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From Network to Network Infrastructure Emergence

V. Marbukh National Institute of Standards and 100 Bureau Drive, Stop 8920 Gaithersburg, MD 20899-8920 [email protected]

selfish in the network security [4] can be Abstract – This paper suggests that evolutionary models of affected if network users and providers have an option of network infrastructure in a can be derived from the underlying selfish behavior of users and providers of network disconnecting from providers who create or propagating services in the same way as non-equilibrium thermodynamics is security risk. Another example is selfish investment in the derived from the underlying statistical physics of interacting bandwidth within existing infrastructure. The inefficiency particles. This approach may be useful for overcoming of the competitive equilibrium in the bandwidth pricings restrictions of existing models failing to account for the effect of and offerings [5] may be a result of inefficient network the details of user/provider selfish behavior on the infrastructure infrastructure, which prevents provider . In evolutionary path. Network security considerations may be a both examples provider ability to modify infrastructure part of this user/provider behavior. Our main assumption is may significantly affect the emerging network that “almost ” keeps the close to infrastructure. the “social optimum”. This paper suggests that infrastructure evolutionary Keywords—network, , infrastructure, emergence. models can be derived from the underlying game-theoretic model of selfish user/provider behavior in the same way as non-equilibrium thermodynamics is derived from the I. INTRODUCTION underlying statistical physics of interacting particles. We In a market economy network evolution is driven by assume time scale separation between “fast” convergence to technological constraints and self- of the user/provider equilibrium for a given network infrastructure participants. Each user generates demand in attempt to and “slow” infrastructure evolution modeled by a Markov maximize its , which is the difference between the process. We consider the situation of almost perfect utility of obtaining and the service while each competition in which a large number of providers compete service provider attempts to maximize its , which is for demand generated by the same users [6]. Based on the difference between its revenue and expenses. We results from economics [6], as well as results for some model a network by a capacitated graph with links/nodes particular networks [7]-[9], we conjecture that selfish owned by different providers. The key feature of our user/provider behavior maximizes the aggregate user utility model is that providers not only can expand/reduce subject to provider profitability. network capacity within the infrastructure represented by Exploiting similarities between utility maximization the network graph, but also have the option to modify the and entropy maximization in a closed physical system with network infrastructure by adding/eliminating network fixed energy [10], we propose to model the infrastructure links. While capacity expansion within a fixed evolution by a time-reversible Markov process. The infrastructure can be done at the of the proposed model includes a possibility of selfish investment , growing network infrastructure by adding in network security and is open to various generalizations. links/nodes requires initial and typically significant The paper is organized as follows. Section II discusses investment, e.g., for putting fiber underground. user/provider competitive and socially optimal equilibria. This initial investment typically destroys convexity Section III proposes network infrastructure models and causes a multiplicity of Nash equilibrium consistent with the underlying selfish user/provider infrastructures in the natural game-theoretic formalization interactions. Finally, Section IV briefly summarizes and of the selfish user/provider behavior, making interpretation outlines directions of future research. of this game-theoretic model difficult. This difficulty has led to using game-theoretic models of selfish user/provider II. NETWORK behavior only for fixed network infrastructure when the corresponding game typically has a unique Nash This Section describes economics driving user/provider equilibrium in demand, , , and strategic interactions. Subsection A describes the network capacities. Separate modeling of infrastructure growth model. Subsection B describes the bandwidth typically relies on phenomenological random graph models supply/demand model. [1]-[3]. However, for the effect of user/provider A. Network economic incentives on the network infrastructure We model the network by a directed capacitated graph evolution may be essential. For example, the result of with set of nodes N and set of links

1 = ∈ ∈ ∈ ≤ * φ = L {Lnk : n N,k N \ n} where links l Lnk directly Cnn C corresponding to processing cost nn (C) 0 if connect node n to node k . The network infrastructure is C ≤ C* , and φ (C) = ∞ if C > C* . In this paper we = nn determined by network graph g (N, L) and ownership of uniformly refer to network nodes and links as network = elements. network elements. Let snk dim{Lnk } be the number of links directly connecting node n to node k , and Cnk be the B. Bandwidth Users and Providers aggregate capacity of these links. We model the effect of investment in the network security Limited link capacities impose constraints on the feasible by assuming that traffic of rate x traversing network element end-to-end throughputs x from nodes n ∈ N to nodes (n,k ) l faces security risk xh (q) , where q = (q ,l ∈ L) is the ∈ l l k N \ n vector of levels of investment in the security of network = , (1) x(n,k ) ∑ xr elements l is. We assume functions hl (q) to be decreasing ⊂ r Rnk in all vector q components due to positive , i.e., improving security of a network element not only improves = ≤ , (2) Yij ∑ xr Cij this element security; but also is beneficial for the security of ∈ ⊂ r:(i, j) r Rnk the entire network [4]. ∈ where Ynk is the aggregate rate through links l Lnk , vector Assuming that the owner of a network element l = ∈ x (xr ,r R) characterizes end-to-end flow rates on all charges users price pl for a unit of this element's feasible routes R = R , and the set of feasible routes bandwidth, the profit generated by network element l is n,k nk U f = ( p − q ) x −φ (c ) (4) from node n ∈ N to node k ∈ N \ n is R . l l l ∑ r ll l nk r:l∈r For simplicity of exposition we assume that all links Generally, there are S service providers, with service ∈ provider s ∈ S owning network elements l ∈ M . Each l Lnk directly connecting node n to node k are identical s with respect to their capacities: c = C s and carried network element is owned by some provider: M ≡ L . l nk nk Us∈S s = ∈ load: yl Ynk snk . In our economic model these Each provider s S attempts to maximize its aggregate profit assumptions are justifiable when the cost of providing = Fs ∑ fl (5) φ ∈ l∈M bandwidth c , is the same nk (c) , for all links l Lnk . In s this case bandwidth providers are inclined to charge the same We assume that each user is uniquely identified by the ∈ origin-destination pair (n,k),n ∈ N,k ∈ N \ n . Let price for bandwidth pnk on all links l Lnk and users are ∈ unk (x) be user (n,k) ’s utility of obtaining end-to-end indifferent to using links l Lnk [8]-[9]. Generalization to a case when the cost of providing bandwidth on different links bandwidth x from node n to node k . Utility functions ∈ u (x) are assumed to be monotonously increasing and l Lnk may be different is straightforward. nk ≥ We assume that bandwidth costs φ (c) are non- concave in x 0 at least in the case of file transfers. For nk example, weighted (α,w) - fair bandwidth allocation [12] is decreasing in c ≥ 0 . Communication bandwidth costs based on user φ (c); n ∈ N,k ∈ N \ n are concave in c ≥ 0 , e.g., −α nk ⎧w x1 (1−α) if α ≠ 1 u(xα,w) = (6) φ = + ∈ ∈ ⎨ α = nk (c) ank bnkc, n N,k N \ n (3) ⎩ wln x if 1 α > ≥ where ,w 0 are some parameters. In a particular case of where ank 0 characterizes the fixed cost, e.g., cost of proportional fairness, when α = 1, parameters w = w can putting the fiber underground and b ≥ 0 characterizes the nk nk be interpreted as user (n,k) ’s willingness to pay for end-to- marginal capacity cost. end bandwidth x from node n to node k . Our model can be easily generalized to incorporate User (n,k) net utility is its gross utility of having end-to- constraints imposed by limited node processing capacity end bandwidth minus expenses and expected losses due to , ∈ . It is natural to assume that costs of the Cnn n N security risks: φ ∈ ≥ processing bandwidths nn (C),n N are convex in C 0 ⎛ ⎞ = ⎜ ⎟ − []+η due to technological constraints. For example, considered in U nk unk ∑ xr ∑∑xr pl nk hl (q) (7) ⎜ ∈ ⎟ ∈∈ [11] is the case of hard processing bandwidth constraints ⎝ r Rnk ⎠ rrRnk l

2 where parameter η ≥ 0 quantifies user (n,k) ’s sensitivity nk B. Competitive Equilibria to the security risk. Each user (n,k) attempts to maximize Consider a non-cooperative game of providers ∈ its net utility (7) over the vector of flow rates (x ,r ∈ R ) , s S r st attempting to maximize their profits = ∈ * * given security risks h (hl ,l L) and = (15) Fs ( p,q) ∑ fl ( p,q) = ∈ l∈M p ( pl ,l L) . s over pricing pl , and investment in security ql of their III. USER/PROVIDER EQUILIBRIA ∈ * l M s , where fl ( p,q) are given by (14). Typically, this This Section introduces notions of user/provider game has multiple Nash equilibria corresponding to different equilibria. Subsection A describes user/provider best network infrastructures defined by a combination of network responses. Subsection B discusses multiplicity of the topology and ownership of network elements. These Nash competitive (Nash) equilibria of the user/provider game. equilibria are associated with attractors of the following Subsection C conjectures that under perfect competition evolutionary/learning provider adjustments [13]: these competitive equilibria approach the social optimum. q = ∂F * (q, p) ∂q , p = ∂F * (q, p) ∂p l ∈ M (16) A. User/Provider Best Responses &l s l & l s l s = * It is easy to see that user net utility (7) maximization Note that since best user response (8)-(12) x x (q, p) is results in minimum cost routing where the adjusted cost of discontinuous in a case of equal minimum cost multipath a route is the sum of the adjusted costs of network elements routing, the differential equations (16) may have comprising the route discontinuous right-hand sides. It can be shown that in this η = η case system (16) should be understood as a differential dr ( ) ∑dl ( ) (8) inclusion [14] describing sliding modes along the l∈r discontinuity hyperplanes. and network element l adjusted cost is Bertrand's model of user/provider equilibria [9], [15] d (η) = p −ηh (q) (9) assumes that in "fast" time users/providers achieve l l l = ∈ Assuming sufficient capacity, the entire user (n,k) equilibrium over user demands x (xr ,r R) , = bandwidth pricing p ( pl ) and investment in security * = [ − * ] x(nk ) argmax unk (x) dnk x (10) q = (q ) subject to the capacity constraints (2) for fixed x≥0 l = is sent over minimum cost routes capacities c (cl ) . On a "slow" time scale bandwidth * = = * ∈ Rnk {r : dr dnk ,r Rnk } , where pricing and investment in security become functions of the * * * = = d = min d (η ) (11) network element capacities: p p (c) and ql q (c) . nk ∈ r nk r Rnk = * Users are indifferent between feasible routes of minimum cost. Nash equilibrium capacities c c are associated with attractors of the following bandwidth adjustments [9]: * = * If there is a single minimum cost route: Rnk rnk , the entire = ∂ * * * ∂ ∈ ∈ c&l Fs [q (c), p (c)] cl l M s , s S (17) user (n,k) demand is carried on this route: Figure 1 demonstrates multiplicity of provider equilibria ⎧x* if r = r* achieved by adjustment process (17) on an example of two * = nk nk xr ⎨ (12) providers competing for the same demand [9]. ⎩ 0 otherwise In the particular case of user utilities (6): = α C2 unk (x) u(x nk , wnk ) , user (n,k) aggregate demand is α * = ()* 1 st x(nk ) wnk dnk (13) Further we assume that users always generate the best response demand, and thus element l generates profit * = * − −φ ⎡ * ⎤ fl ( p,q) pl ∑ xr ( p,q) ql l ⎢∑ xr ( p,q)⎥ (14) r:l∈r ⎣r:l∈r ⎦ Expression (14) implies that the optimal provider responses ensure tightness of the capacity constraints (2). Indeed, providers have incentive to eliminate any spare capacity to C1 reduce expenses while in a case of insufficient capacity the provider can increase its revenue by raising the price. Fig. 1. Capacity adjustments by two competing providers.

3 ⎡ ⎤ Three equilibria are possible: one equilibrium with both = − −φ ⎛ ⎞ ≥ FM ∑∑⎢( pl ql ) ∑ xr l ⎜ xr ⎟⎥ 0 (20) providers supplying bandwidth and two equilibria with only ∈∈∈ lrM s ⎣ r:l r ⎝ r:l ⎠⎦ one provider supplying bandwidth and another provider driven Note that for bandwidth cost function (3) constraints (20) take out of business. the following form:

C. Perfect Competition: Social Optimum ⎡ + + − ⎤ ≤ ∑∑⎢al (bl ql pl ) xr ⎥ 0 (21) ∈∈ In competitive equilibrium providers ensure profitability lrM s ⎣ r:l ⎦ by charging users a competitive price which is higher than the It is easy to see that due to the assumed concavity of the marginal bandwidth price. It is known from economics [6] communication bandwidth costs φ (c); n ∈ N,k ∈ N \ n , that as the number of providers competing for the same nk demand increases, the competitive prices drop approaching the = at a social optimum (18)-(20) no more than one link l lnk corresponding marginal prices. This increase in competition ∈ ∈ squeezes the provider profit margins but increases user net connects node n N to node k N \ n . This observation = utilities as shown in Figures 2 and 3 respectively for N allows us to assume in (18)-(20) that at most one link l lnk parallel links competing for demand generated by the same connects a node n ∈ N to a node k ∈ N \ n . However, it users [8]-[9]. poses a fundamental question since the social optimization (18)-(20) views system performance from the user perspective, assuming that providers only recover their expenses without making any profit, and this may occur only close to perfect competition with a large number of providers competing for demand generated by the same users [6]. The rest of this Subsection qualitatively demonstrates that low barriers to entering the market are necessary for this situation to occur and for the optimization problem (18)-(20) to approximate the competitive equilibria. Consider N providers competing for the same demand generated by a user with utility function u(x) = wlog x disregarding network security (η = 0) . Each provider

charges the user a competitive price pN x for bandwidth x , Fig. 2. Provider profit vs. number of competing providers N while the provider’s own expense for supplying this bandwidth is a + bx . Since the user’s individual optimization results in = demand x w pN equally divided among all N providers, > + the provider profitability condition pN x N a b x N < − ↑ takes the form N (w a)(1 b pN ) . Since pN b as N ↑ ∞ (see [9]), the profitability condition imposes an upper bound on the number of competing providers N < N * . Optimization problem (18)-(20) implies a large number of providers N * driving provider profit down to zero, and it is easy to see that in our simple case this can occur only if the barrier for a provider to enter the market a is much less than user willingness to pay w : a << w . It can be shown that this Fig. 3. User net utility vs. number of competing providers N qualitative conclusion holds for general topology networks.

One may expect that in the limit of perfect competition IV. INFRASTRUCTURE MODELS with a large number of providers competing for the same users Given set of network nodes N , we characterize the competitive equilibria will approach a social optima which network infrastructure by a binary vector maximizes the aggregate user utility δ = δ = ∈ ∈ δ = ( l : l (n,k),n N,k N \ n) , where nk 1 if maxUΣ (x, p,q) (18) = (x, p,q) directed link l lnk from node n to node k exists and = δ = 0 otherwise. This Section proposes equilibrium and U Σ (x, p,q) ∑U nk (x, p,q) (19) nk (n,k ) evolutionary models for δ = δ (t) under assumptions that subject to provider profitability “almost perfect” competition keeps the network infrastructure

4 close to socially optimal. The proposed models are based on aggregate provider expenses: the underlying economic model of user/provider interactions. ⎡ ⎛ ⎞⎤ * δ = − + φ Subsection A introduces infrastructure social utility W ( ) , W (x,q) UΣ (x,q) ∑∑⎢ql l ⎜ xr ⎟⎥ (24) which is the result of maximization of the social welfare over lr⎣ ⎝ r:l∈ ⎠⎦ bandwidth demand/supply for a given vector δ . Subsection B In a case of bandwidth cost function (3) social welfare (24) suggests an entropy-based equilibrium model for vector δ takes the following form: with W *(δ ) playing the role of the negative energy. Subsection C suggests an approximation of infrastructure δ = − ⎡ + + ⎤ + W ( ,q, x) ∑∑⎢al (bl ql ) xr ⎥ δ = δ δ =∈ evolution by a time-reversible Markov process (t) used lr::l 1 ⎣ r l ⎦ in physics to describe evolution of interacting spins. (25) ⎡ ⎛ ⎞ ⎤ ⎜ ⎟ −η A. Infrastructure Utility ∑∑∑⎢unk ∑ xr nk xr hl (q)⎥ ⎜ ∈ δ ⎟ ∈∈δ ()nR,k )⎣⎢ ⎝ r Rnk ( ) ⎠ rrnk ( l ⎦⎥ Optimization problem (18)-(20) is non-convex due to the assumed concavity of the communication bandwidth cost where the set of available routes from node n ∈ N to node φ ∈ ∈ ∈ δ functions nk (c); n N,k N \ n . It is known that this k N \ n conditioned on infrastructure is non-convexity leads to multiple local maxima of the aggregate δ = δ δ = ⎧ δ = ⎫ utility (19). However, given the vector (x) ( l (x)) , Rnk ( ) Rnk ⎨r : ∏ l 1⎬ . (26) I ∈ > ⎩ l r ⎭ ⎪⎧1 if ∑ xr 0 δ = ∈ δ l (x) ⎨ r:l r , (22) Define the social utility of infrastructure to be the ⎩⎪0 otherwise maximum of social welfare (20) achievable for this describing network infrastructure, the aggregate utility infrastructure: maximization (18)-(20) typically becomes a convex * optimization problem. The corresponding maximal aggregate W (δ ) = maxW (δ ,q, x) (27) q,x * utility is a function of vector δ (x) = δ : UΣ (δ ) assuming Infrastructure utility (25) can be evaluated effectively and even that infrastructure δ (x) = δ can support provider explicitly under realistic assumptions. Due to limited space we only note that in a case of network security insensitive users * δ profitability (20). It is natural to interpret UΣ ( ) as the social (η = 0) with utilities u (x) = w log x the δ nk nk nk utility of infrastructure . Of course actual of infrastructure utility (25) is * UΣ (δ ) is computationally infeasible, and in the rest of this subsection we briefly discuss some approximations. * δ = ⎛ ⎞ − W ( ) log⎜ wnk min bl ⎟ Consider the Lagrangian of the optimization problem ∑∑r∈R (δ ) (nr,k ) ⎝ nk l∈ ⎠ (18)-(20) (28) ⎡ ⎤ ⎡ ⎛ ⎞⎤ − + ⎛ ⎞ W = UΣ − γ ( p − q ) x −φ ⎜ x ⎟ (23) ⎢al bl ⎜ wnk min bl ⎟⎥ M ∑∑⎢ l l ∑ r l r ⎥ ∑∑r∈R (δ ) ∑ ∈∈∈ l:δ =1 ∈ * δ nk l∈r lrM s ⎣ r:l r ⎝ r:l ⎠⎦ l ⎣⎢ (n,k ):l rnk ( ) ⎝ ⎠⎦⎥ γ = γ where M M ( p,q) are the Lagrange multipliers solving where user (n,k) optimal route is the corresponding dual problem. Let W *(δ ) be the * δ = rnk ( ) arg min bl (29) maximum of function (23) over ( p,q, x) for a given vector r∈R (δ ) ∑ nk l∈r δ . Due to non-convexity, solutions to the primal (18)-(20) and the corresponding dual problems may differ, and thus B. Equilibrium Infrastructure * δ − * δ ≠ UΣ ( ) W ( ) const (utilities are defined up to additive We model unavoidable present in large constant). Nevertheless, the function W * (δ ) can be viewed as networks by assuming that vector δ is a binary random an approximation of the infrastructure utility. The variable with some probability Ω(δ ) and entropy * computational advantage of the function W (δ ) over = − Ω δ Ω δ (30) H ∑δ ( )log[ ( )] * * UΣ (δ ) is that to evaluate W (δ ) one has to solve only one Social optimization either maximizes the infrastructure optimization problem rather than solve an optimization expected social utility δ * δ * = * δ Ω δ (31) problem for each vector to evaluate function UΣ ( ) . W ∑δ W ( ) ( ) In a case of low barriers to entering the market, further subject to low bound on entropy (30) or, equivalently, simplification is possible since (23) can be approximated by maximizes entropy (30) subject to low bound on the expected the difference between the aggregate user utility (18) and infrastructure utility (31).

5 Solution to these dual optimization problems is connectivity to a specific node against cost of this connection. − β * δ It can be shown that, after some simplifying assumptions, Ω*(δ ) = Z 1e W ( ) (32) network growth based on infrastructure utility gain Δ W *(δ ) where the parameter β > 0 is determined by the l takes the form of “heuristically optimized” [3]. corresponding optimization constraint and Z is the → normalization constant. As decreases: H 0 or V. CONCLUSION AND FUTURE RESEARCH equivalently β → ∞ , the distribution (32) recovers the Assuming that the network infrastructure is represented by optimal network infrastructure. Similar to approach [2] we the network graph and ownership, this paper suggests that define an equilibrium graph distribution by the entropy network infrastructure modeling should be based on the maximization procedure. The difference is that in our case the network microeconomics. The paper argues that graph energy E(δ ) = −W * (δ ) is derived from user/provider methodologies used in transitions from statistical physics of a microeconomics. This opens a possibility of exploring the large number of interacting particles to thermodynamics of a effect of microeconomics, including network security small number of macro-variables may prove applicable to the considerations, on the network infrastructure emergence. networking domain. The proposed approach assumes that “almost perfect competition” keeps the system close to social C. Infrastructure Emergence optimum, which maximizes the aggregate user utility subject to provider profitability. Future research will be concentrated We assume that competitive pressures may result in on demonstrating the practical validity of the proposed elimination of some network elements approach and extending this approach to competitive δ = → δ = user/provider equilibria far from perfect competition. l : ( l 1) ( l 0) due to un-profitability and/or adding δ = → δ = some network elements l : ( l 0) ( l 1) due to REFERENCES l potential profit. Let vector δ be the result of flipping [1] A.L. Barabasi and R. Albert. Emergence of scaling in random networks. Science, 286(15), pp. 509-512, 1999. component δ δ → − δ l in vector δ , i.e., infrastructure l : l 1 [2] J. Park and M. E. J. Newman The statistical mechanics of networks, described by vector δ l contains (does not contain) link l if the Phys. Rev. E 70, 066117, 2004. δ [3] A. Fabricant, E. Koutsoupias, and C.H. 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