On a Network Creation Game
*t
Alex Fabrikant
alexf @ cs. berkeley.edu
Ankur Luthra
ankurl @cs.berkeley.edu
Elitza Maneva
elitza @ cs.berkeley.edu
Scott Shenker §
shenker@ icsi.berkeley.edu
Christos H. Papadimitriou
christos @ cs.berkeley.edu
ABSTRACT
Because the Internet is the product o f - and an arena for
- - the interaction of many economic agents, it may not be
optimized in any conventional sense. How costly is this lack
of coordination? In Theoretical Computer Science, we have
in the past confronted such questions by developing research
frameworks based on ratios: the approximation ratio measures the cost of the lack of tractability; the competitive ratio, the cost of information (often in a distributed setting).
In [4], it was proposed t h a t the ratio of the social costs of
the worst-case Nash equilibrium and the social optimum,
the so-called price of anarchy, may be an informative measure of the lack of coordination in situations in which agents
interact by pursuing each their own interest.
We introduce a novel game t h a t models the creation of
Internet-like networks by selfish node-agents without central design or coordination. Nodes pay for the links t h a t
they establish, and benefit from short paths to all destinations. We study the Nash equilibria of this game, and prove
results suggesting t h a t the "price of anarchy" [4] in this context (the relative cost of the lack of coordination) may be
modest. Several interesting: extensions are suggested.
Keywords
Distributed network design, game-theoretic models, Nash
equilibria, price of anarchy
The concept of Nash equilibrium is in some sense the analog of centralized optimal design in the context of multiple
distributed selfish agents. But it is not without its conceptual and practical problems. The concept is declarative (as
opposed to algorithmic), providing no guidance on how it
can be reached (besides some weak and unconvincing convergence results); it is also delightfully and intriguingly nondeterministic (due to the existence of multiple Nash equilibria), something explored and delimited in the current line of
work.
1. INTRODUCTION
The Internet is the first computational artifact t h a t was not
designed by one economic agent, but emerged from the distributed, uncoordinated, spontaneous interaction (and selfish pursuits) of many. Today's Internet consists of over
12,000 subnetworks ("autonomous systems"), of different
sizes, engaged in various, and varying over time, degrees
of competition and collaboration.
The Internet is also the first object studied by computer
scientists that must be approached with humility and puzzlement, and studied by measurement, experiments, and the
development of models and fMsifiable theories - - very much
like the cell, the universe, the brain, and the market.
Since [4], there has been much progress in understanding
the price of anarchy in more and more general situations,
in which individual users choose routes selfishly, and suffer
from the created congestion [5, 7, 1, 6]. In the Internet,
however, users do not choose routes; the situation is much
more complex. Routes are chosen by the interaction of packets with touters, users adjust their usage to the resulting
congestion, while autonomous systems add bandwidth and
hardware to the resulting hot spots. Can this much more
complex situation be illuminated as successfully as the congestion game? What is the price (in both hardware costs and
*Computer Science Division, UC Berkeley, Berkeley, CA
94720-1776
~'Supported by the Fannie and John Hertz Foundation
:~Supported by a DARPA grant, an NSF ITR grant, and by
research awards from IBM and Microsoft Research.
§International Computer Science Institute, Berkeley, CA
94704. Supported in part by NSF grants ITR-0205519, ANI0207399, ITR-0121555, I'rR-0081698, ITR-0225660 and
ANI-0196514.
quality of service) off the Internet's open architecture?
Permission to make digital or hard copies of all or part of this work for
personal or classroom use is granted without fee provided that copies are
not made or distributed for profit or commercial advantage and that
copies bear this notice and the fidl citation on the first page. To copy
otherwise, or republish, to post on servers or to redistribute to lists,
requires prior specific permission and/or a fee.
PODC'03, July 13-16, 2003, Boston, Massachusetts, USA.
Copyright 2003 ACM 1-58113-708-7/03/0007... $5.00.
347
In this paper we propose a simple game-theoretic model of
network creation. The agents are nodes, and their strategy
choices create an undirected graph. Each node chooses a
(possibly empty) subset of the other nodes, and lays down
edges to them. The edges are undirected, in that, once installed, they can be used in both directions, independently
of which node paid for the installation. The union of these
sets of edges is the resulting graph. (The union may not be
disjoint; t h a t is, it may be t h a t an edge is paid for by both
the Internet topology. We also outline in t h e last section
several extensions t h a t may make t h e model a little more
realistic, while being still tractable in interesting ways.
of its endpoints. However, this obviouly will never be the
case at equilibrium.)
The cost to each node of such a combination of choices has
two components: t h e total cost of t h e edges laid down by
this node (the n u m b e r of edges times a constant a > 0,
the only p a r a m e t e r in this model), plus t h e sum of the distances from the node to all others. Our game tries to capture aspects of the Internet related to a u t o n o m o u s systems
peering and otherwise agreeing to communicate. Each such
agreement may be costly, b u t pays off in quality of service
improvements. T h a t is, our model takes into account b o t h
hardware costs and quality of service costs; however, for the
latter it ignores congestion, t h e focus of much other work.
The M o d e l
T h e game we consider has players {0, 1 , . . . , n - 1} - - we
denote this set by [n]. T h e strategy space of player i is the
set Si = 2 In]-{~}. Given a combination of strategies s =
( s o , . . . , s n - l ) £ So × . . . × S n - 1 , we consider t h e graph G[s],
the underlying undirected graph of G0[s] = (In], U~_~0a ({i} ×
si)). The cost incurred by p]ayer i under s is defined to be
n--1
c~(s) = a-Is~l + ~ dG[sl(i,j),
j=O
W h a t are the (pure) Nash equilibria of this game, and how
do they compare to the social o p t i m u m (the combination
of strategies with smallest sum of costs)? (It follows easily
from our results t h a t t h e game does have pure equilibria.)
where dG[s] (i, j ) is the distance b e t w e e n nodes i and j in the
graph G[s].
A (pure) Nash equilibrium in this game is an s such that,
for each player i, and for all s' t h a t differ from s only in the
ith component, e~(s) > c~(s').
For small values of a (under 2) t h e situation is fairly straightforward: the social o p t i m u m is the clique, and the star is the
worst Nash equilibrium. T h e worst-case "price of anarchy"
is then 4 (see Section 2).
One may try to discover Nash equilibria by starting from
one strategy combination and repeatedly replacing a player's
strategy by its best response. T h e following is cautionary in
this regard:
For a > 2, the plot thickens. T h e star is always the social
o p t i m u m (Section 2), and there is a lower b o u n d of 3 - e
for the price of anarchy ( T h e o r e m 2). As for upper bounds,
we prove t h a t the price of anarchy is O ( x / ~ ) (Theorem 1).
This is t h e best general upper b o u n d t h a t we have.
PROPOSITION l. It is NP-hard, given s 6 So × . . . × S n - 1
and i 6 [n], to compute the best response of i.
In our constructions, as well as our experiments, we have
almost always come up with Nash equilibria t h a t are trees.
T h e only known exception t h a t remains an equilibrium for
a > 2 (and even then, only for a < 4), the Petersen graph,
is a transient 1 equilibrium. For a < 2, experiments yield
a wide variety of non-tree equilibria, but T h e o r e m 1 guarantees t h a t the price of anarchy of all of these is at most
constant. (Incidentally, our e x p e r i m e n t s are hindered by
t h e fact t h a t c o m p u t i n g the best response in this game is
NP-hard, Proposition 1.)
S k e t c h : Reduction from DOMINATING SET.
Player i is given the configuration of t h e rest of the graph,
including the edges coming in, and has to pick a subset of
vertices such t h a t when t h e edges to t h e m are built, c~ is
minimized. For any 1 < a < 2, if t h e r e are no incoming
edges, the strategy is a d o m i n a t i n g set for the rest of the
graph, since the d i a m e t e r of G must be at most 2, and making more t h a n t h e m i n i m u m number of links would only
improve t h e distance t e r m of ci by 1. Hence the costs are
minimized (and t h e utility maximized) when the size of the
subset is minimized.
These considerations have led us to fornmlate t h e tree conjecture, stating t h a t , for ct above some constant A, all Nash
equilibria are trees. Finally, we show (Theorem 3) t h a t any
tree Nash equilibrium is less t h a n 5 times more costly t h a n
the social o p t i m u m - - hence, under the tree conjecture, t h e
price of anarchy (for non-transient equilibria) is at most a
constant, d e p e n d e n t only on A.
2.
BASIC RESULTS
The social cost is simply the sum of all players' costs, which,
for any situation where no connection is paid for by both
endpoints (a constraint satisfied by all Nash equilibria), is:
Despite its simplicity, our model may be a useful analytical
tool for shedding light on t h e complex processes t h a t create
the topology and other reality of t h e Internet. For example,
our analysis can be seen as justifying t h e prominence of the
star and the clique as i m p o r t a n t primitives in the s t u d y of
c ( c ) : ~ c, : -IEI + ~ dG(i, j)
i
1 By transient, we mean a weak Nash equilibrium (i.e.
one in which at least one player can change his strategy
without affecting his payoff), from which there exists a sequence of single-player strategy changes which do not alter
t h e changer's payoff leading eventually to a non-equilibrium
position. For t h e Petersen graph, t h e r e are several such
chains, all leading to positions from which all further chains
of single-player changes strictly beneficial to the changer
lead to a tree.
i,j
Since every pair of vertices t h a t is not connected by an edge
is at least distance 2 apart, the following is a lower bound
for the social cost:
C(G)
348
>_ ~[E[ + 21E[ + 2 ( n ( n - 1) - 21El)
= 2 n ( n - 1) + ( a - 2)[EI.
(1)
T h i s b o u n d is achieved by a n y g r a p h of d i a m e t e r at m o s t 2.
infinity or < 2 * d i a m ( G ) < 4 x / ~ (where d i a m ( G ) d e n o t e s
t h e d i a m e t e r of G). We consider t h e s e two cases separately.
W h e n a < 1, by Eq. 1, t h e social o p t i m u m is achieved w h e n
[E I is m a x i m u m . T h e r e f o r e it is t h e c o m p l e t e graph. A n y
N a s h equilibrium h a s to be 2 of d i a m e t e r 1, which implies
t h a t t h e c o m p l e t e g r a p h is also t h e only N a s h equilibrium.
If Ti a n d v are c o n n e c t e d in G ~ = (V, E - ei), t h e n for every
u E T~
dG,(v,u) - dG(v,u) < 4x/~.
W h e n 1 < a < 2, t h e social o p t i m u m is still achieved for t h e
c o m p l e t e g r a p h (even t h o u g h it is no longer a N a s h equilibrium). A n y N a s h e q u i l i b r i u m is of d i a m e t e r at m o s t 2, so
t h e social cost is e x a c t l y equal to t h a t in e q u a t i o n 1. It is
worst w h e n IEI is m i n i m u m , w h i c h is n - 1 for a c o n n e c t e d
graph. T h u s t h e worst N a s h equilibrium is t h e star. T h e
price of a n a r c h y is then:
T h e total i m p r o v e m e n t is ~ u e T ~ (dG,(v, u) - - d e ( v , u)) > a.
T h i s implies t h a t [Ti[ = ~(vZ~). So we found ~ ( x / ~ ) vertices, such t h a t (u, v) is n o t a n edge. Such n o n - e d g e s will be
c o u n t e d at m o s t twice (from b o t h sides).
If Ti a n d v are not c o n n e c t e d in G' t h e n G' h a s two compon e n t s a n d we c a n c o u n t [T~]- I + [ V - T i I - 1 : n - 2 = ~ ( v / ~ )
non-edges - t h o s e incident on v or w a n d t h e o t h e r component, where ei = (v,w). T h e s e are also c o u n t e d at m o s t
twice.
C ( s t a r ) _ (n - 1 ) ( a - 2 + 2n)
C(Kn)
n ( n - 1 ) ( ~ - ~ + 2)
4
4 - 2a
2+ a
n ( 2 + c~)
4
< - - - -4< 2+a - 3
T h i s c o m p l e t e s t h e proof t h a t t h e n u m b e r of edges is O ( - ~ )
of t h e total n u m b e r of v e r t e x pairs, a n d t h e proof of t h e
theorem. []
4.
W h e n c~ > 2, t h e social o p t i m u m is achieved for m i n i m u m
tel, so it's t h e star. T h e s t a r is also a N a s h equilibrium, b u t
t h e r e m a y be worse N a s b equilibria.
3.
rium of the network game with the price of anarchy greater
than 3 - e.
AN U P P E R B O U N D
PROOF. For a n y k _> 4, d > 2, consider t h e family of
complete k-ary directed trees of d e p t h d, Tk,d, with all edges
going from p a r e n t nodes to their children. Let n be t h e
n u m b e r of n o d e s in Tk,d, a n d set, w i t h foresight, a = ( d -
Observe t h a t if c~ > n 2 t h e N a s h equilibrium is a tree, because, unless t h e d i s t a n c e to a node is infinity, a player h a s
no interest in building a n edge. In t h a t case, t h e price of
a n a r c h y is trivially O(1).
On.
THEOREM 1. For c~ < n :~ the price of anarchy f o r our
LEMMA 1.
model is O ( x / ~ ).
Tk,d i8 a Nash equilibrium of the network game.
PROOF. Every non-leaf node i m u s t link at least once to
t h e s u b t r e e of each of its children; else t h e g r a p h b e c o m e s
disconnected, which would carry an infinite p e n a l t y for ci.
If only one link is m a d e to a p a r t i c u l a r child's s u b t r e e (i.e.
t h e set of t h e child's d e s c e n d a n t s , including t h e child itself),
it is clearly o p t i m a l for it to link to t h e child directly - - linking to a g r a n d c h i l d ' s s u b t r e e brings t h e node closer to t h a t
s u b t r e e b u t f u r t h e r away from t h e child a n d at least t h r e e
o t h e r g r a n d c h i l d r e n ' s subtrees, t h u s strictly increasing t h e
d i s t a n c e t e r m . T h i s m a k e s t h e total c o n t r i b u t i o n of t h a t
s u b t r e e to t h e d i s t a n c e t e r m of ci be at m o s t (d - 1)n/k,
since t h e s u b t r e e c o n t a i n s at m o s t n / k nodes, a n d t h e y are
all at m o s t d - 1 hops from t h e node we're e x a m i n i n g . Since
c r e a t i n g 2 or m o r e links to t h a t s u b t r e e would require an
additional cost of ( d - 1)n, b u t would n o t increase t h e dist a n c e from our node to a n y o t h e r by m o r e t h a n d - 1, t h a t
will not h a p p e n .
PROOF. T h e price of a n a r c h y is:
p(G) = 0
A LOWER BOUND
THEOREM 2. For any ~ > O, there exists a Nash equilib-
(c~[EI+ ~,,~d a ( i , j ) ~
/
Notice t h a t d v ( i , j ) < 2 x / ~ for every i a n d j , since otherwise i would connect to j to m a k e itself closer to all nodes
more t h a n half way to j along t h e s h o r t e s t p a t h from i to j.
Therefore it suffices to prove t h a t IE] = O ( -n~
~).
Consider t h e edges o u t of v e r t e x v: e l , e 2 , . . . , et. For a n y
edge we will c o u n t vertices u for which (v, u) is not in t h e
graph. In o t h e r words, we will associate several n o n - e d g e s
with each edge. Ideally, we w a n t t h e ratio of t h e n u m b e r of
edges to t h e n u m b e r of n o n - e d g e s to be 1 : x/~Let Ti : {u C V: t h e s h o r t e s t p a t h from v to u goes t h r o u g h
ei }. We e n s u r e t h a t Ti are disjoint by considering a canonical s h o r t e s t p a t h for each vertex. Before edge ei was built,
t h e alternative s h o r t e s t p a t h from v to u 6 Ti was either
Every non-root node i, at d e p t h 5 > 1, is already connected
to all nodes outside its s u b t r e e via a link from its parent.
Note t h a t (1) if s u c h a node links to t h e root, it will become
closer to every o t h e r node by no m o r e t h a n 5 - 1, for a total
possible s a v i n g s of no m o r e t h a n (d - 1)(n - 1); a n d (2) if
such a node links to a child of its p a r e n t (its sibling), it will
b e c o m e closer to every o t h e r node by no m o r e t h a n 1, for a
total s a v i n g s of no m o r e t h a n n - 1. Consider s o m e o t h e r
node j which is n o t a d e s c e n d a n t of i (addressed above) a n d
2Note t h a t here, a n d at several points t h r o u g h o u t , we rely
on t h e fact t h a t a n y N a s h e q u i l i b r i u m c a n n o t be m i s s i n g
edges whose a d d i t i o n would reduce t h e s u m inter-node dist a n c e s by m o r e t h a n a , w h i c h follows i m m e d i a t e l y from t h e
definition of t h e cost function.
349
referred to as i's subtrees. A n a r b i t r a r y one is chosen for
L(i) if t h e r e is m o r e t h a n one s u b t r e e of i of t h e m a x i m u m
size. W e first note t h a t T h a s at least one center node z - - a
node s u c h t h a t ]L(z)l < n/2. T h i s m a y be seen by s t a r t i n g
at an a r b i t r a r y node zo, a n d following a sequence of nodes
zo, z t , . . , until a center is reached; z~ is o b t a i n e d from (noncenter) z,-1 by following t h e edge b e t w e e n z,-1 a n d L(z~-l).
If we consider t h e s u b t r e e s of zi, all b u t t h e one c o n t a i n i n g
zi-1 will be strict s u b s e t s of L(zi-1), while t h e s u b t r e e cont a i n i n g zi-1 will be t h e c o m p l e m e n t of L(z~-l) and, since
IL(zi_l)l > n/2, will c o n t a i n at m o s t n/2 nodes. Hence,
IL(z,)l < m a x { n / 2 , I L ( z i - 1 ) l - 1}, w h i c h clearly g u a r a n t e e s
t h a t t h e s e q u e n c e will e v e n t u a l l y reach a center node.
not its p a r e n t (trivially wasteful to link to). Let j0 be t h e
sibling of i which is a n a n c e s t o r of j , or root, if t h e r e is no
such siblinga; t h e n t h e simple p a t h j o , j l , . . . ,jx = j passes
t h r o u g h no node linked to i. W e know t h a t t h e savings from
linking to j0 are strictly less t h a n c~ - - now simply note t h a t
t h e savings from linking to j y + l are even less t h a n t h o s e from
linking to j~, since i is b r o u g h t closer by 1 to t h e nodes in
j y + l ' s subtree, a n d m o v e d f u r t h e r by 1 from t h e n o d e s in t h e
s u b t r e e s of at least 2 o t h e r children of jy (the ones t h a t are
t h e a n c e s t o r s of neither j nor i, a n d hence do n o t have a n y
shorter p a t h s to i). Inductively, t h e s a v i n g s from linking to
j are strictly less t h a n c~ as well. F u r t h e r m o r e , since m a k i n g
a n y one e x t r a link alone is n o t w o r t h it, m a k i n g m o r e t h a n
one c a n n o t be worthwhile either, since t h e n e t savings in
t h e d i s t a n c e c o m p o n e n t are no g r e a t e r t h a n t h e s u m of t h e
savings yielded by a d d i n g a n y link alone. Since none of t h e
edges i is p a y i n g for c a n be r e m o v e d either, Tk,d is a N a s h
equilibrium. []
Let d > 2 be t h e d e p t h of T w h e n rooted at z (if d = 1,
we have a star, for which t h e t h e o r e m holds trivially). T h a t
is, s o m e leaf l at d e p t h d chose n o t to link to z. Since
t h e s u b t r e e of z w i t h i n which l lies c o n t a i n s at m o s t n/2
nodes, t h e r e are at least n/2 - 1 n o d e s s u c h t h a t p a t h s from
l to t h e m go t h r o u g h z. T h u s a link from l to z would
yield a s a v i n g s of at least (d - 1 ) ( n / 2 - 1) for cl. Hence,
a > ( d - 1 ) ( n / 2 - 1).
T h e total cost of t h e social o p t i m u m (the star, for sufficiently
high values of d a n d n) is a ( n - 1) + 2 n ( n 1) = ( d +
1 ) n ( n - 1). By c o u n t i n g d i s t a n c e s b e t w e e n leaves alone,
which n u m b e r at least n(k - 1)/k, we g e t C(Td,k) > o~(n -1)+2d(n(kk-1) --1) 2. T h u s , limk,d~o¢ p(Td,k) = limd~oc((d-1 ) n ( n -- 1) + 2 d ( n - 1 ) 2 ) / ( ( d + 1 ) n ( n - 1)) = 3.
5.
THE
TREE
T h e d i a m e t e r o f T is at m o s t 2d, so d i a m ( T ) < 2 + 4 a / ( n - 2 ) .
T h e n , since 2 ( n - 1) ordered pairs of n o d e s are at d i s t a n c e
1 from each other,
-C(T)_<~(n-1)+
CONJECTURE
N u m e r o u s e x p e r i m e n t s a n d a t t e m p t e d c o n s t r u c t i o n s for c~ >
2 have t h u s far only yielded N a s h equilibria t h a t are trees,
w i t h t h e sole e x c e p t i o n of t h e P e t e r s e n g r a p h , which is a
t r a n s i e n t e q u i l i b r i u m for 1 _< a < 4. F u r t h e r m o r e , it is
s o m e w h a t intuitive to s u p p o s e t h a t n o n - t r e e equilibria are
less likely to a p p e a r as a l p h a grows a n d r e d u n d a n t links
b e c o m e costlier 4. W e t h u s p o s t u l a t e t h e following:
+2(n-l)
( 2 + ~ _ 24 a
) (n-
=ha(n-1)+2(n-
2)(n-
1)+
(2)
1) 2 .
T h u s , t h e price of a n a r c h y is
p(T) < 5c~(n - 1) + 2(n - 1) 2 < 5
a(n - 1) + 2n(n - 1)
[]
CONJECTURE 1
(THE TREE CONJECTURE). There exists a constant A, such that for all a > A, all non-transient
Nash equilibria are trees.
F r o m T h e o r e m 1 a n d T h e o r e m 3, we directly obtain:
COROLLARY 4. If the Tree Conjecture holds for some value
off A, then for all non-transient Nash equilibria G, p(G) =
A r m e d w i t h t h i s conjecture, we c a n exploit t h e s t r u c t u r e
of trees to o b t a i n a c o n s t a n t u p p e r b o u n d on t h e price of
anarchy.
O(./A).
It s h o u l d be n o t e d t h a t t h e above proof relies crucially on t h e
existence of a "center" node, w h i c h s e e m s to be t h e p r i m a r y "
barrier to e x t e n d i n g t h e proof to m o r e general graphs.
THEOREM 3. For any tree Nash equilibrium T, p(T) < 5.
PROOF. Let n be t h e n u m b e r of n o d e s in T , and, for a
node i, let L(i) be t h e set of n o d e s w h i c h forms t h e largest
c o n n e c t e d c o m p o n e n t of t h e g r a p h o b t a i n e d if i a n d all edges
adjacent to it are r e m o v e d - - t h e s e c o m p o n e n t s are hereafter
Note also t h a t t h e n o n - t r a n s i e n c e c o n s t r a i n t in t h e conjecture a n d t h e c o n s e q u e n t r e s u l t s is optional, as we have not
found even t r a n s i e n t n o n - t r e e equilibria for c~ > 4. It is included above b e c a u s e it w e a k e n s t h e c o n j e c t u r e a n d s e e m s
to provide a m o r e n a t u r a l n o t i o n of a n equilibrium.
3Note t h a t this includes t h e case w h e n i a n d j s h a r e a nonroot a n c e s t o r o t h e r t h a n a parent.
4Note t h a t , for a p o s i t i o n to be a N a s h equilibrium, t h e r e
m u s t be no single n o d e w i t h a different s t r a t e g y t h a t lowers
its costs, no m a t t e r how different t h a t s t r a t e g y is from its
c u r r e n t strategy. E.g., it m a y at first a p p e a r t h a t a directed
n-cycle for n > 5 (i.e. a s i t u a t i o n where node i links to i + 1
only, w i t h n - 1 linking to 0), is a N a s h equilibrium. T h i s is
n o t t h e case, since node 0 would lower its costs if it i n s t e a d
linked to node 2 only.
W e would love, of course, to see a proof of t h e Tree Conjecture, as well as a n y o t h e r c o n s t a n t u p p e r b o u n d for t h e
price of anarchy.
6.
DISCUSSION
AND
EXTENSIONS
T h e r e are s o m e i n t e r e s t i n g variations on o u r m o d e l t h a t m a y
be w o r t h y of study:
350
7.
• In our model, edges paid for by a node can be used by
others. In the directed version an edge paid by i can
only be used in the direction from i to j. The equilibria
in this model are much more complex and numerous,
although this variant seems to be less applicable to real
situations.
REFERENCES
[1] A. Czumaj, P. Krysta, and B. VScking. Selfish traffic
allocation for server farms. In Proc. of 2002 STOC,
pages 287-296. ACM, May 2002.
[2] M. Faloutsos, P. Faloutsos, and C. Faloutsos. On
power-law relationships of the internet topology. In
Proc. of 1999 SIGCOMM, pages 251-262. ACM,
September 1999.
• In our model, a node must either pay for an edge in
full, or not at all (and hope that the other endpoint
will decide to pay for it). In the fractional model, a
node may pay for a fraction of an edge, and the edge
will exist if the sum of the two fractions exceed one
(naturally, at equilibrium this sum will be either zero
or one). This model may allow for a greater variety of
equilibria.
[3] J. Feigenbaum, C. Papadimitriou, R. Sami, and
S. Shenker. A bgp-based mechanism for lowest-cost
routing. In Proc. of 2002 PODC, pages 173-182. ACM,
July 2002.
[4] E. Koutsoupias and C. H. Papadimitriou. Worst-case
equilibria. Lecture Notes in Computer Science,
1563:404-413, 1999.
• Stars (even though prevalent in the real Internet) are
problematic network,,; because of congestion. It would
be interesting to introduce this consideration to our
model; the challenge again is, in doing so, to keep it
relatively tractable.
[5] M. Mavronikolas and P. Spirakis. The price of selfish
routing. In Proc. of 2001 STOC, pages 510-519. ACM,
July 2001.
[6] T. Roughgaxden. Selfish Routing. PhD thesis, Cornell
University, 2002.
• We assume that all pairs of nodes have the same traffic,
weighing distances between all node pairs equally. A
more detailed model would use a traffic matrix [tij] >
0~ and the term dc(s)(i,j) of cost function would be
multiplied by ti3.
[7] T. Roughgarden and E. Taxdos. How bad is selfish
routing? In Proc. of 2000 FOCS, pages 93-102. IEEE,
November 2000.
• The previous extension uses n 2 parameters, a little too
many. In the rank-one special case we assume that
tij = wi. wj, where w~ are node weights capturing the
strength (or importance, or market share) of the nodes.
It would be very interesting to identify ways in which
the weights affect the degrees of the nodes at equilibrium; such an approach may shed light to phenomena
of heavy-tailed distributions recently observed [2] (it
would be a much more primitive assumption, consistent with experience from other markets, to postulate
that the weights are so distributed).
• With our colleague Kunal Talwax, we have been considering a network creation game in which node costs
reflect Vickrey payments - - a proposal from [3]. The
nature of the equilibria of such a game could explain
the modest Vickrey overcharges observed in the real
Internet in [3].
• A more realistic model would assume a large but not
infinite cost to i when j is disconnected from the network. Such an assumption may disallow certain absurd
equilibria that are the result of "blackmail by j."
• Finally, it would be more realistic to look at networks
that axe developed in stages, where new nodes arrive
at each stage (old nodes can add links), and all stages
should be equilibria.
While some of these variants and extensions seem analytically intractable, we beliew~ that research in some of these
more sophisticated models may result in increased understanding of the fascinating phenomenon that is the Internet.
351