A Game-Theoretic Model for Analysis and Design of Self-Organization Mechanisms in IoT

0.49 MB·

by  Vahid Behzadan

A Game-Theoretic Model for Analysis and
Design of Self-Organization Mechanisms in IoT
Vahid Behzadan and Banafsheh Rekabdar
Department of Computer Science and Engineering
University of Nevada, Reno
7 1664 N Virginia St, Reno,NV 89557
1 [email protected],[email protected]
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Abstract. WeproposeaframeworkbasedonNetworkFormationGame
n
for self-organization in the Internet of Things (IoT), in which heteroge-
a
J neous and multi-interface nodes are modeled as self-interested agents
7 whoindividuallydecideonestablishmentandseveranceoflinkstoother
1 agents. Through analysis of the static game, we formally confirm the
emergenceofrealistictopologiesfromourmodel,andanalyticallyestab-

] lish the criteria that lead to stable multi-hop network structures.
T
G Key words: Internet of Things, Topology Control, Self-Organization,
. Game Theory
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c
[

1 Introduction and Motivation
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v
2 Throughthepastdecade,thenumberofinternet-enableddeviceshasbeengrow-
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ingatanunprecedentedrate.TheparadigmofInternetofThings(IoT)envisions
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an even more immersive and pervasive exploitation of internet connectivity by
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0 enabling more objects and devices to connect. Emerging applications of this
. move towards ubiquitous connectivity are wide and vast [1], ranging from do-
1
mestic monitoring and smart home solutions to healthcare solutions [2], smart
0
7 grids[3], and disaster monitoring [4]. It hence follows that instances of IoT will
1 be comprised of a great number of various devices, each with unique require-
v: ments and capabilities, leading to heterogeneity both in terms of function and
i communications.
X
The inevitably high degree of heterogeneity and scalability of IoT, dim the
r
odds of feasibility and scalability for centralized control approaches [5]. An al-
a
ternative to centralized architectures for IoT are those that rely on autonomic
management of connectivity and resources through self-configuration [6]. Such
solutions modelthe networkasa systemcomprisedofindividualagents,eachof
whichaimstoretainconnectivitywiththenetworkwhileoptimizingtheirobjec-
tives,suchasenergyconsumptionandthroughput.Eventhoughthismulti-agent
abstractionpresentsa promisingapproachtowardsscalability,the decentralized
nature of self-configuring IoT gives rise to many critical challenges in mecha-
nism design. Of the most critical of these challenges is the problem of topology
2 VahidBehzadan et al.
Fig. 1: Applications of IoT
control, which is further complicated by the heterogeneity of IoT devices. Ow-
ing to the similarity of distributed IoT and Ad Hoc networks, the literature on
self-organizationandtopologicalanalysisofIoTaremainlyfocusedonadopting
techniquesthatareoriginallydevelopedforgenericdistributednetworkssuchas
WirelessSensorNetworks(WSNs)[6].Yet,uniquefeatures,suchastheimmense
diversityincapabilitiesandrequirementsinallaspectsofIoTpresentmajordis-
tinguishingfactorsthatnecessitatethedevelopmentoftechniquesspecifictothe
challenges of this emerging technology.
Themulti-agentmodelofIoTis comprisedofopportunistic devicesthataim
to maximize their success infulfilling their individual objectives,suchas preser-
vation of connectivity to the network, minimization of energy consumption and
maintenance ofaminimum Quality ofService(QoS).The inherentlimitationof
resourcesavailableto suchopportunisticagentsinanyreal-worlddeploymentof
IoT gives rise to a competitive environment, which motivates a game theoretic
investigationofinteractionsinself-organizingIoT.The applicationofgamethe-
orytodistributedtopologycontrolandself-organizationhasbeenanactivearea
of research in recent years. Some of the notable literature in this area include
theworkofEidenbenz,et.al.[7]ontheanalysisofequilibriaintopologycontrol
games, Nahir, et. al.’s detailed investigationof applying game theory to various
problems of topology control [8], and Saad et. al. proposal’s of a game theo-
retic algorithm for cooperative relaying in [9], based on their earlier analysis of
the formation of hierarchical topologies in multi-hop networks [10]. The models
presented in these and many other topology control games, one critical limita-
tion is the assumption on homogeneity of the network. Recently, Meirom et. al.
proposed a model of topology control games for heterogeneous AS-Level net-
Game-Theoretic Model of Self-Organization in IoT 3
works[11] [12], which considerssome degreeof heterogeneity,but only accounts
homogeneous link costs.
Based on the inevitable emphasis on the connectivity aspects of IoT net-
works, this paper builds on the aforementioned models to provide a framework
for analysis and design of distributed topology control mechanisms in IoT. The
proposedframeworkisbasedonmodelingofself-organizationasaNetworkFor-
mationGame[13],inwhichtheactionsofplayersareestablishmentorseverance
of links with other nodes in the IoT. Contrary to previous models, we consider
heterogeneity in both communications and link cost. The proposed model also
accounts for nodes equipped with multiple communication interfaces, thus sup-
porting modern devices such as smart phones. We provide an analytical deriva-
tion of the criteria required for formation of a clique topology between nodes
that are directly connected to the internet, and further develop this analysis to
present the necessary criteria which lead to formation of hierarchical and star
topologies between internet-connected nodes and the rest of the network.
The remainder of this paper is organized as follows: Section 2 details the
model of IoT networks, followed by the formulation of network formation game
in Section 3. Emergence of stable IoT topologies and their criteria is discussed
in Section4,andthe results of a numericalsimulationis presentedin 5. Finally,
Section 6 concludes the paper with remarks on future areas of work.
2 IoT Network Model
The genericdefinition of IoT has givenrise to numerousmodels for the network
structure and architecture [5]. In this work, IoT is considered to be a network
formed with the objective of enabling direct or relayed connectivity of hetero-
geneous nodes to the internet (or other backbone networks). Heterogeneity of
nodesentailsdiversehardwareandsoftwareparametersthroughoutthenetwork,
such as the number and type of communication interfaces (e.g. WLAN, LTE,
Ethernet, etc.), energy constraints, and bandwidth requirements.
Accordingly,wemodelthe IoTasa networkG(P)ofN nodes P ={P |∀i∈
i
{1,2,...,N}}, each with an arbitrary number of single channel radio interfaces.
Thisdefinitionmaybeseamlesslyextendedtocovermulti-channelradiosaswell,
via representing each as a group of single-channel radios. It is assumed that all
interfaces of a node can be active simultaneously, but as detailed in Section 3,
the effects of activating each additional interface on undesired aspects such as
co- and cross-interference,channel congestion, and energy consumption may be
suitably captured in the system cost function. The presented model also allows
that some, or all of the interfaces in nodes may remain idle throughout the
analyzed operation.
As the focus of this study is on topological properties, it is assumed that
nodes are static relative to each other. Also, we consider the case that every
node in the network is aware of its distance in terms of number of intermediate
hops with every other node in the network. This can be justified by reliance on
routing tables obtained from proactive network layer protocols such as OLSR
4 VahidBehzadan et al.
[14].Theextentofanode’sknowledgeoftheoverallnetworktopologyisassumed
to be limited to its directly connected neighbors.
Nodes are classified in two categories: Those with direct connectivity to the
internet, such as WiFi Access Points and 3G/LTE Enabled Devices, and those
which need to be connected to the internet via the nodes in the former group,
such as Bluetooth/Zigbee sensors.Let the set of Internet Connected (IC) nodes
G ∈ P denote the set of nodes with direct connection to the internet, and
I
the set of non-ICs G ∈ P\G is the set of nodes that do not have a direct
S I
connection to the internet. The emerging network is thus hierarchical with at
least two tiers: a higher tier formed by IC nodes, and a lower tier comprised
of non-IC nodes who aim to connect to the higher tier. Hence, an important
objective of IoT network controllers, whether centralized or distributed, is to
enable the connection to the internet to the non-IC node, via linking them to
oneormoreICnodes.Inline withpracticalnetworkprotocols,afurtherlimitis
imposed to the maximum number of hops that may exist between each pair of
nodes, denoted by h . The following section provides the details of one such
Max
controllerbased on a game theoretical frameworkknownas Network Formation
Games.
3 Game Formulation
Formation of macro-scaletopologies in distributed networks is the collective re-
sult of the individual decisions made by each nodes on which set of nodes to
connect with, and which links to severe. With the assumption that every such
nodeaimsatgainingmoreutilityfromitsdecisionsandconsequentactions,this
interaction of multiple decision makers can be formulated as a Network Forma-
tion Game [13]. Such games are comprised of competing agents who controlthe
set ofnodes they areconnected to,with the commonobjective offorming coali-
tions of nodes that is most profitable for the deciding agent. It is evident that
the game being consideredin this workis ofthe non-cooperativetype, since the
decisions are made independently. Another assumption adopted in our proposal
is that a link between two nodes is established if, and only if, both nodes con-
sentto its establishment. This assumptionemulates the real-worldphenomenon
that occurs in cost-optimizing distributed networks. A simple, yet realistic ex-
ample is depicted in figure (2. This figure illustrates a network formation game
in which the objective of all players is to minimize their cost while maintaining
their reachability from any other player by at most one intermediate hop - a
property that we shall label as one-hop-reachable. The cost incurred to each
player of this game is the cost of establishing their immediate links (denoted by
edgeweightsinfigure(2),whichisassumedtobethesameforbothofthelinked
nodes.Iftwonodesarenotone-hop-reachable,theircostissettobeinfinity.For
instance,thecostincurredtonodeB isthecostofestablishingthelinkBC plus
the cost of establishing BD, i.e. 2+4 = 6. As is shown in the figure, for node
C to be one-hop-reachableto node A,the minimum costis obtainedby relaying
through node D. Yet for node D, establishment of a link to node C does not
Game-Theoretic Model of Self-Organization in IoT 5
B
C
A
4
D
Fig. 2: Example of the mutual consent in Myerson games
bring any utility but losses, as node D has already established a cheaper path
to C via node B, and is directly connected to node A. Hence, node D will not
consent to spending its limited bandwidth and energy to relay a transmission
that gains him no benefits. Consequently, nodes A and C settle on establishing
an expensive direct link to avoid the infinite cost of unreachability.
Networkformationgamesthatarebasedonconsensualestablishmentoflinks
areknownasbilaterallinking orMyersongames[15],whichis a widely adopted
model in game theoretic distributed topology control, mainly due to its agree-
ment with the opportunistic behavior of agents in decentralized networks. Our
proposedframeworkbuildsatopofthepreviousworkonbilaterallinkformation
byextendingtheapplicationoftheMyersonmodeltoconsiderationsbeyondthat
of minimizing energy consumption as the sole objective of the game, replacing
the abstracted link establishment parameters with those of real wireless inter-
facecharacteristicsandpropagationmodel,andfilling the gapinself-organizing
IoTs by providing a novel cross-layer framework for analytical design and eval-
uation of protocolsand parameters involvedin the distributed formationof IoT
topologies.
Eventhoughtherealphenomenonofnetworkformationinadhoccommuni-
cations networks is of a dynamic nature, this work concentrates on the analysis
ofastaticbilaterallinkinggame,withtheaimofgaininginsightsonthecharac-
6 VahidBehzadan et al.
teristics ofemergingstable topologies,alongwith the criteriathatleadsto their
emergence. Similar to every other game, our proposed Myerson game is formed
of players, set of strategies, and a payoff/cost structure, the details of each are
presented in this section.
3.1 Players
LetP ={p ,p ,...p }denoteagroupofN agents.Eachagentp ischaracterized
1 2 N i
by the following features:
– Ordered set of its radio interfaces R , where |R | is the number of interfaces
i i
and R (r ∈ {1,2,...,|R |}) ∈ {0,1} is a binary value, indicating whether the
i i
interface is currently being used or not.
– Frequency of operation for each radio interface f
i,r
– Maximum bandwidth for each radio interface b
i,r
– Minimum required bandwidth bMax
i
– Maximum transmit power for each radio interface τ
i,r
– Receiver sensitivity for each radio interface §
i,r
– Maximum antenna gain for each radio interface x
i,r
– 2-D Position γ =(x ,y )
i i i
– Feature tuple for each interface w =(f ,b ,bMax,τ ,S ,x ,γ )
i,r i,r i,r i i,r i,r i,r i
Define the network topology G = {g : i,k ∈ P,i 6= j}. If a bidirectional
ij
link is established between p and p , then g = (r ,r ), where r ∈ R is the
i j ij i j i i
interface chosenby the node i to communicate with the correspondinginterface
in node j, i.e. r ∈R . If there is no directlink between i and j, g =(−1,−1).
j j ij
3.2 Strategies
Let C denote the cost function for every node p . Any node p ∈ P may form
i i i
a link g =(r ,r ) with any Node p in the neighborhood M(i), defined as the
ij i j j
set of all nodes that fall within the maximum communications range of i, if:
1. Nodesmusthaveatleastonetypeofradiointerfaceavailableandincommon,
i.e. :
2. ∆C(p ,G+(r ,r ))<0
i i j
3. ∆C(p ,G+(r ,r ))<0
j i j
Where∆C(p ,G+(r ,r ))=C(p ,G∪{(r ,r )})−C(p ,G)isthedifference
K k l k k l k
between the total cost to node p by establishing the link (r ,r) and the total
k k l
cost to p without the establishment of this link.
k
Agentp mayremovealink with agentp in M′ ⊂Rc{R (k)6=(−1,−1)}if:
i j k i
∆C(p ,G−(r ,r ))<0
i i j
Where∆C(p ,G−(r ,r ))=C(p ,G\{(r ,r )})−C(p ,G)isthedifference
k k l k k l k
between the cost incurred by node p removing the link (r ,r ) and the cost
k k l
incurred by maintaining this link.
Game-Theoretic Model of Self-Organization in IoT 7
3.3 Payoff Structure
For each node p , the payoff of forming a direct link is dependent on the set of
i
objectives listed below:
1. Minimize the total cost of link establishment deg(pi)L′(z)
Pz=1 i
2. Minimize the hop distance to all nodes in the network, with priority over
minimizing distance to the nodes directly connected to the internet.
3. Minimize energy consumption by avoiding excessive relay transmissions
The corresponding cost function for each node is thus formulated as:
deg(pi)
C(i,G)=Ci = X L′i(z)+Γ X h(i,j)
z=1 j∈GI
+ X h(i,k)+Bi (1)
k∈GS
WhereL′(z)isthecostofestablishingthez-thlinkofp ,withdeg(p )denot-
i i i
ing the number of links established by p . Let L (z) be the link between nodes i
i i
and z. To model the link cost, the following factors are considered:
– L (z) is directly proportional to the minimum transmission power required
i
for z to receive the signal. The transmission power depends on the fading
model and noise on the channel, which generally is inversely proportional to
the Euclidean distance between nodes, their antenna gains, and the receiver
sensitivity.Everyinterfacehasamaximumbudgetedtransmitpower,beyond
which L (z)=∞
i
– L−i(z) is directly proportional to the number of connections established on
interfaceR (r).Themorethisnumberis,themorecongestionisexpectedand
i
hence the throughput suffers.
– L (z) is inversely proportional to bandwidth. Higher the bandwidth, higher
i
the throughput will be.
Hence, a generic formulation for L (z) is constructed as:
i
deg(pi) |Ri| σ
X L′i(z)=XRi(r) X α.ρi.βir (2)
ir
z=1 r=1 z∈Ps.t.giz=(Ri(r),o)
Whereαisaconstantfactoringtheeffectofeachadditionallinkoninterface
R (r), ρ istherelativeimportanceofpreservingenergytoachievingthe desired
i i
throughput, σ is the power transmitted by p on this link, and β is the ratio
ir i ir
of the available bandwidth to the required bandwidth, i.e.:
b
ir
β = (3)
ir bMin
i
8 VahidBehzadan et al.
ThefactorΓ ≥1istheweightingfactorfortuningtheemphasizeonminimiz-
ing the shortesthop-distance h(i,j) to everyIC node j ∈G . B is the bridging
I i
coefficient of node p , estimating the local burden of bridging communities and
i
thus modeling the relative amount of relay transmissions that p may have to
i
handle for its neighbors. It is shown in [16] and [17] that the higher values of
bridging coefficient represent a higher risk of congestion, as well as collisions.
Bridging coefficient is calculated as:
1
deg(i
B = (4)
i 1
Pj∈{k:gik6=(−1,−1)} deg(j)
4 Equilibrium Topologies in Static game
This section investigates the criteria which enable the emergence of stable and
efficient topologies from the proposed network formation mechanism. Having a
gametheoreticabstractionoftheproblem,westudythecharacteristicsofstable
networksbyanalyzingtheequilibriaofourmodel.Oneofthemostintuitivetypes
of equilibrium is the Nash equilibrium, defined as strategy profiles at which no
player can increase its profit by unilaterally deviating from that profile, hinting
at a stable outcome. Yet, Nash equilibrium is shown to be a weak notion for
stabilityinnetworkformationgames[18].Consideringthebilateralnatureoflink
formation in such games, stability of outcomes is characterizedmore accurately
by considering bilateral deviations. To satisfy this requirement, we consider the
notion of pairwise stability [18]. A strategy profile is said to be pairwise stable
if no unilateral or bilateral deviations could increase the utility of the players.
Formally, a topology G is pairwise stable if the following conditions are met:
1. ∀i,ij ∈G,C(i,G)≤C(i,G−ij)
2. ∀i,j ∈/ G,ifC(i,G+ij)<C(i,G)thenC(j,G+ij)>C(j,G)
Inthefollowingsubsections,weutilizepairwisestabilityintheformalanalysis
of stable topologies that can emerge from the proposed model.
4.1 Formation of Cliques
Anotablenumberofrecentliteratureonbilaterallinkformationgamesarebased
onmodelsthatresultinsystematicallimitationofpairwisestabilitytoforestand
tree topologies (e.g.[19], [20]) This property greatly neuters the applicability of
such models to IoT. As discussed in Section 2, nodes in IoT are categorized as
either Internet-Connected (IC) or non-IC. It is intuitive to assume that each
IC node is directly connected to every other IC nodes through the internet
connection, thereby the set of all IC nodes inherently forms a clique. Therefore,
ifthecostoflinkestablishmentisboundedbyacriticalvalue,itisexpectedthat
the clique remains stable.In the following theorem,we provethatunder certain
criteria, this topology is indeed pairwise stable.
Game-Theoretic Model of Self-Organization in IoT 9
Theorem 1 Let L (k) be the maximum cost for any internet-connected node
i
p ∈G to establish a link with node p ∈G . If L (k)<Γ −1, then the nodes
i I k I i
in G form a clique.
I
Proof. Assume a node p that is yet to establish connections to any node in G.
i
For any node p ∈G , the cost difference of establishing a link is given by:
k I
C(p ,G+g )=C(p ,G∪{r ,r )})−C(p ,G)
i ik i i k k
=L (k)+Γ(−1)+0+∆B (5)
i i
Where
1
deg(i)+1
∆B =
i 1 +1
Pj∈{∀ζ|giζ6=(−1,1)} deg(j)
1
deg(i)
− (6)
1
Pj∈{∀ζ|giζ6=(−1,1)} deg(j)
Consideringtheminimumandmaximumvaluesofdeg(i)and 1 },
Pj∈{∀ζ|giζ6=(−1,1)} deg(j)
it is trivial to show that:
0<∆B <1
i
Hence, the maximum valid value of the cost difference is given by:
∆C(p ,G+g )=L (k)−Γ +1 (7)
i ik i
For this cost difference to be feasible for all nodes in G , the following con-
I
dition must be satisfied:
∆C(p ,G+g )<0
i ik
⇒L (k)−Γ +1<0
i
⇒L (k)<Γ −1 (8)
i
Ifthisconditionholdstrue,establishmentofalinkbetweenanypairofnodes
in G decreases the cost for both nodes, hence leading to a clique topology.
I
Inversely, severing any link in the resulting clique by any node i ∈ G would
I
impose a higher costto i than gain.Therefore,this criteria leads to cliques that
are pairwise-stable.
4.2 Formation of Stars and Hierarchies
Having established the criteria for the proposed model to result in a realistic
stable topology for IC nodes, we study the topologies that emerge under this
criteria for non-IC nodes. First, we derive the conditions that result in every
non-IC node being linked to at most one of the IC nodes. Then, we derive the
necessaryconditionsforformationofstarclustersbetweennon-ICnodesandIC
nodes.
10 VahidBehzadan et al.
Theorem 2 IfL (k)<Γ−1,themaximumnumberoflinksbetweenanynon-IC
i
node j ∈G and the set of Internet-connected nodes G is 1.
S I
Proof. Assuming there already exists a link between i ∈ G and j ∈ G , the
I S
maximum cost difference of establishing a second link from another node i′ ∈
G \{i} to j is:
I
∆C(i′,G+gi′j)=Li′(j)−Γ +0+1 (9)
For this link to be feasible for i′, the following condition must be met:
∆C(i′,G+gi′j)<0
⇒Li′(j)<Γ −1 (10)
Therefore, if the minimum cost of connection to a node j ∈ G satisfies
S
Li′(j)>Γ −1, every non-IC node is connected to at most one IC node.
In the following theorem, we derive the conditions under which every non-
IC node is directly connected to an IC node, thus forming star-shaped clusters
whose centers are IC nodes.
Theorem 3 Let Li(k)<Γ−1 and Li′(j)>Γ−1, the maximum degree of any
non-IC node j ∈G is 1 iff ∀j′ ∈G \j,L (j′)> 1.
S S j 2
Proof. Theorem 2 proves that under the aforementioned conditions, the maxi-
mum number of links between any non-IC node and all IC nodes is 1. Assume
thatj establishesis asecondlinkto anode j′ ∈G .The costdifferenceis given
S
by:
1
∆C(j,G+gjj′)=Lj(j′)+0−1+
2
(11)
Forthisactiontobeinfeasible,thecostdifferencemustbepositive.Therefore:
1
L (j′)+0−1+ >0
j
2
1
⇒L (j′)> (12)
j
2
As a corollary of Theorem 3, it is worth noting that if G is connected and
the conditions of theorems 1 and 2 are satisfied, but condition of theorem 3 is
not,thenthe resultingtopologycontainsnodesthat haveone link to the IC set,
butareconnectedtooneormorenon-ICnodes.Suchnodesactasgatewaysand
relays for other non-IC nodes connected to them, and the emerging topologies
havemorethanthe original2 levelsofhierarchy,namelyIC andnon-IC.Conse-
quently, this model allows for resource planning by determination of nodes that
are bound to become relays, and therefore require higher communications and
processing capabilities.