# Modeling and Performance Studies of Data Communication Networks using
Dynamic Complex Networks

Suchi Kumari  Department of Computer Science and Engineering,
National Institute of Technology,
Delhi, India
email<EMAIL_ADDRESS>Anurag Singh  Department of Computer Science
and Engineering,
National Institute of Technology,
Delhi, India
email: Corresponding author<EMAIL_ADDRESS>
###### Abstract

All the existing real world networks are evolving, hence, study of traffic
dynamics in these enlarged networks is a challenging task. The critical issue
is to optimize the network structure to improve network capacity and avoid
traffic congestion. We are interested in taking user’s routes such that it is
least congested with optimal network capacity. Network capacity may be
improved either by optimizing network topology or enhancing in routing
approach. In this context, we propose and design a model of the time varying
data communication networks (TVCN) based on the dynamics of in-flowing links.
Newly appeared node prefers to attach with most influential node present in
the network. In this paper, influence is termed as reputation and is applied
for computing overall congestion at any node. User path with least betweenness
centrality and most reputation is preferred for routing. Kelly’s optimization
formulation for a rate allocation problem is used for obtaining optimal rates
of distinct users at different time instants and it is found that the user’s
path with lowest betweenness centrality and highest reputation will always
give maximum rate at stable point.

###### keywords:

Data Communication Networks, Time varying Networks model, System Utility,
Routing

##  1 Introduction

Many real world networks, such as data communication network, World Wide Web,
transportation network etc., follow heavy tailed power law degree distribution
results in scale free behavior [1]. These networks are evolving, hence,
dynamics of the networks have been studied in past few years. In communication
networks, data packets are transported from some source ($S$) to destination
($D$) via a path. The critical issue concerned with the communication networks
is to optimize the network structure to improve searching or routing
efficiency and avoid traffic congestion. The next issue is related to
efficient usage of the capacity of the networks. It is observed that few nodes
become congested and then, later it results in propagation of congestion in
the whole network. It happens because these nodes are used most in the
communication path as they are more central. Network capacity may be improved
using two methods: (i) Methods to optimize network topology, and (ii)
Enhancement in routing approach.

###  1.1 Optimization of network topology

Optimization of network topology corresponds to reorganization of the network
for minimizing congestion and getting improved network capacity. Zhao et al.
[2] have enhanced network capacity by redistributing the load of heavily
loaded nodes to other nodes in the network. Links are sorted in descending
order according to their end node degrees and links are removed in the
descending sorted list [3]. For heterogeneous networks, Zhang proposed a
method in which nodes having high betweenness centrality are removed first
[4]. Networks are growing during the whole of the life span and nodes are
connected via preferential attachment (BA model) [5]. In BA model, appearance
of new connections totally depends on the addition of new nodes to the system.
But, in real systems, links are added and updated continuously. Tadic [1] has
developed a model for dynamics of directed networks. Degree distribution of
both the outgoing and incoming links follow power law degree distribution and
there is a correlation between them. It suggests that local structure of the
network is different in comparison with the case of without updates. There is
a need to include time parameter with the nodes and links of the basic network
structure so, many models have been proposed for these time-varying graphs
(TVGs) [6, 7, 8, 9]. Model developed by Kostakos et al. [10] is based on the
concept of quick transfer and delayed transaction of data between distinct
nodes in the communication network. Kim et al. [8] focus their work on
discovering closest and most betweenness central node in the dynamically
changing networks. A series of static graphs (i.e., the snapshots) are used to
represent the network at a given time instant [11]. The snapshot model makes
an external assumption that the network must be capable of retaining the link
transitivity during the period of disconnection. Recently, Wehmuth et al. [6]
have proposed a new unifying model for representing finite discrete TVGs. TVG
nodes are considered as independent entities at each time and it is analyzed
the proposed TVG is isomorphic to a static graph. It is an important result
because of the use of the isomorphic directed graph as a tool to analyze both
the properties of a TVG and the behavior of dynamic processes over a TVG. A
framework to represent mobile networks dynamically in a Spatio-temporal
fashion is designed and algebraic structural representation is also provided
[12].

###  1.2 Enhancement in routing approach

Network restructuring cost is more, hence most of the researchers focus on
finding better routing strategies. Traditionally, packets are sent along the
shortest paths in the term of hop count or minimal sums of weighted links
[13]. Most of the shortest paths are passing through most central nodes and
hence, sending data only through shortest path is not efficient. Yan et al.
[14] have focused their work on searching efficient path, instead of searching
of shortest paths. In communication networks, two rates are associated with
each node: packet generation rate $\lambda$ and packet forwarding rate $\mu$.
There is a critical rate of packet generation $(\lambda_{c})$, below which
network will be free from congestion. Two models are proposed by Zhao et al.
[2] for finding packet delivery rate of a node, (i) based on node’s links and
(ii) number of the shortest path passing through the node. In later model,
critical rate is independent of network’s size and topology, unlike, previous
model. Du et al. [15] have developed a model in which packets are generated
with a rate $\lambda$ between random source and destination pairs. Packets are
propagated into the network such that load is minimized and will be delivered
by Shortest Remaining Packet First (SRPF) algorithm. They also studied the
behavior of congested node with the arrival of large number of packets. A
heuristic routing strategies are used by Jiang et al. [16], in which routing
process is divided into $N$ (size of network) steps. All the routes from the
source node are based on the betweenness centrality and node’s degree
information. Pu et al. [17] have formed a network model based on Barabasi
Albert (BA) model and routing between pair of nodes are done such that cost
function is minimized. Robustness and network efficiency are calculated
against cascading failure. Networks are changing rapidly with time, hence,
acquiring global information of the network using global routing strategy is
not useful for longer time. Various local routing strategies are proposed by
Lin et al. [18]. Proposed methods include node duplication avoidance,
searching next nearest neighbors and restricted queue length. All the proposed
strategies are used to enhance the performance of the system, by increasing
packet generation rate of the nodes and reducing packet transmission time.

###  1.3 Rate control mechanism

Some nodes as well as links are getting added or removed but there is a net
increment in the size of the networks with time. As network is expanding day
by day so design and control of such network is a challenging work. Structure
of networks is dependent on distribution of links and directly affects the
accessibility of particular node. Topology is changing and due to change in
topology, the performance of the network is also affected. Communication
network can respond to the randomly changing traffic but modern communication
networks face multiple challenges at different layers. Real life communication
networks are extremely volatile and making connections in the volatile
environment with optimal network utility is a challenging task. Modeling their
rate control behavior [19] with volatile and dynamic connectivity helps us in
getting optimal utilization of the system. La et al. [20] have extended
Kelly’s work by introducing a suitable pricing scheme. An algorithm is
proposed for adjusting network’s price and users’ window size is re-framed to
achieve optimal system utility by maintaining weighted proportionally
fairness. Priya Ranjan et al. [21] have investigated Kelly’s optimization
framework and used for finding stability conditions with the consideration of
arbitrary delays. Effect of the non responsive traffic on the system stability
is studied. It is shown that ubiety of such elements help in improving the
stability of the system. Recently, progress is made on the earlier work of
modeling of TVCN and a new Lyapunov function is being formulated for rate
control scheme and long term stability is obtained [22].

In previous researches, various routing strategies are proposed to avoid
congestion in the communication networks. Here, work is focused on congestion
and network utility both and the routing of the users are established with
minimum congestion and maximum network utility along them. Network utility is
calculated by summing up all individual user’s utilities, and it will be
maximum when, each user send data to their destinations with optimal rate.
Once network is designed then, users select their routes for data
communication with their destined nodes. In this paper, network is designed
with the consideration of expansion, rewiring and removal of the links. A
model is proposed for removing infrequently used links in the network apart
from addition and rewiring of links. Addition and rewiring of links are based
on preferential attachment. The removed links increase unnecessarily
maintenance cost, hence, the removal of these links will reduce the cost. In
the proposed model, the links attached to a node having lower degree, will be
preferred to remove. Optimality of the routes are checked by analyzing Kelly’s
optimization approach [19] for a rate allocation problem in communication
networks at different time instants.

Centrality plays an important role for congestion in the network, especially
betweenness centrality. Betweenness centrality of a node $v$ is equal to the
number of shortest paths from all node pairs pass through that node $v$ and is
given by,

$g(v)=\sum_{S\neq v\neq D}\frac{\sigma_{S\rightarrow
D}(v)}{\sigma_{S\rightarrow D}}.$

If larger number of shortest paths are passing through a node then,
betweenness centrality and degree of that node will be more but, vice-versa is
not true [23]. Connectivity distribution of both betweenness and degree
centrality of the real world networks follow power law distribution with
different but, correlated exponents [24]. Nodes with maximum betweenness
centrality is the most congested node in the network [2]. In the proposed
routing strategy, shortest paths with lowest total sum of betweenness
centrality of nodes are selected for sending data packets.

Some nodes are endorsed by the congested nodes and cause increase in load at
that node. Hence, these nodes may be avoided for reducing data traffic in the
network. Eigen vector centrality is used to check influence of one node to
others. In this paper, influence is measured in terms of congestion in the
networks. A node with high Eigen vector centrality is highly congested and it
is undesirable to connect with that node. We have used a term, “reputation”,
for finding overall congestion at any node. Reputation of a node is inversely
related to Eigen vector centrality of the particular node, hence, most Eigen
vector central node is least reputed. Both, betweenness centrality and
reputation are collectively used for avoiding congestion in the network. Among
all the shortest paths, the path having lowest betweenness central and most
reputed are desirable for data communication between user pairs. These
desirable paths are taken as an input for Kelly’s optimization framework and
are the achievers of optimal data rates $x^{*}$.

Section 2 states background information about the related work and describe
the mathematical model used in the analysis of rate control behavior of the
user route. Section 3 introduces model for growth dynamics of TVCN,
formulation of scaling exponent and the proposed routing strategies for
communication network. Section 4 presents theoretical and simulation results,
and in Section 5, conclusion and future research are discussed.

##  2 Background and Related work

In communication network, each node can generate and forward packets with
distinct rate, $\lambda$ and $\mu$ respectively. There are three possible
relationships between $\lambda$ and $\mu$: (i.) $\lambda<\mu$ implies that
system in free flow state, (ii.) $\lambda=\mu$ shows boundary case for
congestion and (iii.) $\lambda>\mu$ allows system in congestion. Zhao et al.
[2] have proposed a model in which packet forwarding rate $\mu$ of a node $i$
for model: (i) based on degree of a node, $\mu_{i}=1+\lfloor\beta
k_{i}\rfloor$ and (ii) based on betweenness centrality of a node,
$\mu_{i}=1+\lfloor\frac{\beta g_{i}}{N}\rfloor$. Here, $k_{i}$ = degree of
node $i$, $g_{i}$ = betweenness centrality of node $i$, $N=$ size of the
network and $\beta$, $0<\beta<1$. By theoretical estimation, the value of
critical rate is given by $\lambda_{c}=\frac{\mu_{Lmax}(N-1)}{g_{Lmax}}$. The
node with maximum packet forwarding capacity and betweenness centrality are
denoted by $\mu_{Lmax}$ and $g_{Lmax}$ respectively.
An order parameter $\zeta(\lambda)$ [25], is used to describe the traffic and
given by,

$\zeta(\lambda)=lim_{t\rightarrow\infty}\frac{\mu}{\lambda}\frac{\langle\Delta
Num_{p}\rangle}{\Delta t}$

$\Delta Num_{p}=Num_{p}(t+\Delta t)-Num_{p}(t)$, $Num_{p}(t)$ is number of
packets at time $t$ and $\langle.\rangle$ shows that average value over the
time window $\Delta t$. $\langle\Delta Num_{p}\rangle=0$ shows that no packet
in the network and when $\eta=0$, system is in free flow state. Yan et al.
[14] have found average betweenness centrality $g(k)$ for the nodes with same
degree $k$:

$g(k)=\frac{1}{N(k)}\sum_{v:k_{v}=k}g(v)$

where, $N(k)$ is number of nodes with degree $k$. For Scale Free networks,
$g(k)\sim k^{\alpha}$. An efficient path $P(i\rightarrow j)$ between node $i$
and $j$ is calculated such that the sum $L(P(i\rightarrow
j:\alpha)=\sum_{i=0}^{N-1}k(x_{i})^{\alpha}$ should be minimized for an
optimal value of $\alpha$. Critical rate, $\lambda_{c}$ also depends on $\mu$
and hence, $\lambda_{c}(\beta)=\frac{N(N-1)}{g_{Lmax}^{\beta}}$. Liu et al.
[3] have studied network traffic by considering local and global routing
strategies. Betweenness Centrality (globally calculated) of a link whose end
nodes are $m$ with degree $k_{m}$ and $n$ with degree $k_{n}$, is linearly
correlated with multiplication of degree of end nodes, $k_{m}k_{n}$ (locally
evaluated). To avoid congestion in the network, links having maximum
$k_{m}k_{n}$ values are closed. After closing few links, $\lambda_{c}$,
average path length $(L_{avg})$ and average travel time $(\langle T\rangle)$
are studied. Jiang et al. [16] have found heuristic method for routing and
used an incremental approach for adding new paths in the network. As addition
of new paths cause change in betweenness and degree of each node.

$L^{s}(P(i\rightarrow j))=\sum_{v=0}^{v=N-1}f^{S}(n_{v})$

where, $f^{S}(n_{v})=(g_{n_{v}}^{S}Deg_{n_{v}}^{S})^{\alpha}$ is dynamic
weight of node $v$, $n_{v}$ for source node $S$, $g_{x_{v}}^{S}$ is dynamic
betweenness centrality of $n_{v}$ and $Deg_{n_{v}}^{S}$ is dynamic degree of
node $v$. Hence, weight of each link $(i,j)$ is
$(g_{i}^{S}Deg_{i}^{S})^{\alpha}+(g_{j}^{S}Deg_{j}^{S})^{\alpha}$. All the
previous works discussed till now are focused on routing and optimizing
network structure. Tadic [1] has discussed about restructuring of directed
networks by considering growth and rearrangement at unique time scale. At each
time unit $t$ a new node is added to the network (growth) and a number $X(t)$
and fraction $\beta$ are used to find newly added and rearranged links in the
network. (i.) A portion $f_{add}(t)=\beta X(t)$ of new links are out-flowing
links from the new added node $i=t$ and (ii.) the remaining part
$f_{update}(t)=(1-\beta)X(t)$ are the updated links at other nodes in the
network. Optimization of network topology and developing efficient routing
strategies help us to avoid congestion and to improve overall network
performance. In communication network, various users want to establish
connection between specious node pairs via an optimal route. Routes are
selected such that user can send data with optimal rate to their destinations.
Kelly [19] has formulated the rate allocation problem into optimization
problem for a static network. We have used the formulation of rate control
algorithm and proposed a mathematical model for dynamic communication
networks.

###  2.1 Mathematical model

A brief description is presented about mathematical representation of the
dynamic networks. Network consists of $N$ nodes and $E$ links and life span of
the network is $T$. A set of $R$ users are willing to send data in the
network. Any link $e_{mn}$ connects node $m$ with node $n$ can send maximum
$C_{e_{mn}}$ units of data through it, where, $C_{e_{mn}}$ is capacity of the
link $e_{mn}$ and $e_{mn}\in E$. In TVCN, each user $r$ is assigned a route
$r\in R$ for a particular time instant $t_{i}\in T$. At the end of
$t_{i}^{th}$ time, a zero-one matrix $A$ of the size $(N\times N)_{t_{i}}$ is
defined where, $A_{n_{i},n_{j},t_{i}}=1$, if node $n_{i}$ and $n_{j}$ are
connected at time $t_{i}$ otherwise zero. When the user $r$ is assigned a rate
$x_{r,t_{i}}$ then, utility of user $r$ at rate $x_{r,t_{i}}$ is given as
$U_{r,t_{i}}(x_{r,t_{i}})$ is increasing, strictly concave function of
$x_{r,t_{i}}$ over the range $x_{r,t_{i}}\geq 0$. Aggregate utility is
calculated by summing up all utilities of user $r$ with rate $x_{r,t_{i}}$ and
is denoted as $\sum_{r\in R,t_{i}}U_{r,t_{i}}(x_{r,t_{i}})$. Rate allocation
problem can be formulated as the following optimization problem.

$\displaystyle SYSTEM(U_{t_{i}},A_{t_{i}})$ $\displaystyle maximize\sum_{r\in
R,t_{i}}U_{r,t_{i}}(x_{r,t_{i}})$ (1) $\displaystyle
A_{t_{i}}^{T}x_{t_{i}}\leq C_{t_{i}}\mbox{ and }x_{t_{i}}\geq 0$

$A_{t_{i}}$ is a matrix at time interval $t_{i-1}$ to $t_{i}$. The given
constraint states that a link can not send data more than its capacity [19].
It is difficult and unmanageable to allocate a suitable utility function and
optimal rate to distinct users in complex networks. Hence, Kelly has divided
this problem into two simpler problems named as user’s optimal problem and
network’s optimal problem [19]. Let user $r$ decide an amount
$\mathcal{P}_{r}(t_{i})$ to pay per unit time, while $r$ is demanded a price
$\eta_{r}$ for unit flow. Hence, user will acquire a flow,
$x_{r}(t_{i})=\mathcal{P}_{r}(t_{i})/\eta_{r}$ at time $t_{i}$. User’s optimal
price for sending $x_{r}(t_{i})$ at time $t_{i}$ can be obtained using the
optimization problem as

$\displaystyle User_{r}(U_{r}(t_{i}),\eta_{r}(t_{i})),$ $\displaystyle\mbox{
maximize }U_{r}(x_{r}(t_{i}))-\mathcal{P}_{r}(t_{i}),$ (2)
$\displaystyle\mathcal{P}_{r}>0$

On the other hand, network wants to maximize weighted log function of
$\mathcal{P}_{r}(t_{i})$. Therefore, network optimization problem can be
formulated as

$\displaystyle NETWORK(A_{t_{i}},p_{t_{i}}),$ $\displaystyle
maximize\sum_{r\in R,t_{i}}\mathcal{P}_{r}(t_{i})log(x_{r}(t_{i})),$ (3)
$\displaystyle A_{t_{i}}^{T}x_{t_{i}}\leq C_{t_{i}}\mbox{ and }x_{t_{i}}\geq
0.$

As the network is time varying so, the values of $\eta_{r}$ and $x_{r}$ are
varying with time. Each user $r\in R$, initially computes the price per unit
flow by using the Eq. ((2)) and it is willing to pay,
$\mathcal{P}_{r}(t_{i})$. Utility of each user $r$ is assumed as a strictly
concave function of user’s rate $x_{r}(t_{i})$ at that time ($t_{i}$). At each
time instant if user wants to access a link $e_{mn}\in E$ then, the cost will
depend on the total data flow through the link at that time and is given by
$\psi_{e_{mn}}(t_{i})=\varsigma_{e_{mn}}(\sum_{r:e_{mn}\in E}x_{r}(t_{i}))$
where, $\varsigma_{e_{mn}}(\bullet)$ is growing if it comes in large number of
user’s path and $\varsigma_{e}(y)$ is given by

$\varsigma_{e_{mn}}(y)=c_{e_{mn}}.(y/C_{e_{mn}})^{\omega}$

where, $c_{e_{mn}}$ is constant and assumed with value one, $C_{e_{mn}}$ is
the capacity of link $e_{mn}\in E$. Now, consider the following system of
differential equation for getting optimal data rate ($x^{*}$)

$\frac{dx_{r}(t_{i})}{dt_{i}}=\vartheta_{r}(\mathcal{P}_{r}(t_{i})-x_{r}(t_{i})\sum_{e_{mn}\in
r}\psi_{e_{mn}}(t_{i}))$ (4)

Here, $\vartheta_{r}$ is proportionality constant. Each user first computes
it’s willingness to pay as $\mathcal{P}_{r}(t_{i})$ then, it adjusts its rate
based on the response provided by the links in the network and trying to
balance its willing to pay and total price. Eq. ((4)) consists of two
components: a steady increase in the rate proportional to
$\mathcal{P}_{r}(t_{i})$ and steady decrease in the rate proportional to the
response $\psi_{e_{mn}}(t_{i})$ provided by the network.

##  3 Time varying communication network model and routing strategies

Network topology and routing are responsible for congestion and network
efficiency. An optimal network structure is generated using an efficient TVCN
model. Proposed model considers all the aspect of growth and alteration
(rewiring & removal) of links in the network. In the established network,
users establish dedicated connection between the nodes with whom they want to
communicate. These connections are able to manage traffic efficiently, which
results minimized congestion in the network.

###  3.1 Time varying communication network model

Time varying data communication networks are designed by maintaining specific
set of rules and nodes are connected via directed links. Directed links may
have two categories: links which come out from a node (out-flowing) and other
links are incidental to a node (in-flowing). Degree distribution of both out-
flowing and in-flowing links follow a heavy tail power law distribution with
distinct power law exponents. In the proposed model, the preferences are
assigned to in-flowing links. The newly incoming node will be attached to the
node endorsed by the highest number of nodes [1, 5].

####  3.1.1 Network Dynamics

A communication network model is proposed by using the Barabasi-Albert (BA)
model [5] and the model using the dynamics of directed graphs [1]. The
proposed directed network model has following properties:

  1. 1.

Network consists of directed links for connecting nodes.

  2. 2.

Network expansion, link rewiring and removal are done at unique time scale.

At each time instant $t_{i}\in T$, a new node $n_{i}$ is added to the network
(expansion) and a number $X(\leq n_{0})$ is selected for network expansion,
rewiring and removal, where $n_{0}$ is initial number of nodes present in the
seed network. Links are divided into three categories: newly added, rewired
and removed links. Distribution of the links are done using the given set of
rules [26].
Notations:

  1. (a.)

$\beta=$ Fraction of the evolving (appear from the new incoming node and
appear/disappear in the existing network) links at any time instant. It
informs about the establishment of new connections from the new incoming node
$n_{t}$ at time $t$ , $0<\beta<1$.

  2. (b.)

$\gamma=$ Fraction of the links, those are rewired in the existing network,
$0.5<\gamma\leq 1$.

Using the above notation, following set of rules may be defined:

  1. (i)

Total number of new out-flowing links from the new appeared node, $i(=t)$ with
the nodes existing in the network at $(t-1)$ based on preferential attachment
is given by,

$f_{add}(t)=\beta X$

  2. (ii)

Few links are rewired in the existing network. $\gamma$ fraction of the
available $X$ links are chosen for rewiring,

$f_{rewire}(t)=\gamma(X-f_{add}(t))=\gamma(1-\beta)X$

  3. (iii)

Last part of the remaining segment of $X$ are used for deleting most
infrequently used links.

$f_{delete}(t)=X-f_{add}(t)-f_{rewire}(t)=(1-\gamma)(1-\beta)X$

The parameter $\delta$ is the ratio of altered and added links in the model
and is given by,

$\displaystyle\delta={}$
$\displaystyle\frac{f_{delete}(t)+f_{rewire}(t)}{f_{add}(t)}$ (5)
$\displaystyle={}$
$\displaystyle\frac{(1-\gamma)(1-\beta)X+\gamma(1-\beta)X}{\beta X}$
$\displaystyle={}$ $\displaystyle\frac{(1-\beta)}{\beta}$

Here, $\delta$ is independent of the number $X$ and is known as correlation
parameter.

While studying the behavior of communication networks, the concept of
preferred linking to a node is driven by the demand of the node to outflow the
data into the network. Only few influential nodes have the right to rewire
their links. Nodes rewire their infrequently used links and connect it to
preferred nodes for data communication. The preference is given to the
infrequently used links for the removal of links. Hence, the links attached to
a node having lower degree will be preferred for removal.

####  3.1.2 Degree Distribution

The scale free behavior is found, after studying the scaling properties of the
time varying network model. A mean field theoretical approach [5] is used to
anticipate the growth dynamics of distinct nodes, which is later applicable
for analytical computation of connectivity distribution and scaling exponents.
Network is growing with time hence, some of the previously existing nodes will
get more time to acquire links and are being the holders of highest degrees.
Degree $k_{i}$ of the node $i$ is changing continuously with time, so
probability $\Pi(k_{i})$ is interpreted as rate of change of $k_{i}$. At each
time stamp, fractions $\beta$ and $\gamma$ decide the number of links chosen
for expansion (addition), rearrangement and removal, and is deciding
parameters of degree of the node $i$. Analysis is given using following steps:

  1. 1.

A fraction $\beta$ of the number $X$ links are newly added links at time $t$.

$\left(\frac{dk_{i}}{dt}\right)_{add}=\beta X\frac{k_{i}}{\sum_{j}k_{j}}$ (6)

Effect of newly added links on the degree of node $i$ is written on the left
hand side of the equation and on the right side $\beta X$ links are formed
using preferential attachment.

  2. 2.

A fraction $\gamma(1-\beta)X$ links are re-arranged at time $t$.

$\left(\frac{dk_{i}}{dt}\right)_{rewire}=\gamma(1-\beta)X\left[\frac{1}{n_{0}+t}+\left(1-\frac{1}{n_{0}+t}\right)\frac{k_{i}}{\sum_{j}k_{j}}-\frac{1}{n_{0}+t}\left(1-\frac{k_{i}}{\sum_{j}k_{j}}\right)\right]$
(7)

When few links are rewired then, change in degree of node $i$ depends on three
terms: first term shows random selection of nodes, second term corresponds to
linking with other existing nodes having high preferential attachment
probability and third term shows removal of link having low preferential
attachment value.

  3. 3.

A fraction $(1-\gamma)(1-\beta)X$ links are removed from the network at time
$t$.

$\left(\frac{dk_{i}}{dt}\right)_{delete}=-(1-\gamma)(1-\beta)X\left[\frac{1}{n_{0}+t}+\left(1-\frac{1}{n_{0}+t}\right)\left(1-\frac{k_{i}}{\sum_{j}k_{j}}\right)\frac{1}{n_{0}+t}\right]$
(8)

Removal of links affect the degree of node $i$ and it is shown in above
equation. First term shows random selection of a node and that link will be
removed which has low preferential attachment value.

At time $t$, the sum of degrees of nodes in the network will be:

$\sum_{j}k_{j}=2t\left[\beta X+(1-\beta)\left(\gamma
X-(1-\gamma)X\right)\right]=2Xt\left[\beta+(1-\beta)(2\gamma-1)\right]$

Let $c=\beta+(1-\beta)(2\gamma-1)$.
Now, combining the eqns. (6) to (8) we get the change in degree of node $i$
with respect to time $t$.

$\displaystyle\frac{dk_{i}}{dt}={}$ $\displaystyle\frac{\beta
k_{i}}{2ct}+\frac{\gamma(1-\beta)k_{i}}{2ct}$
$\displaystyle-(1-\beta)(1-\gamma)X\left[\frac{2}{n_{0}+t}-\frac{1}{(n_{0}+t)^{2}}-\frac{k_{i}}{2ct(n_{0}+t)}+\frac{k_{i}}{2ct(n_{0}+t)^{2}}\right]$
$\displaystyle={}$
$\displaystyle\left(\frac{\beta}{2c}+\frac{\gamma(1-\beta)}{2c}\right)\frac{k_{i}}{t}-\left((1-\beta)(1-\gamma)X\right)\frac{1}{t}$
(for large t )

Let us assume $\theta_{1}=\frac{\beta}{2c}+\frac{\gamma(1-\beta)}{2c}$ and
$\theta_{2}=-(1-\beta)(1-\gamma)X$
Hence, the above equation can be re-written as

$\frac{dk_{i}}{dt}=\theta_{1}\frac{k_{i}}{t}+\theta_{2}\frac{1}{t}$ (9)

The solution of the Eq. ((9)) is derived by taking initial condition that node
$i$ appears at time $t_{i}$ with $X$ connections $k_{i}(t_{i})=X$ and is given
as:

$\displaystyle k_{i}(t)={}$
$\displaystyle-\frac{\theta_{2}}{\theta_{1}}+\left(k_{i}(t_{i})+\frac{\theta_{2}}{\theta_{1}}\right)\left(\frac{t}{t_{i}}\right)^{\theta_{1}}$
$\displaystyle={}$
$\displaystyle-\frac{\theta_{2}}{\theta_{1}}+\left(X+\frac{\theta_{2}}{\theta_{1}}\right)\left(\frac{t}{t_{i}}\right)^{\theta_{1}}$

At each time stamp, a node is added into the network hence, Probability
density of $t_{i}$ is:

$P_{i}(t_{i})=\frac{1}{n_{0}+t}$

The probability that a node has total $k_{i}(t)$ connections and which is
smaller than $k$, $P(k_{i}(t)<k)$, can be written as:

$\displaystyle P(k_{i}(t)<k)={}$ $\displaystyle
P(-\frac{\theta_{2}}{\theta_{1}}+\left(X+\frac{\theta_{2}}{\theta_{1}}\right)\left(\frac{t}{t_{i}}\right)^{\theta_{1}}<k)$
$\displaystyle={}$ $\displaystyle
P\left(t_{i}>\left(\frac{X+\frac{\theta_{2}}{\theta_{1}}}{k+\frac{\theta_{2}}{\theta_{1}}}\right)^{\frac{1}{\theta_{1}}}t\right)$
$\displaystyle={}$ $\displaystyle
1-P\left(t_{i}\leq\left(\frac{X+\frac{\theta_{2}}{\theta_{1}}}{k+\frac{\theta_{2}}{\theta_{1}}}\right)^{\frac{1}{\theta_{1}}}t\right)$

The probability density of $k$ is $P(k)$ and can be written as:

$\displaystyle P(k)={}$ $\displaystyle\frac{\partial P(k_{i}(t)<k)}{\partial
k}$ (10) $\displaystyle={}$
$\displaystyle\frac{t}{\theta_{1}(n_{0}+t)}\left(X+\frac{\theta_{2}}{\theta_{1}}\right)^{\frac{1}{\theta_{1}}}\left(k+\frac{\theta_{2}}{\theta_{1}}\right)^{-(1+\frac{1}{\theta_{1}})}$

Now, assume $\alpha=-(1+\frac{1}{\theta_{1}})$.
The value of the scaling exponent of the real world network lies between $2$
and $3$, hence, for dynamic communication network, exponent ($\alpha$) must
lie within $2<\alpha\leq 3$. Constraint on $\alpha$ will be fulfilled if
$0.5<\theta_{1}<1$. As $P(k)$ is always positive so,
$X+\frac{\theta_{2}}{\theta_{1}}$ should be positive.

###  3.2 Routing Strategies

Each node in the communication network generates packets with a certain rate
and can deliver packets according to the capacity of that particular node.
Capacity of the network ($C$) is calculated by summing up the capacity of all
the individual nodes in the communication network. Aggregate values of the
packet generation rate of all the nodes are termed as the load of the network.

$\displaystyle L={}$ $\displaystyle\sum_{v=0}^{N-1}l_{v}$ $\displaystyle\mbox{
here, }l_{v}=\mbox{ load at node }v$

When, load ($L$) exceeds the capacity ($C$) of the network then, it will
become congested with the packets. If all the packets are sent through the
shortest paths to their destinations then, some nodes may appear frequently in
the formation of shortest path and it will become congested. In the large
communication networks, multiple users want to establish a connection between
a source ($S$) and destination ($D$). There may exist multiple shortest paths
$\sigma_{z}(S\rightarrow D)$ from $S$ to $D$, where $z={1,2,...,\chi}$ and
$\chi$ is total number of shortest paths between $S$ and $D$. In the shortest
path, a node may lie with high betweenness centrality. Sending data through
the shortest path will lead to the congestion in the network (most of the user
pairs will try to follow the same path). Therefore, it is important to
investigate a shortest path $\sigma(S\rightarrow D)$ between the user pairs
such that overall betweenness centrality of the nodes appears in the path

$\displaystyle\theta_{g}[\sigma_{z}(S\rightarrow D)]={}$
$\displaystyle\sum_{v:v\in\sigma_{z}(S\rightarrow D)}g_{v}$
$\displaystyle\mbox{ here, }g_{v}=\mbox{ betweenness of node }v$

should be minimum. Hence, we want to find out a path whose nodes should not be
the part of large number of other shortest paths (low betweenness centrality).
Hence, it is defined by,

$min\mbox{ }\\{\forall z:\theta_{g}(\sigma_{z}(S\rightarrow D))\\}$

Apart from routing strategy using betweenness centrality of the nodes, a local
strategy is used by considering influence of the neighbor nodes. If a heavily
loaded node is connected with some other nodes then, it will make all the
succeeding nodes congested. Heavily congested preceding nodes have negative
impact on successor nodes. The impact of one node to others is measured by
Eigen vector centrality of the node $v$,

$x_{v}=\frac{1}{\kappa}\sum_{j\in N}a_{vj}x_{j}$

Here, $\kappa$ is largest Eigen value of the matrix $A$ and if node $v$ and
$j$ are connected then, $a_{vj}=1$. If a node is most Eigen vector central,
then it will be highly congested and hence, least reputed. So, among all the
shortest path, that path will be reputed if aggregate value of the reputation
of the path $\theta_{r}[\sigma(S\rightarrow D)]$

$\displaystyle\theta_{r}[\sigma_{z}(S\rightarrow D)]={}$
$\displaystyle\sum_{v:v\in\sigma_{z}(S\rightarrow D)}\frac{1}{x_{v}}$
$\displaystyle x_{v}=\mbox{ Eigen vector centrality of node }v$

is maximized and and the function is defined by,

$max\mbox{ }\\{\forall z:\theta_{r}(\sigma_{z}(S\rightarrow D))\\}.$

Optimal path between $S$ and $D$

$\displaystyle\theta_{gr}\sigma_{z}(S\rightarrow D)]={}$
$\displaystyle\sum_{v:v\in\sigma_{z}(S\rightarrow D)}{g_{v}}{x_{v}}$

is calculated by considering both betweenness centrality and reputation. A
path is optimal if it’s elements will have least betweenness central as well
as reputed and is given by,

$min\mbox{ }\\{\forall z:\theta_{gr}(\sigma_{z}(S\rightarrow D))\\}.$

Figure 1: Block diagram

A snapshot of the proposed method is discussed in Fig. 1. Two users, $U_{1}$
and $U_{2}$ want to send data from node $9$ to node $4$ and from node $10$ to
node $5$, respectively. Their respective routes are $9-1-4,9-5-4$ and
$10-1-5,10-2-5$. Both the users avoid congested routes hence, $U_{1}$ avoids
the route $9-1-4$ and $U_{2}$ prefers $10-2-5$. From Fig. 1, it is evident
that node $1$ is highly congested node in the network, so both users avoid to
send data through node $1$ and select other possible options available for
sending data to their destined node. Packet generation rate (PGR) of a node
depends on the packet forwarding rate and betweenness centrality of the node.
Packet forwarding rate depends on the capacity of a node, that affects
bandwidth of a link.

Capacity of a link $C_{e_{mn}}$ is dependent on the degrees $k_{m}$ and
$k_{n}$ of node $m$ and $n$. It can be approximated by a power-law dependence
[27]

$C_{e_{mn}}=b(k_{m}k_{n})^{\alpha}$

$\alpha$ is the degree influenced exponent which depends on the type of
networks and $b$ is a positive quantity. Three possible relations between
capacity $C_{e_{mn}}$ and exponent $\alpha$: (i) if $\alpha>0$ then, data is
transmitted through high degree nodes (ii) when $\alpha<0$ then, data will be
sent through low degree nodes and (iii) $\alpha=0$ shows degree independent
transmission. The value of $\alpha$ in the proposed model is taken positive
value with constant $\alpha=1$ and $b$ is also equal to $1$. Reputation and
betweenness centrality are deciding parameter for evaluating congestion.
Reputation of a node is inverse of the Eigen vector centrality of a node.
While, congestion is proportionally related to betweenness centrality and
inversely related with reputation.

Algorithmic steps are given for expansion, rewire and removal of links in the
TVCN (Algorithm 1).

1: Input: A number of nodes $(n_{0})$ for creating seed network, $n(\leq
n_{0})$, $\beta$, $\gamma$ and $T$.

2: Output: Time varying data communication networks.

3: while $t\leq T$ do

4: Expand the network with one node at each time instant $t$.

5: for $x$: 1 to $f_{add}(t)$ do

6: Attach the newly added node with node having highest inflowing link
attachment probability in the existing network.

7: end for

8: for $y$ : 1 to $f_{rewire}(t)$ do

9: Remove the infrequently used link of an influenced node and attach to the
node having higher inflowing link probability.

10: end for

11: for $z$ : 1 to $f_{delete}(t)$ do

12: Remove the infrequently used link of a randomly selected node.

13: end for

14: end while

Algorithm 1 Network Evolution

Steps involved in Algorithm 1, are used to design TVCN. In the time varying
network, various users want to communicate data from a source $S$ to
destination $D$. User gives information about the source and destination, and
accordingly $S\rightarrow D$ pairs are generated. Increments in the number of
users lead to the network into congested state. Therefore, our aim is to find
efficient routing paths such that maximum number of users are getting
benefited with a unique stable value of the optimal data sending rate
$x_{r}^{*}$ and corresponding convergence vectors will be
$x^{*}={x_{r}^{*},r\in R}$ using equation Eq. (4). Shortest path
$\sigma(S\rightarrow D)$ with lowest betweenness centrality and highest
reputation $(\theta_{gr}[\sigma(S\rightarrow D)])$ are assumed as efficient.
After finding such paths of all the users, optimal data sending rate of each
user is calculated by using rate allocation equation Eq. (4). Detailed
description of selecting shortest path with lowest (highest) betweenness
centrality and highest (lowest) reputation along with optimal rate is given in
Algorithm 2.

1: Input: All source destination $S-D$ pairs $\mathcal{N}_{SD}$ in the
network, designed through Algorithm 1, $a$ and $b$ such that $a>0$ & $0<b<1$.

2: for $i:=1\mbox{ to }length(\mathcal{N}_{SD})$ do

3: Evaluate all shortest paths $\chi_{i}$ of user $i$.

4: if $\chi_{i}>1$ then

5: for $m:=1\mbox{ to }z$ do

6: Calculate $\theta_{gr}[\sigma_{m}(S\rightarrow D)]$.

7: end for

8: Select shortest paths $\chi_{i}(S\rightarrow D)$ having maximum and minimum
value of $\theta_{gr}[\sigma_{m}(\chi_{i}(S\rightarrow D))]$.

9: else

10: Calculate $\theta_{gr}[\sigma(S\rightarrow D)]$.

11: end if

12: for $j:=1\mbox{ to }N$ do

13: Frequency of occurring of node $j$, $\omega_{j}$ in user’s path decides
initial data rate of each node.

14: $x_{j}=\frac{C_{j}}{\omega_{j}}$;

15: end for

16: end for

17: for $i:=1\mbox{ to }length(\mathcal{N}_{SD})$ do

18: for $d:=1\mbox{ to }length(\mathcal{N}_{SD}(i))$ do

19: Update network feedback $\psi_{d}$ for each element $d$.

20: end for

21: $x_{r}=min(x_{\mathcal{N}_{SD}(i)})$;

22: $A(r)=rand(1,10)$;

23: $\mathcal{P}_{r}=x_{r}*(\frac{a}{x_{r}+b})$;

24: $\psi_{r}=\psi_{d}\\{\forall d:d\in\mathcal{N}_{SD}(i)\\}$;

25: end for

26: Use the value of $x_{r}$, $A(r)$, $\mathcal{P}_{r}$ and $\psi_{r}$ to find
the rate of convergence of each user.

Algorithm 2 Finding shortest path having lowest (highest) betweenness
centrality with highest (lowest) reputation and optimal rate for each user

##  4 Simulation and results

For dynamics of TVCN model, the simulation starts by establishing the
infrastructure of the network followed by algorithmic steps in Algorithm 1 and
the rules are given in the Section 3.1.1. The proposed TVCN model is one of
the example of real world networks, hence, degree distributions must follow a
heavy-tailed power law distribution: $P(k)\sim k^{-\alpha}$, where
$2<\alpha\leq 3$, known as scale-free networks. Scale-free networks are non-
homogeneous, numerous nodes with few links coexist with a few hub nodes,
connected with thousands or even millions of links. In this paper, the
parameters are set to be seed node $n_{0}=5$, number $X=5$, fraction of newly
added links $\beta$ range in $(0,1)$, fraction of rewired links $\gamma$ is in
the range of $(0.5,1)$, with network size ranging from $N=10^{2}$ to
$N=10^{4}$. Any nodes can be included in the user’s $S-D$ sets or may
participate in routing also. Data forwarding capacity of a node $n$, $C_{n}$
is equal to the in-flowing degree $k_{n}$ of the node $n$. Capacity of a link
$C_{e_{mn}}$ is obtained by multiplying the degree $k_{m}$ and $k_{n}$ of end
nodes $m$ and $n$ respectively. At each time stamp, degree of the nodes will
be different, hence capacity of the nodes as well as links change accordingly.

In the Section 3.1.1, the value of correlation parameter,
$\delta=\frac{1-\beta}{\beta}$, where, $0<\beta<1$ , implies that
$\infty<\delta\leq 0$. The value of $\delta$ will be minimum with value $0$ or
maximum with value $\infty$, for maximum value of $\beta=1$, or minimum value
of $\beta=0$. The following cases are possible for the value of the $\delta$,
depending on altered (rewired & removed) and newly added links in the network,

  1. (a)

$\delta=1$, when both the values of altered (rewired & removed) and added
links are same.

  2. (b)

$\delta<1(\beta>0.5)$, more number of new links are getting added than
altered.

  3. (c)

$\delta>1(\beta<0.5)$, more number of links are altered than added.

  4. (d)

$\delta=0(\beta=1)$, only expansion, no alteration in the existing network.

$\begin{array}[]{cc}\includegraphics[width=195.12767pt,height=180.67499pt]{2000digdistpr.eps}&\includegraphics[width=195.12767pt,height=180.67499pt]{10000_3_1_1.eps}\\\
\mbox{(a)}&\mbox{(b)}\\\
\includegraphics[width=195.12767pt,height=180.67499pt]{10000_1_3_1.eps}&\includegraphics[width=195.12767pt,height=180.67499pt]{10000_5_0.eps}\\\
\mbox{(c)}&\mbox{(d)}\end{array}$

Figure 2: Degree distribution of the network when number of nodes are average
ratio of altered and newly added links are (a) $N=10^{4}\mbox{ ,
}\delta=1.00$, (b) $N=10^{4}\mbox{ , }\delta=0.67$, (c) $N=10^{4}\mbox{ ,
}\delta=3.00$ and (d) $N=10^{4}\mbox{ , }\delta=0.00$

The degree distribution is plotted of the generated TVCN (Fig. 2 $(a-d)$) for
number of nodes, $N=10^{4}$. Different values are considered for parameters
$\beta$, $\gamma$ and $\delta$ (Table 1 $(a-d)$). The degree distribution is
found power law in nature with all the considered values of the parameters
(Fig. 2). A fraction $(1-\beta)$ of the number $X$ is used for alteration of
links and $\gamma$ part of the altered links are used for rewiring. To
maintain the growing nature of TVCN, the value of $\gamma$ is considered in
the range $(0.5,1)$. Once the value of $\beta$ is increased it implies that
more number of new incoming links appear with the newly added node. Therefore,
in the network a node may acquire more number of links. Hence, the tail of the
distribution is found longer and increasing slope of degree distribution of
TVCN (Fig. 2 $(a-d)$). The power law exponent $\alpha$ is also increasing
($2<\alpha\leq 3$) for all the possible values of correlation parameter
$\delta$ (Table 1), with the increasing value of $\beta$.

Table 1: Values of power law exponent $\alpha$, for different values of $\delta$ at $N=10^{4}$. | $\beta$ | $\gamma$ | $\alpha$
---|---|---|---
$\delta=1$ | $0.50$ | $0.50$ | $2.486$
$\delta<1$ | $0.60$ | $0.50$ | $2.455$
$\delta>1$ | $0.25$ | $0.70$ | $2.065$
$\delta=0$ | $0.90$ | $0.90$ | $2.664$

As per proposed mathematical model for calculating the degree distribution of
the generated TVCN, power law exponent $\alpha$ is dependent on the value of
$\theta_{1}$ and may be written as $\frac{1}{\alpha+1}$. For real world
networks, range of $\alpha$ is $(2,3]$ hence, $\theta_{1}$ must lie between
$0.5$ and $1$. The density function is always positive, hence,
$X+\frac{\theta_{2}}{\theta_{1}}$ must be positive. For different values of
$\delta$, both conditions are fulfilled (Table 2).

Table 2: User’s optimal rate through the shortest routes having different betweenness centrality values (maximum and minimum), when number of nodes $N=10^{2}$ | $\beta$ | $\gamma$ | $\theta_{1}$ | $\theta_{2}$ | $\left(X+\frac{\theta_{2}}{\theta_{1}}\right)$
---|---|---|---|---|---
$\delta=1$ | $0.5$ | $0.6$ | $0.75$ | $-1.25$ | $3.336$
$\delta<1$ | $0.6$ | $0.5$ | $0.667$ | $-1$ | $4.351$
$\delta>1$ | $0.25$ | $0.7$ | $0.7045$ | $-1.125$ | $3.4019$
$\delta=0$ | $0.9$ | $0.5$ | $0.5278$ | $-0.25$ | $4.526$

The degree distribution of the in-flowing links in TVCN is calculated and
compared with all the possible cases of the parameter $\delta$, by using mean
field theoretical approach and through simulation, for the network of size
$N=5*10^{2}$. The obtained values of the power law exponent $\alpha$ are very
proximate to each other (Fig. 3) with corresponding values (Table 3). Hence,
proposed mathematical model for degree distribution for TVCN is validated.

$\begin{array}[]{cc}\includegraphics[width=216.81pt,height=180.67499pt]{x1y11.eps}&\includegraphics[width=216.81pt,height=180.67499pt]{x2y21.eps}\\\
\mbox{(a)}&\mbox{(b)}\\\
\includegraphics[width=216.81pt,height=180.67499pt]{x3y31.eps}&\includegraphics[width=216.81pt,height=180.67499pt]{x4y41.eps}\\\
\mbox{(c)}&\mbox{(d)}\\\ \end{array}$

Figure 3: Simulated and analytical results of degree distribution of in-flowing links for the network of $N=5*10^{2}$ nodes for $\delta=1$ (a), $\delta<1$ (b), $\delta>1$ (c) and $\delta=0$ (d). Table 3: Comparative study of the values of power law exponent $\alpha$ obtained through simulation and theoretical approach, for different values of $\delta$ for network size $N=5*10^{2}$. | $\beta$ | $\gamma$ | $\alpha$ | $\alpha$
---|---|---|---|---
|  |  | (Simulation) | (Theoretical)
$\delta=1$ | $0.50$ | $0.50$ | $2.750$ | $2.619$
$\delta<1$ | $0.60$ | $0.50$ | $2.616$ | $2.500$
$\delta>1$ | $0.25$ | $0.70$ | $2.021$ | $2.419$
$\delta=0$ | $0.90$ | $0.90$ | $2.741$ | $2.895$

In the communication networks, many paths are available for sending packets
for each user, between the desired source and destination. All routes are
assumed as equally weighted hence, users can select any of the available
routes for sending the packets. Each user can send data along one of the
shortest paths to the destination with a maximum flow rate of individual
links. The data sending rate is reduced as multiple users want to share the
common resources. Furthermore, user may not send more data with maximum rate.
User’s rate depends on two parameters; it’s own willingness to pay
$\mathcal{P}_{r}(t_{i})$ and network’s feedback $\psi_{r}(t_{i})=\sum_{e\in
r}\psi_{e}(t_{i})$. Using rate control theorem given in Eq. ((4)), an optimal
data sending rate of each user is obtained (Fig. 4). Optimal data rate of two
users, User1 and User2 are shown at different time instants. Interval of the
two time instants $\Delta t$ is taken as $\Delta t=5*10^{2}$. Size of the
network is changing but number of users are kept same as in initial network.
Size of the network increases but optimal rates are decreasing to a stable
point. User rates depend on the demand of particular resources coming in the
shortest route. If demand is high then, data sending rate will be less.

Figure 4: Conservation of data sending rates of User1 and User2 at different
time instants

Multiple users want to establish connections between a distinct pair of nodes
through a shortest possible communication path. More than one shortest routes
are possible for making those connection, so data flow rates will be less
through the paths having highest betweenness centrality and lowest reputation
summation of all the existing nodes. A network is formed with the network
network of size $N=5*10^{2}$ with considering $20$ source to destination $S-D$
pairs along with their betweenness centrality and reputation values and
optimal rate ($x^{*}$) (Table 4).

Table 4: Optimal rate $x_{r}^{*}$ of $20$ users through the shortest routes with minimum and maximum $\theta_{gr}[\sigma(S\rightarrow D)]$ (maximum and minimum), when size of network $N=5*10^{2}$ | Source | Destination | path | $\theta_{gr}[\sigma(S\rightarrow D)]$ |  | $\theta_{gr}[\sigma(S\rightarrow D)]$ |
---|---|---|---|---|---|---|---
|  |  | length | (Minimum) | ($x_{r}^{*}$ ) | (Maximum) | ( $x_{r}^{*}$)
User1 | 29 | 14 | 3 | 0.0156 | 8.9969 | 0.0987 | 3.3189
User2 | 77 | 1 | 2 | 0.0192 | 8.0567 | 0.0552 | 3.8823
User3 | 74 | 73 | 4 | 0.0025 | 4.8526 | 0.0509 | 2.2117
User4 | 72 | 1 | 2 | 0.0183 | 8.7746 | 0.0544 | 4.0416
User5 | 58 | 73 | 3 | 0.0118 | 4.5454 | 0.0518 | 2.7566
User6 | 34 | 84 | 3 | 0.0115 | 8.8328 | 0.0560 | 4.4339
User7 | 94 | 25 | 3 | 0.0028 | 10.8234 | 0.0508 | 3.1719
User8 | 96 | 36 | 2 | 0.0005 | 4.8525 | 0.0008 | 3.6785
User9 | 66 | 85 | 4 | 0.0027 | 6.1431 | 0.0566 | 2.4510
User10 | 20 | 2 | 3 | 0.0023 | 9.2059 | 0.0569 | 4.4499
User11 | 58 | 16 | 3 | 0.0121 | 8.0484 | 0.0554 | 3.6219
User12 | 3 | 52 | 2 | 0.0415 | 8.5054 | 0.0457 | 5.2154
User13 | 14 | 26 | 3 | 0.0100 | 9.2094 | 0.0646 | 4.6149
User14 | 95 | 74 | 5 | 0.0006 | 5.7869 | 0.0517 | 2.7312
User15 | 70 | 58 | 3 | 0.0025 | 6.2374 | 0.0559 | 3.5818
User16 | 31 | 69 | 3 | 0.0024 | 7.0966 | 0.0578 | 4.9822
User17 | 28 | 12 | 3 | 0.0072 | 7.2207 | 0.0582 | 2.7948
User18 | 22 | 7 | 3 | 0.0236 | 6.6195 | 0.0852 | 2.7941
User19 | 89 | 100 | 3 | 0.0030 | 8.6138 | 0.0503 | 3.7978
User20 | 57 | 31 | 3 | 0.0025 | 8.0771 | 0.0535 | 4.0460

Now, the number of users are equal to the network size $N=5*10^{2}$. Here, in
this network $39$ users’ source destination pairs are taken and for each pair
the highest and lowest summation of the betweenness centralities for all the
available shortest path are calculated. Optimal data sending rates of the $39$
users with highest and lowest betweenness centrality are calculated and the
result are shown in Fig. 5(a). Apart from betweenness centrality, there is one
more parameter named as reputation effects optimal data rate $x^{*}$ of each
user. If path is formed by takings over all sum of most reputed nodes then,
rate of the user will be maximum. Results of optimal data rates ($x^{*}$) of
$39$ users with aggregate sum of most and least reputed nodes are displayed in
Fig.5(b).

$\begin{array}[]{cc}\includegraphics[width=216.81pt,height=202.35622pt]{250YB.eps}&\includegraphics[width=216.81pt,height=202.35622pt]{250YR.eps}\\\
\end{array}$

Figure 5: User’s optimal rates of user paths with (a) highest & lowest
betweenness and (b) reputation (highest & lowest)

Calculation of betweenness centrality and reputation individually is not
sufficient for obtaining optimal rates of users. Hence, we have considered
both betweenness and reputation of the nodes in the networks. Those paths are
selected. A path considered as efficient for routing if contains those nodes,
which is least betweenness central as well as reputed. Simulation results in
Fig. 6 show that the data sending rate of the user path having lowest or
highest value for the aggregate sum of betweenness and highest or lowest total
sum of reputation, is always maximum or minimum.

Figure 6: User’s optimal rates at time $t=5*10^{2}$ by considering both
betweenness and reputation ( $\theta_{gr}[\sigma(S\rightarrow D)]$) with
maximum & minimum value

##  5 Conclusions and future work

A framework is designed to represent time varying communication networks
(TVCN) where links may appear/disappear randomly with time and nodes are added
continuously. A model is proposed to design an optimal network with the
consideration of all the aspects of dynamicity (addition, rewiring and
removal) of links. At each time step, a new node is added into the network,
and a number $X$ is taken for establishing, rewiring and removal of
connections. The values of the fraction $\beta$ and $\gamma$ determine the
number of added and altered (rewired and removed) links. Various experiments
were performed for finding the topological structure for the TVCN with
different values of correlation parameter, $\delta$. It is observed that with
increasing value of $\beta$ in the range $(0,1)$ leads to decrement of the
correlation parameter $\delta$ in the interval $(\infty,0)$. Furthermore, the
scaling exponent increases in the range $2<\alpha\leq 3$. It is found that the
value of power law exponents obtained from the simulation and mean field
theoretical analysis of connectivity distribution are proximate to each other.
In TVCN, multiple users select their paths such that nodes in the path should
have least betweenness centrality and most reputed. Once a path is
established, optimal data sending rate of the user is obtained by using rate
control theorem proposed by Kelly [19]. Interval of the two time instants
$(\Delta t)$ is taken as $\Delta t=5*10^{2}$. Number of users are kept same as
in initial network despite of change in network size. At the end of
$(t_{i}+\Delta t)^{th}$ time instant, $x^{*}$ of users are evaluated and it is
found that size of the network increases but optimal rates are decreasing to a
stable point, $x^{*}$. At each time step, optimal rate of the users are
calculated and it is found that data sending rate of the user’s are different
with time. We have also studied the case in which number of users increase
with network growth. Shortest path with lowest or highest betweenness and
highest or lowest reputation are chosen for routing and observed that the
value of optimal rate will always be the maximum or minimum.

In this paper, system utility is calculated without consideration of delays in
the network. Utility function, itself can be replaced by using game
theoretical approach. We have not considered the resilience under cascaded
failure and targeted attack in the network.

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