# Failure-tolerant Distributed Learning for Anomaly Detection in Wireless
Networks

Marc Katzef, Andrew C. Cullen, Tansu Alpcan, Christopher Leckie, and Justin
Kopacz Marc Katzef and Tansu Alpcan are with the Department of Electrical and
Electronic Engineering, The University of Melbourne, Australia.Andrew C.
Cullen and Christopher Leckie are with the School of Computing and Information
Systems, The University of Melbourne, Australia.Justin Kopacz is with the
Northrop Grumman Corporation, USAManuscript received Month Day, Year; revised
Month Day, Year.

###### Abstract

The analysis of distributed techniques is often focused upon their efficiency,
without considering their robustness (or lack thereof). Such a consideration
is particularly important when devices or central servers can fail, which can
potentially cripple distributed systems. When such failures arise in wireless
communications networks, important services that they use/provide (like
anomaly detection) can be left inoperable and can result in a cascade of
security problems. In this paper, we present a novel method to address these
risks by combining both flat- and star-topologies, combining the performance
and reliability benefits of both. We refer to this method as “Tol-FL”, due to
its increased failure-tolerance as compared to the technique of Federated
Learning. Our approach both limits device failure risks while outperforming
prior methods by up to $8\%$ in terms of anomaly detection AUROC in a range of
realistic settings that consider client as well as server failure, all while
reducing communication costs. This performance demonstrates that Tol-FL is a
highly suitable method for distributed model training for anomaly detection,
especially in the domain of wireless networks.

###### Index Terms:

Distributed Learning, Reliable, Deep Learning, Software-Defined Radio

##  I Introduction

Improving wireless network security through anomaly detection has attracted a
significant research effort, with new methods presented in lockstep with a
changing technological landscape. As this landscape has evolved to include
networks with an increasing number of devices, approaches to detect anomalies
have become more reliant on automated methods offering pattern recognition.
Machine Learning (ML) has found use in rapidly-evolving, critical applications
like distributed network security because of its flexibility and ability to
process high dimensional data [1, 2].

ML-based anomaly detection models can identify if unusual (and possibly
harmful) activity is present in an observed network. Observations of a
wireless network may be produced by devices scattered over a large spatial
area, producing partially overlapping observations. This data is readily
available at each wireless device, along with storage and computational
resources. However, in many contexts, the data cannot be easily centralised
for use in ML training [2]. Recognising the risks and resource usage imposed
by transferring data from distributed devices to a centralised point, recent
research has focused upon the development of schemes in which locally-trained
models are exchanged with their peers [3] One popular example of such edge
computing is Federated Learning (FL) [4], which combines distributed
computation with centralised aggregation, thus removing the need for
exchanging datasets [4].

A single round of FL’s core algorithm consists of selecting a subset of
clients, sending to them the most recent model parameters, and allowing these
clients to further train the model on local data before averaging their
locally-updated models. While such a scheme has significant advantages in
terms of both parallel processing and data privacy, it has an oft-overlooked
weakness: that its inherent star topology introduces systemic fragility, due
to the risk of failure of the centralised server. A common assumption with
prior distributed ML works is that the server is highly reliable and that the
main failure or attack modes are through the client devices instead [4, 5, 6].
This hardware dependence is highlighted by Figure 1, in which there is a clear
distinction between star- and flat-topology training, with star-topologies
depending on a central device. Due to FL’s inherent need for a server, the
robustness of such a training scheme to device failures has received little
attention. This is a significant omission, as critical infrastructure should
always be considered in context of the question what happens when devices
fail? Such a consideration is especially important in distributed training due
to the large set of potential failure modes, including those relating to
connectivity, device behaviour, and resources or bandwidth. While distributed
training schemes have the potential to continue training in the event of
device failure [7, 1], a level of robustness and redundancy is required in the
training scheme to enable the remaining devices to seamlessly resume training
where a faulty/compromised device failed. Such robustness is especially
important for critical applications, such as those involving network security.

Prior works have suggested building on FL with extensions that improve
training speed, such as FedGroup [8], IFCA [9], and FeSEM [10] (all of which
train multiple models in parallel in the network). To the authors’ knowledge,
however, this paper is the first to apply a clustered training scheme
consisting of both FL and a flat-topology, round robin learning scheme to the
wireless network anomaly detection field. In this work, we propose a novel
machine learning training scheme that was designed specifically for wireless
network applications, and analyse the impact of client or server failure.

Figure 1: Star and flat topologies can coexist as parts of a single network,
as indicated by dashed borders. Each cluster (star topology) comprises a
server and multiple clients, and the flat topology comprises the servers of
the star topologies in the network.

This paper addresses a distributed training research gap of strengthening such
methods for the case of crucial devices failing while training an anomaly
detection model. We address this problem by first presenting an algorithm
(Tol-FL) along with a theoretical analysis of its ability to continue training
after any one device is removed. After establishing this method, Tol-FL, we
compare the performance of this training algorithm with existing methods on
the task of anomaly detection when exposed to a simulated network scenario
that contains examples of resource misuse and novel network traffic patterns
as well as random device failures, representing a range of realistic settings.
The contributions of this paper are:

  1. 1.

A novel ML training scheme—we call Tol-FL—that combines hierarchical and flat
network topologies

  2. 2.

A new parametrisation, $k$ that allows for Tol-FL to smoothly transition
between flat and hierarchical forms—balancing the performance benefits of each

  3. 3.

A complexity analysis of the failure tolerance and convergence speed of FL
variants in non-ideal networks in which devices or communication links may
fail throughout training.

##  II The Problem of Failure-Tolerant Distributed Anomaly Detection

Centralising data collected by distributed devices inherently presents both
privacy and security risks, while also introducing communication costs. To
circumvent these limitations, a range of specialised ML variants have been
proposed, in which models are trained over distributed devices that have
access to only parts of the full dataset. While some candidate training
schemes like FL appeared in literature [4], they bring new risks and
challenges that must be addressed. One major risk is the presence of a central
server—typically a critical component for the training algorithm to operate.
The risk that the central server poses is one of reliability, wherein, a
server failure brings any ongoing tasks to a halt. With ML being increasingly
used in tasks like network anomaly detection, such an outage can affect the
security of remaining devices.

In this paper, we consider the case where individual devices or a server that
is participating in a distributed training scheme may be brought offline at
random, either due to a targeted attack, or simply due to environmental
conditions. Through such an outage, any one networked device may become
unreachable at any stage during network operation and—our main focus—during
the training phase of an anomaly detection system. For star-based topologies,
if the failing entity is a server, this event can prevent communication
amongst all the remaining devices. This worst-case scenario separates the
remaining devices, their computational resources, and their datasets. With the
possibility of client or server failure, we seek to maximise the performance
of the resulting model without sacrificing performance in other scenarios or
increasing resource usage.

##  III Failure Tolerant Federated Learning

Tol-FL has been designed to reduce the dependence on a single device that is
found in star-topology schemes by combining FL with an entirely flat topology
training scheme. Tol-FL is the combination of two opposing distributed
training schemes: FL and a completely flat-topology alternative to FL that has
been used as an example in prior works such as [11, 12]. To the authors’
knowledge, this flat-topology technique has not yet been assigned a name; to
simplify further discussion, we refer to this flat, multi-device, round robin-
based ML training scheme as Super-Batch Training (SBT). Through this
combination, the system is hardened against the potential failure of
individual clients or servers—minimising any degradation to performance or
resource usage under such conditions.

Tol-FL has been designed to balance the benefits and drawbacks inherent in
star and flat topologies. While FL and other star-topology based schemes can
offer high levels of parallel processing (speeding up training time), this
scheme comes with the cost of relying on a highly connected and dependable
server being live without failure. SBT by comparison, trains a model in
sequence by iterating over all of the participating devices’ local datasets as
if they were mini-batches on a server, multiple times (further detailed in
Appendix A-A). This flat-topology training approach divides the tasks of a
central server over many non-critical devices, increasing training robustness.
However, this improved training robustness comes at the cost of an increased
number of sequential events (rather than parallel) and thus an increase in
training time.

Tol-FL addresses the respective shortcomings of SBT and FL by combining the
two algorithms to provide the structure in Figure 1. Instead of using a flat
topology or a star topology exclusively, Tol-FL is the composition of FL and
SBT in which FL is performed within subgroups of devices in the network (using
an arbitrary member device as the server) and SBT is applied over these
subgroups. Through this combination, Tol-FL benefits from FL’s training speed
and SBT’s tolerance to server/device failure. In this combined structure, we
introduce a new design parameter—the number of clusters $k$ in the
network—which allows us to smoothly transition between FL and SBT to provide a
required balance of robustness and speed. This parameter and other required
terms in Table I appear in the formal definition of Tol-FL given in Algorithm
1.

0: $D,k,\theta_{t}$ // Devices, number of clusters, current model parameters

// In each cluster, in parallel:

for i = 1, 2, …, $k$ do

$n_{t,i},g_{t,i}=\text{FedAvg}(D_{i},\theta_{t})$

end for

// Initialise on the first cluster head in a sequence

$g_{t}=\textbf{0}$ // Mean gradient

$n_{t}=0$ // Sample count

// In each cluster, in sequence:

for i = 1, 2, …, $k$ do

$n_{t}\leftarrow n_{t}+n_{t,i}$ // Record the number of samples in cluster

$r\leftarrow\frac{n_{t,i}}{n_{t}}$ // Calculate a weighting factor

$g_{t}\leftarrow rg_{t,i}+(1-r)g_{t}$ // Update the mean gradient

end for

// On the last cluster head of the sequence, apply gradient update

$\theta_{t+1}=\theta_{t}-\alpha g_{t}$

// Broadcast the latest parameters $\theta_{t+1}$

Algorithm 1 One round of Tol-FL training for a network containing $N$ devices,
$D=\\{d_{i}\\}_{i=1}^{N}$, each assigned to one of $k$ clusters
$D_{i},i\in\\{1..k\\}$ that are non-overlapping ($D_{i}\cap
D_{j}=\emptyset\forall i\neq j$ and $\bigcup_{i=1}^{k}D_{i}=D$).

The two main steps in Tol-FL consist of a parallel processing stage in each of
$k$ clusters, followed by sequential aggregation of each cluster’s output.
These two steps appear in Algorithm 1 as two iterative processes; one
involving FL’s FedAvg algorithm, and the other using the SBT algorithm. In
each round of Tol-FL training, every device optimises a copy of the global
model on their local dataset before passing their updated model to an elected
device (the cluster head). The cluster head aggregates all the updated models
from devices in its cluster using FedAvg to determine $g_{t,i}$ and $n_{t,i}$
(the per-cluster gradient and number of samples used) before transferring the
aggregated model along with the number of samples used in this update to a
neighbouring cluster head. These cluster head messages can be combined easily
in sequence using SBT, before finally broadcasting the updated model to all
participating devices. Each iteration of the model is generated using the
well-known [13] model update form

$\theta_{t+1}=\theta_{t}-\alpha g_{t}$

however, to obtain $g_{t}$, our iterative update method is used over clusters
$i$ through $k$ to combine their cluster head messages as

$g_{t}=\sum_{i=1}^{k}\left(r_{i}g_{t,i}+\left(1-r_{i}\right)\sum_{j=1}^{i-1}g_{t,j}\right)$

where $r_{i}={n_{t,i}}/{\sum_{j=1}^{i}n_{t,j}}$ weights each gradient by the
number of samples used to calculate it.

This iterative training is performed as per the sequence shown in Figure 2. A
notable property of this sequence is how it varies with the number of
clusters, $k$; as $k$ varies, Tol-FL’s behaviour falls into one of three
categories. With $k=1$, all devices belong to the same cluster, and so FL’s
FedAvg algorithm is performed over the entire network—this results in standard
FL with high parallelisation but low robustness due to the cluster head device
taking on the role of an FL server. With $k=N$, all clusters consist of just a
single device and as a result, Tol-FL is reduced to standard SBT. In the
remaining case of $1<k<N$, we see a balance between parallel processing and
robustness. Notably, in every value for $k$, the global model is updated with
respect to every data sample in the network (with equal weighting). By
construction, this property of Tol-FL means that model updates from a round of
training are independent of $k$ and result in identical outputs (ignoring low
levels of numerical error, which would arise in practice). With the model
output left unchanged, the parameter $k$ allows us to change the division of
tasks amongst devices and profoundly change the expected reliability,
communication costs, and processing times.

Figure 2: Tol-FL sequence diagram showing all communication steps involved in one global model update, where each cluster is initialised with $\theta_{t,1,k}$, and updates it to $\theta_{t+1,1,k}$ (the next iteration that considers training sets from clusters $1$ through $k$) TABLE I: Collaborative training scheme notation. Symbol | Meaning
---|---
$N$ | Total number of devices in distributed case
$N_{i}$ | Number of devices in cluster $i$
$k$ | Number of clusters
$n_{t,i}$ | Number of samples at time $t$, device $i$
x | Data sample
$\hat{\textbf{x}}$ | Reconstructed data sample
$\theta_{t,i}$ | Model parameters at time $t$, device $i$
$E$ | Number of local epochs
$\alpha$ | Learning rate
$g_{t,i}$ | Gradients at time $t$, device $i$

##  IV Analysis of Tol-FL Robustness and Resource Usage

In this section we analyse the resource usage of Tol-FL during training and
its robustness to random client or server failure.

###  IV-A Tol-FL’s Training Rate

As each topology requires a different sequence of communications and
processing, the amount of time they need for their training to converge
varies. This time is determined by task sequencing, available data rates, and
CPU resources across the set of devices. To evaluate the resource requirements
of each of the competing methods, we present the resource usage of competing
techniques in Table II.

TABLE II: Resource usage of Tol-FL, SBT and existing methods during training, where $s$ is the number of samples, $d$ is the number of devices, and $k$ is the number of clusters. Note, communication resources are further broken down by sources and destinations with, for example, C2S standing for Client to Server. Method | Comp. (Server/s) | Comp. (Client/s) | Comms (S2S) | Comms (C2S) | Comms (C2C) | Time
---|---|---|---|---|---|---
Batch | $\mathcal{O}(s)$ | $-$ | $-$ | $-$ | $-$ | $\mathcal{O}(s)$
FL | $\mathcal{O}(d)$ | $\mathcal{O}(s)$ | $-$ | $\mathcal{O}(d)$ | $-$ | $\mathcal{O}(s+d)$
Super-Batch | $-$ | $\mathcal{O}(s)$ | $-$ | $-$ | $\mathcal{O}(d)$ | $\mathcal{O}(s+d)$
Tol-FL | $\mathcal{O}(d)$ | $\mathcal{O}(s)$ | $\mathcal{O}(k)$ | $\mathcal{O}(d)$ | $-$ | $\mathcal{O}(s+k)$

###  IV-B Tol-FL’s Tolerance to Server Failure

One of Tol-FL’s greatest strengths is its ability to continue training even in
the event of an individual device (client or server) failing during training.
In such an event, a single device going offline cannot force the entire
network to cease training collaboratively—instead, only the subset of devices
that depended on the failing device are removed from training. This new worst-
case scenario can be seen as a method of mitigating a training-time risk that
would ordinarily be enticing to malicious attackers.

The way in which Tol-FL is able to continue training can be seen through
Algorithm 1. This algorithm shows that Tol-FL divides devices into clusters
and that if one device were to fail throughout the network, one of two
scenarios would occur, depending on the device’s role inside the cluster to
which it has been assigned (i.e., cluster member or cluster head). If the
device was a standard cluster member (i.e., not the cluster head) then the
cluster may continue participating in distributed training as usual, only
without the use of the failed device’s resources—data and processing. If,
instead, the device was designated as the cluster head—the one member of the
cluster responsible for inter-cluster communication—then the entire cluster
would be absent from subsequent training. Subsequently, the worst-case
scenario for Tol-FL would be losing a cluster head, which would result in
losing access to one entire cluster $D_{i}$. The devices in the remaining
$k-1$ clusters $D_{j},\forall j\neq i$ may continue training collaboratively.

Tol-FL’s continued training is in stark contrast to standard FL, in which the
following failure modes exist: if a FL client fails, subsequent training can
no longer use that device’s data or processing resources; if the FL server
fails, distributed training is terminated and client devices can no longer
benefit from other devices’ resources. From these FL failure scenarios, the
worst-case failure mode is again from server failure, leaving the remaining
devices with the task to make alternative arrangements such as assigning a new
FL server—a non-trivial task given that the FL server is often much more
powerful than any one client—or training locally on each client. Notably, such
arrangements were not defined in the original FL formulation and are often
overlooked in FL research. By comparing standard FL with Tol-FL, we see that
the worst-case scenario for a device failure is more detrimental to FL. How
this translates into real-world performance is explored in Section V.

Performance in practice depends on the likelihood of different failure events
occurring. This performance can be approximated from device failure scenarios
that are formulated to display the best- and worst-case device failure modes
in the compared training methods. Expected performance (e.g., loss or anomaly
detection score, $J$), however, is expressed as $\sum_{s\in S}p_{s}\cdot
J_{s}$. Depending on the probability of any client or server failing and the
performance each method achieves in that scenario, either batch, FL, or Tol-FL
may be most suited to the application.

##  V Tol-FL Experimental Evaluation

The following sections present a series of experiments that have been designed
to measure the performance of an anomaly detection scheme’s training with and
without a client or server failure. These experiments have been designed to
mimic realistic scenarios that can result in performance drops due to devices
becoming unavailable during training—the real-world performance drops that
Tol-FL aims to ameliorate.

###  V-A Experimental Setup and Baseline

Tol-FL’s performance was measured and compared with existing methods of batch
training, FL, FedGroup, IFCA, and FeSEM in multiple scenarios, using multiple
datasets which have been assigned to devices for a realistic representation of
network security tasks. The task in question is to train an autoencoder
anomaly detection model on a range of datasets that are representative of
wireless network environments. The autoencoder model trained in these
experiments consists of a fully-connected artificial neural network for both
the encoder and decoder, trained to minimize reconstruction error
$J(\textbf{x})=||\textbf{x}-\hat{\textbf{x}}||_{2}^{2}$. Each model was
designed with three hidden layers with $64$ to $128$ neurons per layer and a
code vector of length $32$. ReLU was used as the activation function for
hidden layers and linear was used for output. Following [14], dropout was used
during training for all hidden layers using a probability of $0.2$. When
presented with a new sample $\mathbf{x}$, this state-of-the-art method’s
reconstruction error $J(\mathbf{x})$ indicates how similar this sample is to
previous samples, leading to $J(\mathbf{x})$ being used as the anomaly scoring
function.

The selected anomaly detection model was trained using the above methods (Tol-
FL, FL, FedGroup, IFCA, and FeSEM) on the authors’ publicly available dataset
generator—Comms-ML [15]—as well as a collection of commonly-used reference
datasets described in Appendix A-B. Together, these datasets give an accurate
representation of wireless networks both in terms of sample structure and in
terms of data layout—distributed over multiple devices. For datasets that are
not naturally divided into region/device datasets, the data was subdivided in
multiple arrangements based on class to test both centralised and
decentralised method performance. For all decentralised experiments, these
data divisions occurred by separating the overall dataset into approximately
equally-sized subsets of length $|D_{i}|=N_{i}\leq\lceil\frac{N}{k}\rceil$ for
each cluster. Within each cluster, this division method is performed yet again
to allocate the sub-dataset (of any one cluster) to devices that belong to
that sub-dataset’s cluster.

Using Comms-ML, dividing a dataset across multiple devices is easily achieved
by defining custom networks, giving a flexible way to investigate the scenario
of wireless communications. Generated through simulations, the Comms-ML
dataset contains raw in-phase/quadrature data as well as network statistics
that would be available to devices in an SDR network in which multiple devices
were simulated to communicate in multiple, distinct geographic regions using
Wi-Fi. The communications within these networks are recorded in two formats—a
compact transcript of the transmitted data and a realistic, wide-band waveform
received from each of the devices in the region. These data types both appear
in each sample as either raw readings or signal statistics as in Figure 3.

Using this wireless signal generator, anomalies have been introduced into the
network by adjusting the communication patterns of existing devices (e.g.,
more or less frequent transmissions) or adding novel devices to the
environment (e.g., Bluetooth devices entering a previously WLAN-only network).
Each of these anomaly types changes the underlying data distribution in
different ways, and should be identified by a well-trained anomaly detector.
Table III confirms that all competing training methods achieve strong
performance across all datasets when they are free to train without any
server/device failure occurring. While most results are clustered, FL and Tol-
FL are seen to be the top two methods in this case, showing promise for use in
ideal, failure-free scenarios—scenarios that we will use as a baseline. In the
next two sections, we explore the scenarios in which client or server failures
occur partway through training.

Figure 3: An excerpt of (20 out of 112 features of) wireless network activity showing the mean values of statistics-based features (indices 0–11) and raw physical signals (indices 11 onward) for one class of the Comms-ML dataset. Successful anomaly detection requires distinct values or patterns to be found between typical and anomalous classes. TABLE III:  Anomaly detection AUROC results without server or device failure. Here, † denotes schemes that train multiple instances of the same model and ∗ denotes the top-performing instance from such schemes.  Dataset | Tol-FL | FedGroup∗ | IFCA∗ | FeSEM∗ | FL | Batch | FedGroup† | IFCA† | FeSEM†
---|---|---|---|---|---|---|---|---|---
Comms-ML | $\mathbf{0.80\pm 0.01}$ | $0.75\pm 0.02$ | $0.76\pm 0.02$ | $0.79\pm 0.01$ | $\mathbf{0.80\pm 0.01}$ | $0.82\pm 0.01$ | $0.84\pm 0.02$ | $0.85\pm 0.03$ | $0.82\pm 0.02$
FMNIST | $0.95\pm 0.02$ | $0.88\pm 0.04$ | $0.89\pm 0.03$ | $0.92\pm 0.03$ | $\mathbf{0.96\pm 0.02}$ | $0.96\pm 0.02$ | $0.97\pm 0.02$ | $0.97\pm 0.02$ | $0.96\pm 0.02$
CIFAR-10 | $0.71\pm 0.05$ | $0.69\pm 0.04$ | $0.70\pm 0.05$ | $0.71\pm 0.04$ | $\mathbf{0.73\pm 0.04}$ | $0.72\pm 0.04$ | $0.74\pm 0.03$ | $0.74\pm 0.04$ | $0.73\pm 0.05$
CIFAR-100 | $0.75\pm 0.04$ | $0.73\pm 0.04$ | $0.72\pm 0.04$ | $0.73\pm 0.05$ | $\mathbf{0.77\pm 0.05}$ | $0.78\pm 0.03$ | $0.80\pm 0.02$ | $0.77\pm 0.03$ | $0.76\pm 0.03$

###  V-B Client Failure

Client failures have been considered in multiple past works, including the
paper introducing FL [4]. This failure scenario consists of the server
continuing with training as usual, in the absence of any lost clients. In this
scenario, clients may be ignored as they are plentiful when compared with the
solitary server and so the performance degradation is estimated to be low;
losing only the computational power and dataset of a single device out of the
entire network.

The failure of a client represents the ‘best case’ device failure outcome; in
other words, should a device failure occur, the loss of a client device is the
preferred mode of failure, and is likely to have the smallest impact on the
training in terms of both time and performance. To evaluate this scenario, the
failure-free reference experiments described in Section V-A were repeated with
client failures introduced at specified times in training to illustrate the
impact in performance that any one client can have in the event of
failure/harm that prevents the device from participating in training.

Client failures were introduced into our experiments by eliminating a device
from training past a certain epoch—the same device at the same epoch in
training for each method. In doing so, the same subset of data and
computational resources are removed from the network at the midpoint of
training. As each client contributes a subset of training data, we expect the
loss of one client to increase the test loss, which contains elements from all
classes (including any present on the failed device). This increase in loss
occurs for the remaining devices’ model as determined by the importance of the
client’s data; if the client held unique data that is significant for the
testing phase, the network will use the remaining epochs to train on only a
subset of the necessary data to arrive at the failure-free loss. In our client
failure experiments (Table IV), this degradation in training losses results in
consistently lower performance than the baseline presented in Section V-A.

A series of experiments were run for $10$ repetitions each, taking the
performance of the global model or (in the case of FedGroup, IFCA, and FeSEM)
the best-performing model throughout the network. The results of these client
failure experiments show that the trends in training losses after a client
device failure are similar across training methods. These results demonstrate
that either FL or Tol-FL always outperforms the remaining methods, likely due
to the fact that these methods train an individual model for a shared task
(representing all data classes) whereas other methods specialise their models
to perform better on a subset of data (one cluster’s data). For wireless
anomaly detection, such specialisation can be seen as overfitting to that
cluster’s data, resulting in a higher false positive rate on data from other
clusters.

TABLE IV:  Anomaly detection AUROC results with client failure occurring after epoch $50$. Here, † denotes schemes that train multiple instances of the same model and ∗ denotes the top-performing instance from such schemes.  Dataset | Tol-FL | FedGroup∗ | IFCA∗ | FeSEM∗ | FL | Batch | FedGroup† | IFCA† | FeSEM†
---|---|---|---|---|---|---|---|---|---
Comms-ML | $0.78\pm 0.02$ | $0.74\pm 0.03$ | $0.75\pm 0.02$ | $0.78\pm 0.02$ | $\mathbf{0.79\pm 0.02}$ | $0.80\pm 0.02$ | $0.81\pm 0.02$ | $0.82\pm 0.02$ | $0.80\pm 0.02$
FMNIST | $0.92\pm 0.02$ | $0.86\pm 0.03$ | $0.87\pm 0.04$ | $0.90\pm 0.03$ | $\mathbf{0.93\pm 0.03}$ | $0.93\pm 0.03$ | $0.94\pm 0.03$ | $0.94\pm 0.02$ | $0.93\pm 0.03$
CIFAR-10 | $\mathbf{0.71\pm 0.04}$ | $0.68\pm 0.03$ | $0.69\pm 0.05$ | $0.69\pm 0.05$ | $\mathbf{0.71\pm 0.05}$ | $0.71\pm 0.05$ | $0.72\pm 0.04$ | $0.72\pm 0.04$ | $0.71\pm 0.04$
CIFAR-100 | $0.73\pm 0.05$ | $0.71\pm 0.04$ | $0.71\pm 0.05$ | $0.72\pm 0.04$ | $\mathbf{0.75\pm 0.04}$ | $0.76\pm 0.05$ | $0.79\pm 0.03$ | $0.76\pm 0.04$ | $0.74\pm 0.04$

###  V-C Server Failure

Through Section IV-A we have shown that under normal operating conditions,
batch training, FL, and Tol-FL converge to optimal solutions (supported by
Table III) and that FL and Tol-FL achieve this after using the same
computational budget. This section explores the largest deviation from normal
operating conditions—the case of server failure. The key difference between FL
and SBT—which are Tol-FL with $k=1$ and $k=N$, respectively—is succinctly
shown through the cases of server and device failure when training on the
MNIST dataset in Figure 4.

Figure 4:  Worst-case training result for FL and SBT—isolated or collaborative
training, respectively. These configurations are special cases of Tol-FL: FL
is Tol-FL with $k=1$, SBT is Tol-FL with $k=N$. Results shown are the average
of the remaining $N=9$ devices’ independent training.

Compared to client failure, server failure has seen less detailed analysis due
to the fact that in many scenarios, the server is essential. In a fixed, star
topology, loss of the server does not merely reduce training performance, it
terminates all collaborative training (leaving only the option of local
training). It is this restriction that Tol-FL addresses through its
hierarchical, distributed structure. Instead of simply terminating training,
Tol-FL can endure any one device failure including the “servers”/cluster heads
of each cluster. This section demonstrates the impact this tolerance has on
training performance.

To each method’s anomaly detection performance in the event of a server-
failure, the reference experiments detailed in Section V-A were repeated with
a single server/cluster head failure occurring at a predetermined epoch. For
batch training (which is a centralised training method), such a failure
results in the model’s training loss plateauing at the final value reached
before failure. For Tol-FL, this failure results in the worst-case Tol-FL
performance, in which an entire cluster is removed from training. For FL, the
remaining devices are left with a decision: to train independently on their
own datasets or to cease training and keep the final performance in the same
way as batch training.

The results of server failure experiments (in Table V) consistently place Tol-
FL’s anomaly detection performance above the competitors. This performance in
the event of server failure is due to Tol-FL’s distributed structure, in which
no single server is essential to proper performance. Instead, Tol-FL’s worst-
case performance occurs when a single cluster head device—a device performing
FL locally with the members of that cluster—is removed from training. In this
event, the members of that cluster are removed from training, but all other
clusters can continue unaffected. Compared to previous failure modes, the loss
of a large portion of data explains why performance is lowest for server
failures. Comparing between methods in this failure mode, the lack of a
central server explains why Tol-FL is the least affected and achieves the
highest performance. The performance improvement is most noticeable with the
Fashion MNIST dataset, in which Tol-FL outperforms its closest competitors by
$8\%$.

TABLE V:  Anomaly detection AUROC results with server failure occurring after epoch $50$. Here, † denotes schemes that train multiple instances of the same model and ∗ denotes the top-performing instance from such schemes.  Dataset | Tol-FL | FedGroup∗ | IFCA∗ | FeSEM∗ | FL | FedGroup† | IFCA† | FeSEM†
---|---|---|---|---|---|---|---|---
Comms-ML | $\mathbf{0.75\pm 0.03}$ | $0.71\pm 0.02$ | $0.70\pm 0.04$ | $0.71\pm 0.03$ | $0.64\pm 0.06$ | $0.75\pm 0.02$ | $0.76\pm 0.03$ | $0.80\pm 0.03$
Fashion MNIST | $\mathbf{0.85\pm 0.03}$ | $0.79\pm 0.04$ | $0.79\pm 0.03$ | $0.78\pm 0.04$ | $0.65\pm 0.05$ | $0.88\pm 0.02$ | $0.91\pm 0.01$ | $0.89\pm 0.02$
CIFAR-10 | $\mathbf{0.68\pm 0.04}$ | $0.65\pm 0.05$ | $0.66\pm 0.05$ | $\mathbf{0.68\pm 0.04}$ | $0.59\pm 0.06$ | $0.70\pm 0.03$ | $0.71\pm 0.03$ | $0.70\pm 0.03$
CIFAR-100 | $\mathbf{0.73\pm 0.02}$ | $0.70\pm 0.03$ | $0.68\pm 0.04$ | $0.70\pm 0.04$ | $0.67\pm 0.05$ | $0.74\pm 0.03$ | $0.72\pm 0.04$ | $0.73\pm 0.03$

###  V-D Resource Usage

While the previous sections show Tol-FL matches or exceeds performance of
prior works in different failure cases, any additional functionality like
failure tolerance must not only offer strong performance outright, but also
suitably low resource usage in order to be practical. To verify that Tol-FL
meets this requirement, we measure processing time and communication usage—two
critical factors to consider in a wireless device’s operation.

Regarding processing time, the training speed of Tol-FL, FL, and SBT (three
opposing approaches that are expected to give the widest range) were compared
under similar conditions to real-world applications. This comparison was
performed with each method being set to train an anomaly detection model for
our wireless networking scenario until a predetermined training loss was
reached. The wall-clock time taken and number of epochs needed by each scheme
to converge to the same loss as centralised (batch) training are shown in
Figure 5. Compared to the theoretical estimates in Section IV-A, the trends in
training time are as expected. With FL and Tol-FL both allowing for parallel
processing, these methods were shown to outperform centralised training in
terms of time taken. The differences between the distributed methods are
attributable to the proportion of training that is carried out sequentially as
opposed to in parallel, where Tol-FL is required to send aggregation messages
to prime each device for parallel processing.

Figure 5: Training time comparison—numbers of epochs and the wall-clock
duration for each training method to reach the centralised (batch) training’s
converged loss (within 5%, averaged over 5 trials).

Regarding computational cost, the results of training for $100$ epochs in edge
cases of Tol-FL (spanning FL to SBT) are shown in Table VI. The results show
that we achieve a lower communication cost than competing methods. This
reduction is expected due to the greater proportion of communications that can
occur within a cluster (without any feedback needed within each cluster)
removing a substantial proportion of messages that occur in FL. To arrive at
the same low resource usage, a limit on communications cost in FL would
require a restricted number of epochs, worsening performance. Exploring this
deterioration of performance is left to be addressed in future work.

TABLE VI: Distributed training communication costs Method | Expected | Actual (MB/epoch)
---|---|---
FL | $\mathcal{O}(2N)$ | $28.3$
SBT | $\mathcal{O}(N)$ | $12.8$
Tol-FL | $\mathcal{O}(N+k)$ | $21.0$

##  VI Related Work

Tol-FL is not the first method to build on FL. However, as the following
subsections will explain, Tol-FL addresses an oft-overlooked shortcoming in
the standard FL definition, and explores possible improvements for the
application of distributed anomaly detection.

###  VI-A Existing Methods for Distributed Anomaly Detection

Within a distributed anomaly detection problem, the task of identifying
anomalies considers the collected samples at each distributed device, and
defining what constitutes a typical sample. For the case of network monitoring
in an SDR context, these samples can be network readings from different
geographic regions or segments of the frequency spectrum; any sample drawn
from a different distribution from the training set should be identified as an
anomaly. The resulting problem is often viewed as a pattern recognition
problem or a data representation task—learning what features are present in
typical data samples [16, 17, 18].

The state-of-the-art literature for anomaly detection tasks consist of both
geometric and ML approaches. Previous studies have explored finding relevant
features [19, 20, 21, 22] or training a ML-based anomaly detection model using
distributed devices [23, 24, 25]. Both FL and gossip-based techniques have
been shown to approach theoretical performance limits [26, 27]. Given these
similarities in performance, a consideration of secondary features like
resource usage and fragility becomes more important.

###  VI-B Prior Federated Learning Research

The FL training paradigm has been analysed [28, 6], adopted [29, 30, 31], and
improved upon in research and industry alike—seeing use in mainstream software
like such as Google’s Android Keyboard. In such applications, the task of
model training is moved to the devices at the edge of a network (such as
smartphones) to collect local results through on-device training before
aggregating multiple updates at a central location. The underlying algorithm
is FL’s FedAvg, which utilises a star topology for computing which while
computationally efficient, introduces risks relating to both device failure
and data security, which recent works have attempted to rectify [5, 3, 32].

Existing studies have sought to improve the robustness of FL with new features
that consider crash failures and Byzantine failures [29, 33] and support more
adaptable schemes such as run-time task assignment [34]. These works, however,
consider crashes of the client devices instead of the server; the server is
assumed to be infallible and active for the duration of an FL simulation.
These techniques ignore the realistic threat of server failure during model
training, the failure that Tol-FL remedies through its use of a hybrid
topology.

Previous works that have considered collaborative training and server failure
include [35, 36] in the context of FL and [12, 32, 37] that consider gossip-
based training schemes. While gossip-based methods rely on random walks of
data, the former, FL modification methods opt for a clustering method from
which Tol-FL draws its inspiration. Ignoring the constraints of local
communication, the FL-based methods [35, 36, 11] determine a natural grouping
scheme over all of the available devices based on the similarity between their
datasets. This modified scheme discards the speed benefit that arises through
single-hop communications and practical considerations of link availability
and instead forming virtual clusters that may include devices with large
communications delays. Other alternatives include hierarchical structures such
as [38] which performs similar processing to Tol-FL but requires manual, time-
consuming initial setup, and following a rigid schedule.

##  VII Conclusion

Device failure can mean the difference between a smoothly-running
collaborative system and a segregated set of devices left to perform sub-
optimally. Throughout this paper, we have explored the effect of client and
server failures in the context of collaborative ML training, and proposed a
novel training method—Tol-FL—that extends the de-facto standard FL so that a
level of tolerance is introduced against server failure. As FL does not
feature any redundancy for its server, a method such as Tol-FL is needed for
training to persist in the event of a server failure; continuing training with
the existing clients without downtime, reconfiguration, or exposing any
additional security risks.

Our work shows that in the event of server failure, Tol-FL outperforms FL in
terms of model losses (and subsequently, anomaly detection), while maintaining
similar performance in the case of client failure or standard, failure-free
operation. Furthermore, we show that Tol-FL achieves this increased failure
tolerance while maintaining similar or reduced resource requirements compared
to FL. From this performance, Tol-FL shows an avenue forward for failure-
tolerant FL to meet the needs that arise in volatile or hazardous environments
in which neither clients nor server(s) are guaranteed to remain operational.

## Acknowledgment

This work was supported in part by the Australian Research Council Linkage
Project under the grant LP190101287, by Northrop Grumman Mission Systems’
University Research Program, and by The University of Melbourne’s Research
Computing Services and the Petascale Campus Initiative.

##  Appendix A Appendix

###  A-A Super-Batch Training

A completely flat alternative to FL has been proposed without a name in prior
work. For simplicity, we dub this scheme “Super-Batch Training” (SBT) due to
its use of whole device datasets as if they were merely mini-batches. In SBT,
the global gradient, $g_{t}$, can be accurately obtained for model updates by
iterating over the participating devices and combining their calculated
gradients with the previous device’s output, following a classical round robin
scheme [39]. This combination is a weighted sum between the local result and
the mean that was calculated from previous samples, weighted based on sample
sizes. While batch training iterates over the entire dataset before updating
weights, mini-batch training only considers a subset of the dataset. In
contrast, “Super-Batch Training” (SBT) iterates over not just the full local
dataset, but instead uses the full dataset across all devices before updating
weights. SBT updates a ML model only after passing over the data of multiple
devices in a loop.

0: $D,\theta_{t}$ // Devices and Model parameters at the current time step

// On each device, in parallel:

for i = 1, 2, …, N do

$g_{t,i}=\nabla_{\theta}J(X_{i},\theta_{t})$ {Calculate gradients}

end for

// Initialise on the first device in a sequence

$g_{t}=\textbf{0}$ // Mean gradient

$n_{t}=0$ // Sample count

$N=|D|$ // The number of devices present

// On each device, in sequence:

for i = 1, 2, …, N do

$n_{t}\leftarrow n_{t}+|X_{i}|$ // Record the number of local samples

$r\leftarrow\frac{|X_{i}|}{n_{t}}$ // Calculate a weighting factor

$g_{t}\leftarrow rg_{t,i}+(1-r)g_{t}$ // Update the mean gradient

end for

// On the last device of the sequence, apply gradient update

$\theta_{t+1}=\theta_{t}-\alpha g_{t}$

// Broadcast the latest parameters, $\theta_{t+1}$

Algorithm 2 One round of SBT, $SuperBatch(D,\theta_{t})$

Algorithm 2 shows how SBT is performed, using the same steps of batch training
but with additional communication between devices. In this algorithm, the task
given to each participating device is simply to calculate training gradients
on their local dataset with the model parameters $\theta_{t}$. After
calculating local gradients, these gradients are sequentially exchanged with
peers in a communication-efficient and flexible scheme. By delaying all
parameter updates until all selected devices have been traversed, all
gradients are calculated for the same model parameters. Local gradients can
then be seen as equivalent to those constructed using batch training on a
single device.

While SBT mirrors the weighting scheme of FL, its flat architecture introduces
two key differences: the lack of need of a server; and the increase in
communication time spent transferring models in sequence instead of in
parallel, highlighted in Figure 6. The inherent trade off in these properties
makes the two schemes more appropriate for different network topologies, with
SBT more suited to mesh networks and structures where there are concerns
regarding the robustness of the centralised server. This increased failure
tolerance introduces an overhead in terms of the scheduling and coordination
of the gradient updates, due to their sequential nature. In contrast, FL is
more suited to star-topologies.

Figure 6: SBT sequence diagram showing all communication steps involved in one
global model update from $\theta_{t}$ to $\theta_{t+1}$

Super-Batch Training performs model updates that are identical to centralised,
batch training. From first principles, using $\hat{f}_{a,b}$ to represent the
mean of samples $f_{a}$ through $f_{b-1}$ or

$\hat{f}_{a,b}=\underset{i=a,...,b-1}{\mathbb{E}}[f_{i}]=\frac{1}{b-a}\sum_{i=a}^{b-1}f_{i},$
(1)

which allows the global mean to be iteratively calculated by

$\hat{f}_{a,c}=\frac{1}{c-a}\left((b-a)\hat{f}_{a,b}+(c-b)\hat{f}_{b,c}\right)$
(2)

until the entire dataset is covered. Using the global gradient $g_{t}$ as a
function that depends on devices $1$ through $n_{t}$ $\hat{f}_{1,n_{t}}$, each
new device extends the range of samples included in the mean
($n_{t}\rightarrow n_{t+1}$), and efficiently adjust the mean ($g_{t}$) to
account for the new entries.

###  A-B Reference Datasets

Each of the datasets described in Table VII was cast as an anomaly detection
task by designating one or more classes within each dataset as “anomalous”
data. The remaining data classes were divided amongst the client devices in a
simulation. Where clusters are present, this data was assigned as one class
per cluster.

TABLE VII: The datasets used to test Tol-FL training. Dataset | Sample shape | # Classes | # Samples/class
---|---|---|---
Fashion-MNIST | $28\times 28$ | $10$ | $7,000$
CIFAR-10 | $32\times 32$ | $10$ | $7,000$
CIFAR-100 | $32\times 32$ | $100$ | $500$
Comms-ML | $112\times 1$ | $4$ | $3,000$

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# Maximum Mass of the Hot Neutron Star with the Quark Core

T. Yazdizadeh1 and G. H. Bordbar2,3 1Islamic Azad University, Bafgh Branch,
Bafgh, Iran
2Department of Physics, Shiraz University, Shiraz 71454, Iran
3Research Institute for Astronomy and Astrophysics of Maragha,
P.O. Box 55134-441, Maragha, Iran

###### Abstract

We have considered a hot neutron star with a quark core, a mixed phase of
quark-hadron matter, and a hadronic matter crust and have determined the
equation of state of the hadronic phase and the quark phase, we have then
found the equation of state of the mixed phase under the Gibbs conditions.
Finally, we have computed the structure of hot neutron star with the quark
core and compared our results with those of the neutron star without the quark
core. For the quark matter calculations, we have used the MIT bag model in
which the total energy of the system is considered as the kinetic energy of
the particles plus a bag constant. For the hadronic matter calculations, we
have used the lowest order constrained variational (LOCV) formalism. Our
calculations show that the results for the maximum gravitational mass of the
hot neutron star with the quark core are substantially different from those of
without the quark core.

###### pacs:

21.65.-f, 26.60.-c, 64.70.-p

##  I Introduction

A hot neutron star is born following the gravitational collapse of the core of
a massive star just after the supernova explosion. The interior temperature of
a neutron star at its birth is of order $20-50\ MeV$ burrows . Therefore, the
high temperature of these stages cannot be neglected with respect to the Fermi
temperature throughout the calculation of its structure. This shows that the
equation of state of the hot dense matter is very important for investigating
the structure of a newborn neutron star. Depending on the total number of
nucleons, a newborn neutron star evolves either to a black hole or to a stable
neutron star strobel . Hence, calculation of the maximum mass of hot neutron
star is of special interest in astrophysics.

As we go from surface to the center of a neutron star, at sufficiently high
densities, the matter is expected to undergo a transition from hadronic
matter, where the quarks are confined inside the hadrons, to a state of
deconfined quarks. Finally, there are up, down and strange quarks in the quark
matter. Other quarks have high masses and do not appear in this state.
Glendenning has shown that a proper construction of the hadron-quark phase
transition inside the neutron stars implies the coexistence of nucleonic
matter and quark matter over a finite range of the pressure. Therefore, a
mixed hadron-quark phase exists in the neutron star and its energy is lower
than those of the quark matter and nucleonic matter glen1 . These show that we
can consider a neutron star as composed of a hadronic matter layer, a mixed
phase of quarks and hadrons and, in core, a quark matter. Recent Chandra
observations also imply that the objects RX J185635-3754 and 3C58 could be
neutron stars with the quark core prakash .

Burgio et al. have investigated the structure of neutron stars with the quark
core at zero burgio1  and finite temperature burgio2  with the Brueckner-
Bethe-Goldstone formalism to determine the equation of state of the hadronic
matter, they have used . We have calculated the structure properties of the
cold neutron star by considering a quark phase at its core b1  and compared
the results with our previous calculations for the neutron star without the
quark core bh . In these works, we have employed the lowest order constrained
variational (LOCV) method for the hadronic matter calculations. In the present
paper, we intend to extend these calculations for the hot neutron star with
the quark core.

##  II Equation of State

As it was mentioned in the previous section, we consider a neutron star
composed of a hadronic matter (hadron phase), a mixed phase of quarks and
hadrons, and a quark core (quark phase). Therefore, we should calculate the
equation of state of these phases separately as follows.

###  II.1 Hadron Phase

For this phase of the neutron star matter, we consider the total energy per
nucleon as the sum of contributions from the leptons and nucleons,

$E=E_{lep}+E_{nucl}\cdot$ (1)

The contribution from the energy of leptons (electrons and muons) is

$E_{lep}=E_{e}+E_{\mu},$ (2)

where $E_{e}$ and $E_{\mu}$ are the energies of electrons and muons,
respectively,

$E_{i}=\frac{m_{i}^{4}c^{5}}{\pi^{2}n\hbar^{3}}\int_{0}^{\infty}\frac{\sqrt{1+x^{2}}}{1+Exp\\{\beta[m_{i}c^{2}\sqrt{1+x^{2}}-\mu_{i}]\\}}x^{2}dx.$
(3)

Here $\mu_{i}$ and $m_{i}$ are the chemical potential and mass of particle
$i$, $\beta=\frac{1}{k_{B}T}$ ($T$ is the temperature), $n$ is the total
number density of nucleons ($n=n_{p}+n_{n}$), $c$ is speed of light and $x$ is
as follows,

$x=\frac{\hbar k}{m_{i}c}.$ (4)

In our calculations, the equation of state of hot nucleonic matter is
determined using the lowest order constrained variational (LOCV) method as
follows b2 ; b3 ; b4 ; b5 ; b6 ; b7 ; b8 ; b9 . We adopt a trail wave function
as

$\psi=F\phi,$ (5)

where $\phi$ is the Slater determinant of the single-particle wave function
and $F$ is the correlation function which is taken to be

$F={\cal S}\prod_{i>j}f(ij).$ (6)

${\cal S}$ is a symmetrizing operator. For the energy of nucleonic matter, we
consider up to the two-body term in the cluster expansion,

$E_{nucl}=E_{1}+E_{2}.$ (7)

The one body term $E_{1}$ for the hot asymmetrical nucleonic matter that
consists of $Z$ protons and $N$ neutrons is simply the fermi gas kinetic
energy,

$E_{1}=\sum_{i=1,2}{\cal E}_{i}$ (8)

Label 1 and 2 are used instead of proton and neutron, respectively, and ${\cal
E}_{i}$ is

${\cal E}_{i}=\sum_{k}\frac{\hbar^{2}k^{2}}{2m_{i}}f_{i}(k,T,n_{i}),$ (9)

where $f(k,T,n_{i})$ is the Fermi-Dirac distribution function fetter ,

$f(k,T,n_{i})=\frac{1}{e^{\beta[\epsilon_{i}(k,T,n_{i})-\mu_{i}(T,n_{i})]}+1}.$
(10)

In the above equation, $n_{i}$ are the number densitis and $\epsilon_{i}$ are
the single particle energies associated with the protons and neutrons,

$\epsilon_{i}(k,T,n_{i})=\frac{\hbar^{2}k^{2}}{2m_{i}^{\ast}(T,n_{i})},$ (11)

where $m_{i}^{\ast}$ are the effective masses.

The two-body energy, $E_{2}$, is

$E_{2}=\frac{1}{2A}\sum_{ij}<ij|\nu(12)|ij-ji>,$ (12)

where

$\nu(12)=-\frac{\hbar^{2}}{2m}[f(12),[\nabla^{2}_{12},f(12)]]+f(12)V(12)f(12).$
(13)

$f(12)$ and $V(12)$ are the two-body correlation and inter-nucleonic
potential.

We note that the conditions of charge neutrality and beta stability impose the
following constraints on the number densities and chemical potentials,

$\displaystyle n_{p}=n_{e}+n_{\mu}$ (14)
$\displaystyle\mu_{n}-\mu_{p}=\mu_{e}=\mu_{\mu}.$ (15)

The procedure to calculate the nucleonic matter has been fully discussed in
the Refs. b2 ; b3 .

###  II.2 Quark Phase

We use the MIT bag model for the quark matter calculations. In this model, the
energy density is the kinetic energy of quarks plus a bag constant (${\cal
B}$) which is interpreted as the difference between the energy densities of
non interacting quarks and interacting ones farhi1 ,

${\cal E}_{tot}={\cal E}_{u}+{\cal E}_{d}+{\cal E}_{s}+{\cal B},$ (16)

where ${\cal E}_{i}$ is the kinetic energy per volume of particle $i$,

${\cal
E}_{i}=\frac{g}{2\pi^{2}}\int_{0}^{\infty}{(m_{i}^{2}c^{4}+\hbar^{2}k^{2}c^{2})^{1/2}}{f(k,T,n_{i})}{k^{2}dk}.$
(17)

In above equation, $g$ is the degeneracy number of the system and $n_{i}$ is
the number density of particle $i$,

$n_{i}=\frac{g}{2\pi^{2}}\int_{0}^{\infty}{f(k,T,n_{i})}{k^{2}dk}.$ (18)

For the quark phase, the Fermi-Dirac distribution function, $f(k,T,n_{i})$, is
given by

$f(k,T,n_{i})=\frac{1}{Exp\\{\beta((m_{i}^{2}c^{4}+\hbar^{2}k^{2}c^{2})^{1/2}-\mu_{i})\\}+1}.$
(19)

We assume that the up and down quarks are massless, the strange quark has a
mass equal to $150\ MeV$ and ${\cal B}=90\ MeVfm^{-3}$. Now, by applying the
beta stability and charge neutrality conditions, we get the following
relations for the chemical potentials and number densities,

$\mu_{d}=\mu_{u}+\mu_{l},$ (20) $\mu_{s}=\mu_{u}+\mu_{l},$ (21)
$\Rightarrow\mu_{d}=\mu_{s},$ (22)
$\frac{2}{3}n_{u}-\frac{1}{3}n_{d}-\frac{1}{3}n_{s}-n_{l}=0,$ (23)
$n_{B}=\frac{1}{3}(n_{u}+n_{d}+n_{s}),$ (24)

where $n_{l}$ and $\mu_{l}$ are the leptonic number density and chemical
potential, and $n_{B}$ is the baryonic number density.

The pressure of the system is calculated from free energy using the following
equation,

$P=\sum_{i}{n_{i}\frac{\partial{\cal F}_{i}}{\partial n_{i}}-{\cal F}_{i}},$
(25)

where the Helmholtz free energy per volume (${\cal F}$) is given by

${\cal F}={\cal E}_{tot}-T{\cal S}_{tot}.$ (26)

The entropy of quark matter (${\cal S}_{tot}$) can be written as follows,

${\cal S}_{tot}={\cal S}_{u}+{\cal S}_{d}+{\cal S}_{s},$ (27)

where ${\cal S}_{i}$ is the entropy of particle $i$,

$\displaystyle{\cal S}_{i}(n_{i},T)$ $\displaystyle=$
$\displaystyle-\frac{3}{\pi^{2}}k_{B}\int_{0}^{\infty}[f(k,T,n_{i})\ln(f(k,T,n_{i}))$
(28) $\displaystyle+(1-f(k,T,n_{i}))\ln(1-f(k,T,n_{i}))]k^{2}dk.$

###  II.3 Mixed phase

For the mixed phase, where the fraction of space occupied by quark matter
smoothly increases from zero to unity, we have a mixture of hadrons, quarks
and electrons. In the mixed phase, according the Gibss equilibrium condition,
the temperatures, pressures and chemical potentials of the hadron phase (H)
and quark phase (Q) are equal glen1 . Here, for each temperature we let the
pressure to be an independent variable.

The Gibss conditions implies that

$\mu^{Q}_{n}=\mu^{H}_{n},$ (29) $\mu^{Q}_{p}=\mu^{H}_{p},$ (30)

where $\mu^{H}_{n}$ and $\mu^{Q}_{n}$ ($\mu^{H}_{p}$ and $\mu^{Q}_{p}$) are
the neutron (proton) chemical potentials in the hadron phase and the quark
phase, respectively,

$\mu_{n}=\frac{\partial{\cal E}}{\partial{n_{n}}},$ (31)
$\mu_{p}=\frac{\partial{\cal E}}{\partial{n_{p}}}.$ (32)

In above equations, ${\cal E}$ is the energy density of the system,

${\cal E}=n(E+mc^{2}).$ (33)

To obtain $\mu^{H}_{p}$ and $\mu^{H}_{n}$ for the hadronic matter in mixed
phase, we use the semiempirical mass formula kutschera ; langris ; wiringa ,

$E=T(n,x)+V_{0}(n)+(1-2x)^{2}V_{2}(n),$ (34)

where $x=\frac{n_{p}}{n}$ is the proton fraction. $T(n,x)$ is kinetic energy
contribution and the functions $V_{0}$ and $V_{2}$ represent the interaction
energy contributions which are determined from the energies of the symmetric
nuclear matter $(x=\frac{1}{2})$ and pure neutron matter $(x=0)$. We calculate
$V_{0}$ and $V_{2}$ using our results for the LOCV calculation of nucleonic
matter with the $UV_{14}+TNI$ nuclear potential which is discussed in section
II.1. Now, we can obtain the chemical potentials of neutrons and protons from
Eqs. (31)-(34) as follows,

$\displaystyle\mu^{H}_{p}$ $\displaystyle=$ $\displaystyle
T(n,x)+n\frac{\partial{T(n,x)}}{\partial{n}}+\frac{\partial{T(n,x)}}{\partial{x}}+V_{0}(n)+n{V_{0}}^{\prime}(n)$
(35)
$\displaystyle+(-3+8x-4x^{2})V_{2}(n)+(1-2x)^{2}n{V_{2}}^{\prime}(n)+mc^{2},$

$\displaystyle\mu^{H}_{n}$ $\displaystyle=$ $\displaystyle
T(n,x)+n\frac{\partial{T(n,x)}}{\partial{n}}-\frac{\partial{T(n,x)}}{\partial{x}}+V_{0}(n)+n{V_{0}}^{\prime}(n)$
(36) $\displaystyle+(1-4x^{2})V_{2}(n)+(1-2x)^{2}n{V_{2}}^{\prime}(n)+mc^{2}.$

For the quark matter in mixed phase, we have

$\mu_{p}^{Q}=2\mu_{u}+\mu_{d},$ (37) $\mu_{n}^{Q}=\mu_{u}+2\mu_{d}.$ (38)

At a certain pressure, we calculate $\mu_{u}$ for different $\mu_{d}$ under
the condition that the densities yield this certain pressure. By calculating
$\mu_{u}$ and $\mu_{d}$, we obtain $\mu^{Q}_{p}$ and $\mu^{Q}_{n}$.

Now, we plot $\mu_{p}$ versus $\mu_{n}$ for both hadron and quark phases, the
cross point of the two curves satisfies the Gibss conditions. In the mixed
phase, as the chemical potentials determine the densities, the volume fraction
occupied by quark matter, $\chi$, can be obtained by the requirement of global
charge neutrality,

$\chi(\frac{2}{3}n_{u}-\frac{1}{3}n_{d}-\frac{1}{3}n_{s})+(1-\chi)n_{p}-n_{e}=0.$
(39)

Finally, we can calculate the baryonic density of the mixed phase (M),

$n_{B}=\chi n_{Q}+(1-\chi)n_{H},$ (40)

and then the total energy density of mixed phase is found,

${\cal E}_{M}=\chi{\cal E}_{Q}+(1-\chi){\cal E}_{H}.$ (41)

###  II.4 Results

We have shown our results for the energy densities of hadron phase, quark
phase and mixed phase in Figs. 1 and 2 at two different temperatures. Figs. 1
and 2 show that at low densities the energy density of the hadronic matter is
lower than those of other phases. However, as the density increases, at first
the energy of mixed phase and finally the energy of quark phase is lower than
those of other phases. We also see that there is a mixed phase for a range of
densities. Below (beyond) this range, we have the pure hadron (quark) phase.
By comparing Figs. 1 and 2, we see that for a given value of the density, the
energies of all phases increases by increasing the temperature.

Using the above calculated energy density, we can determine the equation of
state and finally the structure of the hot neutron star with the quark core
which is discussed in the next section.

##  III Structure of the Hot Neutron Star with the Quark Core

The structure of neutron star is determined by numerically integrating the
Tolman-Oppenheimer-Volkoff equation (TOV) shapiro ; glen2 ; weber ; alder ,

$\frac{dP}{dr}=-\frac{G[{\cal E}(r)+\frac{P(r)}{c^{2}}][m(r)+\frac{4\pi
r^{3}P(r)}{c^{2}}]}{r^{2}[1-\frac{2Gm(r)}{rc^{2}}]},$ (42)

$\frac{dm}{dr}=4\pi r^{2}{\cal E}(r),$ (43)

where $P$ is the pressure and ${\cal E}$ is the total energy density. For a
given equation of state in the form $P({\cal E})$, the TOV equation yields the
mass and radius of star as a function of the central mass density.

In our calculations for the structure of hot neutron star with the quark core,
we use the following equations of state: (i) Below the density of $0.05\
fm^{3}$, we use the equation of state calculated by Baym baym . (ii) From the
density of $0.05\ fm^{3}$ up to the density where the mixed phase starts, we
use the equation of state of pure hadron phase calculated in section II.1.
(iii) In the range of densities in which there is the mixed phase, we use the
equation of state calculated in section II.3. (iv) Beyond the density of end
point of the mixed phase, we use the equation of state of pure quark phase
calculated in section II.2. All calculations are done for ${\cal B}=90\
MeVfm^{-3}$ at two different temperatures $T=10$ and $20\ MeV$. Our results
are as follows.

The gravitational mass as a function of the central mass density for the hot
neutron star with the quark core at two different temperatures has been
presented in Figs. 3 and 4. It is seen that for both relevant temperatures,
the gravitational mass increases by increasing the central mass density and
finally reaches a limiting value (maximum mass). In Figs. 3 and 4, our results
for the case of neutron star without the quark core have been also given for
comparison. We see that by including the quark core for the neutron star, our
results for the gravitational mass are substantially affected. For the neutron
star with the quark core, our results for the gravitational mass at three
different temperatures ($T=0$, $10$ and $20\ MeV$) have been compared in Fig.
5. It is seen that the gravitational mass increases by increasing the
temperature.

Figs. 6 and 7 show the gravitational mass versus the radius for both cases of
the neutron star with and without the quark core at two different
temperatures. At each temperature, it is seen that there is a reasonable
difference between the mass-radius relations of these two cases of the neutron
star. However, for both cases, we see that the radius decreases as the mass
increases. By comparing Figs. 6 and 7, we can see that the decreasing rate of
the radius versus the mass is substantially different for different
temperatures.

Our results for the maximum gravitational mass of the hot neutron star with
the quark core and the corresponding values of radius and central mass density
have been given in Tables 1 and 2 for two different temperatures. Our results
for the case of hot neutron star without quark core have been also presented
for comparison. For different temperatures, it is seen that the inclusion of
the quark core considerably reduces the maximum mass of the hot neutron star.
This is due to the fact that by including the quark core for the neutron star,
the equation of state becomes softer than that without the quark core.
However, we do not see any substantial changes for the radius and central mass
density of these two cases of the hot neutron star.

##  IV Summary and Conclusion

For the hot neutron star, from the surface toward the center, we have
considered a pure hadronic matter layer, a mixed phase of quarks and hadrons
in a range of densities which are determined by employing the Gibbs
conditions, and a pure quark matter in the core, to calculate its equation of
state at finite temperature. For calculating the equation of state of the hot
hadronic matter, we have applied the lowest order constrained variational
(LOCV) method at finite temperature. The equation of state of the hot quark
matter has been computed using the MIT bag model with the bag constant ${\cal
B}=90\ MeVfm^{-3}$. Using this equation of state, we have solved the TOV
equation by numerical method to determine the structure properties of the hot
neutron star with the quark core at $T=10$ and $20\ MeV$. Then, we have
compared the results of these calculations with those for the hot neutron star
without the quark core. It is found that our results for the maximum
gravitational mass of the neutron star with a quark core are less than those
of the neutron star without the quark core.

###### Acknowledgements.

This paper is derived from research project entitled: Structure of neutron
stars with a quark core at finite temperature. Financial support from Islamic
Azad University, Bafgh Branch, Research council is gratefully acknowledged. G.
H. Bordbar wishes to thank the Shiraz University Research council. G. H.
Bordbar also wishes to thank the Research Institute for Astronomy and
Astrophysics of Maragha for its financial support.

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Table 1: Maximum gravitational mass $(M_{\max})$, and the corresponding radius ($R$) and central mass density $(\varepsilon_{c})$ of the hot neutron star without (NS) and with (NS+Q ) the quark core at $T=10\ MeV$. | $M_{\max}(M_{\odot})$ | $R(Km)$ | $\varepsilon_{c}(10^{14}gr/cm^{3})$
---|---|---|---
NS | 2.07 | 10.22 | 26.94
NS+Q | 1.76 | 10.45 | 27.38

Table 2: As Table 1 but at $T=20\ MeV$. | $M_{\max}(M_{\odot})$ | R(Km) | $\varepsilon_{c}(10^{14}gr/cm^{3})$
---|---|---|---
NS | 2.09 | 10.64 | 27.01
NS+Q | 1.78 | 11 | 27.37

Figure 1:  Energy density versus the baryonic density at $T=10\ MeV$

for the hadron phase (solid curve), quark phase (dotted curve) and mixed phase
(dashed curve).

Figure 2: As Fig. 1 but at $T=20\ MeV$. Figure 3: Gravitational mass versus
the central mass density for the neutron star with (dotted curve) and neutron
star without (solid curve) the quark core at $T=10\ MeV$. Figure 4: As Fig. 3
but at $T=20\ MeV$. Figure 5: Gravitational mass versus the central mass
density for the neutron star with the quark core at $T=0$ (dotted-dashed
curve), $10$ (solid curve) and $20\ MeV$ (dotted curve). Figure 6:  Mass-
radius relation for the neutron star with (dotted curve) and without (solid
curve) the quark core at $T=10\ MeV$. Figure 7: As Fig. 6 but at $T=20\ MeV$.