Wireless Network Coding with Local Network Views: Coded Layer Scheduling
Abstract
One of the fundamental challenges in the design of distributed wireless networks is the large dynamic range of network state. Since continuous tracking of global network state at all nodes is practically impossible, nodes can only acquire limited local views of the whole network to design their transmission strategies. In this paper, we study multilayer wireless networks and assume that each node has only a limited knowledge, namely 1local view, where each SD pair has enough information to perform optimally when other pairs do not interfere, along with connectivity information for rest of the network. We investigate the informationtheoretic limits of communication with such limited knowledge at the nodes. We develop a novel transmission strategy, namely Coded Layer Scheduling, that solely relies on 1local view at the nodes and incorporates three different techniques: (1) per layer interference avoidance, (2) repetition coding to allow overhearing of the interference, and (3) network coding to allow interference neutralization. We show that our proposed scheme can provide a significant throughput gain compared with the conventional interference avoidance strategies. Furthermore, we show that our strategy maximizes the achievable normalized sumrate for some classes of networks, hence, characterizing the normalized sumcapacity of those networks with 1local view.
I Introduction
In dynamic wireless networks, optimizing system efficiency requires information about the state of the network in order to determine what resources are actually available. However, in large wireless networks, keeping track of the state for making optimal decisions is typically infeasible. Thus, in the absence of centralization of network state information, nodes have limited local views of the network and make decentralized decisions based on their own local view of the network. The key question then is, how do optimal decentralized decisions perform in comparison to the optimal centralized decisions which rely on full network state information.
In this paper, we consider multisource multidestination multilayer wireless networks and seek sumrate optimal transmission strategies when sources have only limited local view of the network. To model local views at the nodes, we use a generalization of the hopcount based model that we introduced in [2] for singlelayer networks. In the hopcount based model each source knows the channel gains of those links that are up to certain number of hops away and beyond that it only knows whether a link exists or not. The hopcount based model was appropriate for singlelayer networks where all destinations are within onehop from their respective sources. For multilayer networks, a more scalable approach is to model local views based on the knowledge about sourcedestination (SD) routes in the network (instead of sourcedestination links). The motivation for the routebased model stems from coordination protocols like routing which are often employed in multihop networks to discover SD routes in the network. Hence, a reasonable quanta for network state information is the number of such endtoend routes that are known at the source nodes. In this paper, we consider the case where each SD pair has enough information to perform optimally when other pairs do not interfere. Beyond that, the only other information available at each node is the global network connectivity. We refer to this model of local network knowledge as local view.
Since each channel gain can range from zero to a maximum value, our formulation is similar to compound channels [3, 4] with one major difference. In the multiterminal compound network formulations, all nodes are missing identical information about the channels in the network, whereas, in our formulation, the local view results in asymmetric information about channels at different nodes.
In this paper, our metric to measure the performance of transmission strategies is normalized sumcapacity as defined in [2], which represents the maximum fraction of the sumcapacity with full knowledge that can be always achieved when nodes only have partial knowledge about the network.
Ia Contributions
Our main contribution is a new transmission scheme, named Coded Layer (CL) Scheduling, which only requires local view at the nodes and combines coding with interference avoidance scheduling. Developed as a graph coloring algorithm on a routeextended graph, coded layer scheduling is a combination of three main techniques: (1) per layer interference avoidance, (2) repetition coding to allow overhearing of the interference, and (3) network coding to allow interference neutralization.
We characterize the achievable normalized sumrate of the CL scheduling as the solution to a new graph coloring problem and analyze its optimality for some classes of networks. In particular, we show that coded layer scheduling achieves the normalized sumcapacity in singlelayer and twolayer folded chain networks (defined in Section V). Furthermore, by considering nested foldedchain networks (defined in Section V), we show that the gain from CL scheduling over interference avoidance scheduling can be unbounded.
We also investigate network topologies in which with local view at the nodes, interference avoidance scheduling is informationtheoretically optimal. More specifically, we consider another class of networks, i.e. networks, which is a flow network where all intermediate layers have only relays. We show that for this class, a simpler scheme based on only interference avoidance techniques, named Independent Layer (IL) scheduling, is optimal and coding is not required to achieve normalized sumcapacity with local view. In our limited experience, coding across timeslots can provide gains when there is some regular topological structure in the network and/or there is dense connecvitity. However, the general connection between network topology, partial information and optimal schemes remains a largely open problem.
IB Related Work
In any network state learning algorithm, network state information is obtained via a form of message passing between the nodes. Since the channels through which the communication takes place are noisy and have delay, imprecise network information at the nodes becomes an important problem. Many models for imprecise network information have been considered for interference networks. These models range from having no channel state information at the sources [5, 6, 7, 8, 9], delayed channel state information [10, 11, 12, 13] or analog feedback of channel state for fullyconnected interference channels [14]. Most of these works assume fully connected network or a small number of users. A study to understand the role of limited network knowledge, was first initiated in [15, 16] for general singlelayer networks with arbitrary connectivity, where the authors used a messagepassing abstraction of network protocols to formalize the notion of local view of the network at each node, such that the view at different nodes are mismatched from each others’. The key result was that localviewbased (decentralized) decisions can be either sumrate optimal or can be arbitrarily worse than the globalview (centralized) sumcapacity.
The initial work in [15, 16] was strengthened for arbitrary user singlelayer interference network in [2, 17, 18], where the authors proposed a new metric, normalized sumcapacity, to measure the performance of distributed decisions. Further, the authors computed the normalized sumcapacity of distributed decisions for several network topologies with onehop, twohop and threehop localview information at each source. In this paper, we investigate the performance of decentralized decisions for multilayer wireless networks.
The rest of the paper is organized as follows. In section II, we will introduce our network model and the new model to capture partial network knowledge and we define the notion of normalized sumcapacity. In Section III, via a number of examples, we motivate our transmission strategies. In Section IV, we present our main result, i.e. coded layer scheduling, and we charaterize its performance for multilayer networks. In Section V, we prove the optimality of our strategies (in terms of achieving normalized sumcapacity) for some networks. Finally, Section VI concludes the paper and presents some future directions.
Ii Problem Formulation
In this section, we introduce our models for channel, network, and network knowledge at the nodes. We further define the notions of normalized sumcapacity introduced in [2], which will be used to measure the performance of the strategies with partial network knowledge.
Iia Network Model and Notations
In this subsection, we will describe two channel models that will be studied in the paper, namely the linear deterministic model [19], and the Gaussian model. In both models, a network is represented by a directed graph , where is the set of vertices representing nodes in the network, is the set of directed edges representing links among the nodes, and represents the channel gains associated with the edges.
We consider a layered network in this paper, i.e. the nodes in this network can be partitioned into subsets . Out of nodes in the network, are denoted as sources and are destinations. We label these source and destination nodes by and respectively, . We set and . The remaining nodes are relay nodes which facilitate the communication between sources and destinations. We denote a specific relay in by , and . Without loss of generality, we can also refer to a node in simply as , .
The layered structure of the network imposes the following constraint on the edges in the network,
(1) 
The two channel models used in this paper are as follows.

The Linear Deterministic Model [19]: In this model, there is a nonnegative integer, , associated with each link , which represents its gain. Let be the maximum of all the channel gains in this network. In the linear deterministic model, the channel input at node at time is denoted by . The received signal at node at time is denoted by , and is given by
(2) where is the shift matrix and the operations are in . If a link between and does not exist, we set to be zero.

The Gaussian Model: In this model, the channel gain is denoted by . The channel input at node at time is denoted by , and the received signal at node at time is denoted by given by
(3) where is the additive white complex Gaussian noise with unit variance. We also assume a power constraint of at all nodes, i.e. .
A route from a source to a destination is a set of nodes such that there exists an ordering of these nodes where the first one is , last one is , and any two consecutive nodes in this ordering are connected by an edge in the graph.
Definition 1.
An induced subgraph is a subgraph of with its vertex set being the union of all routes from source to a destination , and its edge set being the subset of all edges in between the vertices of .
We say that SD pair and SD are noninterfering if and are two disjoint induced subgraphs of .
The indegree function , is the number of incoming edges connected to node . Similarly, the outdegree function , is the number of outgoing edges connected to node . Note that the indegree of a source and the outdegree of a destination are both equal to . The maximum degree of the nodes in is defined as
(4) 
We also need the following definitions that will be used later in this paper.
Definition 2.
At any node , we define the index set as follows
(5) 
In other words, is the set of indices of those SD pairs that have on a route between them.
Definition 3.
The routeexpanded graph associated with a layered network with sources in and destinations in is constructed by replacing each node with nodes represented by where , and connect them according to
We define , (i.e. all the duplicates of node ) and we refer to it as a supernode (or equivalently a superrelay if is a relay).
For an illustration of the routeexpanded graph see Figure 1. For simplicity, we have represented each pair with a shape, i.e. , , and for SD pairs , , and respectively. The routeexpanded graph of the network in Figure 1(a) is illustrated in Figure 1(b). Each relay is on a route for two SD pairs, hence each superrelay contains two nodes.
IiB Model of Partial Network Knowledge
In this subsection, we describe the model of [1] for partial network information that will be used in this paper. We first define the routeadjacency graph of , which is an undirected bipartite graph consisting of all sources on one side and all destinations on the other side; see Figure 2 for an example. A source and a destination are connected in , if there exists a route between them in . More formally, where and .
We now define the model for partial network knowledge that will be used in the paper, namely local view, as the following:

All nodes have full knowledge of the network topology, , i.e., which links are in , but not their channel gains. The network topology knowledge is denoted by side information .

Each source, , knows the gains of all those channels that are in a route from source to destination , such that and are at most hops away from in . The hop channel knowledge at a source is denoted by .

Each node (which is not a source) has the union of the information of all those sources that have a route to it, and this knowledge at node is denoted by .
Note that this model is a generalization of the hopbased model for partial network knowledge in single layer networks [2]. While the partial information model is general, we will focus on the case where . In other words, each SD pair has enough information to perform optimally when other pairs do not interfere (i.e., it knows the channel gains of all links that are in a route to its own destination). However beyond that, each pair only knows the connectivity in the network (structure of interference). We are interested to find if one can outperform interference avoidance techniques with such limited knowledge. In the following subsection, we define the metrics we use to measure the performance of transmission strategies with local view.
IiC Normalized SumCapacity
We now define the notion of normalized sumcapacity, which is our metric for evaluating network capacity with partial network knowledge [2, 18]. Normalized sumcapacity represents the maximum fraction of the sumcapacity with full knowledge that can be always achieved when nodes only have partial knowledge about the network, and is defined as follows.
Consider the scenario in which source wishes to reliably communicate message to destination during uses of the channel, . We assume that the messages are independent and chosen uniformly. For each source , let message be encoded as using the encoding function , which depends on the available local network knowledge, , and the global side information, .
Each relay in the network creates its input to the channel , using the encoding function , which depends on the available network knowledge, , and the side information, , and all the previous received signals at the relay . A relay strategy is defined as the union of of all encoding functions used by the relays, , and .
Destination is only interested in decoding and it will decode the message using the decoding function , where is the destination ’s network knowledge. Note that the local view can be different from node to node.
Definition 4.
A Strategy is defined as the set of: (1) all encoding functions at the sources; (2) all decoding functions at the destinations; and (3) the relay strategy for , i.e.
(6) 
An error occurs when and we define the decoding error probability, , to be equal to . A rate tuple is said to be achievable, if there exists a set of strategies such that the decoding error probabilities go to zero as for all network states consistent with the side information. Moreover, for any SD pair , denote the maximum achievable rate with full network knowledge by . The sumcapacity , is the supremum of over all possible encoding and decoding functions with full network knowledge.
We will now define the normalized sumrate and the normalized sumcapacity.
Definition 5 ([2]).
Normalized sumrate of is said to be achievable, if there exists a set of strategies such that following holds. As goes to infinity, strategy yields a sequence of codes having rates at the source , , such that the error probabilities at the destinations, , go to zero, satisfying
for all the network states consistent with the side information, and for a constant that is independent of the channel gains.
Definition 6 ([2]).
Normalized sumcapacity , is defined as the supremum of all achievable normalized sumrates . Note that .
Iii Motivating Examples
Before diving into the main results in Section IV, we will use a sequence of examples to arrive at the main ingredients of the proposed coded layer scheduling. The key point of the discussion is to understand the mechanisms that allow outperforming interference avoidance with only local view.
As defined earlier, local view means that each SD pair has enough information to perform optimally when other pairs do not interfere. However, beyond 1local view, each node only knows the connectivity in the network (structure of interference). So, at first glance it seems that the optimal strategy is to avoid interference between the SD pairs and at each time, schedule as many noninterfering pairs as possible. Through an example, we investigate the performance of the above strategy which maximizes spatial reuse while avoiding interference at each node.
Consider the network depicted in Figure 3(a) with local view. From the routeadjacency graph of this network depicted in Figure 3(b), we can see that SD pairs and are noninterfering. We implement an achievability strategy described as follows. We split the communication block into two timeslots of equal length and represent each timeslot with a color, namely black and white. SD pairs and communicate over timeslot black, whereas, SD pair communicate over timeslot white. With this coloring, we have effectively seperated induced subgraphs of interfering pairs, see Figure 4. Now, since each pair can communicate interferencefree over half of the communication block length, it can achieve half of its capacity with full network knowledge. Hence, we achieve a normalized sumrate of .
This scheduling strategy can be viewed as a specific coloring of nodes in the routeexpanded graph (defined in Section II). Consider the routeexpanded graph of this example, as shown in Figure 5. The aforementioned scheduling strategy can be viewed as a coloring of nodes in the routeextended graph, such that (1) all nodes of the same shape, i.e. pair ID, receive the same color, and (2) any two nodes with different shapes, i.e. different pair IDs, that are connected to each other should have different colors. Note that since the nodes inside the same supernode are connected to the same nodes in and they have different shapes, they will be assigned different colors. Figure 5 illustrates such coloring of nodes in this example by using only two colors, B and W. In other words, we have assigned different colors to the induced subgraphs of interfering SD pairs. Therefore, each SD pair gets a chance to communicate over its induced subgraph interferencefree during the timeslot associated with its color.
As we will see in the next lemma, which is proved in Appendix A, is also an upper bound on the normalized sumcapacity of this network. Hence, scheduling noninterfering pairs performs optimally in this example. More generally, the following upper bound on exists for a general class of multilayer networks.
Lemma 1.
In a user multilayer network (linear deterministic or Gaussian) with local view, if there exists a path from to , for some , then the normalized sumcapacity is upperbounded by .
While the aforementioned interference avoidance strategy performed optimally in the network depicted in Figure 3(a), we now illustrate an example where it is not optimal. A key observation is that the scheduling described above, ignores the available knowledge of interference structure in each layer of the network and it only schedules pairs that are noninterfering over all layers. To see how this knowledge of interference structure can be exploited, consider the network depicted in Figure 6(a) with local view. Applying the previous scheduling to this network, we achieve a normalized sumrate of . However, we show that it is possible to go beyond and achieve a normalized sumrate of for Figure 6 example.
We implement an achievability strategy described as follows. Similar to the previous example, we split the communication block into two timeslots of equal length and represent each timeslot with a color, namely black and white. Unlike the previous case where we assigned a color to each SD pair, in this example, the color assignment is carried on in each layer seperately. We let sources and to communicate in the first layer over timeslot black and source over timeslot white. However, in the second layer, relays communicate to destinations and in timeslot black and to destination in timeslot white, see Figures 7(a) and 7(b). With this strategy each SD pair can communicate over its induced subgraph interferencefree during half of the communication block length, see Figures 7(c), 7(d) and 7(e). Hence, we achieve a normalized sumrate of . By Lemma 1, we know that is also an upperbound on the normalized sumrate of this network, hence, we have achieved it normalized sumcapacity.
This strategy can be viewed as a modification of our previous coloring of the nodes in the routeexpanded graph as follows. Nodes with the same shape can be assigned different colors at different layers, however, still any two nodes with different shapes that are connected to each other should have different colors. In other words, we assign colors such that the induced subgraphs of different SD pairs have different colors in each layer only if they are interfering at that layer. Figure 8 illustrates such coloring of nodes in this example by using only two colors, B and W. Since the induced subgraphs of interfering pairs have different colors in each layer, each SD pair has a chance to communicate over its induced subgraph interferencefree during half of the communication block.
The scheduling developed for the network depicted in Figure 6, illustrates a major deficiency of the scheduling developed for the network depicted in Figure 3, which is the restriction of applying the same scheduling to all nodes on a route between and . By exploiting the available information of interference structure and scheduling nodes in different layers separately, we outperformed the first scheduling. In Section IV, we will formally define this new scheme and refer to it as Maximal Independent Layer (MIL) scheduling.
So far, our proposed transmission strategies are based on interference avoidance either in an endtoend manner or in a perlayer manner (MIL scheduling). But, can we go beyond interference avoidance with such limited knowledge at the nodes? To answer this question, first consider the singlelayer network depicted in Figure 9(a). Since the conflict graph of this network is fully connected, using MIL scheduling we can only achieve . However, we now show that it is possible to achieve by employing a coding strategy that only requires local view.
Consider the linear deterministic model. By using repetition coding at the sources (as in [2]), we show that it is possible to achieve . Consider the induced subgraphs of all three SD pairs, as shown in Figures 9(b). We show that any transmission strategy over these three induced subgraphs can be implemented in the original network by using only two timeslots, such that all nodes receive the same signal as if they were in the induced subgraphs. This would immediately imply that a normalized sumrate of is achievable.^{1}^{1}1Since any transmission strategy for the diamond networks can be implemented in the original network by using only two timeslots, we can implement the strategies that achieve the capacity for any SD pair with full network knowledge, i.e. , over two timeslots. Hence, we can achieve . On the other hand, we have . As a result, we can achieve a set of rates such that , and by the definition of normalized sumrate, we achieve .
To achieve , we split the communication block into two timeslots of equal length and represent each timeslot with a color, namely black and white. Sources and transmit the same codewords as if they are in the induced subgraphs over timeslot black. Destination will receive the same signal as if it is only in the induced subgraph without any interference and destination receives interference from source . Over timeslot white, source transmits the same codewords as if they are in the induced subgraphs, and source repeats its transmitted signal from timeslot black. Destination will receive its signal interferencefree. Now, if destination adds its received signals over two timeslots, it recovers its intended signal interferencefree, see Figure 10. In other words, we have used interference cancellation at destination . Therefore, all SD pairs can effectively communicate interferencefree over two timeslots.
Again, we can view this strategy as a modification of the previous colorings of the nodes in the routeexpanded graph as follows. Each shape, i.e. pair ID, can be assigned a subset of colors such that any two nodes with different shapes that are connected, have either different colors, or if they share a color, one them has a different color in its subset. Figure 11 illustrates such coloring of nodes in this example by using only two colors, and . The subset assigned to source represents repetition coding, i.e. the transmitted signal in timeslot white is the same as the one transmitted in timeslot black. Since the interference can be cancelled out as described before, each SD pair has a chance to communicate over its induced subgraph interferencefree during half of the communication block.
This example illustrated that with only 1local view it is still possible to take advantage of (repetition) coding at the sources and go beyond interference avoidance. This raises a natural question: can we also exploit network coding at the relays with only local view? If so, what is a systematic procedure for doing that?
To shed light on the aforementioned questions, consider a multilayer network as depicted in Figure 12(a). Assume linear deterministic model for the channels. It is straightforward to see that by using interference avoidance, we can atmost achieve normalized sumrate of . We now show that by using repetition coding at the sources and linear coding at the relays, it is possible to achieve . Consider the induced subgraphs of all three SD pairs, as shown in Figures 12(b), 12(c), and 12(d). We now show that any transmission strategy over these three induced subgraphs can be implemented in the original network by using only two timeslots, such that all nodes receive the same signal as if they were in the diamond network. Therefore, a normalized sumrate of is achievable.
Consider any strategy for SD pairs , , and as illustrated in Figures 12(b), 12(c), and 12(d). In the first layer, we implement the achievability strategy of Figure 10 and we have illustrated it in Figure 13. As it can be seen in this figure, at the end of the second timeslot, each relay has access to the same received signal as if it was in the diamond networks of Figures 12(b), 12(c), and 12(d).
In the second layer, during timeslot black, relays and transmit and respectively, whereas, relay transmits , see Figure 14. Destination receives the same signal as in Figure 12(b). During timeslot white, relays and transmit and respectively, whereas, relay transmits . Destination receives the same signal as as in Figure 12(c). If destination adds its received signals over the two timeslots, it recovers the same signal as in Figure 12(d). Therefore, each destination receives the same signal as if it was only in its corresponding diamond network, over two timeslots. Hence, the normalized sumrate of is achievable. By Lemma 1, we know that is also an upperbound on the normalized sumrate of this network, hence, we have achieved it normalized sumcapacity.
This strategy can be viewed as a new coloring of the nodes in the routeexpanded graph as follows. Each shape, i.e. pair ID, can be assigned two subsets of colors. Figure 15 illustrates such coloring of nodes in this example by using only two colors, and . The subset assigned to source represents repetition coding as before. The second subset of colors can be interpreted as the time instants from which we can add (or subtract for the Gaussian model) the codewords to perform network coding. To clarify, consider the first superrelay in Figure 15, node circle communicates the codeword of SD pair over timeslot . Over timeslot , node square within this superrelay adds the codeword of SD pair transmitted by the other node in the same supernode over timeslot to the codeword of SD pair it has to send. Similar interpretation can be used for the other tuple of colors in this routeexpanded graph. Since the interference can be cancelled out as described before, each SD pair has a chance to communicate over its induced subgraph interferencefree during half of the communication block.
In the following section, we incorporate all the ideas developed for the examples in this section to define a transmission strategy, i.e. coded layer scheduling, which outperforms interference avoidance techniques with local view. We also characterize its performance and later in Section V, we evaluate its performance for some network topologies.
Iv Coded Layer Scheduling
Via the examples presented in Section III, we saw that multiple ideas can be incorporated to enhance the achievablility scheme in multilayer networks with local view: (1) per layer interference avoidance, (2) repetition coding to allow overhearing of the interference, and (3) network coding to allow interference neutralization. In this section, we define a general transmission strategy, named coded layer scheduling to incorporate all the aforementioned ideas. This scheduling can be represented by a specific coloring of nodes in the routeexpanded graph (defined in Section II). We refer to this coloring as the Coded Layer coloring and it is defined as follows.
Consider a multilayer wireless network , and its corresponding routeexpanded graph . A Coded Layer coloring of with distinct colors assigns to any node ,

a transmit color set, denoted by , which represents the time instants in which will be transmitting for SD pair using repetition coding,

a coding color set, denoted by , which represents the time instants from which node will use the transmit signal to perform network coding for SD pair ,

a receive color set, denoted by , which represents the time instants in which it is listening.^{2}^{2}2We refer to the transmit color set, the coding color set, and the receive color set of source , , by , , and respectively, similar notations hold for destinations, i.e. , , and for destination , .
To describe the conditions that these color assignments should satisfy, we need a few definitions.
Definition 7.
At any node , a node is called an interferer if and

, i.e. an interferer should have a SD pair ID different from ,

, i.e. the colors used by an interferer are not used by any node in the same supernode that has SD pair ID and performs network coding, otherwise its transmit signal will be neutralized.

, i.e. an interferer transmits during a time instant that some node with SD pair ID is transmitting to and is listening, the set of all such time instants is denoted by .
Definition 8.
We define as the set of all nodes in that have pair ID and are connected to , i.e.
(7) 
The conditions on the assignment of , and of a coded layer coloring are as follows.
C.1: The transmit color sets assigned to the nodes that belong to the same supernode are disjoint, i.e.
(8) 
C.2: If a node is performing network coding, it only transmits once, i.e. if , then .
C.3: The coding color set includes at most one color from each transmit color set of a node within the same supernode who is not performing network coding, i.e.
(9) 
C.4: The receive color set includes at least one color from each such that , i.e.
(10) 
C.5: The receive color set includes each such that , i.e.
(11) 
C.6: If , then .
C.7: At each node either there are no interferers, or all interferers share a common color in their transmit color sets, which is in , i.e.
(12) 
moreover, the color that the interferers share should be exclusive to them, i.e., for ,
(13) 
Based on the coded layer coloring of nodes in , we now define the coded layer scheduling of nodes in as follows:
The transmission is broken into blocks of size time instants. At the beginning of the block, ,

Source , , creates a signal , which is a function of its message ( for the linear deterministic model and for the Gaussian model such that it satisfies the average power constraint at transmit nodes). The choice of this function depends on the specific strategy that each source picks,

Each relay node creates a signal for each SD pair , which is a function of its received signals and the global side information. The choice of this function depends on the specific strategy that each relay picks.
During the block,

Source , , will transmit
(14) where .

Each relay node will transmit
(15) where . Note that subtraction in is the same as XOR operation.
Finally, each destination , , will decode based on its received signals and the global side information.
We next state our main result for the coded layer scheduling.
Theorem 1.
For a multilayer network (linear deterministic or Gaussian) with local view, if there exists a coded layer coloring of with colors as defined above, then a normalized sumrate of is achievable by coded layer scheduling.
Proof.
We first prove the theorem for the linear deterministic model. Assume that there exists a coded layer coloring of nodes with colors , denoted by , and . Suppose has SD pairs and consider the induced subgraphs of all SD pairs, i.e. , . We will show that by using the coded layer scheduling, any transmission snapshot over these induced subgraphs can be implemented in the original network over time instants, such that all nodes receive the same signal as if they were in the induced subgraphs.
Consider a transmission snapshot in the induced subgraphs where

Node in the induced subgraph transmits ,

Node in the induced subgraph receives
(16)
Transmission strategy: At any time instant , node will choose and will transmit
(17) 
and it will receive
(18) 
where summation is carried on in .
Constructing the received signals: Based on the transmission strategy described above, we need to show that at any node , the received signal can be obtained. At any node , , we create as follows,
(19) 
where is given by (18). We will show that . We have
(20) 
Consider a node such that ; we face 2 cases:
Case 1: , then from condition C.1 and (17), we get
(22) 
Then based on condition C.7, we have 2 subcases:
Case 1a: There are no interferers. In this case since no interfering signal is received during time instants associated with colors in , hence,
(23) 
Then, we have
(24) 
Case 1b: All interferers share a common color that is in but not in . In this case, the transmit signal of any interferer appears exactly twice in time instants associated with colors in , once during the time instant associated with the color that all interferers share, see (IV), and once during a different time instant, see Definition 7. Hence, when adding the received signals over all time instants associated with colors in , the transmit signal of any interferer gets canceled. Moreover, condition C.6 guarantees that the desired signal, i.e. , is transmitted exactly once during time instants associated with colors in . Hence, from (22) and the argument presented above, we have
(25) 
Case 2: . Note that according to condition C.3 the colors in can only appear in where . As a result, each one of those codewords added to as in (17), are also transmitted during time instants corresponding to colors in . Hence, if ,
(26) 
Again based on condition C.7, we have 2 subcases:
Case 2a: There are no interferers. In this case since no interfering signal is received during time instants associated with colors in , hence,
(27) 
Then, we have
(28)  
(29) 
Case 2b: All interferers share a common color that is in but not in . The argument is similar to that of case 1b, i.e. the transmit signal of any interferer appears exactly twice in time instants associated with colors in , once during the time instant associated with the color that all interferers share, see (IV), and once during a different time slot, see Definition 7. Hence, when adding the received signals over all time instants associated with colors in , the transmit signal of any interferer gets canceled. Moreover, condition C.6 guarantees that the desired signal, i.e. , is transmitted exactly once during time instants associated with colors in . Hence, from (26) and the argument presented above, we have
(30) 
Therefore, we have shown that in all cases we have , which as described before proves that . As a result, at any node , the received signal can be obtained interferencefree over time instants. This implies that the transmit signal , which is only a function of and the global side information, can be created over time instants as well. Hence, any transmission snapshot over the induced subgraphs can be implemented in the original network over time instants, such that all nodes receive the same signal as if they were in the induced subgraphs.
Moreover, any transmission strategy with block length for the induced subgraphs can be implemented in the original network over time instants in a similar manner. Hence, we can implement the strategies that achieve the capacity for any SD pair with full network knowledge, i.e. , as over time instants. Therefore, by choosing ’s according to the optimal transmission strategies and creating as in (17), we can achieve . On the other hand, we have . As a result, we can achieve a set of rates such that , and by the definition of the normalized sumrate, we achieve .
We will next prove the theorem for the Gaussian model. We will show that by using the coded layer scheduling, any transmission snapshot over the induced subgraphs , , can be implemented in the original network over time instants, such that all nodes receive the same signal as if they were in the induced subgraphs.
Consider a transmission snapshot in the induced subgraphs where

Node in the induced subgraph transmits ,

Node in the induced subgraph receives
(31)
where is the additive white complex Gaussian noise with variance . We also assume a power constraint of at the transmit nodes in the induced subgraphs.
Transmission strategy: At any time instant , node will choose and will transmit
(32) 
note that number of transmit signals at each time instant is less than and due to the power constraint of in the induced subgraphs, the power constraint in the original network is satisfied. At any time instant , node