1 of 7

AUTOMATED TOPOLOGY CONTROL FOR WIDEBAND DIRECTIONAL LINKS IN AIRBORNEMILITARY NETWORKS

Daniel J. Van Hook, Mark O. Yeager, John D. LairdMIT Lincoln Laboratory

Lexington, MA

ABSTRACT

Future airborne military operations will rely on emergingwideband directional RF and optical links for machine tomachine networking. Management and control for suchdirectional wireless links on mobile platforms is a signifi-cant challenge that does not arise in fixed terrestrial net-works. As a consequence, new protocols and approachesare needed. This paper presents an architectural frame-work for control of wideband directional links in airbornemilitary networks. Several distributed algorithms forautomated topology management that we have devised andprototyped are described and compared. Results and expe-rience from simulations and emulations are presented.

INTRODUCTION AND BACKGROUND This paper focuses on algorithms for forming an airbornenetwork from high capacity directional links. Recent re-lated work can be found in [1-5].

Airborne networking is an important capability for futurewarfighting. The U.S. Department of Defense (DoD) isinvesting heavily in this area through a number of pro-grams including the Joint Tactical Radio System (JTRS),Family of Advanced Beyond Line-of-Sight Terminals(FAB-T), Multi-Platform Common Data Link (MP-CDL),and Transformational Communications (TC). A funda-mental goal is to improve warfighting effectiveness bygetting the right information where it is needed at the righttime. The DoD’s vision is a flexible, universal airbornenetwork capability similar to that provided by the Internet.IP technology (Internet Protocol) is a key enabler for thisvision.

This airborne network vision is a departure from tradi-tional airborne communications. Historically, radios havebeen dedicated to each sensor or command and controlapplication. In general, radios have not been shared bysensors or applications. This is in contrast to the DoD’s

This work is sponsored by the United States Air Force under Air Force Con-

tract #FA8721-05-C-0002. Opinions, interpretations, recommendations andconclusions are those of the authors and are not necessarily endorsed by theUnited States Government.

airborne network vision. The airborne network of the fu-ture will supply communications services for multiple sen-sors and applications.

The architecture of the airborne network will likely includea backbone capability established between widebody air-craft [6]. This backbone will be formed from high capacityRF and optical point-to-point links. The backbone willprovide data transport services for dynamically constitutedsmall combat networks, both legacy and emerging. Thebackbone will also provide access to satellite networkingcapabilities for smaller aircraft that cannot afford suchconnectivity.

Establishing and maintaining an airborne network back-bone composed of directional, high capacity links is a sig-nificant challenge. Issues that need to be considered in-clude:

• Topology choices, i.e., which links to form and whichradios to use

• Outages due to blockage by aircraft parts and terrain orby clouds in the case of optical links

• Mobility induced topology changes• Disruption of large, time critical data flows• Range limitations• Impact of topology changes on routing protocols, trans-

port protocols, and applications• Vulnerability to both malicious attacks and stochastic

failuresIn essence, the problem is one of dynamically provisioningreliable connectivity for time critical applications ontimescales on the order of minutes or less without signifi-cant disruption. This is in contrast with the terrestrial In-ternet in which fiber is laid and lighted on timescalesranging from weeks to months and transitions can gener-ally be prepared for and scheduled well in advance.

In this paper we address the problem of establishing andmaintaining an airborne network topology composed ofhigh capacity directional links. First, we describe an ar-chitectural approach and framework for this problem. Us-ing this architectural approach, we then describe algo-

2 of 7

rithms we have developed for controlling airborne networktopologies. Finally, we present experiments and results.

ARCHITECTURE AND APPROACH

In this section we describe an architecture and approachfor topology algorithms for forming an airborne network.The architectural context for our work is 4xComm. [7] The4xComm architecture concept is depicted in Figure 1.

Figure 1. 4xComm Architecture Concept

The focus of 4xComm is a unified IP-based airborne net-work composed of wideband (WB) and narrowband (NB),line-of-sight (LOS) and beyond-line-of-sight (BLOS)links. The “4x” part of the 4xComm moniker is derivedfrom the four combinations WB/LOS, WB/BLOS,NB/LOS, and NB/BLOS. Key architectural components of4xComm shown in Figure 1 are:

• Switched data network overlay. User data from appli-cations is carried over a network formed from the avail-able communications resources.

• Orderwire network overlay. The orderwire is a protocolfor transporting network control information. Examplesof network control information include link status, nodepositions, and traffic flow specifications. The orderwiredoes not depend on a dedicated radio; instead, the or-derwire is an overlay on top of existing communica-tions resources. The orderwire supports a distributeddatabase model. Data objects are kept consistent acrossthe system using highly resilient and low data rate pro-tocols.

• Brokering service. The brokering service is imple-mented by a collection of layered broker agents thatmanage and control radios, links, routers, and othercommunications devices. Broker agents exchange net-work control information using the orderwire service.

• Middleware. A middleware layer provides servicesthrough which applications and the brokering servicecan interact. Examples include resource reservation,flow specification, and status monitoring.

Two salient aspects of the 4xComm concept are automatedcontrol and self-organization of the airborne network. Webelieve that automation and self-organization are essentialfor meeting the need for highly dynamic machine-to-

machine communication and for leveraging scarce andvaluable operator expertise. We have prototyped initialversions of the 4xComm components and successfullydemonstrated their operation in conjunction with the JointExpeditionary Force Experiment (JEFX) in 2004.

The topology control function is allocated to the 4xCommbrokering service in our architecture concept. The primaryactivities related to topology control are 1.) informationgathering including node positions, link status, and flowspecifications; 2.) topology calculation; and 3.) topologyimplementation, which consists of setting up/tearing downlinks by pointing antennas and selecting transmit/receivemodes. Topology control is a continuous, ongoing processthat must, in general, account for changing node positions,link state, traffic demands, network membership and otherfactors.

The architecture for our topology control algorithms isdepicted in Figure 2. This figure shows a layered controlstructure consisting of an upper deliberative layer and alower reactive layer. This layering permits a useful parti-tioning between longer time-scale, more globally optimalprocessing in the deliberative layer and shorter timescale,more localized processing in the reactive layer. The delib-erative layer calculates topologies based on orderwire in-formation such as node positions, link state, and poten-tially traffic flows. The role of the lower reactive layer isto implement the deliberative layer’s topology choices andto directly control the radios by coarse antenna pointingand transmit/receive mode selection. Refined pointing,acquisition and tracking are left to individual radios. Fig-ure 2 also alludes to exploitation of higher level applica-tion data such as plans, environment (weather, terrain),tactical situation, and policy in the deliberative controller.These are subjects of future development efforts and havenot been implemented, as indicated by the dashed box.

Figure 2. Deliberative/Reactive Controller Layering

3 of 7

Our algorithm approach is fully distributed and requires nocentralized server. We employ the concept of distributedreplicated computation on identical data. Although the or-derwire is designed to maintain consistent network controldata across the system, our algorithms are tolerant of tran-sient discrepancies. In our approach, each broker agentindependently calculates a topology consistent with otherbroker agents and then implements its portion by control-ling its own radios. To ensure that all broker agents calcu-late the same topology our algorithms exploit a number oftechniques. These include time stamping all orderwireevents and data, gridding positions to minimize positionerrors, accessing data structures consistently throughoutthe system (facilitated by lexicographic ordering of objectidentifiers, for example), and consistent seeding of randomnumber generators. In addition, our topology algorithmsare frame based and run at periodic intervals. Finally, weassume system clocks synchronized to levels straightfor-wardly available via NTP, the Network Time Protocol.

As an aside, this approach to distributed replicated com-putation is analogous to that employed in the widely de-ployed and successful OSPF routing protocol. UnderOSPF, each router independently executes the same algo-rithm on the same data to arrive at the same result. Wehave adopted this type of approach for the topology con-trol problem reasons that include

• Robustness, no single point of failure• Low overhead for exchanging network control infor-

mation over the orderwire• Rapid future growth in computational power relative to

wireless communication capacityIn addition, this approach can offer good scalabilitythrough hierarchy.

TOPOLOGY ALGORITHMS

In this paper we consider three topology algorithms usingthe architecture approach outlined in the previous section.The first uses a genetic algorithm to choose topologies thatmaximize delivered data. The second is based on heuristicsthat try to achieve a well-connected network. The thirdforms random topologies. We have chosen these three al-gorithms for this paper because they offer contrasts withrespect to computational cost, assumptions about informa-tion available to the algorithm, and underlying approach.

All three algorithms have been implemented in C++ aspart of the 4xComm Broker software component. All threealgorithms support the fully distributed topology controlapproach outlined in the previous section. In addition tosoftware-based stand-alone testing such as that describedlater in this paper, we have exercised these algorithms as

components of larger airborne network emulations incor-porating off-the-shelf commercial routers, protocols, ap-plications, and hardware in the loop.

Genetic Algorithm Topology Algorithm (GAOPT)

We have developed a Genetic Algorithm (GA) based algo-rithm for calculating network topologies. A GA [8] can beviewed as a way to search for an optimal solution. In ourcase we use a GA to optimize the network topology fordelivering maximum data. GAs are modeled after theprocess of biological evolution. GAs maintain a populationof candidate solutions to which mutation and crossoveroperators are applied to generate new population members.The fitness of any particular population member is deter-mined by a fitness function, sometimes called an objectivefunction. A GA proceeds until either the solution con-verges and no further improvement is obtained or until amaximum generation count has been exceeded.

We have based our implementation on a downloadableopen source library GAlib [9], which provides a frame-work for executing a GA. To use GAlib, a user defines aproblem-specific representation for the population mem-bers, implements mutation, crossover, and fitness func-tions, and selects from a menu of selection schemes andalgorithm choices supported by the open source library. Inour context, we have defined a graph data structure used torepresent a population of network topologies.

We have defined mutation and crossover operators thatmodify members of the population. The mutation operatorrandomly selects two radios in a population member andforms a new link between them. The crossover operatorexchanges randomly selected subgraphs of two populationmembers. For both of these operators only feasible linksare retained. A feasible link is one for which the candidateradios are within range. The implementation also endeav-ors to link any radios that become unlinked as a result ofapplying mutations or crossover operators.

We presently use only a single objective – delivered data –in our fitness function. The fitness function ranks membersof the population according to the ability of the corre-sponding topology to deliver data. Topologies better ableto deliver data are considered more fit. We calculated de-livered data using a heuristic because an exact solution istoo computation intensive for real-time operation in apractical system. Our heuristic method calculates a shortestpath routing, aggregates flows traversing each link, deter-mines the proportion of each flow that is delivered basedon the most congested link (the “bottleneck”) along itspath, and finally forms a sum of the data delivered. Thisfitness function evolves members of the population to-

4 of 7

wards network topologies delivering higher proportions ofoffered traffic load.

We find a GA-based approach attractive because it is rela-tively simple and mechanical to implement, given theavailability of a good GA framework such as GAlib. Care-ful attention must be paid to algorithm tuning to get goodresults, however. We also believe that future availability ofever cheaper and more powerful processing resources willmake GA-based approaches even more attractive in com-ing years.

Mesh Topology Algorithm (MESH)

We have designed MESH with the goal of efficientlyforming highly connected networks. MESH is so namedbecause it attempts to form a well-connected mesh. Thisalgorithm applies a sequence of heuristic operations thatwe have found to work well. We have designed MESH tobe highly efficient. In contrast with GAOPT, MESH usesno flow information to determine better links.

MESH executes the sequence of steps outlined below. Thealgorithm gives priority to linking nodes that are not wellconnected. MESH also attempts to minimize redundantlinks (links between nodes that are already directly linked).In all cases, only feasible links are considered. Links areconsidered feasible only if the associated nodes are withinrange. The algorithm proceeds by considering all nodes ateach step before going to the next step.

• Nodes are put into a node pair list sorted by distanceapart. Node pairs whose distance apart exceeds themaximum range for the radios are ignored.

• If there is an existing topology then break redundantlinks.

• Traverse the entire node pair list closest pair first look-ing for nodes that have zero links. Form links betweennode pairs for which both nodes have zero links.

• Traverse the entire node pair list closest pair first look-ing for nodes that have zero links. Form links betweennode pairs for which one node has zero links and theother node has one link.

• Traverse the entire node pair list closest pair first look-ing for nodes that have zero links. Form links betweennode pairs for which one node has zero links and theother node has two or more links.

• Traverse the entire node pair list farthest pair firstlooking for nodes that have one link. Form links be-tween node pairs for which each node has one link.Form the second link per node only if it is not redun-dant, but not if the other node is within two hops (toavoid small loops).

• Traverse the entire node pair list farthest pair firstlooking for nodes that have one link. Form links be-tween node pairs for which each node has one link.Form the second link per node only if it is not redun-dant, and allow the link if the other node is within twohops.

• Traverse the entire node pair list closest pair first. Formanother link per node but only if the new link is not re-dundant.

• Traverse the entire node pair list closest pair first. Formanother link per node allowing redundant links.

• Traverse the entire node pair list closest pair first. Formas many links as possible allowing redundant links.

Random Topology Algorithm (RANDOM)

For purposes of comparison with the two previous algo-rithms we have implemented a random topology generator.RANDOM forms topologies by randomly pairing radiosand forming links. Only feasible links are retained, i.e.,those for which the associated nodes are within range.

RANDOM is very simple and requires only a very limitedamount of knowledge. It can be viewed as a stand-in forad-hoc topology formation approaches that form links us-ing reactive mechanisms only. We use it as a baseline casefor comparison purposes in the next section.

EXPERIMENTS AND RESULTS

We have compared the topology algorithms described inthe previous section using a variety of metrics. Our com-parison addresses the quality of the topologies calculatedas well as computational cost. We employ a flexible soft-ware framework through which we set up and executemultiple iterations of each algorithm using a range of pa-rameter values. Although our topology algorithms are con-structed to operate in the fully distributed architecture out-lined in the previous section we run only a single instanceat a time for this comparison. This comparison does notaddress algorithm dynamics over time or the systems is-sues of the fully distributed topology algorithm architec-ture previously outlined.

The variables we have employed in our topology algorithmcomparison are

• Number of nodes: 5 through 40, in increments of 5• Number of radios per node: 2, 3, 4, or 5• Geographic extents, specified as the diagonal of a

square operating area: 1, 2, or 3 times the maximumline-of-sight range of the radios

We assume all nodes to be identical with the same numberof identical radios per node. Radios are modeled as capa-

5 of 7

ble of forming bidirectional point-to-point links of fixedcapacity and limited range. Nodes are randomly placedwithin the bounds defined by the geographic extents.

In addition, we have used four different traffic models inour study, listed below. Although idealized, the differentcharacteristics of these traffic models highlight variousaspects of our topology algorithms. We have selected traf-fic levels that result in heavy network loading because wehave found that all the algorithms perform similarly withrespect to data delivery under lightly loaded conditions.

• Uniform. Each node transmits to all other nodes.• Half-uniform. Each node transmits to half the other

nodes (a randomly selected subset).• One-to-one. Each node transmits to exactly one other

node (randomly selected).• Hub-spoke. All nodes transmit to the same node (ran-

domly selected).All flows for a given traffic model are the same size. Inaddition, we hold the sum of all flows emanating fromeach node constant by scaling the transmit rate as we varythe total number of nodes or the number of radios per nodeacross test runs.

We have selected a diverse collection of metrics to char-acterize and compare our topology algorithms. The metricswe calculate are:

• Delivered data. We use a linear program to calculatemaximum delivered data, given a shortest path routingof the flows. The linear program constraints we use arethat link capacities are not exceeded and that the actualvalue of each flow is bounded by 0 and its maximum.

• Hop count. We calculate the number of hops along theshortest path route for each flow and form an averageover all the flows.

• Execution time. We collect execution time by instru-menting the code with gettimeofday() system calls.

• All Terminal Reliability (ATR). ATR [10][11] countsthe number of spanning trees in a network. A spanningtree is a connected, acyclic graph containing all verticesof a graph but not necessarily all edges. The morespanning trees a network has the less likely it is thatremoving an edge will leave some vertices/nodes un-able to communicate with the rest. Larger ATR valuesimply lower vulnerability to stochastic and maliciousfaults.

• Betweenness Centrality (BC). BC methods [12][13]rank the nodes or links of a network by the number ofshortest paths that go through each node or links. BChas been developed by social scientists to identify keyindividuals social networks. In our context, we believe

BC provides a measure of how vulnerable a network isto failures. Larger BC values indicate that particularnodes or links carry a greater portion of the traffic.Larger BC values imply higher vulnerability to sto-chastic and malicious faults. We collect three BC met-rics, all normalized by the number of nodes: the meanvalue; generalized vulnerability (standard deviation di-vided by mean); acute vulnerability (maximum dividedby mean). We use the algorithm developed by Brandes[14] as implemented in the JUNG (Java Universal Net-work/Graph) framework [15] to compute BC metrics.

Our experiment framework consists of a collection of pa-rameter-driven scripts and programs. We have verifiedcorrectness and repeatability through a set of known testcases. The framework sequentially steps through ranges ofparameter variables to generate input data for the topologyalgorithms. Each topology algorithm is applied to the sameset of inputs. The resulting topologies are analyzed to gen-erate and log metrics. The graphs in Figures 3 through 6show a small subset of our accumulated analysis results.Each point in these graphs represents the mean value of 20iterations for three radios per node, the one-to-one trafficmodel, and diagonal geographic extents of twice the as-sumed range of the radios.

Figure 3 shows results for the delivered data metric. Thegraph shows that GAOPT calculates topologies capable ofdelivering more data under heavily loaded conditions thaneither MESH or RANDOM. We performed a one-wayANOVA on the delivered data metric and determined thatthere is a statistically significant difference with a p-valueof less than 0.05. Our general observation from thisevaluation is that GAOPT is more effective with respect tothe delivered data metric for smaller numbers of radios (2or 3) and/or less dense networks and/or smaller numbers ofnodes. MESH is generally more effective than RANDOMbut only for small numbers of nodes. These results illus-trate that careful choice of topology is more critical underconstrained conditions than when connectivity is poten-tially richer. Under constrained conditions, best perform-ance can be achieved by matching topologies to dataflows.

Figure 4 shows results for the hop count metric. The graphshows that GAOPT calculates topologies yielding averagehop counts approximately one fewer than that for MESHand RANDOM. We believe that this advantage is a by-product of GAOPT’s fitness function, which only takesaccount of delivered data. Maximum data is delivered byselecting topologies that minimize congestion and loss.Paths in these topologies tend to be shorter. In addition,MESH and RANDOM yield comparable results.

6 of 7

Figure 3. Comparison of Delivered Data

Figure 4. Comparison of Average Hop Count

Figure 5. Comparison of Computation Time

Figure 6. Comparison of All Terminal Reliability

Figure 5 shows computation time for our three algorithms.GAOPT clearly requires significantly more computationthan either MESH (up to tens of milliseconds) orRANDOM (up to hundreds of milliseconds), especially forlarger network sizes. We fit power law functions to thedata and found that computation time for GAOPT scalesapproximately as N2.5, MESH approximately as N, andRANDOM approximately as N2 where N is the number ofnodes. R-squared values in all cases were greater than0.99. While these results raise questions about the practi-cality of real-time, online optimization using an algorithmlike GAOPT, we note that the approximate scale of an air-borne backbone is on the order of 10 nodes. For this sizenetwork, computation times are a few seconds using off-the-shelf computers. We expect future improvements toaccrue from Moore’s Law, parallel implementations, hier-archical algorithm approaches, and/or specialized hard-ware.

Figure 6 shows results for the ATR metric, which countsspanning trees. Our observation is that MESH performsslightly better than RANDOM while GAOPT is much lesseffective for larger network sizes. This behavior is inde-pendent of the number of radios. Our interpretation is thatMESH and RANDOM produce less vulnerable topologiesthan GAOPT. We explain these differences between thealgorithms by noting that MESH intrinsically drives to-wards a well connected network which yields large num-bers of spanning trees. The fitness function used inGAOPT, however, selects only for topologies deliveringmaximum data and ignores measures of connectivity. Wenote that extending the GAOPT fitness function to incor-porate connectivity and vulnerability measures is feasible.

7 of 7

The BC metric counts shortest paths traversing each node.We believe BC measures vulnerability of a topology. Ourresults for BC show that increased numbers of radios yieldimprovements. This effect is independent of traffic modelfor MESH and RANDOM, though MESH performs betterfor smaller numbers of nodes. GAOPT, in contrast, showsa strong additional dependence on traffic model withasymmetric traffic patterns (one-to-one, hub-spoke) per-forming significantly less well than symmetric patterns(uniform, half-uniform). We believe these results are con-sistent with previous results showing that increased so-phistication has a higher payoff when there are more con-straints, i.e., when the choices that are made are morecritical.

SUMMARY AND CONCLUSIONS

In this paper we have described a fully distributed archi-tecture approach for airborne network topology controlalgorithms. The approach does not rely on a central servernode for topology calculations. We have implemented keycomponents of this architecture and tested it both in labo-ratory emulations with hardware in-the-loop and in a liveflight test.

We have presented three topology algorithms with con-trasting characteristics including computational cost, as-sumptions about available information, and underlyingapproach. The first algorithm (GAOPT) uses a genetic al-gorithm approach to construct topologies that maximizedelivered data. The second algorithm (MESH) constructstopologies using heuristics that provide good connectivitybut without regard for data flows. The third algorithm(RANDOM) forms random topologies.

We observed that GAOPT generally delivers more datawith better delay characteristics under heavy loads. Allthree algorithms perform equally with respect to thesemeasures on lightly loaded networks. GAOPT advantagesare most evident for constrained scenarios, i.e., sparsenode distributions and/or few radios (2 or 3 per node). Weobserve that such constrained scenarios are likely for afuture deployed airborne network backbone. For thissituation the careful matching of topology to applicationdata flows afforded by GAOPT can be advantageous.

MESH and RANDOM are significantly less computation-ally costly than GAOPT. For large numbers of nodes wequestion the practicality of GAOPT or similar algorithms.However, we note that for anticipated sizes of an airbornenetwork backbone (perhaps 10 or so nodes), the computa-tional load for GAOPT is reasonable. Considering Moore’sLaw and the potential for parallel implementation, hierar-chical algorithm approaches, and specialized hardware, we

believe that further consideration of extensions to GAOPTfor airborne network backbones is warranted.

MESH yields topologies that are significantly less vulner-able to stochastic and malicious faults than GAOPT andslightly superior to RANDOM. We believe this is a sideeffect of focusing on rich connectivity on the part ofMESH and RANDOM. We note that criteria related tovulnerability could be added to the GAOPT fitness func-tion to select for less vulnerable topologies.

We believe that hybrid approaches combining the bestcharacteristics of heuristic approaches such as MESH andoptimization approaches such as GAOPT are attractive.For example, an optimization algorithm selecting potentialtopologies and parameter values running in the back-ground could supply choices for a highly efficient and re-sponsive heuristic algorithm running in the foreground.

REFERENCES

[1] Aniket Desai, Jaime Llorca, Stuart Milner, “Autonomous Recon-figuration of Backbones in Free Space Optical Networks,” IEEE-MILCOM 2004.

[2] Christopher C. Davis, Igor I. Smolyaninov, Stuart D. Milner,“Flexible Optical Wireless Links and Networks,” IEEE Commu-nications Magazine, vol. 41, no. 3, March 2003.

[3] Jifang Zhuang, Michael J. Casey, Stuart D. Milner, Steven A.Gabriel, Gregory B. Baecher, “Multi-Objective OptimizationTechniques in Topology Control of Free Space Optical Net-works,” IEEE-MILCOM 2004.

[4] Aniket Desai, Jaime Llorca, Stuart Milner, “Obscuration Minimi-zation in Dynamic Free Space Optical Networks Through Topol-ogy Control,” IEEE-MILCOM 2004.

[5] Stuart D. Milner, Christopher C. Davis, “Hybrid Free Space Op-tical/RF Networks for Tactical Operations,” IEEE-MILCOM2004.

[6] Leonard J. Schiavone, “Airborne Networking – Approaches andChallenges,” IEEE-MILCOM 2004.

[7] Daniel J. Van Hook, Stephen M. McGarry, James O. Calvin,“The 4xComm Architecture Concept for Wireless Airborne Net-works,” IEEE-MILCOM 2003.

[8] David E. Goldberg, “Genetic algorithms in search, optimization,and machine learning”Addison Wesley, 1989.

[9] Matthew Wall, GAlib Genetic Algorithm Library,http://lancet.mit.edu/ga

[10] J.M. Anthonisse, “The Rush in a Directed Graph,” TechnicalReport BN 9/71, Stichting Mathematisch Centrum, Amsterdam,1971.

[11] C.J. Colbourn, “The Combinatorics of Network Reliability,”Oxford: Oxford University Press, 1987, 0-19-504920-9.

[12] N. Fard and T.H. Lee, “Spanning Tree Approach in All-TerminalNetwork Reliability Expansion,” Computer Communications,vol. 24, pp. 1348-1353, 2001.

[13] L.C. Freeman, “A Set of Measures of Centrality Based on Be-tweenness,” Sociometry, 40:35-41, 1977.

[14] U. Brandes, “A Faster Algorithm for Betweenness Centrality,”Journal of Mathematical Sociology, vol. 25, pp. 163-177, 2001.

[15] J a v a U n i v e r s a l N e t w o r k / G r a p h f r a m e w o r k ,http://jung.sourceforge.net/.