Interference networks with local view: A distributed optimization approach
ABSTRACT In practice, a node in a network learns the channel through local message passing and obtains a local view of the network. Pure wireless message passing as well as mixed wireless and wireline message passing are considered in this paper. We study the distributed optimization of sumrate for a class of deterministic interference networks with local view. A connection based utility function is designed for each user to exploit the local knowledge. This utility design turns out to be a potential game with sumrate as the potential function. For the onetomany channel with 1.5 wireless rounds of message passing, we show that there is a unique Nash equilibrium and using this strategy, the sum capacity can be achieved. We provide a sufficient condition for which a topology does not have unique Nash equilibrium. Then we consider the scenario that the network size and the users IDs are provided to each user. For various mixed wireless and wireline message passing patterns, including wireline at transmitter/receiver side and sequential/concurrent message passing scheduling, we identify whether a threeuser interference network can achieve the sum capacity in a distributed fashion. Compared with the 1.5 pure wireless rounds of message passing, the results show that 2.5 mixed wireless and wireline rounds of message passing can significantly improve the system performance of threeuser interference networks. We also derive some sufficient conditions for general Kuser interference networks such that the sum capacity can not be achieved based on each user's local view.

Conference Paper: Feedback via message passing in interference channels
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ABSTRACT: In distributed wireless networks, nodes often do not know the topology (network size, connectivity and the channel gains) of the network. Thus, they have to compute their transmission and reception parameters in a distributed fashion. In this paper, we consider the information required at the nodes to achieve globally optimal sum capacity. Our first result relates to the case when each of the transmitter know the channel gains of all the links that are atmost twohop distant from it and the receiver knows the channel gains of all the links that are threehop distant from it in a deterministic interference channel. With this limited information, we find that distributed decisions are sumrate optimal only if each connected component is in a onetomany configuration or a fullyconnected configuration. In all other cases of network connectivity, the loss can be arbitrarily large. We then extend the result to see that O(K) hops of information are needed in general to achieve globally optimal solutions. To show this we consider a class of symmetric interference channel chain and find that in certain cases of channel gains, the knowledge of a particular user being odd user or even user is important thus needing O(K) hops of information at the nodes.Signals, Systems and Computers, 2009 Conference Record of the FortyThird Asilomar Conference on; 12/2009
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Interference Networks with Local View: A
Distributed Optimization Approach
Jun Xiao1, Vaneet Aggarwal2, Ashutosh Sabharwal3, Youjian Liu1
1Department of ECEE, University of Colorado at Boulder
2Department of Electrical Engineering, Princeton University
3Department of ECE, Rice University
Abstract—In practice, a node in a network learns the channel
through local message passing and obtains a local view of
the network. Pure wireless message passing as well as mixed
wireless and wireline message passing are considered in this
paper. We study the distributed optimization of sumrate for a
class of deterministic interference networks with local view. A
connection based utility function is designed for each user to
exploit the local knowledge. This utility design turns out to be
a potential game with sumrate as the potential function. For
the onetomany channel with 1.5 wireless rounds of message
passing, we show that there is a unique Nash equilibrium and
using this strategy, the sum capacity can be achieved. We
provide a sufficient condition for which a topology does not
have unique Nash equilibrium. Then we consider the scenario
that the network size and the users IDs are provided to each
user. For various mixed wireless and wireline message passing
patterns, including wireline at transmitter/receiver side and
sequential/concurrent message passing scheduling, we identify
whether a threeuser interference network can achieve the sum
capacity in a distributed fashion. Compared with the 1.5 pure
wireless rounds of message passing, the results show that 2.5
mixed wireless and wireline rounds of message passing can
significantly improve the system performance of threeuser
interference networks. We also derive some sufficient conditions
for general Kuser interference networks such that the sum
capacity can not be achieved based on each user’s local view.
I. INTRODUCTION
Although the general capacity region still remains un
known even for the twouser case, there has been an ex
tensive literature on interference networks regarding achiev
able region, outer bound and degree of freedom, etc. [1]–
[4]. Considering the nature of interference networks that
each transmitter has independent message only intended
for corresponding receiver, it is more interesting to study
the distributed optimization rather than a centralized one.
Noncooperative game model is widely used to tackle the
distributed optimization problem. Each user in the network
is viewed as a selfish player and is only interested in
maximizing its own utility function. Due to the lack of
capacity achieving strategy, most existing game theory works
in the context of interference channel treat the interference
as Gaussian noise and hence are suboptimal in general
[5], [6]. Recently, Berry and Tse [7] analyzed the non
The work was supported in parts by USNSF Grants CCF0728955 and
ECCS0725915.
cooperative game in the twouser deterministic interference
channel, which has a more information theoretic taste.
One key assumption in almost all previous works is that
each node has the full knowledge of the whole network
topology, defined as the network connectivity and the channel
gain of each link. In practical systems, however, obtaining
the full channel knowledge for each user may consume
too much resource such as time, bandwidth, power, etc.
Also for a large distributed network, it is reasonable to
assume that the network is not fully connected due to the
size of the network and the heterogeneous nature of nodes.
Therefore, it is more interesting to consider the scenario that
each node has different partial knowledge about the network
topology obtained by local message passing. We call this
partial knowledge as local view. The work in [8] considered
this problem by letting the node incrementally learn the
topology through a wireless message passing algorithm. The
performance of Z and doubleZ deterministic interference
networks with local view are characterized.
We consider the deterministic channel model [9] through
out this paper. Deterministic model can give significant
insight towards the Gaussian model in certain cases [10],
[11]. We propose a connection based utility function which
turns out to be a potential game with the sumrate as
the potential function. With full knowledge, potential game
guarantees that there exist Nash equilibria (NE) achieving
the sum capacity for any topology. Then a local information
game approach is used to study the distributed optimization
problem with local view. Specifically, each user assumes
the incomplete topology as the true topology and bases its
decision on the NE of this known topology. For the Kuser
onetomany channel with 1.5 wireless rounds of message
passing, we show that there exists a unique NE and the
system can achieve the sum capacity even with incomplete
information of the network topology. When the unique NE
is efficient, which means that it achieves the sum capacity,
we denote it as distributedly optimal strategy. A sufficient
condition that a topology with certain local view does not
have distributedly optimal strategy is provided. However,
we observe that overwhelming majority of topologies have
multiple NE. The multiplicity of equilibria makes the sum
capacity nonachievable in the absence of the predetermined
strategy or extra cooperation.
With the help of predetermined strategy, the authors of [12]
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FortySeventh Annual Allerton Conference
Allerton House, UIUC, Illinois, USA
September 30  October 2, 2009
9781424458714/09/$26.00 ©2009 IEEE
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proved that only fully connected topology and onetomany
topology can achieve the sum capacity distributedly with
1.5 wireless rounds of message passing. More information
is needed if we want to improve the system performance.
In a practical system, nodes are not always physically iso
lated, such as connected base stations in a cellular network.
Therefore, based on the wireless message passing algorithm
proposed in [8], we take a further step to consider wireline
message passing at the transmitter or receiver side. The ex
isting works of interference channel with transmitter/receiver
cooperation only involve transmitted information bits or
functions of it [13], [14]. Here we assume the nodes only
exchange channel information via linear wireline topology.
Various mixed wireless and wireline message passing pat
terns, including wireline connection at transmitter or receiver
side and sequential or concurrent scheduling, are considered.
Assuming the network size and users IDs are given to each
user, a network topology with certain local view can achieve
the sum capacity distributedly if it has a universally optimal
strategy [11]. We show that, with one wireless round followed
by one sequential/concurrent wireline round at transmitter
side and half forward wireless round, at least 59/54 out of
64 wireless topologies of threeuser interference networks
have universally optimal strategy. Also, at least 59/54 out
of 64 wireless topologies of threeuser interference channels
have universally optimal strategy with half forward wireless
round followed by one sequential/concurrent wireline round
at receiver side, a half backward wireless round and a
half forward wireless round. Compared with the fact that
only 14 topologies have universally optimal strategy with
1.5 pure wireless rounds, above results show that wireline
message passing can significantly improve the performance
of threeuser interference networks. We also derive some
sufficient conditions for Kuser interference networks where
universally optimal strategy does not exist.
The rest of the paper is organized as follows. We formulate
the problem in Section II. Local information game and
the topologies that have distributedly optimal strategy are
discussed in Section III. In Section IV, we identify whether
there exists a universally optimal strategy for a topology with
wireline cooperation. Conclusions are provided in Section V.
II. PROBLEM FORMULATION
Consider an interference network with K transmit
ter/receiver pairs. The transmitters 1,2,...,K have inde
pendent messages M1,M2,...,MK intended for receivers
1,2,...,K, respectively. The network topology can be rep
resented by a K × K channel matrix H, where the (i,j)th
entry nij is the channel gain from transmitter i to receiver
j. We consider the deterministic interference channel model
[9], [10], where nijis the number of binary connections from
transmitter i to receiver j. Note that ni,j= 0 implies there is
no link between transmitter i and receiver j. We assume that
none of the nodes in the network has a priori information of
channel matrix H.
1
2
K
…...
Figure 1. Linear wireline topology.
A. Wireless message passing
We adopt the wireless message passing algorithm proposed
in [8] to let each node learn a local view of the network
topology. This message passing algorithm proceeds in rounds
between transmitters and receivers, which is analogous to
the iterative decoding of LDPC codes [15]. There are two
major differences compared with iterative decoding. First, the
messages are broadcasted due to the wireless nature. Second,
the messages are scheduled one by one for each node to avoid
collision. The algorithm is as follows [8].
1) A half forward round: Each transmitter broadcasts
training signal so that the receivers can learn the
channel gains of the connected links.
2) A half backward round: Each transmitter/receiver is
assumed to have a unique ID. Receivers broadcast a
message which is the set of learned channel gains
and the corresponding IDs, {(nij,i,j)}, so that the
transmitters can learn part of the channel matrix H.
The above two steps complete the first round.
3) In the round t > 1, both transmitters and receivers only
broadcast channel information that are new to at least
one listener.
4) Stop when no node has new updates, i.e. all nodes have
the complete knowledge of the network. Or, for low
complexity operation, the message passing can stop
after a half forward round so that the receivers know
what the transmitters know to enable decoding.
After certain rounds of message passing, each node in the
network has its own local view of the network topology.
Formally, the local view of node k is defined as Lk =
{(nij,i,j)?s received by node k}. The local view of a user
may be different compared with those of other users.
B. Wireline message passing
With limited topology knowledge, the performance of the
system sometimes is unboundedly inferior to the optima
assuming global channel knowledge. However, completing
the wireless message passing in a sparse connected networks
requires a large number of wireless rounds. In the practical
system, nodes are not always isolated. For example in the
cellular system, base stations may be connected via back
bones to speed up information propagation. In this paper we
consider a simple linear wireline topology.
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Definition 1: (Linear wireline topology) A linear wireline
topology is defined as a topology that all transmitters [re
ceivers] are connected by a line, i.e. all nodes have degree
two except the two degree one end nodes.
Starting from an end node, we label all nodes in increasing
order, as shown in Figure 1. In this paper we assume that
the information exchanged through the wireline link is the
channel states. Note that the exchanged information could be
user strategies or the information messages, which is beyond
the scope of this paper [13], [14]. Assume that after certain
rounds of wireless message passing, the ithuser already has
the local view Li. Two wireline message passing schedulings
are considered as follows.
1) Sequential wireline message passing
a) The Kthuser sends its local view LK to neigh
boring node K − 1.
b) At the middle nodes, after receiving the infor
mation?i+1
Li\Li+1to the (i + 1)thuser.
c) The algorithm ends after the first user receives the
information?2
the ithuser will have local view?i−1
2) Concurrent wireline message passing
Each node simultaneously transmits its local view to
the two neighboring nodes. After that, the ithuser will
have local view Li+1
wireline round.
Although the sequential wireline message passing is ef
ficient in the sense of carrying information, it may take a
long time to finish one round. In comparison, the concurrent
wireline message passing is more efficient in time but each
node has less channel knowledge.
n=KLnfrom the (i + 1)thuser, the ith
user transmits?i
n=KLnto the (i − 1)thuser and
n=KLnfrom the second user and
transmits L1\L2 to the second user. In the end,
n=KLn. This
completes one wireline round.
?Li
?Li−1. This completes one
C. Mixed wireless and wireline message passing patterns
Withtransmitter/receiver
tial/concurrent scheduling, we have four different patterns.
For the transmitter cooperation with sequential/concurrent
scheduling, the local view is already at the transmitter side.
For the receiver cooperation with sequential/concurrent
scheduling, the local view is at the receiver side. Therefore,
another half backward wireless round is needed to convey the
local view from the receivers to the connected transmitters.
Specifically, denote the local view of receiver i after receiver
wireline cooperation as LR
that connect to transmitter i as
cooperationandsequen
i, and denote the set of receivers
Si= {j : 1 ≤ j ≤ K, Hij> 0}.
The local view of transmitter i after this half backward
wireless round message passing is
?
The last action of all messaging passing is always a half
forward wireless round.
(1)
LT
i=
j∈Si
LR
j.
We are interested in the system performance with 1.5 pure
wireless rounds as well as 1.5 wireless rounds and 1 wireline
round, which is denoted as 2.5 mixed rounds. We will analyze
whether the sum capacity can be achieved distributedly by
only using the local information.
III. DISTRIBUTEDLY OPTIMAL STRATEGY WITH PURE
WIRELESS MESSAGE PASSING
In this section we use game theory to distributedly op
timize the system. First we discuss the design of utility
function of each user.
A. Utility design
The signal level allocation policy of transmitter i is denoted
by ai∈ Ai, where Aiis the set of all possible signal level
allocation policies of user i. With some abuse of terminology,
the signal level allocation policy is referred to as the strategy
throughout this paper. Let
A = A1× ··· × AK
be the set of all possible strategy of the network. The
collection of strategies a = (a1,a2,...aK) ∈ A is called
a strategy profile.
In our distributed optimization setting, transmitter i is an
autonomous decision maker that chooses its own strategy
ai∈ Aito maximize its own utility function Ui(a). There
fore, the transmitters in the network face a multiplayer game.
Let a−idenotes the collection of strategy other than ai, i.e.
a−i= (a1,...,ai−1,ai+1,...,aK)
and
A−i= A1× ··· × Ai−1× Ai+1× ··· × AK.
Nash equilibriumis
1,a∗
unilaterally deviate from a∗. It is convenient to denote a
strategy profile a as (ai,a−i) when we discuss the strategy of
transmitter i. Mathematically, a strategy profile a∗is called
a pure Nash equilibrium if
The
(a∗
a strategyprofile
a∗
=
2,...a∗
K) that no transmitter has any incentive to
Ui(a∗
i,a∗
−i) = max
ai∈AiUi(ai,a∗
−i), ∀i.
Our goal is to design the utility function Ui such that
sum rate can be maximized. In general, for a given strategy
(ai,a−i) the utility Uican be designed as
Ui(ai,a−i) =
K
?
n=1
wnrn,
where rnis the transmitting rate of the nthuser, wnis the
associated designable weight. There are two simple ways to
design the wn.
The first one is to assign the sum rate as the utility for every
node, i.e. wn = 1,∀n. There is an obvious drawback that
when the node does not have the full topology information,
it can not calculate its own utility.
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Another trivial utility design is Ui = ri. Although the
authors of [7] proved that there exist NE that achieve sum
capacity in twouser interference channel, we can easily con
struct a counterexample, for example onetomany channel,
to show that no NE can achieve the sum capacity for certain
Kuser interference network.
To overcome the problem mentioned above, we consider
following connection based utility function design principle.
After one wireless round of message passing, each transmitter
learns which receivers are connected with it. Then a local way
to design the utility function of each user is
Ui(ai,a−i) =
?
k∈Si
rk.
(2)
where Siis defined in equation (1).
B. Potential game and local information game
The advantage of studying potential game is that the set
of pure Nash equilibria can be found by simply locating the
local optima of the potential function.
Definition 2: (Potential Game [16]) A game is called
potential game if there is a potential function φ : A → R
such that, for every i, for every a−i∈ A−i, and for every
a
i,a
? ??
i∈ Ai,
Ui(a
?
i,a−i) − Ui(a
Proposition 1: If we use the connection based utility func
tion defined in equation (2), any strategy that achieves the
sum capacity is a NE.
Proof: This is a potential game if φ =
defined as the potential function. Note that the strategy ai
only affects?
K
?
=
??
i,a−i) = φ(a
?
i,a−i) − φ(a
??
i,a−i). (3)
?K
k=1rk is
k∈Sirk. The potential function can be written
as
φ(ai,a−i)=
k=1
?
Ui(ai,a−i) + R(a−i),
k/ ∈Sirkis solely determined by strategy
rk
k∈Si
rk+
?
k/ ∈Si
rk
=
where R(a−i) =?
φ(a
a−i. Then, for ∀ i, ∀ a−i∈ A−i, and ∀ a
?
i,a−i) − φ(a
?
i,a
??
i∈ Ai, we have
?
i,a−i
i,a−i) − Ui(a
??
i,a−i)=
Ui
?
a
?
?
i,a−i
??
+ R(a−i)
?
−Ui
Ui(a
a
− R(a−i)
??
i,a−i).
=
?
The proposition follows according to the property of potential
game.
Every transmitter in the fullyconnected network can learn
the full topology information after one wireless round of
message passing. Therefore there exist NE can achieve the
sum capacity. However, there may exist multiple NE so that
the system has to rely on predetermined strategy or extra
cooperation to pick one of them.
KK
22
11
?... ?...
Figure 2.Onetomany channel.
For topologies that have a unique NE, the local information
game G(·) produces strategy profile ˆ a = (ˆ a1,...,ˆ aK), in
which
ˆ ai= G(Li,i),
(4)
where ˆ ai together with ˆ a−i is the unique NE of the game
with topology Li. The game is assumed to be greedy in that
the unknown channel gains are assumed to be zero. Note
that the game is a function of relative local topology, i.e., it
doesn’t change when the nodes are relabeled by any π(·),
G(Lk,k)
=
G({(nij,π(i),π(j))?s received by node π(k)},π(k)).
Since the local view of one user may be different from that of
other users in general, the game played by each user is not
necessarily the same. As a result, the strategy (ˆ a1,...,ˆ aK)
may not be the NE of the game with full topology H.
As mentioned above, instead of considering complete
information we are more interested in whether the system
has the ability to achieve the sum capacity with incomplete
information. The answer is affirmative for some topologies
as shown in next section.
C. Interference networks with 1.5 wireless rounds
This section considers the interference networks with only
one and half wireless rounds of message passing. For the
onetomany channel as shown in Figure 2, assume there are
a total of K transmitter/receiver pairs. The capacity region
and the corresponding strategy with full topology information
at all nodes can be found in [17].
After the first round of wireless message passing, trans
mitter 1 has local view L1which happens to be the same as
H. Transmitter i, 1 ≤ i ≤ K, has local view written in a
matrix
Li=
0
?
0
n1i
nii
?
,
(5)
which turns out to be a twouser Z channel, where n11 is
assumed to be 0.
Theorem 1: For Kuser onetomany channel, there exists
a unique NE with one and half wireless rounds of message
passing. Moreover, this NE can achieve the sum capacity for
any channel gains. Therefore, one and half wireless rounds
is enough to achieve the sum capacity.
Proof: The local view Liof transmitter i, 2 ≤ i ≤ K,
defined in equation (5), is a special Zchannel with n11= 0.
The unique NE for the ith user, which maximizes the utility
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ri, is to send at full rate nii. After one wireless round of
message passing, transmitter 1’s knowledge L1 happens to
be the same as H. The first transmitter knows that all other
users will send at the full rate. To maximize the U1, which
is the sum rate, it has only one choice that sending at the
levels which will not hurt any other receivers. It can be easily
checked that this unique NE can achieve the sum capacity
for any channel gains. Then the system with one and half
rounds of message passing can achieve the sum capacity.
Note that the transmitters in onetomany channel need
two full wireless rounds of message passing to learn the
channel matrix H completely. Theorem 1 tells us that the
sum capacity in fact can be distributedly achieved without
full information. The case of the fullyconnected topology,
in which all nodes have full information after one wireless
round, is analyzed in Section IIIB. The result suggests that
we need predetermined strategy to choose one efficient NE
among all feasible equilibria.
D. Distributedly optimal strategy and universally optimal
strategy
We define the distributedly optimal strategy using the
above local information game.
Definition 3: (Distributedly optimal strategy) A distribut
edly optimal strategy for a topology with certain local view
is a strategy profile produced from the unique NE of the
local information game using the connection based utility
function, such that it can achieve the sumcapacity of the
whole network for all the choices of the channel gains as if
full information is available at all nodes.
In the following we give a sufficient condition that a
topology with certain local view does not have distributedly
optimal strategy.
Proposition 2: A topology with certain local view does not
have distributedly optimal strategy if it satisfies any one of
the following two conditions.
1) There exist two users who are connected with each
other, and both of them know the four links.
2) There exist two users which form a topology Z (or S)
and both of them know the three links.
Proof: For the first condition, consider a special example
that all other links are zero. Then the networks degrade to a
fully connected twouser interference channel with complete
information at the transmitters. The utility function of each
user is the sum rate. We know that in general there exist
multiple NE.
For the second condition, we again consider a special
example that all other links are zero. Then we have a Z
channel for which multiple NE exist.
From this proposition, one can immediately know that
the networks with full topology knowledge does not have
distributedly optimal strategy, except the situation that there
is no cross link.
The Proposition 2 is so restrictive that most topologies
do not have distributedly optimal strategy. But if a minor
side information, network size, is provided to each user, sum
capacity is still achievable distributedly.
Definition 4: (Universally optimal strategy [12]) A uni
versally optimal strategy for a topology with certain local
view is defined as a set of each user’s strategies that is only
a function of local view, node ID, and network size, such
that the sumrate achieved in this distributed fashion is the
sumcapacity of the whole network for all the choices of the
channel gains as if full information is available at all nodes.
In contrast to the distributedly optimal strategy, the func
tion is allowed to give different output when the nodes are
relabeled. This makes it possible to implement predetermined
strategies. Obviously, distributedly optimal strategies is a
subset of universally optimal strategies.
Aggarwal et al. [12] proved that among all possible
topologies, only onetomany and fully connected topologies
have universally optimal strategy after 1.5 rounds of wireless
message passing. Let’s focus on the threeuser interference
networks. There are at most six cross wireless links, which
makes 26= 64 different wireless topologies. 14 out of these
64 topologies, fall into the category of onetomany channels
or fully connected channels. To achieve the sum capacity of
other 50 topologies while avoiding costly wireless message
passing, we investigate how wireline message passing can
help.
IV. UNIVERSALLY OPTIMAL STRATEGY WITH WIRELINE
MESSAGE PASSING
In this section we focus on the existence of universally
optimal strategy with various wireline message passing pat
terns.
A. Sequential wireline message passing at transmitter side
In this section we consider interference networks with
sequential wireline message passing at transmitter side. We
first consider the threeuser case then extend it to the Kuser
case.
Theorem 2:For the sequential transmitter side wireline
message passing, at least 59 out of 64 wireless topologies of
the threeuser interference channels have universally optimal
strategy with one wireless round followed by one wireline
round and a half forward wireless round. Among the remain
ing 5 topologies shown in Figure 3.(e)(i), topologies 3.(e),
3.(f) and 3.(g) do not have universally optimal strategy.
Proof: As mentioned before, 14 topologies fall into the
category of onetomany or fully connected channels, where
one and half wireless rounds suffice to have universally
optimal strategy. It can be easily verified that the transmitters
of the 41 out of the remaining 50 topologies can obtain
full topology information after one wireless round and one
sequential wireline round because when any two of the
transmitters are grouped by wireline, no wireless link is
more than two hops away from it. If every node has the
full topology information, the sum capacity can be achieved
by a predetermined strategy. The remaining 9 topologies are
shown in Figure 3. We analyze these topologies one by one.
In these 9 topologies, the first user and the second user know
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b
33
22
11
a
33
22
11
c
33
22
11
e
33
22
11
d
33
22
11
f
33
22
11
h
33
22
11
g
33
22
11
i
33
22
11
Figure 3.Topologies associated with Theorem 2.
all the channel gains. The third user knows all channel gains
but n11.
3.(a): The universally optimal strategy is that the first and
the third users send at rate n11 and n33, respectively. The
second user backs off according to other users. This strategy
can achieve the sum capacity [17].
3.(b): The universally optimal strategy is that the second
user sends at rate n22. The third user sends at the remaining
levels. The first user sends at the levels which will not
interfere the third user’s signal. By doing this the first and
the second user’s interference are alignment at the third user’s
receiver. It is easy to show that this strategy can achieve the
sum capacity [17].
3.(c): The universally optimal strategy exists in this topol
ogy, as shown in [8].
3.(d): The universally optimal strategy is that the second
user sends at rate n22. The third user backs off according to
the second user. The first user knows all channel gain and
can act accordingly to achieve the sum capacity [8].
3.(e): There does not exist any universally optimal strategy
for this topology. Consider n11 = n22 = n33 = n13 = 6,
n23= n32= 4 and n12= 1. The third user does not know
n11. When n11= 0, the only way to achieve the sum capacity
is the third user transmitting with rate 4. When the third user
transmit with rate 4, we know that R2≤ 4 according to the
constraint of the third user. The first user and the third user
form a Zchannel and hence R1≤ 2 . To sum up, the sum
rate with this strategy is no more than 10. However, the rate
pair (5,6,0) can be achieved with full channel knowledge.
3.(f): The universally optimal strategy does not exist for
this topology. Consider n11= n22= n33= n12= 3, n23=
ii
jj
kk
?...
?...
?...
?...
?...
?...
?...?...
i>j>k
Figure 4.
Kuser interference network which contain topology 3.(f).
n32= 2. The third user does not know n11. When n11= 0,
the only hope to achieve the sum capacity is for the third user
to transmit with rate 2. The first user and the second user
form a Zchannel with sum rate R1+ R2 ≤ 3. Therefore,
the maximum sum rate can not exceed 5. However, the rate
pair(3,0,3) can be achieved with full channel knowledge.
3.(g): The universally optimal strategy does not exist in
this topology. Consider n11= n22= n33= n13= 3, n23=
n32= 2. The third user does not know n11. When n11= 0,
the only hope to achieve the sum capacity is for the third
user to transmit with rate 2. We have R2≤ 2 according to
the constraint of the third user. The first user and the third
user form a Zchannel with R1+ R3 ≤ 3. Therefore, the
maximum possible sum rate is 5. But the rate pair (3,0,3)
can be achieved with full channel knowledge.
3.(h) and 3.(i): We do not know the answer for these two
topologies. Assuming the third user sends at a rate of R3= n,
to prove the existence universally optimal strategy, we would
need to show that the sum rate surface will always intersect
with the R3= n plane in the achievable region. We need to
know the capacity region of this topologies, which haven’t
been solved yet.
The following corollary gives a sufficient condition that
the universally optimal strategy does not exist for Kuser
interference networks.
Corollary 1: For a Kuser interference network with one
wireless round followed by one transmitter side sequential
wireline round and a half forward wireless round, the univer
sally optimal strategy does not exist if three users i,j and k,
where i > j > k, can in order form a wireless connectivity
as shown in Figure 3.(e)(g), and the kthreceiver is not
connected to the mthtransmitter, where i − 1 ≤ m ≤ K.
Proof: Without loss of generality, consider the topology
in the Figure 4, which contains the topology in Figure
3.(e). Because the kthreceiver is not connected to the mth
transmitter, i − 1 ≤ m ≤ K, nkk is at least three hops
aways from user K to i − 1, hence they do not know nkk
after one wireless round. Then we consider one sequential
transmitter wireline message passing. The ithuser will gain
all the knowledge from the Kthuser to the (i − 1)thuser,
which do not contain nkk. Now consider a special channel
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33
22
11
33
22
11
Figure 5.The example of mirror topology.
realization that all direct link gain is zero except nii,njjand
nkk. Then the Kuser interference networks degrade to a
threeuser interference networks which we already analyzed
in the Theorem 2. Therefore, if the Kuser interference
networks contains topologies 3.(e)(g), it can not achieve the
sum capacity when all other direct link gain is zero. Taking
into account this counterexample, the corollary follows.
B. Concurrent wireline message passing at transmitter side
Theorem 3: For the concurrent transmitter side wireline
message passing, at least 54 out of 64 wireless topologies
of threeuser interference channels have universally optimal
strategy with one wireless round followed by one wireline
round and a half wireless round of message passing. Six
topologies, 3.(e)(g) and their mirror topologies, do not have
universally optimal strategy.
Proof: It can be show that 19 out of 50 topologies do
not have the full channel knowledge after the concurrent
transmitter wireline message passing. These 19 topologies in
clude the 3 equivalent manytoone channels and 6 equivalent
doubleZ channels. The same argument in the case 3.(a)(d)
of Theorem 2 applies here. Therefore, the universally optimal
strategies exist for these 9 topologies. For the remaining 10
topologies, 5 of them are exactly the same as topologies 3.(e)
(i) in Figure 3, where the third user does not know n11.
Another 5 topologies are the mirror of topologies 3.(e)(i)
where the first user does not know n33. The mirror means
that the role of the first user and the third user are exchanged.
A mirror example is provided in Figure 5. Therefore, six
topologies, 3.(e)(g) and their mirror topologies, do not have
universally optimal strategy. We do not know the answer for
topologies in Figure 3.(h)(i) and their mirror topologies.
The following corollary gives two sufficient conditions that
the universally optimal strategy does not exist for Kuser
interference networks.
Corollary 2: For a Kuser interference network with one
wireless round followed by one transmitter side concurrent
wireline round and a half forward wireless round of message
passing, the universally optimal strategy does not exist if the
network satisfies one of the following two conditions.
1) Three users i,j and k, where i > j > k, can in order
form a wireless connectivity as in Figure 3.(e)(g), and
the kthreceiver is not connected to the mthtransmitter,
m = i − 1,i + 1.
2) Three users i,j and k, where i > j > k, can in
order form a wireless connectivity as the mirror of the
topology in Figure 3.(e)(g), and the ithreceiver is not
connected to the mthtransmitter, m = k − 1,k + 1.
b
33
22
11
a
33
22
11
c
33
22
11
e
33
22
11
d
33
22
11
f
33
22
11
h
33
22
11
g
33
22
11
Figure 6.Topologies associated with receiver cooperation.
Proof: For the first condition, because the kthreceiver is
not connected to the mthtransmitter, m = i−1,i+1, nkkis
at least three hops aways from user i+1 to i−1, hence they
do not know nkkafter one wireless round. Then we consider
one concurrent transmitter wireline message passing. The ith
user will gain all the knowledge from the the (i − 1)thuser
and the (i + 1)thuser, which do not contain nkk. Along the
same line as in the proof of Corollary 1, we can prove this
corollary. Using the similar method we can prove that the K
user interference networks do not have universally optimal
strategy when the second condition holds.
C. Sequential wireline message passing at receiver side
Theorem 4: For the receiver side sequential wireline mes
sage passing, at least 59 out of 64 wireless topologies of
threeuser interference networks have universally optimal
strategy after a half forward wireless round followed, one
wireline round, half backward wireless round, and a half
forward wireless round. The remaining 5 topologies, 6.(d)
(h), are shown in Figure 6.
Proof: After a half round forward wireless message
passing and one round receiver side sequential wireline
message passing, the receivers of the first user and the second
user will know the full topology. When the receivers feed
back the channel gains, the first transmitter and the second
transmitter will know the full topology. If the third transmitter
has connectivity with the first or the second receiver, then
the third transmitter will also learn full topology. Therefore
we only need to consider the cases that n32 = 0 and
n31 = 0, which include 26−2= 16 topologies. In these
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16 topologies, 8 of them fall into the onetomany or fully
connected topology category. The remaining 8 topologies are
shown in the Figure 6. Let’s examine them one by one. For
all the cases, the first and the second user know the full
topology, the third transmitter does not know the value of
n11and n21.
6.(a): This is a doubleZ channel. The sum capacity can
be achieved by the third user sending at full rate n33while
the first and the second user acting accordingly [8].
6.(b): The same argument as above holds.
6.(c): It is the same case as in case 3.(b) in Theorem 2.
6.(d)(h): We do not know the answer for these topologies.
The same argument of the case 3.(h) and 3.(i) in the Theorem
2 holds here.
D. Concurrent wireline message passing at receiver side
Theorem 5: For the receiver side concurrent wireline mes
sage passing, at least 54 out of 64 wireless topologies of
threeuser interference networks have universally optimal
strategy after a half forward wireless round, one wireline
round, half backward wireless round, and a half forward
wireless round of message passing.
Proof: The proof method is similar to the above and the
details are omitted due to the space limitation.
E. Comparison
The information conveyed by concurrent wireline message
passing is a subset of the sequential one. Therefore, in
terms of the number of topology that has universally optimal
strategy, it is not surprising that the sequential one is better
than the concurrent one in both transmitter and receiver
cooperation. But the drawback of sequential wireline message
passing is also obvious. It costs more time to complete
one round. For the threeuser interference networks with
transmitter cooperation, the difference is only five topologies.
But we believe that the difference will be larger when we
have more users in the network.
V. CONCLUSION
This paper studies the distributed optimization of interfer
ence networks with local view. A local information game
with connection based utility function is proposed. We show
that the onetomany channel with one and half wireless
rounds of message passing has distributedly optimal strategy.
We give a sufficient condition that a Kuser interference
networks can not be distributedly optimized. To improve the
system performance, we consider the wireline cooperation at
transmitter or receiver side. For the threeuser interference
networks, the topologies that have universally optimal strat
egy are identified for various mixed wireless and wireline
message passing patterns.
Completely characterizing the relation between local view
and system performance for general Kuser interference
networks is very complicated but of great interest.
VI. ACKNOWLEDGMENTS
The authors wish to thank An Liu for helpful discussions.
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