inserting this relation into Eq. (15) we obtain the equation of state
parameter of the new agegraphic dark energy

$\displaystyle
w_{D}=-1+\frac{2}{3na}\sqrt{\Omega_{D}}-\frac{b^{2}}{\Omega_{D}}(1+\Omega_{k}).$
(29)

The evolution behavior of the new agegraphic dark energy is now given by

$\displaystyle{\Omega^{\prime}_{D}}$ $\displaystyle=$
$\displaystyle\Omega_{D}\left[(1-\Omega_{D})\left(3-\frac{2}{na}\sqrt{\Omega_{D}}\right)-3b^{2}(1+\Omega_{k})+\Omega_{k}\right].$
(30)

We plot in Figs. 7 and 8 the evolutions of $w_{D}$ and $\Omega_{D}$ of the new
ADE with different interacting parameter $b^{2}$. From Fig. 7 we see that
$w_{D}$ of the new ADE can have a transition from $w_{D}>-1$ to $w_{D}<-1$.
This is in contrast to the original ADE model. Fig. 8 indicates that at the
late time $\Omega_{D}\rightarrow 1$, which is similar to the behaviour of the
original ADE.

Next, we reconstruct the new agegraphic quintessence dark energy model,
connecting the quintessence scalar field with the new agegraphic dark energy.
Using Eqs. (27) and (29) one can easily show that the scalar potential and
kinetic energy term take the following form

$\displaystyle V(\phi)$ $\displaystyle=$ $\displaystyle
m^{2}_{p}H^{2}\Omega_{D}\left(3-\frac{\sqrt{\Omega_{D}}}{na}+\frac{3b^{2}}{2}\frac{(1+\Omega_{k})}{\Omega_{D}}\right),$
(31) $\displaystyle\dot{\phi}$ $\displaystyle=$ $\displaystyle
m_{p}H\left(\frac{2}{na}{\Omega^{3/2}_{D}}-3b^{2}(1+\Omega_{k})\right)^{1/2}.$
(32)

Using Eq. (32), Eq. (31) can be reexpressed as

$\displaystyle V(\phi)$ $\displaystyle=$ $\displaystyle
3m^{2}_{p}H^{2}\Omega_{D}\left(1-\frac{\dot{\phi}^{2}}{6m^{2}_{p}H^{2}\Omega_{D}}\right).$
(33)

We can also rewrite Eq. (32) as

$\displaystyle{\phi^{\prime}}$ $\displaystyle=$ $\displaystyle
m_{p}\left(\frac{2}{na}{\Omega^{3/2}_{D}}-3b^{2}(1+\Omega_{k})\right)^{1/2}.$
(34)

Therefore the evolution behavior of the scalar field can be obtained by
integrating the above equation

$\displaystyle\phi(a)-\phi(a_{0})=\int_{a_{0}}^{a}{\frac{m_{p}}{a}\sqrt{\frac{2}{na}{\Omega^{3/2}_{D}}-3b^{2}(1+\Omega_{k})}da},$
(35)

where $\Omega_{D}$ is now given by Eq. (30). In this way we connect the
interacting agegraphic dark energy with a quintessence scalar field and
reconstruct the potential of the agegraphic quintessence. The reconstructed
quintessence potential $V(\phi)$ and the evolutionary form of the scalar field
are plotted in Figs. 9-12 for different model parameter $n$ with fixed
interacting parameter $b^{2}$, and different $b^{2}$ with fixed $n$. Here the
present fractional matter density is chosen to be $\Omega_{m0}=0.28$.

##  IV Conclusions

In this paper, we have suggested a correspondence between the interacting
agegraphic dark energy scenarios and the quintessence scalar field model in a
non-flat universe. We have demonstrated that the agegraphic evolution of the
universe can be described completely by a single quintessence scalar field. We
have adopted the viewpoint that the scalar field models of dark energy are
effective theories of an underlying theory of dark energy. If we regard the
scalar field model as an effective description of such a theory, we should be
capable of using the scalar field model to mimic the evolving behavior of the
interacting agegraphic dark energy and reconstructing this scalar field model
according to the evolutionary behavior of agegraphic dark energy. We have also
reconstructed the potential of the agegraphic scalar field as well as the
dynamics of the quintessence scalar field which describe the quintessence
cosmology.

###### Acknowledgements.

This work has been supported by Research Institute for Astronomy and
Astrophysics of Maragha, Iran.

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