Effect of correlation on the traffic capacity of Time Varying Communication Network
NNovember 21, 2018 1:51 WSPC/INSTRUCTION FILE ws-mplb˙arxiv
Modern Physics Letters Bc (cid:13)
World Scientific Publishing Company
Effect of correlation on the traffic capacity of Time VaryingCommunication Network
Suchi Kumari
Department of Computer Science EngineeringNational Institute of Technology Delhi, [email protected]
Anurag Singh
Department of Computer Science EngineeringNational Institute of Technology Delhi, [email protected]
Received (Day Month Year)Revised (Day Month Year)The network topology and the routing strategy are major factors to affect the trafficdynamics of the network. In this work, we aim to design an optimal time-varying networkstructure and an efficient route is allocated to each user in the network. The networktopology is designed by considering addition, removal, and rewiring of links. At each timeinstants, a new node connects with an existing node based on the degree and correlationwith its neighbor. Traffic congestion is handled by rewiring of some congested linksalong with the removal of the anti-preferential and correlated links. Centrality playsan important role to find the most important node in the network. The more a nodeis central, the more it can be used for the shortest route of the user pairs and it canbe congested due to a large number of data coming from its neighborhood. Therefore,routes of the users are selected such that the sum of the centrality of the nodes appearingin the user’s route is minimum. Thereafter, we analyze the network structure by usingvarious network properties such as the clustering coefficient, centrality, average shortestpath, rich club coefficient, average packet travel time and order parameter.
Keywords : Time varying communication network; routing; congestion; centrality.
1. Introduction
The structure and dynamics of complex networks attracted much attention fromthe researchers of different areas in recent years. It has been widely accepted thatthe topology and degree distribution of networks have intense effects on the processdynamics on these networks, including disease spreading, information diffusion, traffic movement. Due to the increasing traffic volumes on the networks e.g., roadnetworks, social and data communication networks, fulfillment of user’s demand andminimization of traffic congestion is a challenging task. The performance of thedata communication networks strongly depends on its data forwarding capacity a r X i v : . [ c s . N I] N ov ovember 21, 2018 1:51 WSPC/INSTRUCTION FILE ws-mplb˙arxiv Authors’ Names which is determined by the structure of the underlying network. In this context, anoptimal time-varying communication network model is designed to avoid congestionin the network and user’s route is selected based on centrality information especially,betweenness centrality.It is found that the communication networks are scale-free (SF) and are moresusceptible to traffic congestion than some homogeneous networks. In SF networkslarge degree nodes posses a large volume of data hence, congestion usually starts atthese nodes and then spreads to the whole network. Therefore, researchers proposedvarious strategies, which can be classified into hard and soft strategies in orderto handle traffic congestion and enhance network capacity. The restructuring ofnetwork topology comes under hard strategies. Zhao et al. redistributed a loadof heavily loaded nodes to others, some connections are removed between largedegree nodes, high betweenness centrality nodes are removed first and linksare added between the nodes with long distance. Jiang et al. , assigned capacitydynamically to each link proportional to the queue length of the link. Some fractionof links is rewired based on node’s degree information and betweenness centrality. Chen et al. rewired the link against traffic congestion and proved that the networkshould have a core-periphery structure.Sometimes it is impossible to modify the network topological structure andit also incurs a high cost to change the structure of the network. Hence, a softstrategy based on finding a better routing strategy is preferable to enhance thenetwork capacity. Yin et al. chose an efficient path (EP) for routing. Zhao etal. assigned different routes with different traffic flow priorities and it is shownthat traffic capacity is enhanced by approx 12% compared with the initial EPapproach. In communication networks, two rates are associated with each node:packet generation rate, λ and packet forwarding (delivery) rate, C . Tang et al. considered the λ as a periodic function of time and proposed a mixed routingstrategy to enhance transportation efficiency. For small λ , the shortest paths areused to deliver the packets and when λ is large, an efficient routing method isused and loads are redistributed from central nodes to others. The capacity of thenetwork is maximized using a routing strategy based on the minimum informationpath and average routing centrality degree of the node is calculated to analyze thetraffic load on nodes of different degrees. Some of the researchers studied transportprocesses on multilayer or multiplex networks with emphasis on optimizing thenetwork capacity and transmission efficiency. Yang et al. established a relationshipbetween traffic congestion and network lifetime for the network with moving nodesin a defined area and it is concluded that the network lifetime is inversely relatedto traffic congestion.The previous researches demonstrate the underlying network topologies androuting approaches have significant impacts on the overall network performance.The strategies for optimizing network topology can be divided into two parts:restorative and proactive strategies. Restorative strategies include methods todo some changes in existing networks such that congestion is minimized.
4, 9–11, 13 ovember 21, 2018 1:51 WSPC/INSTRUCTION FILE ws-mplb˙arxiv
Instructions for typing manuscripts (paper’s title) Through proactive strategies, new links are added in such a way that the optimalnetwork structure is formed.
12, 20
A proactive strategy like converts the networkinto the homogeneous network but most of the real-world networks are scale-freenetworks. Therefore, the problem statement for the proposed work may be catego-rize under following categories. (i) Design an optimal network structure by consider-ing both the proactive and the restorative strategy. A time-varying communicationnetwork (TVCN) model is proposed in such a way that the addition of new links arebased on a proactive strategy while removal and rewiring of links follow restorativestrategy. The probability that the new node will be attached to the node in theexisting network is proportional to the degree of the existing node and inverselyproportional to the correlation of the existing node with its neighbor. Few linksof the congested nodes are rewired and connected with the nodes with preferen-tial attachment and having less correlation with its neighbor. Some correlated andanti-preferential links are also removed from the network. (ii) The novelty of theproposed network structure is checked by studying different network parameterssuch as the clustering coefficient ( C l C ), centrality, average path length (APL), andrich club (RC) coefficient. (iii) The route of each user is selected with the help ofcentrality of the node in the network. The betweenness centrality (BC) is used tomeasure the extent to which a node lies on shortest paths between other node pairs.If a node is more betweenness central then it may appear in a large number of users’route. The nodes with maximum BC values are the most congested nodes in thenetwork. Therefore, routes of the users are selected such that the sum of the BCof the nodes appearing in the user’s route is minimum. As a result, it is found thatthe path constructed through small betweenness central nodes is least congestedthan the other shortest paths. Simulation results show that the proposed routingstrategy effectively enhances the transmission capacity and reduce the load of thenetworks.In Section 2 some existing network models and proposed network model aredescribed. Section 3 discusses about the traffic flow models. Section 4 presentsthe simulation results, and in Section 5, conclusions and future research plan arediscussed.
2. Network Models
The concept of evolving network is given by Barabasi-Albert where, a new nodeattaches with an existing node through preferential attachment. In BA model, onlyaddition of nodes and links are considered and a new link will appear only whennew node arrives. But, in real scenario, links may appear or disappear at any time.Hence, two models are discussed: time varying communication network (TVCN)model and disassortative time varying communication network (DTVCN) model. Time Varying Communication Network Model
Time varying communication network (TVCN) model can be represented as G =( N, E, τ ), where, N is the set of nodes, τ is the set of time instants for which theovember 21, 2018 1:51 WSPC/INSTRUCTION FILE ws-mplb˙arxiv Authors’ Names
TVCN is defined, and E ⊆ N × τ × N × τ is the set of links. A dynamic link, e ab between source, a and destination, b in a TVCN, G is defined as an orderedquadruple e ab = ( a, t i , b, t j ), where a, b ∈ N ( G ) while t i , t j ∈ τ ( G ) are the sourceand destination time instants, respectively. In this paper, two types of dynamic linksare considered: temporal link and spatial link. Temporal link is used to connect thesame node at two distinct time instants, e aa is in the form of e aa = ( a, t i , a, t j ),where t i (cid:54) = t j . Spatial links connect two different nodes at the same time instant, e ab is in the form of e ab = ( a, t i , b, t i ), where a (cid:54) = b . t t t Fig. 1. A Time Varying Communication network at t , t and t time instants. In Fig. 1, a network at different time instances is shown. At each time instant,few nodes appear and some links are also getting added into the network. A TVCNmodel is proposed by us where, three operations are performed at each time in-stants; addition of nodes and links, removal and rewiring of links. Addition andrewiring of links are based on the preferential attachment where, probability Πthat a node i will be selected through preferential attachment is proportional to itsdegree and is given by, Π = k i (cid:80) j ∈ N k j . Removal of links are based on anti preferentialattachment and the probability Π (cid:48) of selecting node i with anti-preferential attach-ment is given by, Π (cid:48) = | N |− (cid:16) − k i (cid:80) j ∈ N k j (cid:17) . At each time instant t i ∈ τ , a newnode i is added to the network (expansion) and a number M ( ≤ n ) is selected fornetwork dynamics. Total ϑM links are added from the new node and (1 − ϑ ) M linksare used for alteration (rewiring and removal of links) in the network; 0 < ϑ < Instructions for typing manuscripts (paper’s title) is a network property in which nodes with similar attributes, such as degree, tendto be connected. DDC is used to divide networks into three types: assortative,disassortative and neutral networks. For assortative networks, hubs(small degreenodes) tend to link to other hubs (small degree nodes). In a disassortative network,hubs (small degree nodes) avoid each other, linking instead to small-degree nodes(hubs). While in the neutral network, the number of links between nodes is random.
Disassortative Time Varying Communication Network Model
In the network, a node i may generate packets with a packet generation rate, λ i and may forward packets according to its capacity, C i . For the smaller value ofthe packet generation rate, λ system remains in free flow state as every packet isgetting delivered. But with the increasing value of λ , a point is reached where sys-tem converts into congested phase and this point of phase transition is known ascritical packet generation rate, λ c . The value of λ c is affected by the topology ofthe network. DDC has an important influence on the structural properties of thenetwork and is one of the deciding parameters to find congestion in the network as well. Communication network comes under disassortative network. For that rea-son, in this paper, a disassortative TVCN (DTVCN) model is proposed to achieve ahigher value of λ c . A new node may be attached to the existing nodes by preferringhigher degree and disassortativity with the neighbors of the existing nodes. Somenodes rewire their links and attach with the nodes by preferring degree and disas-sortativity with the nodes’ neighbors. While some fraction of anti-preferential andcorrelated links are removed from the network. In this way, congestion is minimizedand we get the higher value of λ c .A new node, i will establish a new connection with a node v by preferringthe higher degree of the node v and normalized disassortativeness, ζ v of node v with its neighbors, N e ( v ) in the network. The value of ζ v is dependent on thecorrelation r deg ( v ) of node v with its neighbors. The r deg ( v ) measures maximumdisassortativeness of a node v with its neighbor. If a node v is more disassortativeto its neighbor then the selection probability will be increased. The probability Π rv of node i to attach with the node v may be defined as,Π rv = k v (cid:80) u k u ζ v (1)Where, ζ v = r deg ( v ) (cid:80) | Ne ( v ) | n =1 r deg ( v,n ) , r deg ( v ) = min r deg ( v, n ) , ∀ n : n ∈ N e ( v ), and thevalue of r deg ( v, n ) is scaled from the range of [ − ,
1] to [0 , e jk between nodes j and k is removed if the node j shows assortativeness0 < r deg ≤ k . The node j will rewire its connection and connectswith a node u with probability Π ru . The probability Π r (cid:48) v of selecting a node v withanti-preferential attachment and high correlation with its neighbors is given by,Π r (cid:48) v = (cid:32) − k v (cid:80) j k j (cid:33) (1 − ζ v )ovember 21, 2018 1:51 WSPC/INSTRUCTION FILE ws-mplb˙arxiv Authors’ Names .
3. Traffic Model
Traffic flow model is based on different routing algorithms in communication net-works. As the topologies of various networks are different hence, data delivery ca-pacity of each node is considered as different for different nodes, depending on theeffect of the node on the other nodes in the network. Some nodes are endorsed bythe nodes which are already congested and cause an increase in load at that node.The aim of the proposed approach is to assign more capacities to the congestednode in order to reduce data traffic in the network. As Eigenvector centrality isused to define the impact of the node on the the other nodes in the network and isconsidered in the computation of capacity of the node. Therefore, data forwardingcapacity, C i of node i may be defined as, C i = β ˆ x ( v ) | N | . (2)Where, ˆ x ( v ) is Eigenvector centrality of node v , ˆ x ( v ) = κ (cid:80) j ∈ N a ( vj )ˆ x ( j ) and κ is aconstant. The term, β is a considerably modest fractional value and is a controllingparameter for capacity of the nodes. Packets are forwarded through the shortestpaths from the source node to the destination node into the network. Therefore,the probability to pass through a node i is provided as g ( i ) (cid:80) | N | j =1 g ( j ) . At each time step,average number of packets generated is α | N | . Hence, the probability of node i togenerate a packet, λ i , λ i = α D| N | g ( i ) (cid:80) | N | j =1 g ( j ) . (3)Where D is the diameter of the network. The term, α is a small fractional value andis a controlling parameter for λ i for the node i . The capacity of a node increaseswith increase in the value of β . The sum of the capacities C i s of each node i is knownas the capacity C of the network. Similarly, the sum of the packet generation rates, λ i s of each node i is termed as the traffic load, λ , of the network. If traffic loadexceeds the traffic capacity of the network then the system will be in the congestedstate otherwise, it will remain in the free flow state. The point at which the phasetransition occurs from free flow state to congested flow state is known as the criticalpoint as well as the critical packet generation rate ( λ c ). There are three possiblerelationships between λ and C : (i.) λ < C implies the system in free flow state, (ii.) λ = C shows the boundary case for congestion and (iii.) λ > C allows a system incongestion. Local congestion of a node i is calculated according to its C i and λ i . If λ i > C i of node i then the node will be congested and these nodes increase overallnetwork congestion.The node with the maximum BC can be easily congested hence, it is necessaryto consider only the traffic load of this node. The maximum number of packetsthat can be forwarded through node max is C max and the total number of packetsovember 21, 2018 1:51 WSPC/INSTRUCTION FILE ws-mplb˙arxiv Instructions for typing manuscripts (paper’s title) accumulated at maximum BC node is λ c D| N | g ( max ) (cid:80) | N | j =1 g ( j ) . The value of α in Eq. (3) isreplaced by λ c and theoretical estimation of critical packet generation rate is givenby λ c = C max ( | N |− g ( max ) . The nodes with maximum packet forwarding capacity andmaximum BC are denoted by C max and g ( max ), respectively. The average numberof packets for a packet generation rate ( λ ) over a time window ∆ t can be calculatedby using an order parameter θ ( λ ) and is given by, θ ( λ ) = lim t →∞ Cλ (cid:104) ∆ P (cid:105) ∆ t ∆ P = P ( t +∆ t ) − P ( t ), P ( t ) is number of packets at time t and (cid:104) . (cid:105) shows the averagevalue over the time window ∆ t . (cid:104) ∆ P (cid:105) = 0 indicating that there is no packet in thenetwork for θ = 0, system is in free flow state. The ratio of C and λ provides thequantity of undelivered packet over ∆ t in the network.The communication network is shared by a large number of users. But the user’sinformation is not available at the time of design of network structure. If a nodeappears in the path of multiple users then the capacity of the node will be dividedequally among the users. Let us consider a set of R users who want to access thenetwork and each user r selects its source node and destination node randomly.There exist multiple shortest paths σ ru ( s → d ) for each user r , from the source node s to the destination node d for u ∈ [1 , K ] where K is the number of the SPs ( σ u ).Since the value of λ c of the network is affected by the maximum BC node andsometimes, it may happen that maximum BC node appears in user’s route whichleads to congestion in the route. Hence, the probability of selecting the shortestpath σ ru ( s → d ) depends on the value of sum of BC ( W g ( σ ru ( s → d ))) of the nodesappearing in the route of user r and may be defined as, W g ( σ ru ( s → d )) = (cid:88) n : n ∈ σ ru ( s → d ) g ( n ) (5)After calculating W g ( σ ru ( s → d ), we assign weight on each SP from s to d . The SP, σ r with minimum value of W g ( σ ru ( s → d ) is chosen for routing of the data throughthe network and it can be represented as, W g (min) = min u W g ( σ ru ( s → d )) (6)Similarly, the SP with maximum value of W g ( σ ru ( s → d ) can be denoted as W g (max).The topological structure of the BA model is shown in Fig. 3. Different routingstrategies ( W g (min), W g (max) and a random shortest path (SP)) are applied forassigning shortest paths to each user. The sum of BC, W g , of the nodes appearing inthe shortest paths for different routing strategies is computed. In Fig. 3, two users,User1, and User2, want to access the network. The value of W g , for the SP throughred, blue and green color solid lines, is 0 . , . . W g as 0 . , . . Authors’ Names S D S D Fig. 2. A scale-free network generated through the BA model with | N | = 20 and (cid:104) k (cid:105) = 4. Red,blue and green color solid line shows route of User1 and dashed line is for the route of User2 underdifferent routing strategies; W g (max), a random shortest path (SP) and W g (min) respectively. At a particular time instant, only R users want to access the network and fewnodes will be their point of interest. Therefore, instead of considering the wholenetwork to calculate λ c , a subnetwork consisting all the nodes appear in the users’route is only considered. As the size of the network increases, packets are morelikely to be routed to the nodes with higher BC and packets are more likely tobe accumulated at these nodes, resulting in traffic congestion. But, the proposedrouting strategy avoids larger BC nodes hence, we may get a higher value of λ c andthere will be less congestion on the nodes those are the part of the subnetwork.In a network, if a packet is generated then it needs to be delivered. Once thepacket is reached to its destination it is removed from the network. The averagetime (cid:104) T (cid:105) to deliver all the packets to the destination nodes are dependent on theroute of the user. Each node i generates packets with rate λ i . If λ i < C i then allthe packets will be forwarded to the next node towards its destination otherwise itneeds to wait at the end of the queue. After that, packets will be processed on afirst come first serve basis. From this, we can infer that the packets waiting in thequeue increase (cid:104) T (cid:105) and θ as well. The average packet travel time can be formulatedas, (cid:104) T (cid:105) = (cid:88) r : r ∈ R (cid:88) n : n ∈ σ r λ n / min n : n ∈ σ r C n (7)
4. Simulation and Result
For the dynamics of DTVCN model, the simulation starts by establishing the in-frastructure of the network. In this paper, the parameters are set with the value asovember 21, 2018 1:51 WSPC/INSTRUCTION FILE ws-mplb˙arxiv
Instructions for typing manuscripts (paper’s title) seed node n = 5, number M ( ≤ t ), fraction of newly added links ϑ range in (0 , γ is in the range of (0 . , | N | = 10 to | N | = 5 × . Any node can be included in the user’s ( s, d ) sets ormay participate in routing also. The capacity of each node is proportional to theEigen vector centrality of the node. At each time stamp, the degree of the nodeswill be different, hence, data forwarding capacity as well as data generation ratechange accordingly. Here, the range of DDC is scaled from ( − ,
1) to (0 , α and β may be any fractional value. c DTVCN ModelTVCN ModelBA Model
Network Model
Fig. 3. The value of λ c for different values of β . Each result value is the average of 10 independentrealizations of BA model, TVCN model and DTVCN model. In Fig. 3, critical packet generation rate, λ c is evaluated for the networks de-signed through different strategies. β is the controlling parameter for capacity C . As λ c is proportional to the capacity of the maximum BC node hence, it increases with beta . DTVCN model considers congestion at the time of topology design hence, itgives the higher value of λ c . In the TVCN model, addition and rewiring strategiesare based on preferential attachment and it will increase the degree of higher de-gree nodes which leads to increased congestion in the network. Therefore, the modelgives least value of λ c for different β .Network topology and routing strategies both are deciding parameters for theload at the network. If we increase the flow of packets through the congested nodethen, the total number of undelivered packets will increase accordingly. Multipleshortest path exists for the different source node and the destination node. If thepackets are sent through the path with W g (min) then the total number of accumu-lated packets θ is lower for distinct values of λ . The random SP offers the value of θ in between W g (max) and W g (min). The network designed through DTVCN modeland routes selected through W g (min) give least value of θ for different λ and thenetwork designed through TVCN model and the route selected through W g (max)gives highest value of θ . As a result, it is inferred that the network designed throughovember 21, 2018 1:51 WSPC/INSTRUCTION FILE ws-mplb˙arxiv Authors’ Names W g (min) W g (max) SP (c) DTVCN Model(b) DTVCN Model(e) TVCN Model (f) TVCN Model(i) BA Model(g) BA Model(a) DTVCN Model(d) TVCN Model (h) BA Model Fig. 4. The order parameter θ versus the packet generating rate λ under different routing strategies( W g (min), W g (max) and Shortest Path (SP)) and the network designed through DTVCN model,TVCN model and BA model. Red, green, blue, magenta and cyan color blocks correspond tothe simulations of β = 0 . , . , . , . , . DTVCN model outperforms than other models for all routing strategies.Congestion increases the number of the accumulated packet in the route of theuser and the average packet travel time, (cid:104) T (cid:105) . In Fig. 5, (cid:104) T (cid:105) is maximum whenthe routing strategy, W g (max) is applied on the network designed through all themodels (DTVCN model, TVCN model and BA model) while (cid:104) T (cid:105) is minimum for W g (min) routing approach. The value of (cid:104) T (cid:105) for the random shortest path liesin between W g (min) and W g (max) routing approaches. The performance of thenetwork designed through TVCN model for all routing approaches is lower thanthe other models. (cid:104) T (cid:105) for BA model and DTVCN model are approximately samefor their respective routing strategies.In Fig. 6, various network properties such as maximum betweenness centrality( g ( max )), average path length (APL), rich club coefficient ( RC ) and clusteringcoefficient ( C l C ) are studied. All these are the properties of the network as a wholeand not just of the individual node. This allows for the analysis of how the wholenetwork changes and not just the structure around some particular node. Richclub phenomenon is characterized when the hubs are on average more intenselyovember 21, 2018 1:51 WSPC/INSTRUCTION FILE ws-mplb˙arxiv Instructions for typing manuscripts (paper’s title) T W g (min) T W g (max) T SP T T T T T T (c) DTVCN Model(b) DTVCN Model(a) DTVCN Model(d) TVCN Model (e) TVCN Model (f) TVCN Model(g) BA Model (h) BA Model (i) BA Model Fig. 5. Average packet travel time, (cid:104) T (cid:105) versus packet generating rate λ for different value ofcontrolling parameter, β under different routing strategies ( W g (min), W g (max) and ShortestPath (SP)) and the network designed through DTVCN model, TVCN model and BA model.Red, green, blue, magenta and cyan color square blocks correspond to the simulations of β =0 . , . , . , . , .
200 400 600 800 1000 |N|
TVCN Model g(max)APLRCCC
200 400 600 800 1000 |N|
BA Model
200 400 600 800 1000 |N|
DTVCN Model
Fig. 6. Network size | N | , g ( max ), average path length (APL), rich club coefficient ( RC ) andclustering coefficient ( C l C ) for all the models; DTVCN, TVCN and BA when, the network size, | N | varying from 10 to 10 . Each result value is the average of 20 independent realizations ovember 21, 2018 1:51 WSPC/INSTRUCTION FILE ws-mplb˙arxiv Authors’ Names interconnected than the nodes with smaller degrees. When the nodes are with alarge degree than k tends to be more connected than the nodes with the smallerdegree. The rich club phenomenon refers to the tendency of hub nodes to connectwith other higher degree nodes than the nodes with a smaller degree. Presence of RC increases load and congestion on the connecting link between two hubs. Mostof the users want to send data through shortest paths i.e., through hub nodes andcongestion at hub nodes will reduce C and efficiency of the networks. The proposedDTVCN model takes care of congestion and offers the lowest value of RC thanthe other two models; the BA model and TVCN model. The value of g ( max ) forDTVCN model is minimum and TVCN model is maximum for different value of | N | . The average path length (APL) of the network structured through all themodels lie between 2 to 3. The clustering coefficient, C l C decreases with increasingvalue of network size, | N | and the C l C of BA model is minimum and TVCN modelis maximum. The length of the average shortest path of all the models increaseswith increasing value of the | N | . The quantitative value of the network propertiesis shown in Table 1. Table 1. Network size | N | , g ( max ), average path length (APL), rich club coefficient ( RC ) andclustering coefficient ( C l C ) for all the models; DTVCN, TVCN and BA when, the network size, | N | varying from 10 to 10 . Network g ( max ) Average RC Clustering | N | Model Path Length CoefficientDTVCN 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5. Conclusion and Future Direction
In this paper, a time-varying network topology is designed by using preferentialattachment and correlation of the nodes in the network. The network structureconsiders rewiring of some links of the congested node and also the removal ofovember 21, 2018 1:51 WSPC/INSTRUCTION FILE ws-mplb˙arxiv
Instructions for typing manuscripts (paper’s title) some anti-preferential and correlated links in the network. The correlation helpsto mitigate traffic congestion in the network and provides a higher value of thecritical packet generation rate, λ c . After that, user’s data is sent through three typesof shortest paths; a path with a minimum value of W g , the shortest path with amaximum value of W g and a randomly selected shortest path. The proposed routingapproaches are applied to the network designed through different models, namely,the BA model, the TVCN model, and the proposed DTVCN model. Simulationresults show that traffic capacity can be increased considerably and traffic loadsare also reduced by sending data through W g (min). Moreover, the average packettravel time (cid:104) T (cid:105) is reduced compared with the routing approach through W g (max)and by choosing a random shortest path (SP). Further, we analyzed various networkproperties and find that existence of higher betweenness centrality, g ( max ) and richclub (RC) phenomenon have a negative impact on the performance of the networks.In future work, we would like to expend our work to the more realistic envi-ronment and those can be created using network simulators. Network congestioncan be studied for varying packet generation rate on the network designed throughsome other strategies. References
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Scientific reports (2016) 19059.19. X. Yang, J. Li, C. Pu, M. Yan, R. R. Sharafat, J. Yang, K. Gakis and P. M. Pardalos, Physical Review E (2017) 012322.20. L. Zhao, T. H. Cupertino, K. Park, Y.-C. Lai and X. Jin, Chaos: An InterdisciplinaryJournal of Nonlinear Science (2007) 043103.21. S. Kumari and A. Singh, Advances in Complex Systems (2018) 1850006.22. M. E. Newman, Physical review letters (2002) 208701.23. P. Fronczak, The European Physical Journal B (2012) 351. Appendix. List of Symbols
Symbols Meaning N Set of nodes E Set of links T Life span of the networks λ i Packet generation rate of node iC i Packet forwarding rate of node i Π ri probability of a node i will be selected through preferential attachmentand least correlated with its neighborΠ r (cid:48) i probability of a node i will be selected through anti-preferential attach-ment and highly correlated with its neighbor λ c Critical rate of packet generation s, d