Can TCP Metastability Explain Cascading Failures 

and Justify Flow Admission Control in the Internet? 

V. Marbukh 

National Institute of Standards and Technology 

100 Bureau Drive 

Gaithersburg, MD 20899 USA 

E-mail: marbukh@nist.gov 

Abstract—This paper discusses implications of possible 

metastability of TCP-type fair bandwidth sharing under 

random flow arrivals/departures for understanding and 

defending the Internet against cascading failures. Cascading 

failures can be viewed as a process of network transition from 

desirable metastable mode with finite number of flows in 

progress to the congested mode. It is possible to eliminate or 

reduce the possibility of cascading failures with properly 

designed flow admission control which stabilizes the network 

in a close neighborhood of the desirable metastable state. 

Keywords-Internet, TCP, metastability, cascading failures, 

flow admission control. 

I. INTRODUCTION

Flow-level abstraction for fair bandwidth sharing among 

fixed set of flows has been proposed in [1] and a Markov 

model for flow-level performance of bandwidth sharing 

protocols under random flow arrivals/departures has been 

proposed in [2]-[3]. The Markov model assumes 

separation of time scales: the bandwidth sharing protocol 

reaches equilibrium bandwidth sharing much faster than the 

numbers of flows in progress change due to flow 

arrivals/departures. However, the high dimension of the 

corresponding Markov process makes this Markov model 

intractable for practical-size networks. The difficulty 

results from a large number of interacting elements 

(processes) describing numbers of flows with different 

source-destinations, where the interactions occur due to 

sharing link bandwidth by different flows. In the fluid 

asymptotic regime [4] the numbers of flows in progress 

becomes deterministic, making fluid model tractable. 

It has been observed by simulation in [5] that packet 

retransmissions during file transfers may cause instability 

of fair bandwidth sharing under arriving/departing flows 

when Explicit Congestion Notification (ECN) signals are 

represented by packet losses. A goodput-based Markov 

model, which accounts for packet retransmissions by 

assuming that file transmission rates are determined by the 

end-to-end goodputs rather than the corresponding 

throughputs, has been proposed in [6]. The goodput-based 

model is typically unstable for a network with multihop 

routing even under light load when the corresponding 

throughput-based model is stable. This instability is the 

result of the positive feedback created by increase in the 

link packet losses with increase in the number of flows 

carried on the link. Despite this instability, a desirable 

metastable, i.e., persistent, network state with finite 

numbers of flows in progress may still exist for sufficiently 

light exogenous load [6]. An appropriately designed flow 

admission control can transform the desirable metastable 

state into the unique stable state at the cost of small flow 

rejection probability. 

This paper suggests that these theoretical results may 

contribute to understanding of the cascading failures in the 

Internet and how to defend the Internet against such 

failures. We consider TCP Reno type fair bandwidth 

sharing, when ECN signals are delivered to the sources 

through end-to-end packet losses. One may assume that 

due to significant Internet underutilization, metastability 

rather than stability of the observed Internet state may not 

be apparent. Utilization increase due to physical 

infrastructure failures or load increase as a result of a 

fluctuation, denial of service attacks, or flash crowds may 

“push” the network out of the desirable metastable state 

initiating process of cascading failures and leading to a 

congested mode. 

The paper is organized as follows. Section 2 briefly 

introduces a fixed-point performance model of a fair 

bandwidth sharing with fixed set of flows [7]. Section 3 

describes Markov [2]-[3] and fluid [4] performance models 

of fair bandwidth sharing [1] under arriving/departing 

flows. Section 4 discusses stability/instability of these 

Markov and fluid models and the importance of flow 

admission control. Section 4 also suggests possible 

implications of these results for understanding cascading 

failures in the Internet and defending the Internet against 

cascading failures. Section 5 considers specific case of a 

symmetric ring network. Finally, Conclusion summarizes 

our results and outlines directions for future research. 

II. FIXED-POINT MODEL FOR A FIXED SET OF FLOWS

Consider a network with links j ∈ J . Given a directed 

route { ,.. , , ,.. } 1 m m 2 M r j j j j j = + 978-1-4244-2036-0/08/$25.00©2008IEEE passing through link 
j let ( ,.., ) j 1 m r = j j − be the part of the route r lying 

downstream from link j , and ( ,.., ) rj jm+2 jM

+ = be the 

part of route r lying upstream from link j . Given a 

vector of numbers of flows on all feasible routes r ∈ R: 

( )r n = n , this section briefly describes models for 

evaluating the vector of end-to-end throughputs ( )r x = x

and good-puts ( )r g = g . 

Weighted (α,w)-fair bandwidth allocation [1] assigns 

the transmission rate on a route r ∈ R as follows: 

 

1 α















 = ∑

j∈r

r r j x w p , (1) 

where fixed parameters , > 0 α wr characterize fairness, 

and parameters pj represent Explicit Congestion 

Notification (ECN) on links j ∈ J . Further, in this paper 

we assume that pj is the packet loss probability on link 

j ∈ J . When = 1 wr , the cases α → 0 , α →1 and 

α → ∞ correspond respectively to an allocation which 

achieves maximum throughput, is proportionally fair, or is 

max-min fair. Case α = 2 corresponds to TCP-Reno rate 

assignment (1) in a case of small packet losses 

 ≈∑ <<1

j∈r

r j p p . (2) 

Note that a more general rate allocation 

 ( ) α β (1 ) 1

r r r r x = w p − p (3) 

with some β ≥ 0 can be used to describe a case when 

↓ 0 r x as ↑1 r p . The case α = β = 2 has been 

considered in [8]. 

Under the assumption of independent packet losses on 

different links, the end-to-end packet loss probability r p is 

 ∏∈

= − − j r

r rj p 1 (1 p ) (4) 

and the load on a link j resulted from a flow carried on a 

route r is 

 ∏− ∈

= −

j i r

rj r ri x x (1 p ), (5) 

where ri p is the probability that a packet from a flow 

carried on a route r is lost on a link i , and r x is the 

source transmission rate on route r . 

Given a mixture of flows in progress on different 

routes ( ) N = Nr , we assume that the packet loss 

probability on a link j for a flow carried on a route r is 

some known function of the average bandwidth allocated to 

each flow on the link j : 

 p p (N , x , r : j r R) rj = rj r′ r′j ∀ ′ ∈ ′ ⊂ (6) 

Assuming that functions (3) and (6) are known, system (3)-

(6) forms a closed system of fixed-point equations for the 

end-to-end performance (x, p), given a vector of numbers 

of flows in progress on different routes 

N (N , r R) = r ∀ ∈ . Here the vector of source 

transmission rates is x (x , r R) = r ∀ ∈ , and the vector of 

link packet losses for different sources is 

p ( p , j J ) = rj ∀ ∈ . 

Note that while functional form (3) depends on the 

specific TCP implementation, functional form (6) depends 

on the buffer queue management discipline. In this paper 

we assume that link packet loss probability is the same for 

all flows carried on the link: 

 prj = pj , ∀r : j ∈r ⊂ R (7) 

Assumption (7) holds in a bufferless case as well as in a 

case of buffered links with First In First Out (FIFO) packet 

scheduling. In a case (7) 

 











 ≥ = − ∑r

r jr

def

j j p p max 0, 1 1 N x  (8) 

where link packet loss probability pj

 corresponds to the 

fluid limit when link capacity and the number of flows 

carried on the link are large. 

Given a mixture of flows in progress ( ) N = Nr , the 

Fixed Point Model determines the end-to-end packet loss 

probability on a route r

 ∏∈

= − − j r

r j p 1 (1 p ), (9) 

and the corresponding good-put 

 ∏∈

= − j r

r r j g x (1 p ). (10) 

The aggregate good-put on a route r is 

 ∏∈

= = − j r

r r r r r j G N g N x (1 p ) (11) 

III. MODELS FOR ARRIVING/DEPARTING FLOWS

Following [4] assume that the network carries two 

types of flows: file transfers and streaming flows. 

Introduce vector ( , ) N1 N2 , where ( ) N1 = N1r and 

( ) N2 N2r = are the vectors of the numbers of file 

transfers and streaming flows, respectively, carried on all 

feasible routes r ∈ R. We assume that files arrive on a 

route r ∈ R according to a Poisson process of rate 

( , ) Λ1r = Λ1r N1 N2 , and streaming flows arrive on route 

r according to a Poisson process of rate 
( , ) Λ2r = Λ2r N1 N2 , where dependence on the vector of 

flows in progress ( , ) N1 N2 allows for modeling statedependent admission control and routing as well as user 

dissatisfaction with deteriorating good-put as the network 

becomes more congested. The size of a file transmitted on 

a route r ∈ R is distributed exponentially with average 

r b . The holding time of a streaming flow carried on a 

route r ∈ R is distributed exponentially with average θ r . 

All arrival processes, file sizes and durations of streaming 

flows are jointly statistically independent. 

We assume that flow control operates on much faster 

time-scale than the process of flow arrivals/departures. 

Given a vector of flows in progress ( , ) N1 N2 , this 

assumption of separation of time-scales allows us to 

approximate the conditional link losses by the solution to 

the corresponding fixed point system (3)-(6) 

p(N) ( p (N)) = j with a fixed number of flows in 

progress N = N1 + N2 . Under our assumptions, the 

vector ( ( ), ( )) 1 2 N t N t is a homogeneous in time 

t ∈[0,∞) vector Markov process. Probabilities 

( , , ) Pr{ ( ) , ( ) } 1 2 1 1 2 N2 P t N N = N t = N N t = are 

uniquely determined by the following system of linear 

differential Kolmogorov equations: 

[ ]

[ ]

∑

∑

∑

∑

∑

∑

−

−

− −

+ + +

+ + + +

+ − Λ −

+ − Λ −

− +

− Λ + Λ

=

r

r r r

r

r r r r r

r

r r r

r

r r r

r

r r r r r

r

r r

P t N N N

P t N N N g N b

P t N N N

P t N N N

P t N N b g N N N

P t N N N N

P t N N

1

1 2 2

1

1 2 1

1 2 2

1 2 1

2

1

1

1

1 2

1 2 1 2

1 2

( , , 1 )( 1)

( , 1 , )( 1) ( 1 )

( , , 1 ) ( 1 )

( , 1 , ) ( 1 )

( , , ) ( )

( , , ) ( ) ( )

( , , )

θ

θ



 (12) 

 N = N1 + N2 (13) 

supplemented with the normalization condition 

 ( , , ) 1

( , )

1 2

1 2

∑ ≡ N N

P t N N , (14) 

given initial conditions (0, , ) P N1 N2 . In (6), vector r 1

has all components equal zero, except for the component 

corresponding to route r , which is equal to 1. 

If the process ( ( ), ( )) 1 2 N t N t is ergodic, i.e., unique 

proper steady state distribution 

 ( , ) lim ( , , ) 1 2 N1 N2 P N N P t t→∞ = (15) 

exists, it satisfies the corresponding system of linear 

algebraic Kolmogorov steady-state equations, which 

describe the equilibrium of non-steady Kolmogorov 

equations (12). Steady-state performance criteria can be 

expressed in terms of distribution (15), e.g., according to 

the Little formula the average file transfer time on route 

r ∈ R in a case when these files arrive at a constant rate 

Λ1r is 

 ∑

∞

Λ =

= 0

1 1 2

1 1

( , ) 1

N r

r

r

τ r N P N N (16) 

However, due to difficulty of solving steady-state 

Kolmogorov equations for practical-size networks, this 

direct approach to network performance evaluation is 

computationally infeasible. The rest of this section briefly 

introduces the fluid approximation. 

 Consider the fluid asymptotic regime, when link 

capacities C j and flow arrival rates Λir are scaled up 

proportionally to a large parameter −1 ε , while the average 

file sizes r b and average durations of streaming flows θ r

are kept constant: 

 

c O i j J r R

C c

j ir

j j ir ir

= = ∀ ∈ ∀ ∈

= − Λ = − →

, (1), 1,2; ,

, , 0, 1 1

λ

ε λ ε ε

 (17) 

It is possible to show [4] that under the asymptotic regime 

(17) typical numbers of flows carried on all feasible routes 

r ∈ R, Nir , are also scaled up proportionally to the large 

parameter −1 ε : 

Nir = nir nir = O → i∈ r ∈ R − , (1), 0, 1,2; 1 ε ε (18) 

where vector n (n (t),r R) i = ir ∈ evolution on any finite 

time interval t ∈ (0,T ], 0 < T < ∞ is described by the 

following system of ordinary differential equations: 

 ( ) ( ) 1 1 2

1

1 1 1 2 n n n b n g n n r = r + − r r r + −  λ (19) 

 r r r r n n n n2

1

2 2 1 2 ( ) −  = λ + −θ (20) 

IV. METASTABILITY AND CASCADING FAILURES

Informally, bandwidth allocation is stable under 

arriving/departing flows if the number of flows in progress 

remains finite as time progresses. However, even if the 

numbers of flows in progress infinitely grows with time 

t → ∞ , the number of flows in progress may not grow on 

a time scale of practical interest. This possibility arises for 

large-scale networks described by a multi-component 

Markov process ( ( ), ( )) 1 2 N t N t with a large number of 

components. Generally speaking convergence time to the 

steady-state distribution for an egrodic Markov process 

with finite number of states depends on the number of 
states. As the number of states increases, the convergence 

time may infinitely increase creating the possibility of 

metastable, i.e., persistent states. If the “life span” of the 

corresponding metastable state is longer than the time scale 

of practical interest, this metastable state from a practical 

perspective appears to be stable. 

In a case of flow arrival rates independent of the 

number of flows in progress: Λir N ≡ Λir ( ) the necessary 

and sufficient conditions for ergodicity of the conventional 

“throughput-based” Markov model with file transfer rates 

determined by the end-to-end throughputs 

 b j J

C r j r

r r

j

def

j = ∑Λ < ∀ ∈ ∈

1, 1

:

ρ1 1 (21) 

have been obtained in [2]-[4]. Conditions (21) have a 

simple interpretation: aggregate link loads are less than the 

link capacities. 

 However, the “goodput-based” model is typically 

unstable even under light load, when conditions (21) are 

satisfied and the throughput-based model is stable. The 

instability of the goodput-based model is a result of 

aggregate goodput (11) deterioration due to increase in the 

number of packet retransmissions as the number of flows 

carried on the route links increases. The deterioration is a 

result of capacity waste: each link j∈ J wastes certain 

portion of its capacity on transmission of “dead” packets to 

be dropped downstream and then retransmitted by the 

source. These arguments lead to the following necessary 

condition for the network stability: 

 [ ] j J

p N

N b

C r j r

i r

i

r r

j

j

< ∀ ∈ −

Λ ∑

∈ ∏∈ +

1, 1 ( )

1 ( )

:

1 (22) 

Typically, as the number of flows carried on a link j

increases the link packet loss pj increases approaching 

unity: pj ↑1, and thus the necessary stability condition 

(22) is not satisfied for any (arbitrary small) non-zero load 

in presence of multihop routes. Typical aggregate end-toend throughput and goodput for file transfer flows are 

shown in Figure 1. 

Figure 1. Goodput deterioration.

 Despite instability, a desirable metastable network 

state with finite number of flows in progress may still exist. 

One may expect that stability of the network as time 

progresses t → ∞ depends on the attractors and 

topological structure of the system (19)-(20) phase space. 

It can be shown that in a case when system (19)-(20) has a 

unique globally asymptotically stable equilibrium point 

( , )

*

2

*

1 n n with finite components the corresponding 

Markov model is ergodic. Generally, the non-linear system 

of ordinary differential equations (19)-(20) may have 

multiple attractors. It is natural to interpret multiple 

attractors as describing metastable, i.e., persistent, network 

states. According to this interpretation, changes in the 

topological structure of the system (19)-(20) describe phase 

transitions, when network performance characteristics may 

experience abrupt changes with infinitesimally small 

change in the exogenous parameters, e.g., load. 

In particular, equilibrium points to (19)-(20) are given 

by the solution to the following system of fixed point 

equations: 

 ( ) ( ) 1 1 1 2 1 2 n n n b g n n r = λ r + r r + (23) 

 r r r n2 = λ2 (n1 + n2 )θ (24) 

Summarizing equations (23) and (24) we obtain the 

following fixed point equations for the total average 

number of flows r r r n n n = 1 + 2 carried on feasible routes 

r ∈ R: 

 r r r r r r n = λ1 (n)b g (n) + λ2 (n)θ (25) 

In the rest of this section we consider case of flow arrival 

rates independent of the number of flows in progress: 

ir ir λ (n) = λ , i =1,2;r ∈ R . 

We conjecture that system (19)-(20) has the following 

topological structure. For a sufficiently large initial 

number of carried flows the number of flows in progress 

grows infinitely with time: nr (t) → ∞ as t → ∞ . This 

instability is the manifestation of the non-ergodicity of the 

Markov model (12)-(14). In absence of streaming flows: 

0 λ2r ≡ , system (19) has a locally stable, finite 

equilibrium ( ) * *

r n = n if and only if 

 b j J

c r j r

r r

j

def

j = ∑ < ∀ ∈ ∈

1, 1

:

ρ1 λ1 (26) 

Generally, in the presence of streaming flows: 0 λ2r ≥ , 

system (19)-(20) has locally stable finite equilibrium 

( ) * *

r n = n if and only if 

 j J ρ1 j <φ j(λ2r ), ∀ ∈ (27) 

where φ j(0) =1 and the functions ( ) φ j λ2r decrease to 

zero as the streaming flow load λ2r increases. This 

Cl

l∈r

min

G1r

N1r

N1r xr

Λ1rbr
equilibrium ( ) * *

r n = n is the only finite attractor of system 

(19)-(20). The stability domain Ν of the equilibrium * n

shrinks with increase in the exogenous load, completely 

disappearing on the stability boundary of this finite 

equilibrium. 

This behavior of the fluid model suggests that 

equilibrium * n corresponds to a metastable, i.e., persistent, 

state of the Markov model since for sufficiently small 

exogenous load and number of flows in progress, the endto-end goodput is sufficient to keep up with the exogenous 

load. However, eventually the number of flows in progress 

grows infinitely, since for the sufficiently large number of 

flows in progress almost the entire network bandwidth 

becomes occupied with transmitting dead packets to be 

dropped downstream. A flow admission strategy, which 

admits flows if and only if the number of flows in progress 

lies within the stability domain of the equilibrium * n , 

transforms the desirable metastable network state into the 

only stable state at the cost of a small flow blocking 

probability. 

It is known in physics that system transition from a 

metastable to stable mode occurs in two stages. First, a 

supercritical embryo of the stable mode appears due to 

sufficiently large fluctuation of the number of flows in 

progress or some external event such as physical 

infrastructure failure, load fluctuation, flash crowds or 

denial of service attack. Then, this embryo grows 

following the internal system dynamics, eventually 

subsuming the entire system. In our case of transition from 

desirable metastable state to overloaded stable state, this 

second stage is essentially a process of cascading failures. 

A flow admission strategy which admits flows if and only 

if the number of flows in progress lies within the stability 

domain of the equilibrium * n modifies the network 

dynamics eliminating the possibility of cascading failures. 

V. EXAMPLE: RING NETWORK

This section briefly considers a case of weighted (α,w)- 

fair rate allocation (3) with β = 0 for a K -node ring 

network, where nodes k = 1,2,..,K and 

(k +1)mod(K) are connected by a directed link kj of 

capacity C . We consider a homogeneous case, when 

weights wr as well as arrival rates of file transfer flows 

Λ1r and streaming flows Λ2r are route r ∈ R

independent: wr = w, r ∈ R . Here Λir = Λi if r ∈ R

and Λ = 0 ir otherwise, and set of l -link feasible routes is 

 {( , ,.., ): 1,.., } R = jk j(k +1)mod(K ) j(k +l)mod(K) k = K . 

We also assume that all feasible routes carry the same 

number N1 of file transfer flows as well as the same 

number N2 of streaming flows: 

 





 ∈ = otherwise

N if r R

N i

ir 0

For a weighted (α,w)- fair rate allocation (3) with 

β = 0 , source transmission rate is 

 

1 α

1 (1 ) 



 





− − = lp

w

x (28) 

Since the load coming on a link j after already traversing 

k links is thinned by a factor of k (1− p) , we obtain the 

following expression for the link utilization: 

 p

p

C

Nx

p C

Nx l l

k

k 1 (1 ) (1 )

1

0

− − = ∑ − =

−

=

ρ , (29) 

where the number of flows in progress originated from a 

given node is N . 

Combining (28) with (29) we obtain the following 

relation between the average link utilization and link packet 

loss: 

 lp

p

l α

ρ η

1 1 [1 (1 ) ] − − − = , (30) 

where the un-dimensional parameter w l N C α η 1 = . We 

consider the fluid regime (17)-(18), when 

 p = max(0, 1−1 ρ) , (31) 

where “effective link utilization” ρ is given by (19). 

Combining equation (30) and (31), we obtain the following 

fixed point equation for link packet loss: 

 α η

ψ 1 1 [1 (1 ) ] ( ) 1 − − − = = − l

def

p

lp p p (32) 

Function ψ( p) is monotonicly decreasing for p ∈[0,1], 

and ψ(0) = 1, ψ (1) < 1. Equation (32) has unique 

solution (0,1) * p ∈ . It is easy to show the following 

properties of the solution of equation (32): 

 ( ) 0 p* = l

−1

ηα + o ηα as η → (33) 

 = − + → ∞ − − p 1 lη o(η ) as η * 1 1 (34) 

In some particular cases equation (32) can be solved 

explicitly. For example, in a case of proportional fairness, 

i.e., α = 1, equation (32) yields * 1 (1 )

− p = + l η . 

After solving equation (32) one can calculate the endto-end per flow good-put g p x

l = (1− ) and the 

effectiveness of the bandwidth sharing protocol 
 γ = Nl g C . (35) 

Substituting (29) into (31) we obtain 

 l p

p

Nx

C

p 1 (1 )1 − − = − . (36) 

Combining (35) with (36) we obtain the following 

expression for the effectiveness: 

 l

l

p

lp p

1 (1 )

(1 ) 1

− −

− =

−

γ . (37) 

Immediately note that in case l = 1 formula (37) produces 

a plausible result γ = 1, since no bandwidth is wasted on 

transmission of packets blocked downstream. It is directly 

follows from (37) that 

 ( ) 0

2

1 1 + → − = − p o p as p

l

γ (38) 

 (1 ) [(1 ) ] 1 γ = − p l−1

l + o − p l−1as p → (39) 

Combining (38)-(39) with (33)-(34) we obtain: 

 ( ) 0 1 1 + → − γ = − η η η α α o as

l

l

 (40) 

 = + → ∞ − − γ l η o η as η l l l (1 ) 1 1 (41) 

Figure 2 shows solution to the “steady-state” equation 

(25) in two cases: ( ) 2

*

1 1 λ > λ λ and ( ) 2

*

1 1 λ < λ λ , where 

the “metastability threshold” ( ) 2

*

1λ λ for exogenous file 

transfer load 1λ monotonicly decreases from (0) 1 *

1λ = to 

( ) 0 *

1λ ∞ = with increase in the streaming load 2 λ . 

 f (η)

 

( )

( ) :

2

* λ1 λ1 λ

η

>

f 

 

 ( ) : ( ) 2

*

1 1 f η λ < λ λ

 η

0 * η ** η 

Figure 2. Solution to the equilibrium equation. 

If exogenous file transfer load exceeds the metastability 

threshold: ( ) 2

*

1 1 λ > λ λ , equilibrium equation (25) has no 

solution η ≥ 0 and, according to the dynamic equation 

(19)-(20), number of file transfer flows in progress grows 

infinitely with time for any initial η(0) ≥ 0 . If the 

exogenous file transfer load is below the metastability 

threshold: ( ) 2

*

1 1 λ < λ λ , equilibrium equation (25) has two 

solutions * η =η and ** η =η describing respectively 

stable and unstable equilibriums of the dynamic equations 

(19)-(20). Number of file transfer flows in progress 

approaches stable equilibrium if initial number of carried 

flows is sufficiently small: ** η(0) <η . Otherwise, i.e., if 

** η(0) >η then the number of file transfer flows in 

progress grows infinitely with time. Note that thresholds 

( ) 2

*

1λ λ , * η and ** η can easily be evaluated numerically. 

More detailed analysis shows that equilibrium * η

represents the desirable metastable network state with finite 

number of flows in progress. This metastable state can be 

transformed into the stable state with flow admission 

control admitting arriving file transfer flows if and only if 

** η <η . The stabilization is achieved at the cost of small 

flow rejection probability at the fluid asymptotic regime. 

VI. CONCLUSION

This paper discusses a possibility that cascading failures in 

Internet-type networks with fair bandwidth sharing may be 

a result of metastability rather than stability of the observed 

“normal” network state. Sufficient network overload can 

push the network out of the stability region of the normal 

state initiating a process of cascading failures. This 

possibility raises numerous research issues including 

existence, stability margins, and queuing performance of 

the metastable state for a general network topology. Most 

importantly, future research should address the validity of 

the model and its predictions. 

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