From Network Microeconomics to Network Infrastructure Emergence V. Marbukh National Institute of Standards and Technology 100 Bureau Drive, Stop 8920 Gaithersburg, MD 20899-8920 [email protected] selfish investment 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 market economy 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 competition. 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 perfect competition” keeps the system close to infrastructure. the “social optimum”. This paper suggests that infrastructure evolutionary Keywords—network, economics, 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-interests 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 net utility, which is the difference between the process. We consider the situation of almost perfect utility of obtaining service and the service price while each competition in which a large number of providers compete service provider attempts to maximize its profit, 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 marginal cost of the proposed model includes a possibility of selfish investment resources, 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 ECONOMIC MODEL 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, pricing, investments, 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, accounting 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 externalities, 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 utilities 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 prices = (15) Fs ( p,q) ∑ fl ( p,q) = ∈ l∈M p ( pl ,l L) . s over pricing pl , and investment in security ql of their assets III.