#  A Comprehensive Survey On Client Selections in Federated Learning

A. Gouissem1, Z. Chkirbene2, R. Hamila2 1 College of Computing and Information
Technology, University of Doha for Science and Technology, Qatar,  2
Electrical Engineering, Qatar University, Qatar,

###### Abstract

Federated Learning (FL) is a rapidly growing field in machine learning that
allows data to be trained across multiple decentralized devices. The selection
of clients to participate in the training process is a critical factor for the
performance of the overall system. In this survey, we provide a comprehensive
overview of the state-of-the-art client selection techniques in FL, including
their strengths and limitations, as well as the challenges and open issues
that need to be addressed. We cover conventional selection techniques such as
random selection where all or partial random of clients is used for the
trained. We also cover performance-aware selections and as well as resource-
aware selections for resource-constrained networks and heterogeneous networks.
We also discuss the usage of client selection in model security enhancement.
Lastly, we discuss open issues and challenges related to clients selection in
dynamic constrained, and heterogeneous networks.

††publicationid: pubid:  979-8-3503-4760-9/23/$31.00 ©2023 IEEE

##  I Introduction

In recent years, there has been a significant increase in the size and
diversity of data sources available for analysis, including data generated by
social media platforms, internet of things (IoT) devices, and a wide range of
other sources. The growth in data size and sources created new challenges in
terms of storage, processing, and analysis [1].

Classical analytical methods, which rely on statistical analysis and and
human-based decisions, are nowadays limited in their ability to process and
analyze data at scale, particularly when the data is complex or unstructured.
The integration of machine learning (ML) techniques, on the other hand, can
facilitate faster and more precise data analysis, resulting in better
decision-making and increased efficiency. Additionally, it can also help to
uncover previously undetected patterns and trends that may not have been
apparent using other methods. ML attract a lot of attention thanks to its
ability to be applied in a wide range of areas, including image and speech
recognition, relay selection [2], natural language processing, intrusion
detection [3, 4, 5], anti jamming, [6, 7, 8] and even self-driving cars[9].

While ML can facilitate advanced and efficient decision-making, it relies on
the collection and centralization of data, which may not be possible due to
privacy concerns [10]. To address this challenge, Federated learning (FL) is
used to operate on decentralized datasets that are distributed across multiple
devices. In particular, FL is a machine learning method that allows data to be
trained across multiple decentralized devices, such as smartphones or laptops,
without the need to centralize the data[11].

FL is an emerging field in machine learning, with a growing body of literature
exploring its potential benefits and challenges. Review papers such as [11,
12, 13, 14], provide an overview of the concept and its applications, the
state-of-the-art techniques in FL as well as the current challenges and open
issues. However, while these papers have provided a good introduction to the
field, there are still many challenges that need to be addressed. One of these
challenges is the issue of client selection. In Federated Learning, the
selection of clients to participate in the training process is critical to the
performance of the overall system. The clients selected should be
representative of the entire population and should have the necessary
resources and capabilities to contribute to the training process. However,
selecting the appropriate clients is a non-trivial task, particularly in
large-scale networks with a large number of diverse clients.

Another challenge is the issue of privacy and security. Federated learning has
the potential to improve privacy and security for the individuals or
organizations contributing data. However, there are still many open questions
regarding how to protect the privacy and security of the clients and their
data in a federated learning setting. Additionally, there is a need for
techniques that can detect and prevent malicious clients from joining the
network, which can compromise the overall security of the system. Adaptivity
is also a key challenge in Federated Learning (FL) systems. In order to be
effective, FL systems must be able to adapt to changing network conditions and
client capabilities over time. However, many current FL techniques lack this
ability and are unable to adjust accordingly. This limits the effectiveness of
the FL system, and it is an area of active research to improve the adaptivity
of these systems.

This paper is a review of client selection techniques in FL. The goal of this
paper is to provide a comprehensive overview of the state-of-the-art client
selection techniques in FL, including their strengths and limitations, as well
as the challenges and open issues that need to be addressed in this field.
This review aims to provide a comprehensive understanding of the current state
of client selection in FL and to provide insights for researchers and
practitioners working in this field.

##  II Conventional FL Selection

Several studies [15, 16] have made substantial advancements in various aspects
of FL, such as minimizing communication costs and addressing issues caused by
non  independent and identically distributed (i.i.d) data distributions.
However, most of these techniques rely solely on the selection of random
clients which make the model performance non optimal.  In this section, the
random selection as well as full and partial client selection will be
thoroughly examined. The potential drawbacks of each method will also be
discussed.

###  II-A Random Selection

Random selection in federated learning refers to the process of randomly
selecting a subset of devices or participants from a larger group to
contribute data or computation to a FL model. This method of selection aims to
provide a representative sample of data from the entire group and can be used
to improve the generalization of the FL model. The authors in [16] propose a
method for training deep neural networks in a FL manner using iterative model
averaging and conduct a thorough evaluation of its performance using different
datasets. The proposed approach in [16] considers the challenges of client
availability as well as unbalanced and non-IID data. However, the client
selection is performed randomly. In fact, the process begins with the server
selecting a group of clients randomly. Each selected client then uses the
global information and their own data to perform calculations locally, after
which they send the update back to the server. The server takes in these
updates and updates the global information, the cycle is then repeated with
different clients. This increases the processing parallelism, where a larger
number of clients can work independently between each round of communication.
It also increases the computations per client in order to reduce the
communication cost. The simulation results show that this approach is
effective even when dealing with unbalanced and non-IID data and can reduce
the number of communication rounds, by 10-100 times, as compared to other
methods like synchronized stochastic gradient descent.

Consequently, most of the proposed schemes for FL models rely on just
selecting a random subset of participants from a large pool of available
clients in each round of FL. This random selection strategy when combined with
other algorithms and aggregation schemes can solve specific problems related
to security, cost or data distribution.The use of random selection in
Federated Learning (FL) has its limitations. It does not take into account the
variations in data, resources, and systems among clients, which can impact the
efficiency of federated training. By randomly selecting clients, there is a
risk of selecting clients with over-represented data or clients that are slow
due to limited computation speeds or network bandwidths, which can act as
bottlenecks in each round of federated training. This can lead to delays and
inefficiencies in the training process [17].

###  II-B Full and partial State Knowledge Selection

In FL, full state knowledge means that all devices have complete knowledge of
the global model, while partial state knowledge means some devices have
limited knowledge and may not have access to the same information. In [18],
have demonstrated convergence in federated learning models when clients fully
participate in the training process. However, these studies only apply to
unbiased client participation. In [19] evaluated the convergence of federated
learning when clients have the ability to join or leave the training process
or send incomplete updates to the server. Furthermore, most of the client
selection techniques available in the literature assume the knowledge of the
client-side state information[20, 17, 21, 22, 23, 24, 25, 26]. However,
assuming full knowledge of the client-side state information in federated
learning may not be feasible or practical to obtain such information in real-
world scenarios. For example, in a large-scale federated learning system with
thousands of clients, it may be difficult to collect and transmit detailed
state information for each client. Additionally, obtaining such information
may be a privacy concern for clients. This could lead to a low adoption rate,
and less participation from clients.

##  III Performance Aware Selection

In Performance aware selection is a technique that chooses a specific group of
devices based on their expected performance during the training process, these
devices are selected to improve the performance of the model. Thus, the
clients can be selected based on their statistical utility and their potential
to improve the global model accuracy. Such decisions can be made based on the
model updates using various measures [21, 22, 26, 27].  In this section,
various methods for selecting performance-aware federated learning techniques
will be discussed.

###  III-A Active Federated Learning

Active Federated Learning aims to improve the model’s performance by
selectively training on clients with valuable updates. The authors in [22]
highlight that FL training and update process consumes a large amount of the
clients’ bandwidth which makes minimizing the transmission costs crucial to
the success and efficiency of FL. Since the data on each client is highly
variable, Active Federated Learning is proposed in [22] to allow clients to be
selected in each round based on the current model and client data to maximize
efficiency, rather than being chosen uniformly at random. A cost-effective
sampling method is also proposed to reduce the number of required training
iterations by 20-70% while maintaining the same model accuracy. Active
Federated Learning is a method that capitalizes on the stability and
availability of most clients during the training process, by using client-side
state information to guide the selection of clients. One such approach is
ClusterFL [27] that aims to improve the performance of the FL model by
identifying and removing clients that are not performing as well as others or
are less related to other clients within the same cluster. This approach
utilizes the inherent clustering structure among clients based on their local
data distributions to identify these clients and exclude them from the
training process. However, Active Federated Learning can be computationally
expensive to evaluate the model’s performance on a large number of clients
before deciding which clients to select for the next round of training

###  III-B UCB-CS

Upper Confidence Bound for Cooperative Stochastic Bandits ( UCB-CS ) is a
method for selecting clients in Federated Learning that balances exploration
and exploitation using the UCB algorithm to select clients that are expected
to provide the most valuable updates to the model. In [17] a new approach for
selecting clients in FL that is based on the multi-armed bandit algorithm
called UCB-CS is proposed.It addresses issues such as communication
efficiency, noise, variations in local loss values, and error floors that are
present in prior methods. The authors show that UCB-CS can achieve robustness
to error floor, faster convergence, and reduced state estimates of local loss
values, all without requiring additional communication compared to random
selection. Additionally, it ensures fairness among clients by achieving more
uniform performance. The work in [17] is performed under the assumption of
independent local losses for each client which might be unpractical in some
scenarios since clients with similar characteristics may have similar local
losses.  However, UCB-CS assumes that the rewards (or updates) provided by
clients are independent and identically distributed, which may not be the case
in some Federated Learning scenarios.

###  III-C Contribution control mechanisms

Contribution control mechanisms are methods for controlling the participation
of clients to improve the training process. The authors in [26] propose an
adaptive mechanism for selecting clients based on threshold recomputed at each
round. By training the model locally and computing the local contribution of
each device, the device itself can decide whether it should participate or not
in the training according to a preset contribution threshold.

The work in [21] presents a study on the convergence of federated learning
with a focus on biased client selection and its impact on training progress at
each client. The authors found that bias towards clients with higher local
losses improves convergence rates. They propose a new client selection
strategy called Power-Of-Choice which results in faster convergence and better
performance when compared to standard methods. The authors in [28] proposed a
technique named Pyramid that aims to accelerate the FL training process while
maintaining a high model accuracy. PyramidFL performs client selection and
prioritization while taking into account the dissimilarities between chosen
and unselected clients. It also fully utilizes the data and system
heterogeneity within the selected clients to optimize their contribution into
the global model. To achieve this, PyramidFL first selects clients based on
their utility from the global model and then improves the selection by
optimizing client utility locally. The authors also report that PyramidFL was
able to improve final model accuracy and speed at the same time.  However, the
model’s performance in PyramidFL may also be limited by the quality of the
clients in the lower levels of the pyramid, as these clients are likely to
have less computational and communication resources.

##  IV Resource Aware Selection

Resource-aware selection in Federated Learning is a technique that aims to
select clients based on their available computational and communication
resources. The selection can also be performed while aiming to optimize the
limited clients resources [20, 25] and communication constraints [23].

###  IV-A Resource Constrained networks

The authors in [25] proposed a federated learning (FL) protocol called FedCS,
which is designed to mitigate the training bottleneck in mobile edge computing
(MEC) frameworks. In the FedCS protocol, a server coordinates training in a
cellular network comprising participating mobile devices with heterogeneous
resources. The server gathers information about wireless channel states and
computing capabilities from a subset of randomly selected participants, and
then selects the maximum possible number of participants that can complete
training within a prespecified deadline for the subsequent global aggregation
phase. A greedy algorithm is used in [29] to solve the maximization problem by
iteratively selecting participants that take the least time for model upload
and update. Simulation results showed that FedCS can achieve higher accuracy
than an FL protocol that only accounts for training deadline, but FedCS has
only been tested on simple deep neural network (DNN) models. There are
limitations to the use of FedCS for more complex models, such as the
difficulty in estimating the number of participants to be selected and
potential bias towards selecting participants with devices that have better
computing capabilities, which may not hold data representative of the
population distribution.

###  IV-B Reinforcement learning based selection

One of the major challenges in implementing FL in mobile edge networks is that
such networks are very dynamic and hard to predict [30]. Therefore, static
user selection techniques might fail to track the network and constraints
variations. In particular, these participating mobile devices have limitations
in terms of energy, processing power, and wireless bandwidth. The FL server
must therefore determine the most efficient use of the resources to minimize
energy consumption and training time. To solve this issue, the authors in [31]
proposed the use of Deep Q-Learning (DQL) to optimize the allocation of
resources for training deep neural network models. The system includes mobile
devices that collaborate to train the DNN models required by a FL server.
These mobile devices are limited in terms of energy, processing power, and
wireless bandwidth. The Double Deep Q-Network (DDQN) algorithm can also be
used as suggested in [32]. The DDQN algorithm is applied to an optimization
problem in which the FL server acts as the agent, the state space includes the
energy and CPU state of the mobile devices, and the action space includes the
amount of data and energy units that are used from the mobile devices. The
rewards of this optimization problem is computed from factors such as data
accumulation, energy consumption, and training latency. The results of the
simulation studies show that this method can reduce energy consumption by
about 31% when compared with other commonly used methods such as the greedy
algorithm and can decrease training latency by up to 55% compared to random
selection methods.

Other research works, such as [12], have focused on the difficulties of
implementing Federated Learning in mobile edge networks and proposed potential
solutions and use cases. For instance, [33] proposed a framework for effective
resource management for FL in the edge. But, these studies, including [33,
34], did not delve into the key challenges of creating incentives and
optimizing network performance in edge-based FL.

###  IV-C Heterogeneous Networks

The authors in [24] propose a technique named Oort that considers both data
and system heterogeneity at the same time. Such approach creates a compromise
between data usage and system efficiency, and it has been shown to have
superior time-to-accuracy performance compared to random selection. However,
this technique considers the differences between chosen and unselected
clients, and it doesn’t take into account the variations within the chosen
clients.

Another approach, called hybrid-FL, has been proposed in [35] as a way to
generate an approximately IID dataset using only a limited number of privacy-
insensitive participants. This is done by selecting participants whose data
can form an approximately IID dataset, then training a model on the collected
IID dataset and merging it with the global model trained by all participants.
The simulation results showed that this approach can improve classification
accuracy even with a small percentage of participants sharing their data.
However, it is important to note that this approach has the potential to
compromise user privacy and security. To mitigate this, an incentive and
reputation mechanism should be implemented to ensure that only trustworthy
participants are allowed to upload their data.

###  IV-D Game Theory and Incentive Mechanisms

Incentive mechanisms are generally used to motivate a person or a device to
take specific actions. Incentives can be either positive or negative. Positive
incentives encourage actions by offering rewards, while negative incentives
discourage bad behavior by imposing penalties. The use of game theory to
design incentive mechanisms has been extensively studied in other fields such
as crowdsensing [36], edge computing [37], device to device communication
[38], opportunistic networks [39] and others, where users might be regraded
for using or providing specific resources, and penalized for using others.
However, in FL, it is challenging to precisely quantify the value of each
client’s training data and to model the final learning performance of the
system. As a result, it is difficult to model the utility function of each
participant, which makes it hard to directly apply existing incentive
mechanism design works to FL. In contrast, [40] aims to comprehensively review
resource optimization and incentive mechanisms for FL in edge networks.

The authors in [40] propose a novel approach to developing an incentive
mechanism for Federated Learning (FL) using game theory by using the framework
of a Stackelberg game in which FL users are the followers and the base station
is the leader. In this game, FL users can strategically set the number of
local iterations to maximize their own utility. The base station, as the
leader, then uses the best response strategies of the users to optimize FL
performance. This approach offers a new perspective on how to incentivize FL
participants and improve overall FL performance. The paper also identifies key
open research challenges and provides guidelines for FL in edge networks.
However, edge network architecture may be complex and distributed, which can
make it difficult to optimize resources and implement incentive mechanisms.
Moreover, FL based on a Stackelberg game can raise privacy and security
concerns, as data is exchanged between devices and organizations.

##  V FL Client Selection Based Security

FL Client Selection Based Security ensures that only authorized clients and
devices can access protected resources, and that sensitive data remains
confidential. In this section, several techniques for selecting clients using
security concepts in FL will be presented.

###  V-A Poisoning attacks in FL

Federated Learning (FL) is known to be vulnerable to poisoning attacks [41,
42, 43]. The adversary’s goal and capabilities might differ from one model to
another [44]. Based on this goal, poisoning attacks can be classified into
targeted and untargeted attacks. In untargeted poisoning attacks, the
objective is to lower the global model’s accuracy on any test input [42]. In
targeted poisoning, the objective is to reduce the accuracy of some test
inputs while maintaining high accuracy for the remaining inputs. Therfore,
untargeted assaults may drastically degrade the performance of the global
model for any test input. Poisoning may be carried out either directly on the
model (model poisoning) or by changing the training data. In untargeted
poisoning attacks, the goal is to minimize the accuracy of the global model on
any test input [42]. In targeted poisoning attacks, the goal is to minimize
the accuracy on specific test inputs, while maintaining high accuracies on the
rest of the test inputs. Untargeted attacks, however, can completely
deteriorate the global model performance for any test input. The poisoning can
be either performed on the model itself (model poisoning) attacks, [42], or by
manipulating the training data.

Several techniques have been proposed to combat such attacks. The main of
focus of this survey in this regard is the investigation of the different
techniques in the literature that perform the identification and isolation of
the malicious nodes by selecting the most trustworthy ones data owners. In
particular, several techniques exist in the literature to select the
legitimate data owners. The different techniques in Section II and Section III
perform the clients selection before the client training itself to avoid
useless computations. However, poisoning attacks defense mechanisms perform
the selection after the reception of the updates and by trying to identify
anomalies in the reports.

###  V-B Krum

Key-value Retrieval Under Misbehaviors (Krum) aims to improve the robustness
of FL in the presence of Byzantine or adversarial clients. Krum works by
selecting a small number of clients to update the global model, and then using
the median of their updates to update the global model. The authors in [41]
investigate a FL model when there are Byzantine workers present. The paper
presents a Byzantine resilience client selection technique that allow the
isolation of malicious nodes reports. In particular, one way to ensure this
resilience is to consider a majority-based approach, but it has prohibitive
computational costs. An alternative approach is the use of the KRUM metric
that creates a combination of majority-based and squared-distance-based
methods by choosing the vector that is closest to its $n-f$ neighbors. Krum
has the advantage of having a local time complexity of $O(n^{2}d)$ and is
shown to converge in the presence of Byzantine workers.

###  V-C Multi-Krum

An extension of the Krum method entitled Multi-Krum is also presented in [41]
and combines the resilience of Krum with the convergence speed of the standard
averaging and performs well compared to other aggregation rules like the
geometric median. The major issue in both Krum and Multi-Krum is that the
number of nodes to be isolated from the aggregation is setup in advance and
fixed by the administrator whether there is a small or large number of
malicious nodes in the network. Such approach might waste legitimate nodes
time and resources if the number of malicious agents is smaller than the
number of isolated clients. On the other hand, isolating few nodes results in
an increased risk level.

###  V-D Bulyan

Another method used to select benign clients and gradients is proposed in [42]
and called Bulyan. The Bulyan algorithm insurance Byzantine-resilience
aggregation by using recursive rules based on Krum or Brute, Krum, a Medoid,
the geometric median or any other Byzantine–resilient rule. In fact, Among the
set of all the users, the client with closest gradient (based on KRUM or any
other rule) is first selected. This client is removed from the set and the
process is repeated again till obtaining the right number of clients. This
approach is proven to be able to significantly reduce the space in which an
attacker can cause the model to drift towards suboptimal models. Through
experimental testing with different datasets, the authirs demonstrate that
Bulyan successfully avoids convergence to ineffective models, instead
resulting in models that are similar to those produced by a non-attacked
averaging scheme

###  V-E Trimmed-mean

Instead of selecting the same set of clients in all dimensions, the Trimmed-
mean technique proposed in [45] and [46] allows the selection of a different
set of clients at each dimension. In particular, for each coordinate of the
gradient, the $\beta$ smallest and the $\beta$ highest reports are removed and
not considered in the aggregation. According to [45], Trimmed-mean reaches an
error rate that is close to optimal when m is approximately
$m\leq\beta\leq\frac{n}{2}$ for objective functions that are strongly convex.

###  V-F Other techniques

Several other techniques exist in the literature to protect the FL training
from poisining attacks. For example, the authors in [47] investigated a
special type of data poisoning attack where Byzantine nodes may report high
data sizes in order for the model aggregation to be unbalanced in their favor.
A data balancing approach based on weight-truncation is proposed to protect
the FL training. In [48], an enhanced version of the Alternating Direction
Method of Multipliers (ADMM) algorithm is presented to ensure the convergence
of decentralized learning algorithms to a suboptimal solution even when there
is Byzantine data falsification. Additionally, the study includes an analysis
of the speed of convergence of the algorithm. The authors in [49] analyze the
dependency between iterations and aggregated gradients in order to identify
and separate out Byzantine reported updates. A robust version of gradient
descent is presented, which uses geometric median for model aggregation rather
than traditional averaging. The mathematical analysis and simulation results
in the paper support that this proposed method converges uniformly to the true
gradient function. However, all of the above techniques and several others in
the literature process all the received reports to generate a Byzantine-free
report without actually selecting or identifying the malicious ones. Such
approaches might lack accuracy in some scenarios since the malicious nodes are
nodes are not isolated and can still take part of the model direction
decisions.

##  VI Open issues and challenges

In this section, we will discuss the open issues and challenges in FL to
further improve the performance and effectiveness of Federated Learning
systems.

###  VI-A Resource Constraints Challenges

One of the main challenges in FL is dealing with clients that have limited
resources, such as computation power and network bandwidth. Random selection
of clients may result in selecting clients that are not capable of
contributing effectively to the training process due to their limited
resources. In addition, there is a limited number of studies that consider the
dynamic nature of wireless networks. Numerous client selection strategies rely
on static network circumstances, which may not correctly reflect the dynamic
nature of wireless networks. Furthermore, many client selection solutions do
not account for the heterogeneous capabilities of the clients, which might
result in reduced performances. In addition, numerous solutions rely on a
centralized approach, which is sometimes impractical in large-scale networks
due to the substantial computational and communication overhead.

In general, most of the client selection approaches described in the
literature for constrained networks are limited by their inability to adapt to
dynamic network conditions, their disregard for client heterogeneous
capabilities, and their centralized structure, which might be impractical for
large-scale networks.

###  VI-B Security Challenges

Another challenge in FL is ensuring the security and privacy of the clients’
data. Random selection of clients may result in selecting clients that are not
trustworthy or that have vulnerable systems, which can compromise the security
of the FL model. The available client selection techniques in the literature
have several limitations. One limitation is that many techniques rely on a
centralized approach, which can make the system vulnerable to attacks on the
central entity. Additionally, many techniques do not take into account the
varying security capabilities of different clients, which can lead to
suboptimal performance. For example, the trustworthiness of the clients should
be different from one group to another based on their history and contribution
to the global model. Furthermore, many techniques fail to consider the privacy
concerns of the clients, which can lead to leakage of sensitive information.

Another limitation is the lack of consideration for the dynamic nature of
security threats and vulnerabilities, which can change over time.
Additionally, many client selection techniques do not have the ability to
detect or prevent malicious clients from joining the network in advance, which
can compromise the overall security of the network. In particular, most of the
selection techniques in that regard let all the nodes train their models and
then decide which ones to consider which might waster the client resources.

###  VI-C Heterogeneity and Dynamicity Challenges

In FL, clients can have different data distributions, system variations, and
resource constraints. Random selection of clients may result in selecting
clients that have over-represented data or clients that are not representative
of the population, which can negatively impact the performance of the FL
model. In particular, most of the current client selection techniques in
literature have several limitations regarding heterogeneity and adaptivity.
One limitation is that many techniques fail to take into account the dynamic
nature of wireless networks and the dynamic capabilities of different clients,
and rapidly changing network resources which can lead to suboptimal
performance and communication issues. In summary, Selecting clients
dynamically based on the fast changing and heterogeneous state of the system
and clients, the clients’ resources and the model’s performance is still an
open challenge

###  VI-D Performance trade-off Challenges

Most of the client selection techniques in the literature focus on a single
objective such as heterogeneity, security, or resources.  The performance
trade-off challenges in Federated Learning aim to balance between these
objectives.  There was limited efforts in designing a multi-objective model
that can handle several selection problem at once at the same time to create
Performance trade-off by balancing the communication cost, model’s performance
and clients participation.

##  VII Conclusion

The selection of clients to participate in the training process is a critical
factor for the performance of Federated Learning (FL) models. The clients
selected should be representative of the entire population and have the
necessary resources to contribute to the training process. However, choosing
appropriate clients is a non-trivial task, particularly in large-scale
networks with a large number of diverse clients. Poor client selection can
lead to a decrease in the performance of the FL system, and increase in
communication and computation cost. Furthermore, the clients selected should
be trustworthy and reliable, as they may have malicious intent and attempt to
poison the model by providing incorrect or biased data. In this survey, we
have presented the state-of-the-art client selection techniques in FL,
including their strengths and limitations, as well as the challenges and open
issues. We covered conventional selection techniques, performance-aware
selections, resource-aware selections, and model security techniques. We also
discussed open issues and challenges related to resource constraints,
heterogeneity and dynamicity, and performance trade-offs. We believe that
addressing the open issues and challenges related to client selection will
play a crucial role in the success and widespread adoption of FL in various
applications.

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# On The Evans Chain Complex

S. Joseph Lippert

###### Abstract

We elaborate on the construction of the Evans chain complex for higher-rank
graph $C^{*}$-algebras. Specifically, we introduce a block matrix presentation
of the differential maps. These block matrices are then used to identify a
wide family of higher-rank graph $C^{*}$-algebras with trivial K-theory.
Additionally, in the specialized case where the higher-rank graph consists of
one vertex, we are able to use the Künneth theorem to explicitly compute the
homology groups of the Evans chain complex.

##  1 Introduction

In the field of $C^{*}$-algebra classification, $K$-theory computation has
long been a captivating question. An algebra’s $K$-theory groups will fully
classify simple, separable, nuclear, and purely infinite $C^{*}$ algebras
satisfying the Universal Coefficient Theorem by the Kirchberg-Phillips
classification theorem [8, 9]. Further still, the Elliott Invariant, which
consists of the $K$-theory groups as well as information about the trace
simplex, has shown incredible versatility in $C^{*}$-algebra classification
[2, 4, 5].

On the other hand, higher-rank graph $C^{*}$-algebras have been an active
field of research for the last quarter century. These algebras were introduced
in [10] building off work in [12]. They have offered a number of creative
solutions to long standing problems in the field. They were used to show that
every UCT Kirchberg algebra has nuclear dimension zero [14], have been related
to solutions of the Yang-Baxter equations [18, 16], and have reformulated
multipullback quantum odd sphere information in terms of universal properties
[6].

With all of these promising results and many more on the horizon, it seems
crucial to refine our understanding of higher-rank graph algebra $K$-theory. A
landmark paper in this direction was [3] in which the work of [13, 7, 15] was
built out into a spectral sequence converging to the $K$-theory of a higher-
rank graph $C^{*}$-algebra. The goal of this paper is to better understand the
tools developed by Evans and use them to make a number of explicit
computations.

##  2 Preliminaries

Notation: We will assume that $0\in\mathbb{N}$. Further for $1\leq j\leq k$ we
will use $\kappa_{j}:\mathbb{N}\to\mathbb{N}^{k}$ to denote the inclusion into
the $j^{th}$ coordinate and $\iota_{j}:\mathbb{N}^{j}\to\mathbb{N}^{k}$ the
canonical inclusion onto the first $j$ generators. For a small category,
$\Lambda$, we denote the object set as $\Lambda^{0}$. Lastly, for a countable
category, $\Lambda$, we define the group
$\mathbb{Z}\Lambda^{0}:=\bigoplus_{v\in\Lambda^{0}}\mathbb{Z}$.

######  Definition 2.1.

[10, Def 1.1] Let $\Lambda$ be a countable category and
$d:\Lambda\to\mathbb{N}^{k}$ a functor. If $(\Lambda,d)$ satisfies the
factorization property- that is, for every morphism $\lambda\in\Lambda$ and
$n,m\in\mathbb{N}^{k}$ such that $d(\lambda)=m+n$, there exist unique
$\mu,\nu\in\Lambda$ such that $d(\mu)=m$, $d(\nu)=n$, and $\lambda=\mu\nu$\-
then $(\Lambda,d)$ is a $k$-graph (or graph of rank $k$).

As mentioned in the introduction, these objects can be used to build
$C^{*}$-algebras. Specifically this is done using the following relations:

######  Definition 2.2.

[10, Def 1.5] Let $\Lambda$ be a source-free row-finite $k$-graph. Then
$C^{*}(\Lambda)$ is the universal $C^{*}$-algebra generated by a family
$\\{s_{\lambda}:\lambda\in\Lambda\\}$ of partial isometries satisfying:

  1. (i)

$\\{s_{v}:v\in\Lambda^{0}\\}$ is a family of mutually ortogonal projections,

  2. (ii)

$s_{\lambda\mu}=s_{\lambda}s_{\mu}$ for all composable $\lambda,\mu$,

  3. (iii)

$s_{\lambda}^{*}s_{\lambda}=s_{s(\lambda)}$ for all $\lambda\in\Lambda$,

  4. (iv)

For all $v\in\Lambda^{0}$ and $n\in\mathbb{N}^{k}$ we have
${\displaystyle\sum_{\lambda\in r^{-1}(v)\cap
d^{-1}(n)}s_{\lambda}s_{\lambda}^{*}}$.

Many facets of this algebra can be studied by studying the higher rank graph
it is built from. The focus of this article is on the $K$-theory groups of
$C^{*}(\Lambda)$. These groups have a deep relationship with the coordinate
graphs of $\Lambda$ which we define now.

######  Definition 2.3.

[10, Definition 1.9] Let $f:\mathbb{N}^{\ell}\to\mathbb{N}^{k}$ be a monoid
morphism. If $(\Lambda,d)$ is a $k$-graph, the pullback is the $\ell$-graph
$f^{*}(\Lambda)$ defined by

$f^{*}(\Lambda)=\\{(\lambda,n):d(\lambda)=f(n)\\}$

with $d(\lambda,n)=n$, $s(\lambda,n)=s(\lambda)$ and
$r(\lambda,n)=r(\lambda)$.

We call $\kappa_{i}^{*}(\Lambda)$ the $i^{th}$ coordinate graph, and we use
$M_{i}(\Lambda)$ to denote the adjacency matrix of the $i^{th}$ coordinate
graph (simplified to $M_{i}$ when there is no ambiguity).

As alluded to in the introduction, a great tool for $K$-theory computation is
spectral sequences. We now introduce basic definitions, notation, and results
of these tools.

######  Definition 2.4.

A spectral sequence is a set of abelian groups $E_{p,q}^{r}$ together with
differentials $d_{p,q}^{r}:E_{p,q}^{r}\to E^{r}_{p-r,q+r-1}$ such that
$E_{p,q}^{r+1}\cong\ker(d_{p,q}^{r})/\operatorname{im}(d_{p+r,q-r+1})$

A helpful metaphor for these objects is a book where $r$ indicates the page
you are on. The $E^{2}$ page would take the following shape in general:

${\cdots}$${E_{0,1}^{2}}$${E_{1,1}^{2}}$${E_{2,1}^{2}}$${E_{3,1}^{2}}$${E_{4,1}^{2}}$${\cdots}$${\cdots}$${E_{0,0}^{2}}$${E_{1,0}^{2}}$${E_{2,0}^{2}}$${E_{3,0}^{2}}$${E_{4,0}^{2}}$${\cdots}$$\scriptstyle{d_{1,0}^{2}}$$\scriptstyle{d_{2,0}^{2}}$$\scriptstyle{d_{3,0}^{2}}$$\scriptstyle{d_{4,0}^{2}}$$\scriptstyle{d^{2}}$

######  Definition 2.5.

A spectral sequence $E$ is said to stabilize if there exists some $r_{0}$ such
that $E^{r}\cong E^{r_{0}}$ for all $r\geq r_{0}$. This stable page is denoted
$E^{\infty}$. A stable spectral sequence is said to converge to $H_{*}$ (a
family of abelian groups) if for each $n$ there exists a filtration

$0=F_{s}(H_{n})\subseteq\cdots\subseteq F_{p}(H_{n})\subseteq\cdots\subseteq
H_{n}\ $

such that $E^{\infty}_{p,q}\cong F_{p}(H_{p+q})/F_{p-1}(H_{p+q})$.

######  Theorem 2.6.

[3, Theorem 3.15] The Evans Chain Complex: For $p,k\in\mathbb{N}$ with $0\leq
p\leq k$ define the set

$\mathbb{N}_{p,k}:=\\{a=(a_{1},\cdots,a_{p})\in\\{1,\cdots,k\\}^{p}:a_{i}<a_{i+1}\forall
i\\}$

Equip these sets with the standard map such that for $a\in\mathbb{N}_{p,k}$
the element $a^{i}\in\mathbb{N}_{p-1,k}$ is the tuple $a$ with the $i^{th}$
coordinate removed. Additionally, $N_{0,k}:=\\{*\\}$ with $a^{1}=*$ for all
$a\in N_{1,k}$.

For a row-finite, source-free $k$-graph, $\Lambda$, there exists a spectral
sequence $E,d$ converging to $K_{*}(C^{*}(\Lambda))$ with
$E^{\infty}_{p,q}\cong E^{k+1}_{p,q}$ and

$E^{2}_{p,q}\cong\begin{cases}H_{p}(\mathcal{D}),&0\leq p\leq k\wedge 2|q\\\
0,&\text{otherwise}\end{cases}$

where $\mathcal{D}$ is the chain complex

$\mathcal{D}_{p}\cong\begin{cases}\bigoplus_{a\in
N_{p,k}}\mathbb{Z}\Lambda^{0},&0\leq p\leq k\\\
0,&\text{otherwise}\end{cases}$ (1)

with differentials

$\bigoplus_{a\in N_{p,k}}m_{a}\mapsto\bigoplus_{b\in N_{p-1,k}}\sum_{a\in
N_{p,k}}\sum_{i=1}^{p}(-1)^{i+1}\delta_{b,a^{i}}B_{a_{i}}(m_{a})$ (2)

for $1\leq p\leq k$ and $B_{a_{i}}:=I-M_{a_{i}}^{T}$ (which we call the co-
adjacency matrix).

##  3 Block Matrix Presentation

In this section, we will investigate the indexing sets $\mathbb{N}_{p,k}$ more
closely. Using this information, we will partition the maps $\partial$ of
$\mathcal{D}$ into a recursive block matrices. This recursion is then
leveraged to make explicit computations for $k$-graphs with an invertible co-
adjacency matrix and monoid $k$-graphs.

######  Proposition 3.1.

For a fixed $k$, there exists a partition of $N_{p}=N_{p}^{+}\sqcup N_{p}^{-}$
with $N_{p}^{+}:=\\{a\in N_{p}:a_{p}=k\\}$ (the $k$ index has been supressed
for ease of reading).

  1. (i)

For $a\in N_{p}^{-}$ and $1\leq i\leq p$, $a^{i}\in N_{p-1}^{-}$.

  2. (ii)

Define $\psi:N_{p}^{+}\to N_{p-1}^{-}$ by $a\mapsto a^{p}$. The function
$\psi$ is bijective

  3. (iii)

There is a natural bijective inclusion $\varphi:N_{p,k-1}\to N_{p,k}^{-}$.

###### Proof.

(i) Let $a\in N_{p}^{-}$. Since $a_{1}<\cdots<a_{p}$ we conclude $a_{i}<k$ for
all $i$. Thus it is not possible for $a^{i}$ to have a $k$ in its last
position.

(ii) The map $a\mapsto a^{p}$ is injective since $a,b\in N_{p}^{+}$ ensures
they agree in the last position and $a^{p}=b^{p}$ ensures they agree
everywhere else. Surjectivity is also immediate since $b\in N_{p-1}^{-}$
implies $b_{i}<k$ for all $i$. Thus, $(b_{1},\cdots,b_{p-1},k)$ will map to
$b$.

(iii) After observing that this map is well defined, bijectivity can be reason
quickly from the definition of $N_{p,k}^{-}$. ∎

We now use this information to re-index the image of (2).

$\displaystyle\bigoplus_{b\in N_{p-1}^{+}}\sum_{a\in
N_{p}}\sum_{i=1}^{p}(-1)^{i+1}\delta_{b,a^{i}}B_{a_{i}}(m_{a})=$
$\displaystyle\bigoplus_{b\in N_{p-1}^{+}}\sum_{a\in
N_{p}^{+}}\sum_{i=1}^{p}(-1)^{i+1}\delta_{b,a^{i}}B_{a_{i}}(m_{a})$ (3)
$\displaystyle\bigoplus_{b\in N_{p-1}^{-}}\sum_{a\in
N_{p}}\sum_{i=1}^{p}(-1)^{i+1}\delta_{b,a^{i}}B_{a_{i}}(m_{a})=$
$\displaystyle\bigoplus_{b\in N_{p-1}^{-}}(-1)^{p+1}B_{k}(m_{\psi^{-1}(b)})$
$\displaystyle\bigoplus_{b\in N_{p-1}^{-}}\left(\sum_{a\in
N_{p}^{-}}\sum_{i=1}^{p}(-1)^{i+1}\delta_{b,a^{i}}B_{a_{i}}(m_{a})\right)$ (4)

Equalities (3) and (4) come from Prop 3.1 (i) and (ii) respectively.

Notice that our image has split neatly into three pieces. These three pieces
will become the blocks of our matrix. Before writing that down explicitly, we
notice a recursive relationship between $\Lambda$ and the pullback,
$\iota^{*}_{k-1}(\Lambda)$.

######  Proposition 3.2.

Suppose $\Lambda$ is a source-free, row-finite $k$-graph. Define the subgraphs
$\Lambda_{j}:=\iota_{j}^{*}(\Lambda)$. Call the Evans chain complexes of these
subgraphs $\mathcal{D}^{j}$ with differentials $\partial_{p}^{j}$. The
differentials of the Evans chain complex on $\Lambda_{k-1}$ naturally map
$\partial_{p}^{k-1}:\bigoplus_{N_{p,k}^{-}}\mathbb{Z}\Lambda^{0}\to\bigoplus_{N_{p-1,k}^{-}}\mathbb{Z}\Lambda^{0}$
and
$\partial_{p-1}^{k-1}:\bigoplus_{N_{p,k}^{+}}\mathbb{Z}\Lambda^{0}\to\bigoplus_{N_{p-1,k}^{+}}\mathbb{Z}\Lambda^{0}$.

###### Proof.

Since the map of Prop 3.1 (iii) is bijective, we may simply re-index (2).

$\displaystyle\bigoplus_{a\in N_{p,k-1}}m_{a}$
$\displaystyle\mapsto\bigoplus_{b\in N_{p-1,k-1}}\sum_{a\in
N_{p,k-1}}\sum_{i=1}^{p}(-1)^{i+1}\delta_{b,a^{i}}B_{a_{i}}(m_{a})$
$\displaystyle\bigoplus_{\varphi(a)\in N_{p,k}^{-}}m_{a}$
$\displaystyle\mapsto\bigoplus_{\varphi(b)\in
N_{p-1,k}^{-}}\sum_{\varphi(a)\in
N_{p,k}^{-}}\sum_{i=1}^{p}(-1)^{i+1}\delta_{b,a^{i}}B_{a_{i}}(m_{a})$

The mapping of $\bigoplus_{N_{p,k}^{+}}\mathbb{Z}\Lambda^{0}$ by
$\partial_{p-1}^{k-1}$ follows directly from $\psi$ preserving the positioning
of coordinates when mapping $N_{p,k}^{+}\to N_{p-1,k}^{-}$. ∎

Using the above proposition we may write the image of $\partial^{k}_{p}$ as

$\partial_{p-1}^{k-1}\left(\bigoplus_{a\in
N_{p-1}^{+}}m_{a}\right)+\left(\bigoplus_{b\in
N_{p-1}^{-}}(-1)^{p+1}B_{k}(m_{\psi^{-1}(b)})\right)+\partial_{p}^{k-1}\left(\bigoplus_{a\in
N_{p,k}^{-}}m_{a}\right).$

Our main theorem organizes all of this information into the well studied
perspective of block matrices.

######  Theorem 3.3.

For a source-free row finite $k$-graph, $\Lambda$. Let $\mathcal{D}^{j}$
denote the Evans chain complex (1) for $\Lambda_{j}$. The differentials of
$\mathcal{D}^{i}$ have the form:

$\partial^{j}_{p}=\begin{bmatrix}\partial^{j-1}_{p-1}&0\\\
(-1)^{p+1}B_{j}&\partial^{j-1}_{p}\end{bmatrix}$

with $\partial_{p}^{\ell}$ which are not well defined (i.e. $\ell>p$) omitted.
Here, $B_{i}$ acts coordinate-wise on vectors.

###### Proof.

The proof is simply an application of the re-indexing done in earlier
propositions combined with the definition of block matrices. ∎

Before moving on, we will look at some of the differentials for a $4$-graph as
an example to better visualize the pattern of these matrices.

$\begin{array}[]{c|ccc:c}&(2,3,4)&(1,3,4)&(1,2,4)&(1,2,3)\\\
\hline\cr(3,4)&B_{2}&B_{1}&0&0\\\ (2,4)&-B_{3}&0&B_{1}&0\\\
(1,4)&0&-B_{3}&B_{2}&0\\\ \hdashline(2,3)&B_{4}&0&0&B_{1}\\\
(1,3)&0&B_{4}&0&-B_{2}\\\ (1,2)&0&0&B_{4}&B_{3}\end{array}\hskip
24.0pt\begin{array}[]{c|c}&(1,2,3,4)\\\ \hline\cr(2,3,4)&B_{1}\\\
(1,3,4)&-B_{2}\\\ (1,2,4)&B_{3}\\\ (1,2,3)&-B_{4}\\\ \end{array}$ Figure 3.4:
The differential depicted are $\partial_{3}^{4}$ and $\partial_{4}^{4}$
respectively. Each position is made zero if there is no way to delete an
element of the column index and obtain the row index ($\delta_{b,a^{i}}$). If
an even position needs to be deleted, a negative is added ($(-1)^{i+1}$).
Finally, whatever element was deleted determines which $B_{i}$ is put in the
slot ($B_{a_{i}}$).

######  Theorem 3.5.

If $\Lambda$ is a source-free, row-finite, $k$-graph such that $B_{i}$ is
bijective for some $1\leq i\leq k$, then $K_{*}(C^{*}(\Lambda))\cong 0$.

###### Proof.

Since $\Lambda$ is source free and row finite, there exists a spectral
sequence $E,d$ with $E^{2}$ page given by the homology groups of the Evans
chain complex. Further, there exists a pullback $f^{*}(\Lambda)$ which is
isomorphic to $\Lambda$ and swaps $B_{i}$ with $B_{k}$. Thus, without loss of
generality, we suppose that $B_{k}$ is bijective.

We will demonstrate $\operatorname{im}(\partial_{p+1})=\ker(\partial_{p})$ for
$1<p<k$. The special cases of $p=1$ and $p=k$ are not particularly
illuminating, but in the interest of completeness we include them in Lemma
A.1.

Since it is already known that
$\operatorname{im}(\partial_{p+1})\subseteq\ker(\partial_{p})$, we will
demonstrate the reverse inclusion. Since $B_{k}$ is injective, it is
relatively straightforward to check that

$\ker\left(\begin{bmatrix}(-1)^{p-1}B_{k}&\partial_{p-1}^{k-1}\\\
0&I\end{bmatrix}\begin{bmatrix}\partial^{k-1}_{p-1}&0\\\
(-1)^{p+1}B_{k}&\partial_{p}^{k-1}\end{bmatrix}\right)=\ker\left(\begin{bmatrix}-\partial^{k-1}_{p-1}&0\\\
(-1)^{p+1}B_{k}&\partial_{p}^{k-1}\end{bmatrix}\right).$

Since the adjacency matrices commute [10, $\S 6$], we see that the left hand
set simplifies to

$\ker\left(\begin{bmatrix}0&0\\\
(-1)^{p+1}B_{k}&\partial_{p}^{k-1}\end{bmatrix}\right).$

This is exactly the set
$\\{(\alpha,\beta):(-1)^{p+1}B_{k}(\alpha)=-\partial_{p}^{k-1}(\beta)\\}$.
Since $B_{k}^{-1}$ exists, we may instead write
$((-1)^{p+2}B_{k}^{-1}\partial_{p}^{k-1}(\beta),\beta)$.

Lastly, we note that
$B^{-1}_{k}(\beta)\in\bigoplus_{N_{p}^{-}}(\mathbb{Z}\Lambda^{0})=\bigoplus_{N_{p+1}^{+}}(\mathbb{Z}\Lambda^{0})$.
In particular,

$\begin{bmatrix}\partial^{k-1}_{p}&0\\\
(-1)^{p+2}B_{k}&\partial_{p+1}^{k-1}\end{bmatrix}\begin{bmatrix}B^{-1}_{k}(\beta)\\\
0\end{bmatrix}=\begin{bmatrix}(-1)^{p+2}B_{k}^{-1}\partial_{p}^{k-1}(\beta)\\\
\beta\end{bmatrix}=\begin{bmatrix}\alpha\\\ \beta\end{bmatrix}.$

We conclude that $\ker(\partial_{p})=\operatorname{im}(\partial_{p+1})$. This
implies that $H_{p}(\mathcal{D})\cong 0$.

Since $0$ has no nontrivial quotients, we conclude that $E^{\infty}_{p,q}\cong
0$ for all $p,q$ in the Kasparov-Schochet spectral sequence. Then looking at
the definition convergence, we conclude that the only family $H_{*}$ such a
spectral sequence could converge to is the trivial family $H_{*}\cong 0$.
Since $E\implies K_{*}(C^{*}(\Lambda))$, we deduce that
$K_{*}(C^{*}(\Lambda))\cong 0$. ∎

Our final result concerns tensor chain complexes. We begin by defining tensor
complexes and explicitly stating their differential maps. We will then
leverage the similarities between these maps and the Evans differentials.

######  Definition 3.6.

Given two chain complexes of $R$-modules $\mathcal{A}$ and $\mathcal{C}$ we
construct the chain complex $\mathcal{A}\otimes\mathcal{C}$ as follows:

$(\mathcal{A}\otimes\mathcal{C})_{n}=\bigoplus_{i+j=n}\mathcal{A}_{i}\otimes_{R}\mathcal{C}_{j}$

The differential is then defined on elementary tensors $a_{i}\otimes
c_{j}\in\mathcal{A}_{i}\otimes\mathcal{C}_{j}$.

$\partial^{\otimes}(a_{i}\otimes
c_{j})=\partial_{i}(a_{i})\otimes_{R}c_{j}\oplus(-1)^{i}a_{i}\otimes_{R}\partial_{j}(c_{j})$

######  Theorem 3.7.

For a nontrivial monoid $k$-graph, $\lambda$ (i.e. $|\Lambda^{0}|=1$ and
$B_{i}>0$ for some $1\leq i\leq k$), let $\mathcal{D}$ denote the Evans chain
complex. If $\mathcal{C}^{j}$ denotes the chain complex
$\mathcal{C}^{j}:0\to\mathbb{Z}\overset{B_{j}}{\to}\mathbb{Z}\to 0$, then

$\mathcal{D}\cong\bigotimes_{j=1}^{k}\mathcal{C}^{j}$

###### Proof.

Since $k<\infty$, it is enough to show that for
$\Lambda_{k-1}:=\iota_{k-1}^{*}(\Lambda)$ with $\mathcal{D}^{k-1}$ the Evans
chain complex for $\Lambda_{k-1}$ we have
$\mathcal{D}^{k}\cong\mathcal{D}^{k-1}\otimes\mathcal{C}$ with
$\mathcal{C}:0\to\mathbb{Z}\overset{B_{k}}{\to}\mathbb{Z}\to 0$.

First, we investigate the modules of the complex
$\mathcal{D}^{k-1}\otimes\mathcal{C}$. Using Definition 3.6 we have

$(\mathcal{A}\otimes\mathcal{C})_{p}=\mathcal{D}^{k-1}_{p}\otimes_{\mathbb{Z}}\mathbb{Z}\oplus\mathcal{D}^{k-1}_{p-1}\otimes_{\mathbb{Z}}\mathbb{Z}\cong\mathcal{D}_{p}^{k-1}\oplus\mathcal{D}_{p-1}^{k-1}\cong\mathcal{D}_{p}^{k}.$

The first isomorphism is the natural tensor isomorphism given by
multiplication. The second is the re-indexing bijections $\varphi:N_{p,k-1}\to
N_{p,k}^{-}$ and $\psi^{-1}:N_{p-1,k}^{-}\to N_{p,k}^{+}$.

It remains to show that these isomorphisms, $\Psi_{p}$, are chain maps, that
is, they commute with the differentials. To avoid ambiguity, we will denote
the tensor differential $\partial^{\otimes}$ and the Evans differential
$\partial$. Again, we will treat the cases of $p=1$ and $p=k$ separately since
those differentials have a somewhat unique shape (see Lemma A.2).

Let $1<p<k$. The map $\partial_{p}^{k}\Psi_{p}$ acts on elements
$\alpha\otimes x\oplus\beta\otimes y$ with $\alpha\in\mathcal{D}_{p-1}^{k-1}$
and $\beta\in\mathcal{D}_{p}^{k-1}$ in the following way.

$\partial_{p}^{k}\Psi_{p}(\alpha\otimes x\oplus\beta\otimes
y)=\partial_{p}^{k}\begin{bmatrix}x\cdot\alpha\\\
y\cdot\beta\end{bmatrix}=\begin{bmatrix}\partial_{p-1}^{k-1}&0\\\
(-1)^{p+1}B_{k}&\partial_{p}^{k-1}\end{bmatrix}\begin{bmatrix}x\cdot\alpha\\\
y\cdot\beta\end{bmatrix}=\begin{bmatrix}x\cdot\partial_{p-1}^{k-1}\alpha\\\
x(-1)^{p+1}B_{k}\alpha+y\partial_{p}^{k-1}\beta\end{bmatrix}$

Before moving on to the alternative ordering, we recall that
$\partial^{\mathcal{C}}_{0}=0$ and $\partial^{\mathcal{C}}_{1}=B_{k}$. This
will be used to simplify $\partial^{\otimes}$ as we go.

$\displaystyle\Psi_{p-1}\partial_{p}^{\otimes}(\alpha\otimes
x\oplus\beta\otimes y)$
$\displaystyle=\Psi_{p-1}(\partial_{p-1}^{k-1}(\alpha)\otimes
x\oplus(-1)^{p-1}\alpha\otimes B_{k}x+\partial_{p}^{k-1}(\beta)\otimes y)$
$\displaystyle=\begin{bmatrix}x\cdot\partial_{p-1}^{k-1}(\alpha)\\\
(-1)^{p-1}B_{k}x\cdot\alpha\end{bmatrix}+\begin{bmatrix}0\\\
y\cdot\partial_{p}^{k-1}(\beta)\end{bmatrix}$

Finally, since the parity of $p-1$ and $p+1$ align we may conclude that these
maps are equal. ∎

The benefit to phrasing the Evans complex in terms of a tensor complex is that
it allows us to invoke the Künneth theorem

######  Theorem 3.8.

Künneth Theorem:  [17, cf$\sim$Theorem 3.6.3] For a chain complex,
$\mathcal{C}$, with $\mathcal{C}_{p}$ and
$\partial_{\mathcal{C}}(\mathcal{C}_{p})$ flat for all $p$,111Some authors
will substitute $\mathcal{C}$ projective to simplify the initial conditions,
but this is not equivalent and an arbitrary complex $\mathcal{A}$ there exists
a short exact sequence

$\bigoplus_{i+j=p}H_{i}(\mathcal{C})\otimes_{R}H_{j}(\mathcal{A})\to
H_{p}(\mathcal{C}\otimes_{R}\mathcal{A})\to\bigoplus_{i+j=p-1}\operatorname{Tor}_{1}^{R}(H_{i}(\mathcal{C}),H_{j}(\mathcal{A}))$

which splits unnaturally.

From this we may state the $E^{2}$ page of the Kasparov-Schochet spectral
sequence even more explicitly.

######  Theorem 3.9.

For a nontrivial monoid $k$-graph, $\lambda$, define
$g:=\gcd(B_{1},\cdots,B_{k})$ the $E^{2}$ page of the Kasparov-Schochet
spectral sequence has the form:

$E_{p,q}^{2}=\begin{cases}\mathbb{Z}_{g}^{\binom{k-1}{p}},&2|q\\\ 0,&\text{
otherwise }\end{cases}$

###### Proof.

The goal is to use Theorem 3.7 which yields
$\mathcal{D}\cong\bigotimes^{k}\mathcal{C}^{j}$. Then, the Künneth Theorem
with some classical finite group theory results to obtain the necessary
homology groups. As before, we may without loss of generality suppose that
$B_{1}>0$.

We proceed by induction on $k$. Consider the base case
$H_{p}(\mathcal{C}^{1})$. Since $B_{1}\neq 0$, we observe that
$E_{1,2q}=H_{1}(\mathcal{C}^{1})=\ker(B_{1})=0$ and
$E_{0,2q}=H_{0}(\mathcal{C}^{1})=\mathbb{Z}_{B_{1}}$ as desired. Fix some
$j\geq 1$ and define $g_{j}:=\gcd(B_{1},\cdots,B_{j})$. Then suppose:

$H_{p}(\mathcal{C}^{1}\otimes\cdots\otimes\mathcal{C}^{j})\cong(\mathbb{Z}_{g_{j}})^{\binom{j-1}{p}}.$

Consider
$(\mathcal{C}^{1}\otimes\cdots\otimes\mathcal{C}^{j})\otimes\mathcal{C}^{j+1}$.
Since each module in our complex is free which implies projective [17,
cf$\sim\S 2$], we may use the Künneth Theorem and the inductive hypothesis to
obtain

$\displaystyle
H_{p}((\mathcal{C}^{1}\otimes\cdots\otimes\mathcal{C}^{j})\otimes\mathcal{C}^{j+1})\cong$
$\displaystyle(\mathbb{Z}_{g_{j}})^{\binom{j-1}{p-1}}\otimes
H_{1}(\mathcal{C}^{j+1})\oplus(\mathbb{Z}_{g_{j}})^{\binom{j-1}{p}}\otimes
H_{0}(\mathcal{C}^{j+1})\oplus$
$\displaystyle\operatorname{Tor}_{1}^{\mathbb{Z}}\left((\mathbb{Z}_{g_{j}})^{\binom{j-1}{p-2}},H_{1}(\mathcal{C}^{j+1})\right)\oplus\operatorname{Tor}_{1}^{\mathbb{Z}}\left((\mathbb{Z}_{g_{j}})^{\binom{j-1}{p-1}},H_{0}(\mathcal{C}^{j+1})\right).$

There are two cases determined by $B_{j+1}$. First, suppose that $B_{j+1}=0$,
and thus $g_{j+1}=g_{j}$. In this case,
$H_{*}(\mathcal{C}^{j+1})\cong\mathbb{Z}$. This would make the torsion groups
$0$, and

$H_{p}((\mathcal{C}^{1}\otimes\cdots\otimes\mathcal{C}^{j})\otimes\mathcal{C}^{j+1})\cong(\mathbb{Z}_{g_{j}})^{\binom{j-1}{p}}\oplus(\mathbb{Z}_{g_{j}})^{\binom{j-1}{p-1}}\cong(\mathbb{Z}_{g_{j+1}})^{\binom{j}{p}}.$

Suppose that $B_{j+1}\neq 0$. This would mean that
$H_{1}(\mathcal{C}^{j+1})=0$ and
$H_{0}(\mathcal{C}^{j+1})=\mathbb{Z}_{B_{j+1}}$. In particular, this means
that the terms containing $H_{1}(\mathcal{C}^{j+1})$ vanish leaving

$(\mathbb{Z}_{g_{j}})^{\binom{j-1}{p}}\otimes\mathbb{Z}_{B_{j+1}}\oplus\operatorname{Tor}_{1}^{\mathbb{Z}}\left((\mathbb{Z}_{g_{j}})^{\binom{j-1}{p-1}},\mathbb{Z}_{B_{j+1}}\right).$

It is a well known fact [11, cf$\sim\S 6$] that

$\operatorname{Tor}_{1}^{\mathbb{Z}}\left((\mathbb{Z}_{g_{j}})^{\binom{j-1}{p-1}},\mathbb{Z}_{B_{j+1}}\right)\cong(\mathbb{Z}_{g_{j}})^{\binom{j-1}{p-1}}\otimes\mathbb{Z}_{B_{j+1}}\cong(\mathbb{Z}_{g_{j}}\otimes\mathbb{Z}_{B_{j+1}})^{\binom{j-1}{p-1}}\cong(\mathbb{Z}_{g_{j+1}})^{\binom{j-1}{p-1}}$

giving the desired form.

We conclude that $H_{p}(\mathcal{D})\cong\mathbb{Z}_{g}^{\binom{k-1}{p}}$
which allows us to fill in the $E^{2}$ page of the Kasparov-Schochet spectral
sequence with the desired result. ∎

######  Remark 3.10.

In the case where $\Lambda\cong\mathbb{N}^{k}$ (i.e. trivial) then
$C^{*}(\Lambda)\cong C(\mathbb{T}^{k})$ and thus
$K_{1}(C^{*}(\Lambda))\cong\mathbb{Z}^{2^{k-1}}\cong K_{0}(C^{*}(\Lambda))$.
Because the $K$-theory is known, we omitted the case from Theorem 3.9.

The above result for monoid $k$-graphs also sheds some light on Theorem 3.5.
In particular, we find that an invertible co-adjacency matrix is not necessary
for trivial $K$-theory groups.

######  Corollary 3.11.

Let $\Lambda$ be a monoid $k$-graph with $\gcd(B_{1},\cdots,B_{k})=1$. Then,
$K_{*}(C^{*}(\Lambda))=0$.

Corollary 3.11 has a very close relationship to the results of [1]. However,
it is important to note that their result required additional hypotheses on
$\Lambda$.

We conclude this article by investigating monoid $3$ graphs more generally.
Specifically, this will demonstrate that knowing $E^{\infty}$ does not always
give perfect knowledge of $K_{*}(C^{*}(\Lambda))$.

######  Corollary 3.12.

Let $\Lambda$ be a monoid $3$-graph with $|\Lambda^{e_{i}}|>1$ for some $1\leq
i\leq 3$ and define $g=\gcd(B_{1},B_{2},B_{3})$. Then
$K_{1}(C^{*}(\Lambda))=\mathbb{Z}_{g}^{2}$ and there exists a short exact
sequence

$\mathbb{Z}_{g}\to K_{0}(C^{*}(\Lambda))\to\mathbb{Z}_{g}.$

###### Proof.

Consider the map $d_{p,q}^{2}:E_{p,q}^{2}\to E_{p-2,q+1}^{2}$. Observe that
the parity of $q$ meaning that $\operatorname{im}(d_{p,q}^{2})=0$ and
$\ker(d_{p,q})=E_{p,q}^{2}$. We conclude that $E^{2}\cong E^{3}$, this is
indeed true for any $k$-graph. Theorem 3.9 gives that $E^{3}_{p,2q}\cong
E^{2}_{p,2q}\cong\mathbb{Z}_{g}^{\binom{2}{p}}$. In particular, this means
that $d^{3}_{p,q}:E_{p,q}^{3}\to E_{p-3,q}$ must be the zero map. We conclude
as before that $E^{3}\cong E^{4}$, and thus the sequence has stabilized.

First we examine the $K_{1}$ group. We look at the family
$\cdots,E_{-1,2},E_{1,0},E_{3,-2},\cdots$. By Theorem 3.9 $E_{3,-2}\cong 0$ so
all but $E_{1,0}$ are isomorphic to $0$. Moreover, there exists a filtration
of $K_{1}(C^{*}(\Lambda))$, $0\leq\cdots\leq F_{-1}\leq F_{0}\leq F_{1}\leq
F_{2}\leq\cdots\leq K_{1}(C^{*}(\Lambda))$ such that $F_{p}/F_{p-1}\cong
E_{p,1-p}$. The zeros in all positions aside from $E_{1,0}$ allows us to
conclude that $F_{1}\cong K_{1}(C^{*}(\Lambda))$ and $F_{0}\cong 0$. That is,
$K_{1}(C^{*}(\Lambda))\cong F_{1}/F_{0}\cong E_{1,0}\cong\mathbb{Z}_{g}^{2}$.

To determine $K_{0}$, we must look at the family with even total degree
($2|(p+q)$). This family is

$\cdots,0,0,\mathbb{Z}_{g},0,\mathbb{Z}_{g},0,0,\cdots$

We now consider the filtration $0\cong F_{-1}\leq F_{0}\leq F_{1}\leq
F_{2}\cong K_{0}(C^{*}(\Lambda))$. Since $2\nmid 1$, Theorem 1, ensures that
$E_{1,-1}=0$. So, $F_{1}\cong F_{0}$, allowing us to refine to the filtration
$0\leq F_{0}\leq F_{2}$. This ensures that $F_{2}/F_{0}\cong\mathbb{Z}_{g}$
providing the short exact sequence

$0\to\mathbb{Z}_{g}\to K_{0}(C^{*}(\Lambda))\to\mathbb{Z}_{g}\to 0.\qed$

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##  A Appendix

######  Lemma A.1.

Let $\Lambda$ be a $k$-graph with $B_{k}$ invertible. Then
$H_{0}(\mathcal{D})\cong H_{k}(\mathcal{D})\cong 0$.

###### Proof.

The differentials $\partial_{1}$ and $\partial_{k}$ of the Evans chain complex
always take the form

$\displaystyle\partial_{1}$
$\displaystyle=\begin{bmatrix}B_{k}&B_{k-1}&\cdots&B_{1}\end{bmatrix}$
$\displaystyle\partial_{k}$ $\displaystyle=\begin{bmatrix}B_{1}\\\ -B_{2}\\\
\vdots\\\ (-1)^{p+1}B_{k}\end{bmatrix}$

Since $H_{0}(\mathcal{D})=\operatorname{coker}(\partial_{1})$ and
$H_{k}(\mathcal{D})=\ker(\partial_{k})$ we need only demonstrate surjectivity
and injectivity respectively.

Since $B_{k}$ is onto, $\partial_{1}\begin{bmatrix}0\\\ \vdots\\\
\alpha\end{bmatrix}$ is onto $\mathbb{Z}\Lambda^{0}$. Lastly,
$\partial_{k}(\alpha)=\partial_{k}(\beta)$ implies
$B_{k}(\alpha)=B_{k}(\beta)$ and thus by injectivity of $B_{k}$,
$\alpha=\beta$. ∎

######  Lemma A.2.

Under the hypotheses of Theorem 3.7, we have equalities
$\partial^{k}_{k}\Psi_{k}=\Psi_{k-1}\partial^{\otimes}$ and
$\partial^{k}_{1}\Psi_{1}=\Psi_{0}\partial^{\otimes}$.

###### Proof.

Consider $\partial^{k}_{k}\Psi_{k}$ and notice that $N_{k,k}^{-}=\emptyset$.
Since
$(\mathcal{D}^{k-1}\otimes\mathcal{C})_{k}=\mathcal{D}^{k-1}_{k-1}\otimes\mathcal{C}_{1}$,
we let $\alpha\otimes x\in\mathcal{D}^{k-1}_{k-1}\otimes\mathcal{C}_{1}$.

$\displaystyle\partial^{k}_{k}\Psi_{k}(\alpha\otimes x)$
$\displaystyle=\begin{bmatrix}\partial_{k-1}^{k-1}\\\
(-1)^{k+1}B_{k}\end{bmatrix}[x\cdot\alpha]=\begin{bmatrix}x\partial_{k-1}^{k-1}\alpha\\\
x(-1)^{k+1}B_{k}\alpha\end{bmatrix}$
$\displaystyle\Psi_{k-1}\partial^{\otimes}(\alpha\otimes x)$
$\displaystyle=\Psi_{k-1}(\partial^{k-1}_{k-1}(\alpha)\otimes
x\oplus(-1)^{k-1}B_{1}x)=\begin{bmatrix}x\partial_{k-1}^{k-1}\alpha\\\
x(-1)^{k-1}B_{k}\alpha\end{bmatrix}$

For the $p=1$ case we return to general elements $\alpha\otimes
x\oplus\beta\otimes
y\in\mathcal{D}_{0}^{k-1}\otimes\mathbb{Z}\oplus\mathcal{D}_{1}^{k-1}\otimes\mathbb{Z}$.
Additionally, we will utilize that $\partial^{k-1}_{0}=0$.

$\displaystyle\partial^{k}_{1}\Psi_{1}(\alpha\otimes x\oplus\beta\otimes y)$
$\displaystyle=\begin{bmatrix}(-1)^{k+1}B_{k}&\partial_{1}^{k-1}\end{bmatrix}\begin{bmatrix}x\cdot\alpha\\\
y\cdot\beta\end{bmatrix}=\begin{bmatrix}xB_{k}\alpha+y\partial_{1}^{k-1}\beta\end{bmatrix}$
$\displaystyle\Psi_{0}\partial^{\otimes}(\alpha\otimes x\oplus\beta\otimes y)$
$\displaystyle=\Psi_{0}((-1)^{0}\otimes B_{k}x+\partial_{1}^{k-1}\beta\otimes
y)=\begin{bmatrix}xB_{k}\alpha+y\partial_{1}^{k-1}\beta\end{bmatrix}$

∎