Modeling network technology deployment rateswith different network models

Yoo ChungNational Institute for Mathematical Sciences

Daejeon, KoreaEmail: chungyc@nims.re.kr

Abstract—To understand the factors that encourage the de-ployment of a new networking technology, we must be ableto model how such technology gets deployed. We investigatehow network structure influences deployment with a simpledeployment model and different network models through com-puter simulations. The results indicate that a realistic model ofnetworking technology deployment should take network structureinto account.

Index Terms—network, deployment, adoption, model

I. INTRODUCTION

On January 31, 2011, the Internet Assigned Numbers Au-thority allocated the last free blocks of IPv4 addresses to thefive Regional Internet Registries [1]. It will not be long beforeeven the Regional Internet Registries run out of their own IPv4addresses to give out. This is the sort of news which givesurgency to the need for IPv6 to replace IPv4 for its greatlyexpanded address space. And yet IPv6 deployment rates, whilethey are steadily climbing, remain extremely low [2], [3].

Looking to the distant future, clean-slate design approachesare being proposed to solve problems that seem to be insur-mountable or too messy to solve with the design of the Internettoday [4], [5]. Perhaps the technical community may go furtherthan just using a clean-slate design approach for generatingnew ideas and converge on a clean-slate design for a newInternet that can solve all of the problems with the presentInternet. But if IPv6 can hardly get deployed even when thereis a clear perceived need for it today, how can we expect thisclean-slate Internet to replace the by-then obsolete Internet?

In order to figure out how to design new networkingtechnologies so that they are quickly adopted, we need tounderstand the factors that are behind the spread of suchtechnologies. To understand these factors, we need to be ableto model the deployment of these technologies to a reasonabledegree of accuracy.

This work does not attempt to define a comprehensivemodel for the deployment of new networking technologies, nordoes it try to suggest what policies or design principles maypromote the deployment of such technologies. Instead, thiswork focuses on how different network structure may influencethe spread of a new networking technology throughout anetwork, an understanding of which may be important in thedesign of a realistic model. Using computer simulations, we in-vestigate how different network models influence deploymentrates. Other work has focused on the deployment of specific

networking technologies [6] or on the role of “converters”which provide compatibility with an old technology in thedeployment of a new technology [7], [8].

The rest of the paper is structured as follows. Section IIdescribes the deployment model used in the computer simula-tions, which models how each node in the network decidesto make the transition to the new networking technology.Section III describes the different network models we lookedat and shows the growth curves of how a new networkingtechnology spreads with each network model. We end thepaper in section IV, where we discuss how the models mightapply to a real world system, the limitations of the modelingdone here, and some concluding remarks.

II. DEPLOYMENT MODEL

There are a myriad of factors that affect whether a givennode in a network decides to adopt a new networking tech-nology, which makes modeling its spread a very difficultproblem. Our focus here is on how different network modelscan influence the spread of a new networking technology,however, so we use a very simple deployment model as appliedto networks of fixed size.

We also assume that once a node decides to adopt a newnetworking technology, it will not want to revert to the oldertechnology. In fact, we assume that there is only one oldnetworking technology and one new networking technology.To avoid dealing with the influence of converters whichprovide compatibility between the old and new technologies,we assume a dual-use situation, where nodes with the newtechnology can continue to use the old technology.

We model the spreading deployment in a network on a fixedgraph G = (V,E) representing a network, where V representsthe set of nodes and E represents the set of edges betweenthe nodes. The state of deployment throughout the networkchanges in one discrete time step, where each node that has notalready transitioned to the new networking technology decideswhether to transition or not. We define D(t) to be the set ofnodes that have transitioned to the new networking technologyso far at time step t.

How does a node decide whether to transition or not? Anode would be more inclined to transition if more of itsneighbors have already transitioned, since this would allow itto take increasing advantage of the new networking technologyto communicate with others. Contrast this to the case when

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Fig. 1. The logistic function, a type of sigmoid function.

none of its neighbors have transitioned, in which case the oldnetworking technology would have to be used to communicatewith any other node in the network. A concrete examplein the real world would be a single host supporting IPv6inside an IPv4 subnet, in which case the node cannot useIPv6 to communicate with nodes outside the subnet, ignoringtunneling to simplify matters.

A node would also more inclined to transition if lots ofnodes in the network as a whole have already transitioned,sort of not wanting to “fall behind the times”. It is unclearhow this combines with the previous effect to influence thedecision of a node on whether to transition or not, but we willmodel it as multiplicative rather than additive. This would havethe effect of a node being much more inclined to transitionif its neighborhood has already transitioned, given the samegeneral transition rate for the entire network.

On the other hand, transitioning to a new technology isnever free, and the cost associated with a transition woulddiscourage a node from making the transition. We modelthe cost differently for each network model in section III toreflect what the nodes and edges represent in the correspondingmodel, although we do make the simplifying assumption thatthe costs do not change over time.

Given these factors, we define a utility value u(v, t) whichrepresents how inclined a vertex v is towards making atransition at time step t:

u(v, t) =1

α(hG(t)h(v, t)− c(v))

where hG(t) is the total number of transitioned nodes in thenetwork, h(v, t) is the number of nodes adjacent to v that havealready transitioned, c(v) is a cost function that is specific toeach network model, and α is a scaling factor that is usedlater:

hG(t) = |D(t)|

h(v, t) = |{v′ | (v, v′) ∈ E ∧ v′ ∈ D(t)}|

However, a node does not decide whether to transition ornot based on whether the utility value is positive or not. Wetake advantage of the logistic function, which is graphed out inFig. 1, to represent the probability that a vertex v will transitionor not at time step t:

p(v, t) =1

1 + e−u(v,t)+β

Taking advantage of the logistic function ensures that p(v, t)lies between 0 and 1 while having increasing value withincreasing utility value. β is a baseline that shifts the logisticfunction to the right, so that a zero utility value will have lowprobability. The scaling factor α in the definition of the utilityvalue is used to ensure that most of the probabilities lie onthe center slope rather than near 0 or 1. Obviously, u(v, t) andp(v, t) do not apply to nodes that have already transitioned,since we assume that nodes do not revert once they make thetransition.

The deployment model described in this section is a verysimple model of how nodes in a network decide to adopt a newnetworking technology, but hopefully it captures enough thatwe can glean insights to how new networking technologiesspread given different network models.

III. NETWORK MODELS

In this section, we investigate how different network modelsinfluence the deployment of a new networking technology. Thedeployment model of section II is applied to different networkmodels using computer simulations, where we plot out thenumber of nodes which have made the transition as time goesby.

In the rest of the section, all networks have exactly 10000nodes. All of the cases start out with exactly one node havingalready made the transition, and in most cases we run thesimulation until 99% of the nodes have made the transition.We fixed β = 3, but the scaling factor α was adjusted foreach model to prevent the transition probabilities p(v, t) fromclustering around 1

1+eβ≈ 0.047, which would collapse a

growth curve to the uninteresting case of section III-A.Because the constants used in the utility functions and

transition probabilities are somewhat arbitrary, the number ofsteps that a network model uses to attain a certain deploymentrate is rather arbitrary as well. This is why we will ignore theabsolute speed at which different network models saturate, andinstead focus on the features of each deployment growth curve.

A. Independent

Before investigating the other network models, we take alook at how a new networking technology would spread ifevery node decided to make the transition independent ofwhatever the state of the rest of the network might be. In otherwords, we make the exception for the transition probability inthe deployment model so that:

p(v, t) = γ

With γ = 0.05, the growth is shown in Fig. 2, which isunsurprisingly an exponential curve upside down. This caseis included as a comparison for what happens when “networkeffects”, the influence of the rest of the network on a singlenode, are ignored.

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Fig. 2. Deployment growth with independent deployment.

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Fig. 3. Deployment growth with complete graph model.

B. Complete graph

As an example of a simple model that still includes “net-work effects”, we model the network as a complete graph,where every node is connected to every other node. Thenumber of transitioned nodes in the entire network is identicalto the number of transitioned nodes adjacent to a node that hasyet to make the transition, so this is basically the case whereall of the perceived benefit is due to how many nodes havemade the transition globally. Since every node is basically thesame as every other node, we use a constant cost model:

c(v) = γ

With α = γ = 1.25 × 107, the growth for the completegraph model is shown in Fig. 3. Deployment steadily growsuntil it rises exponentially and quickly saturates the network.This roughly replicates the “S curve” that would be expectedfrom technology deployment growth.

C. Random graph

As a slightly more realistic network model than the com-plete graph model of section III-B, we apply the deployment

model of section II to the Erdos-Renyi model [9] of randomgraphs. In this model, every potential edge between nodes ina graph are actualized with probability p, which should notbe confused with the transition probability p(v, t). Basically,nodes in the network are randomly connected to each other.

As for the cost model, we assume that it is more expensiveto make the transition for nodes that are linked to many nodes.In addition, we assume that there is a fixed cost for makingthe transition itself. Thus we model the cost as follows, whered(v) is the number of edges adjacent to v:

c(v) = γ(1 + d(v))

With p = 0.001, α = 10000, and γ = 1562.5, thegrowth for the random graph model is shown in Fig. 4.An alternate way to interpret the Erdos-Renyi model is thatedges are randomly removed from a complete graph, so itshould be no surprise that Fig. 4 looks like Fig. 3, despite thedifference in the cost model. As in the complete graph model,deployment steadily grows until it jumps exponentially andquickly saturates the network.

D. Preferential attachment

One possible scenario for how new networking technologiesare adopted is that an organization upgrades everything atonce, rather than replacing old equipment and software in-crementally. In such a scenario, an organization may be moreinclined to make the transition to new networking technologyif other organizations it directly communicates with have alsomade the transition. The organization would also be moreinclined to make the transition if the rest of the world hastaken up the new technology. For example, an ISP may bemore inclined to take up IPv6 if most of its peer ISPs havealready transitioned to IPv6, and even more so if the rest ofthe Internet has also taken up IPv6.

In such a scenario, the basic entity that makes a transitionwould be an organization, and each organization would beinfluenced by other organizations it has relationships with.In other words, the network model would be a social net-work model where vertexes are organizations and edges arerelationships. We will use the Barabasi-Albert model [10] ofpreferential attachment to model this sort of social network. Inthis model, nodes with larger number of neighbors are morelikely to collect even more neighbors.

A major organization that has more relationships with otherorganizations is also likely to have more equipment andpersonnel, which would raise the cost of making a transition.Thus we model the transition cost as follows, which alsoincludes a fixed cost:

c(v) = γ(1 + d(v))

With an initial network of 100 nodes connected as a ring,α = 3333, and γ = 2500, the deployment growth for thepreferential attachment model is shown in Fig. 5. Superficially,the curve looks like those for the complete and random graphmodels as shown in Fig. 3 and Fig. 4, respectively.

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Fig. 4. Deployment growth with random graph model.

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Fig. 5. Deployment growth with preferential attachment model.

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A closer look reveals a significant difference, however.Whereas deployment increases initially with a steady growthrate in Fig. 3 and Fig. 4, growth slightly flattens in Fig. 5before it skyrockets. The deployment rate in a preferential at-tachment model does not quite reflect the traditional “S curve”ascribed to technology adoption. It is not certain why there isa flattening of the growth rate until a critical mass is reached,but it does suggest that caution should be exercised if oneattempts to estimate how fast a new networking technology isbeing widely deployed by fitting a partial growth curve to an“S curve”.

E. Binary tree

Instead of an organization deciding to make the transitionto a new networking technology all at once, another scenariowould be each piece of equipment being replaced or upgradedone by one. The decision to upgrade a piece of networkingequipment would be influenced by whether neighboring net-working equipment has already made the transition, not tomention an industry-wide trend of adopting the new technol-ogy. In this scenario, actual networking hardware would be thevertexes and the communication links between them would bethe edges. A concrete example would be the router-level graphof the Internet.

The router-level graph of the Internet is built up to maximizetheir utility within the constraints enforced by the hardwaretechnology that is available, which results in a network modelthat is very unlike a scale-free network constructed throughpreferential attachment [11]. One of the features which makesthem very different is that the routers that form the core of theInternet are typically very high bandwidth, low-degree nodesin the network, since supporting very high bandwidth linksmakes it extremely difficult to support more than a few links,while the routers with high degree are typically located at thefringe of the network, connecting low-bandwidth end hosts tothe rest of the Internet.

While we do not try to realistically model the router-levelgraph of the Internet here, we will try to model one aspectof the graph: routers near the core are typically a lot moreexpensive than those at the fringe since they must handle muchhigher bandwidths. The network model itself that we use isa very simple binary tree, while the cost to transition a nodegrows exponentially the nearer it is to the root of the tree.The cost c(v) is defined as follows, where l(v) is the depthof vertex v in the binary tree:

c(v) = γ 2−l(v)

With α = 312.5 and γ = 2000000, the growth for thebinary tree model is shown in Fig. 6. Deployment spreadsexponentially at first and then slows down, saturating thenetwork slowly. This looks more like the traditional “S curve”compared to the other models. The slow saturation at the endseems to be an innate feature of the binary tree model, not anartifact of the constants that we used in the simulation. In fact,a larger scaling factor α shifts the curve generally to the left,which is basically slowing down the saturation even more.

Even more interesting behavior occurs when we use smallerscaling factors. When we set the scaling factor to α = 78,we get growth as shown in Fig. 7. Instead of a single S-likecurve, we get a double-S curve, almost as if there are twogrowth spurts occuring sequentially. The smaller scaling factorincreases the sensitivity of the deployment model to benefitsand costs, so it may be the case that the high transition costof nodes near the root may delay the deployment in a largesubtree in the network, and once the root of this subtree makesits own transition, the rest of the subtree quickly follows. Thiscould explain the double bursts of growth seen in Fig. 7.

Yet smaller scaling factors, which is equivalent to anincreased cost-benefit sensitivity, gives rise to even morecomplex growth curves. When we set the scaling factor toα = 39, we get the growth curve shown in Fig. 8, where wecan see many growth bursts occuring throughout time. Thecurve shows characteristics of a self-similar curve, which ismade clearer when we zoom into a small portion of the curveas in Fig. 9.

While the binary tree model is still an unrealistic modelfor real networks, the unexpected behavior in the growthcurve shows how different network and cost models canresult in qualitatively different behavior in the spread of newnetworking technologies.

IV. DISCUSSION

A. Applicability to real networks

One obvious real-world network we may want to comparethe network models in section III with is the spread of IPv6 in aworld of IPv4. Unfortunately, this is difficult not only becauseIPv6 deployment is still in its initial stages, but also becausemeasuring how widely IPv6 is deployed is hard, althoughthis has more to do with what we actually mean by IPv6being deployed rather than due to a dearth of data. Still, wewill briefly discuss how the models compare with the growthof IPv6 address block allocations to the Regional InternetRegistries, which is shown in Fig. 10.

If one were to look at Fig. 10, one might notice that thegrowth curves for RIPENCC and APNIC are vaguely similarto the growth curve for the preferential attachment model inFig. 5. They start off with the growth rate slowly flatteninguntil they begin to rise again. The growth for the other regionslook more like the other models, although there is not enoughdata to see if the growth curves look more like one modelrather than another.

For RIPENCC and APNIC, the similarity of the growthcurves to Fig. 5 might suggest that organizations are tran-sitioning all at once to IPv6 rather than incrementally, atleast in terms of allocating IPv6 address blocks, and that theyare influenced by other organizations having allocated IPv6address blocks of their own. Or perhaps the similarity is just acoincidence, the initial growth spurt being due to a clarificationof allocation policy which gradually tapered off as suggestedin [2], and the later rise in growth rates being due to theimpending depletion of IPv4 addresses.

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Fig. 10. Growth of prefix allocations for individual registries. Graph reproduced from [2] with permission.

Regardless, the similarities are a tantalizing hint that net-work structure can indeed influence the spread of a newnetworking technology in a real world situation, and in fact,may be an indication of the decision processes by whichorganizations determine their plans concerning the deploymentof IPv6. This may be an issue that deserves a closer look.

B. Limitations

Both the deployment model in section II and networkmodels in section III are very limited in their realism. In thefollowing, we discuss what those limitations are, which alsopoint to directions for further research.

1) Networks grow. We have assumed networks of fixedsize, which does not reflect the reality that even incum-bent networks such as IPv4 are growing. Similarly, theway we model deployment would be like embeddingan IPv6 network inside a fixed IPv4 network, andtechnically, an IPv6 network is entirely separate froman IPv4 network.

2) The decision process for whether a node will make thetransition or not should be modeled more realistically. Inaddition, constants used in an ideal deployment modelshould be based on measurable values that are dependenton underlying factors, rather than being derived by fittingto a curve.

3) The network models should be more realistic. For exam-ple, we may look at actual peering relationships betweenISPs to model the relationships between organization.We could also use more realistic models of communi-cation networks.

4) A look at how new networking technologies spreadthroughout a network. We have only looked at the totalnumber of deployments in a network, and seeing whichnodes actually make the transition as time passes wouldundoubtedly give rise to new insights.

C. Concluding remarks

With a simple model for how new networking technologiesget deployed throughout a network and applying it to differentnetwork models, we have seen how different network struc-tures and cost models can change the qualitative behavior of

how a new networking technology spreads. While the researchis in its infancy such that we are not able to make predictionsor derive insights from existing data with any certainty, wecan make the observation that any realistic model of how newnetwork technologies get deployed will have to account fornetwork structure and related cost models.

ACKNOWLEDGEMENTS

This work was supported as part of a National AgendaProject by the Korea Research Council of Fundamental Sci-ence and Technology. I would also like to thank ElliottKarpilovsky and Jennifer Rexford for graciously allowing theuse of their figure from [2].

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