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Computing Along Routes via Gossiping

Mehmet E. Yildiz, Student Member, IEEE, and Anna Scaglione, Senior Member, IEEE

Abstract

In this paper, we study a class of distributed computing problems where a group of nodes (destinations) is

interested in a function of data which are stored by another group of nodes (sources). We assume that the function

of interest is separable, i.e., it can be decomposed as a sum of functions of local variables, a case that subsumes

several interesting types of queries. One approach to solve this problem is to route raw information from the sources

to the interested destinations by either unicasting or multicasting. The second approach is to compute the function

of interest along some routes while propagating the information from the sources to the destinations. Considering

the second scenario, the goal of this paper is to examine how information should be forwarded to the intended

recipients, computing along the routes by gossiping with selected neighbors. Unlike efficient unicasting/multicasting

problems, nodes are interested in a specific function of the data, rather than the raw data themselves. Moreover,

unlike standard gossiping problems, in our case, the information needs to flow in a specific direction.

Given the underlying network connectivity and the source-destination sets, we provide necessary and sufficient

conditions on the update weights (referred to as codes) so that the destination nodes converge to the desired function

of the source values. We show that the evolution of the source states does not affect the feasibility of the problem,

and we provide a detailed analysis on the spectral properties of the feasible codes. We also study the problem

feasibility under some specific topologies and provide guidelines to determine infeasibility. We also formulate

different strategies to design codes, and compare the performance of our solution with existing alternatives.

Index Terms

Distributed computing, gossiping protocols, data aggregation, duplicate-sensitive data aggregation.

I. INTRODUCTION

A fundamental problem in networks is the transportation and distribution of information from one part of the

network to another [4]–[7]. We focus on a special network computation problem, where a group of destination

nodes is interested in a function that can be decomposed as a sum of functions of local variables stored by another

set of source-nodes. We refer to this problem as the Computing Along Routes (CAR) problem. This particular

problem has a wide range of applications including aggregation queries, distributed detection, content distribution,

decentralized traffic monitoring, distributed control and coordination [5], [8]–[10].

Mehmet E. Yildiz is with the School of Electrical and Computer Engineering of Cornell University, Ithaca, NY. Anna Scaglione

is with the Electrical and Computer Engineering Department of University of California at Davis. Corresponding e-mail addresses are

{mey7@cornell.edu} and {ascaglione@ucdavis.edu}. This work has been funded by National Science Foundation under NSF-ITR-CCR-

042827. Parts of this work was published in Asilomar 2008 [1], ITW 2009 [2] and CAMSAP 2009 [3].

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We focus on the case where the set of source nodes and the set of destination nodes are disjoint, and hence our

case does not include average consensus gossiping [5] as a special case. This particular assumption has strong

practical appeal since, in many cases, the nodes which are responsible for collecting the data and the nodes that

are designated to process these data are disjoint and geographically separated. For instance, in the problem of

distributed detection, only the fusion centers are interested in the outcome of the sensor decisions. In the case of

content distribution, the entity who is interested in the data does not necessarily have access to any part of it before

the distribution occurs. In the case of the leader-follower coordination problem [11], several nodes are to follow a

group of leaders, and the source and destination sets are clearly disjoint.

One solution to the problem we posed is to unicast between each pair of source-destination nodes via the shortest

path joining source and destination [12]. After receiving all the information, each destination node can compute

the desired function independently. Except in very special cases, this strategy is inefficient since the exact same

information flows in the network several times, and it is unreliable since it is severely affected by link failures.

The second approach is multicasting the information, having a single source node transmitting at a time to all

destinations, thereby allowing the computation of the sought results independently [13]. Thanks to network coding

[4], [14], [15], multicasting can be done using the links efficiently. However, this decomposition of the problem is

agnostic to the fact that the nodes do not want the data themselves, but an aggregate result.

In fact, the problem considered in this paper is closely related to the data aggregation and routing problems studied

in the computer science literature, where spatially distributed data is to be collected by a fusion center, utilizing

data aggregation and in-network processing techniques [8], [16]–[19]. The similarity is obvious if one observes

that solutions to the so called duplicate-sensitive data aggregation problem (see e.g. [17]) can be generalized to

find codes that solve our problem. Duplicate-sensitive data aggregation refers to the case where destinations seek

a single copy of data from each source. In the case of a single source and multiple destinations, the duplicate-

sensitive aggregation problem has been studied extensively, proposing energy efficient schemes based on spanning

trees [20], [21], and algorithms that are robust to link/node failures [17]. In the case of multiple source and multiple

destinations, solutions have been proposed including hierarchical structures and minimizing routes costs [18], [19],

[22]. These methodologies are not viable for solving the CAR problem, because one would need to explicitly

consider duplicating the data of all sources at the destinations. Moreover, it falls short of explaining the relationship

between type of topologies and the feasible queries and it does not incorporate feedback.

Observing the fact that the problem has two parts, i.e., the computation of the function and the routing of the

information, we are proposing a joint strategy where the computation of the desired function is achieved along the

routes via gossiping. Our approach incorporates feedback since gossiping based protocols are iterative, thus nodes

continuously exchange their values and create a feedback effect. We focus on separable functions, i.e., functions

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whose synopsis can be written as?

we will consider the scenarios where the function of interest is simply the average, since any other separable

ifi(xi), where the index i is over the set of nodes. Without loss of generality,

function can be calculated by initializing the source nodes’ values with states fi(xi).

As said above (see also Section VI), there are algorithms that result in partially directed solutions for the CAR

problem. However, one of the main interests of this paper is exploring the value of feedback.

Feedback is the trademark of Average Consensus Gossiping (ACG) protocols, which are considered attractive in

wireless sensor applications because the communications among nodes are limited to their immediate neighbors,

they are easily mapped into stationary asynchronous policies, while the network topology can be dynamically

switching. In ACG protocols each node in the network is both a source and a destination [5], [23], [24], [25].

An ACG protocol can directly be applied to our problem: in fact, non-source nodes could set their initial values

to zero, and via ACG the network would converge to a fraction of the average of the source nodes. When the

destination nodes know the number of source nodes and also the network size, the true average can be calculated

by rescaling the ACG result. However, this approach has two main drawbacks: 1) Since the desired information

has to be distributed to the whole network, the convergence will be slower, 2) Every node will have access to the

average at the end of the algorithm, and this is a major weakness since it does not preserve secrecy.

In summary, we are proposing a gossiping based algorithm for jointly routing and calculating the desired function

at the destinations. The innovative aspect of this procedure is that it incorporates feedback, unlike similar problems

that consider a unidirectional flow of data. On the other hand, it does not necessarily distribute the desired value

to the whole network which makes our scheme more secure and flexible.

In this paper, we limit ourselves to non-negative update weights (codes) and disjoint source and destination sets.

Under these assumptions, we formulate our problem in Section II. In Section III, we introduce necessary and

sufficient conditions for the existence of solutions to the prosed problem. In Section IV-A, we investigate spectral

properties of feasible codes and show what classes of code structures leads to feasible solutions. In Section IV-B, we

introduce reductions of the network topology which can be employed to simplify the design problem without loss

of generality. By focusing on stochastic codes, we provide necessary conditions on the topology for the feasibility

and discuss some infeasible cases in Section V. We introduce a formulation for so called partially directed solution

in Section VI, and discuss the complexity and communication cost of the algorithm in Section VII. We compare

the performance of our solution with the existing solutions in Section VIII, and conclude our paper in Section IX.

We note that a setup resembling more to our problem is considered in [26], where the authors have applied the

gossiping algorithm to solve a sensor localization problem, assuming that each node in the network wants to

compute a linear combination of the anchor (source) nodes, to determine their exact locations. Unfortunately, this

analysis can not be utilized to solve our problem since all of the non-source nodes are destinations, and destination

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nodes are not necessarily interested in the same function of the source nodes. A significant work is also due to

Mosk-Aoyama et.al. who have considered distributed computation of separable functions as well as information

dissemination on arbitrary networks [27], [28]. Morever, Benezit et.al. have studied average consensus problem via

randomized path averaging to achieve increased convergence rates [25]. However, our problem differs from these

models in the sense that source and destination sets are disjoint, and there may exist intermediate nodes which are

neither sources nor destinations.

II. PROBLEM FORMULATION

In this paper, we consider a connected network (N,E) with N nodes and the corresponding edge set E which

consists of ordered node pairs (i,j). Given the edge (i,j), i is the tail and j is the head of the edge. We define

the neighbor set of node i as Ni? {j ∈ {1,2,...,N} : (j,i) ∈ E}.

We consider the following problem setup: Each node in the network has an initial scalar measurement denoted by

xi(0) ∈ R where i ∈ {1,...,N}. Let S ? {1,2,... ,N}. There is a set of nodes (destination nodes), denoted as

SD⊆ S, which are interested in the average of a set of nodes (source nodes), denoted as SS⊆ S. We note that

SS∩SD= ∅, i.e., a source node cannot be a destination node. We restrict our attention to a synchronous gossiping

protocol, with constant update weights:

xi(t + 1) = Wiixi(t) +

?

j∈Ni

Wijxj(t), i ∈ S,

(1)

where t is the discrete time index, Wijis the link weight corresponding to the edge (j,i). We note that if j ?∈ Ni,

then Wij= 0. At each discrete time instant, each node updates its value as a linear combination of its own value

and its neighbors values. This type of analysis is usually the first step to investigate more flexible asynchronous

solutions [29]. If we define an N × N matrix W such that [W]ij= Wij and x(t) = [x1(t),x2(t),...,xN(t)]T,

then (1) can be written in the matrix form as:

x(t + 1) = Wx(t) = Wt+1x(0).

(2)

Parallel to the network coding literature, we refer to the matrix W as the code that we are interested in designing.

The equation above implies that:

lim

t→∞x(t) = lim

t→∞Wtx(0) = W∞x(0)

where W∞? limt→∞Wt(assuming that limt→∞Wtexists). One way to solve our problem is assigning zeros

as initial state values for non-source nodes, and running an average consensus algorithm on the network. The

consensus value will be a rescaled version of the source nodes’ average; thus, to determine the desired function,

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the destination nodes have to multiply the consensus value by a scaling factor. For this approach, any code W

that solves the average consensus can be employed [24], [29]–[31]. This approach has two major disadvantages:

First, since the whole network has to converge to a consensus, the converge will be slow which will require more

resources, i.e., energy, time (c.f. Section VIII). Second, at the end of the algorithm, every node will have access to

the average, and this is a major weakness in the case of secret communications.

The approach which we pursue here is to design a W that produces the desired computation irrespective of what

x(0) is, and distributes the value to the destinations only. Since, we are only interested in calculating the average

of the nodes values in SSat all j ∈ SD, we have the following inherent constraints on W∞:

W∞

jk=

|SS|−1

k ∈ SS

0k ?∈ SS

, ∀j ∈ SD,

(3)

where |.| denotes the cardinality of its argument. In other words, the entries of the limiting weight matrix should

be chosen such that destination nodes multiply source nodes’ values by |SS|−1(for averaging), and multiply non-

source nodes’ values by 0. Therefore, the rows of W∞corresponding to destination nodes should have |SS|−1for

the source node columns, and 0 otherwise.

Let us denote the structure above, imposed by sets SDand SS, as F(SD,SS), and as F(E), the structure imposed

by the network connectivity, i.e., Wij= 0 if (i,j) ?∈ E. We denote a code W, which satisfies (3), as an Average

Value Transfer (AVT) solution to the problem of computing along routes (CAR). The following definitions are in

order:

Definition 1. A code W is a feasible AVT solution for CAR on a given network (N,E) and source-destination sets

(SD,SS), if and only if W ∈ F(E) and W∞∈ F(SD,SS) with W∞< ∞.

Definition 2. CAR is AVT infeasible, if there does not exist any AVT solution.

We note that < denotes elementwise strict inequality. In the rest of the paper, unless stated otherwise, we will

omit the word AVT and refer to Definition 1 as the solution and 2 as the feasibility condition. Therefore, our

(in)feasibility definition considers only AVT solutions.

In the rest of the paper, we limit ourselves to codes in the set of nonnegative matrices, i.e., W ≥ 0 and ≥ represents

elementwise inequality, in analogy with the ACG policies [29]. In the following section, we will introduce necessary

and sufficient conditions for a W matrix to be an AVT solution for the CAR problem.

III. NECESSARY AND SUFFICIENT CONDITIONS ON FEASIBLE CODES

In the following, we first give necessary conditions on feasible W and W∞in addition to the structure given in (3).

We will then use the resulting equations to introduce necessary and sufficient feasibility conditions.

We first partition N sensors in the network into three disjoint classes: M sensors that belong to the source nodes set,

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K sensors that belong to the destination nodes set, and L sensors that belong to neither source nodes nor destination

nodes (called intermediate nodes). In other words, |SS| = M, |SD| = K and L = N − M − K. Without loss of

generality, we index the set of source nodes as {1,...,M}, the set of destination nodes as {M +1,...,M +K},

and the set intermediate nodes as {M + K + 1,...,N}. At this point, we can partition the N × N matrix W as:

W =

ADG

BEH

CFP

(4)

such that A ∈ RM×M, E ∈ RK×Kand P ∈ RL×L. Assuming that limt→∞Wtexists, we denote this limit as W∞

and its partitions using the superscript?, i.e., A?,D?,G?, etc. Since the set SDis only interested in the average of

the set SS, then B?= 1/|SS|11Tand E?= H?= 0 by (3). We note that 1 is the all ones vector of the appropriate

dimensions. This inherent structure results in the following necessary constraints on W and W?:

Lemma 1. Given a network F(E) and the sets SSand SD, if W ≥ 0 is a feasible solution to CAR, then:

1) D?= G?= 0,

2) D = G = 0,

3) 1TA = 1T.

Proof: The proof of the lemma is given in Appendix A.

Lemma 1 shows that the information flow between the sources and the rest of the network must be one-way, i.e.,

from the sources to the network. Such a finding is not surprising since the network is only interested in the average

of the source nodes, and the average will be biased if the sources mix their state values with values from the

non-source nodes, whose initial states are arbitrary. Lemma 1 also shows that the row sums of the matrix A are all

equal to 1. Since the matrix A governs the communication among the source nodes, the sum of the source values

must remain constant through the iterations. This result is also intuitive since, otherwise, the average of the source

nodes would change and W would not be a solution for the problem. The fact that the flow from the source nodes

to the rest of the network is directional, has important consequences on what designs are feasible, as we will see

next.

In light of Lemma 1, we will focus on the network codes which have the structure:

W =

A00

BEH

CGP

.

At this point, for mathematical brevity, we repartition the W matrix in four super-blocks recycling the previous

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symbols as follows:

W =

A0

BD

where B =

B

C

,D =

EH

GP

.

(5)

Of note is that, the new B ∈ RN−M×Mand D ∈ RN−M×N−M. Similarly, we can partition the state vector x(t):

s(t) = [x1(t),...,xM(t)]T,r(t) = [xM+1(t),...,xN(t)]T.

We note that s(t) represents the evolution of the source states while r(t) represents the behavior of the rest of

the network including the destination nodes. Similarly, the matrix A governs the communication among the source

nodes, and the matrix D determines the communication structure among the rest of nodes. B instead governs the

flow from the sources to the rest of the network. We expand the general form of the update in (1) using (5) as:

s(t + 1)= As(t),

r(t + 1)= Bs(t) + Dr(t).

The equations above can be rewritten in terms of the initial conditions as:

s(t + 1) = At+1s(0),

r(t + 1) = Dt+1r(0) +

t

?

l=0

Dt−lBAls(0).

The linear system of equations in a compact form is:

x(t + 1) =

s(t + 1)

r(t + 1)

= Wt+1x(0) =

At+1

0

?t

l=0Dt−lBAl

Dt+1

s(0)

r(0)

.

(6)

At this point, we can state the main theorem of the paper which introduces the necessary and sufficient conditions

on the feasible network codes W ≥ 0:

Theorem 1. Given a network F(E) and the sets SSand SD, a W ≥ 0 matrix of the form (5) is an AVT solution

to the CAR problem if and only if:

1) W is in the form of:

W =

A0

BD

,

(7)

where A ∈ RM×M, D ∈ RN−M×N−Mand M = |SS|,

2) limt→∞Wtexists and is finite,

3) limt→∞

?Dt+1?

1:K= 0,

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4) limt→∞

??t

l=0Dt−lBAl?

1:K=

1

M11T,

where [.]1:Kdenotes the first K rows of its argument, and K = |SD|.

Proof: While sufficiency follows from (6) and the problem definition given in Section II, these conditions are

necessary because of Lemma 1 and equation (6).

In the following, we show that one can simplify the fourth constraint in Theorem 1 by exchanging the powers of

A with its limit, i.e., the evolution of the source states does not affect the feasibility of the solution.

Lemma 2. Assuming that conditions (1) − (3) in Theorem 1 hold, then condition (4) holds if and only if

lim

t→∞

?

t

?

l=0

Dt−lBA∞

?

1:K

=

1

M11T,

(8)

where A∞= limt→∞At.

Proof: The proof is given in Appendix B.

Lemma 2 shows that if there exists a limiting matrix A∞which satisfies the constraint in (8), then any A matrix such

that limt→∞At= A∞, satisfies the fourth constraint in Theorem 1. Therefore, given the underlying connectivity,

one can assume that A has already converged to its limit A∞, and then seek a solution for partitions B and D.

Unfortunately, even under this simplification, designing a feasible W is difficult, since the fourth constraint in

Theorem 1 is non-convex with respect to the elements of the partitions D, B and A∞, and also consists of all

non-negative powers of D. Moreover, it is not clear from Theorem 1 what kind of topologies do or do not have

solutions for the CAR problem we posed.

In the following sections, we will both try to simplify the design of an AVT code and determine topologies where

the CAR problem is known to be (in)feasible.

IV. SPECTRAL ANALYSIS AND TOPOLOGY REDUCTIONS

A. Spectral Analysis of Feasible Codes

We start our discussion by introducing some necessary definitions. Motivated by the theory of Markov Chains, we

say that node i has access to node j, if for some integer t, Wt

ij> 0. Two nodes i and j, which have access to

the other, are said to be communicating. Since communication is an equivalence relation, the set of nodes which

communicate forms a class. A class which consists of only source nodes is called a source class. In the following,

we also introduce the graph theoretic definition of a source class.

Definition 3. A set of source nodes forms a source class if and only if the sub-graph consisting of these source

nodes only is irreducible, and including any other source nodes results in a reducible sub-graph.

We note that two definitions are equivalent. At this point, we are ready to introduce our first condition on the rank

of W∞.

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Lemma 3. Given a feasible W which satisfies Theorem 1, the following holds true:

# of source classes ≤ rank(W∞) ≤ # of classes.

Proof: The proof is given in Appendix C.

Let’s denote rank(W∞) as rW∞ and U as the set of linearly independent columns of W∞, implying that |U| = rW∞.

If we denote elements of U as ui, 1 ≤ i ≤ rW∞, then W,W∞satisfy:

Wui= W∞ui= ui, ∀ i ∈ {1,...,rW∞}.

(9)

In other words, the linearly independent columns of the limiting matrix are the dominant eigenvectors of W matrix,

i.e., the eigenvectors corresponding to the eigenvalue one. Therefore, understanding the column structure of the

limiting matrix is crucial to determine the spectral properties of a feasible W and vice versa.

We first note that each source class SC will be represented by a single dominant eigenvector uiin U, which is

equal to [W∞]i ∀i ∈ SC, where [.]idenotes the i-th column of its argument. We note that W∞

kiis the weight of

node i at node k, in the limit. For a given source node i, if W∞

ki> 0, then node k has access to the weighted value

of node i. It is clear that if k is a destination node, then W∞

ki= 1/M, where M is the number of source nodes.

The argument follows from Theorem 1, since W is feasible.

Let’s assume that rW∞ is strictly greater than the number of source classes. Therefore, there exists at least one

class C which is not a source class and ρ(WC) = 1, where WCis the sub-matrix of rows and columns of W that

correspond to the set C. If this is the case, the columns of W∞corresponding to the elements of the class must

have some non-zero values. Thus, there exists at least one i ∈ C, where W∞

ki> 0 ∀k ∈ C. In other words, some

of the nodes in the network will have access to the weighted values of the nodes in class C in the limit. However,

these nodes cannot be utilized in calculating the average of the source nodes at the destinations. In other words,

if there exists a node k which has access to a non-source class C in the limit, the destination nodes cannot have

access to the node k in limit, since otherwise the third condition in Theorem 1 would not hold. We will summarize

result of the discussion above by the following remark:

Remark 1. For a feasible code W and a non-source class C, if ρ(WC) = 1, then removing class C from the

network does not change the feasibility of the AVT algorithm.

Another way to interpret this condition is the following: If an AVT solution exists for a given scenario, then there

exists at least one feasible solution under the extra constraint that there is no non-source class whose corresponding

sub-matrix has spectral radius one. In other words, one can pose the extra constraint that all of the non-source

classes have sub-matrices with spectral radii strictly less than 1, and such constraint does not change the feasibility

of the problem. We also note that since sub-matrices corresponding to the non-source classes are also sub-matrices

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of the partition D of W (since D governs communications among non-source nodes), the eigenvalues of non-

source classes are also the eigenvalues of D. Therefore, the extra constraint that we posed on non-source classes

is equivalent to ρ(D) < 1. At this point, we would like to discuss the physical significance of ρ(D) < 1. We

first note that ρ(D) < 1 implies limt→∞Dt= 0. Moreover, as we have mentioned above, the partition D governs

the communication among non-source nodes. Therefore, in this particular case, non-source nodes will have zero

information about themselves in the limit. Such a result is not surprising since none of these nodes are interested

in the values of non-source nodes. On the contrary, a subset of these nodes (destination nodes) are interested in the

average of the source nodes, while the intermediate nodes do not aim at receiving anything at all. We summarize

our result in the following lemma:

Lemma 4. There exists a feasible code W which satisfies Theorem 1 if and only if there exists a W?which satisfies

both Theorem 1 and ρ(D) < 1.

While adding one more constraint to Theorem 1 may seem like increasing the complexity of an already difficult

problem, we observe the following: A feasible W has to satisfy rW∞ linear equations of the form (9). Moreover,

ρ(D) < 1 implies that the rank of W∞will be equal to its lower bound given in Lemma 3. This, in return,

implicitly shows that a feasible code W has to satisfy fewer equality constraints.

Moreover, since ρ(D) < 1, limt→∞Dt= 0, thus the third condition in Theorem 1 is automatically satisfied. The

condition in Lemma 2 becomes:

?

lim

t→∞

t

?

l=0

Dt−lBA∞

?

1:K

=?[I − D]−1BA∞?

1:K=

1

M11T,

where I is the identity matrix with appropriate dimensions. Therefore, the fourth condition in Theorem 1 is also

simplified.

At this point, we will introduce our last condition on the number of source classes in the network:

Lemma 5. Consider a CAR problem with the sets SS,SD, a network F(E), and a code W with ρ(D) < 1. If W

is a feasible solution, then the following must hold:

# of source classes ≤ 1 + N − K − rank(B).

Proof: The proof is given in Appendix D.

We note that K is the number of destination nodes and the matrix partition B governs the communication between

source nodes and the rest of the network. Lemma 5 shows that if a network code W is feasible, then the number

of source classes and the rank of the partition B have to balance out each other, i.e., their sum has to be less

than or equal to 1 + N − K. Therefore, network topologies which do not satisfy Lemma 5 will be infeasible. In

the following, we will illustrate our point with an example. Fig. 1 shows a network with two sources and two

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S1 S2

D1 D2

Fig. 1.

destinations. Each source node forms a source class by itself, therefore there are two source classes in the network.

An infeasible scenario with two sources and two destinations.

By Lemma 5, the partition B of W has to satisfy: rank(B) ≤ 1. But, the rank of B can be set to one if and only

if one of the two links going from the sources to the destinations is removed. This is not possible since one of the

sources will be unconnected to the destinations. Therefore, there does not exists any feasible AVT code for this

network.

In the following sections, we will propose several claims under the assumption that ρ(D) < 1. In light of Lemma 4,

such statements do not change the generality of the claims, since our assumptions do not change feasibility of AVT

codes for a given problem.

B. Reduction of Topology to Study Feasibility

In this section, we will introduce methods to perform topology reduction, while leaving the feasibility of AVT

codes unchanged, by utilizing our discussions in Sections III and IV-A. In return, we will simplify our design

problem and develop tools with which we can determine, by inspection, the topologies for which AVT codes are

(in)feasible.

In Definition 3, we have given the definition of source class for a given code W. However, for a given network

F(E) and source set SS, source classes are not unique, i.e., one may change the structure of the source classes by

changing the entries of W. An example is given in Fig. 2. Both networks have the same underlying connectivity

F(E). The first network is induced by a W which assigns non-zero weights to the links between source nodes.

Thus, two source nodes form a single source class. On the other hand, the second network corresponds to a W

which assigns zero weights to the links between source nodes. Since these nodes cannot communicate, each forms

a source class by itself.

In Section IV-A, we have also argued that the rank of W∞is closely related with the complexity of the problem

and, under ρ(D) < 1, the rank is equal to the number of source classes in the network. Then, one can also argue

that for CAR, one should cluster source nodes as much as possible, thus reduce the complexity of the problem.

The definition of the minimal source class follows:

Definition 4. For a given network and source set, the minimal source class is such that the total number of source

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S1 S2

S1S2

Network 1Network 2

Fig. 2.

classes is minimized.

Two networks corresponding to the same underlying connectivity but two different network codes, W1 and W2.

This is the case where all of the existing links between source nodes are assigned non-zero weights. The number of

source classes in the minimal class can be obtained by removing all but the source nodes from the network F(E),

and counting the irreducible sub-graphs that remain.

Next, we will prove that focusing on the minimal class does not change the feasibility of the problem, hence we

will conclude that, combined with the design advantages given above, one can focus on minimal source class codes.

Lemma 6. Consider a problem F(SS,SD) and a network F(E). If the problem is feasible, then there exists at

least one solution W which utilizes the minimal source class.

Proof: The proof is given in Appendix E.

The key point of the proof is that activating links among sources without changing an existing solution does not

affect the feasibility of AVT codes. From Lemma 6, the first topology reduction is in order:

Reduction 1. Use the minimal source class and utilize average consensus algorithm within each class.

We note that utilizing average consensus algorithm within each class does not affect the feasibility of our problem.

Such an observation follows from the fact that for a given feasible W, source node i must have equal information

about the source nodes in its class, i.e., W∞ij= αi, ∀j ∈ SC?, where SC? is the class to which node i belongs.

This observation is due to Lemma 1 and since each source class converges to a rank one matrix (c.f. Appendix C).

Therefore one can construct a feasible W?where each class utilizes an average consensus algorithm, by employing

an ACG algorithm among the sources and rescaling some of the entries of the partition B of W matrix, keeping

the partition D unchanged.

As a result of this reduction, each node in a given source class converges to the average of the initial source nodes

in that particular class.

Reduction 2. Treat each source class as a single node whose value is equal to the average of the source nodes

in that particular class. Connect all non-source nodes, which are adjacent to the source nodes in the class, to this

single node.

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We note that the second reduction does not change the feasibility of the AVT code due to Lemma 2.

Reductions 1-2 combined with ρ(D) < 1 simplify our design problem significantly, with respect to the conditions

given in Theorem 1, without changing the feasibility of the problem.

V. A NECESSARY CONDITION ON THE TOPOLOGY

In this section, we focus on AVT codes which satisfies W1 = 1 as well as Theorem 1. Since such condition

combined with non-negativity, implies that W is a stochastic matrix, we will refer these codes as stochastic codes.

In return, we will provide a non-algebraic necessary condition for the existence of a feasible solution, which is easy

to interpret. At the end of the section, we will discuss topologies which are known to have no feasible solutions.

Before introducing the main result of the section, we would like to emphasize the close relationship between

stochastic AVT codes and non-homogenous random walks on graphs with absorbing states. A stochastic AVT

code W represents the transition matrix of a Markov Chain M whose state space consists of the nodes in the

network (N,E). The structure of the chain (locations of possible non-zero transition probabilities) is defined by

the underlying connectivity F(E), i.e., Wijis the probability of jumping from state i to state j in a single step.

Moreover, due to the structure of W in (7), each source class forms an absorbing class by itself. For a given

non-source node i and a source class SC?, the quantity?

is absorbed by the source class SC? given the fact that M has been initialized at node i [32]. Moreover, for a given

j∈SC?W∞

ijwill be equal to the probability that the chain

source node i ∈ SC? and j ∈ SD, the quantity W∞

ij/?

j∈SC?W∞

ijis the frequency that the chain visits node i

given the fact that M has been initialized at node j and it has been absorbed by the source class SC? [32]. Given

the discussion above, we conclude the following:

Remark 2. Constructing a stochastic AVT code is equivalent to designing a transition probability matrix W for a

Markov chain {M(t)}∞

t=0on graph (N,E) with state space S, where each source class forms an absorbing class.

Moreover, for each destination node j ∈ SDand each source class SC?, absorbtion probabilities should be chosen

such that:

P(M(∞) ∈ SC?|M(0) = j) =|SC?|

M

,

where P(.) the probability of its argument. Moreover, for each source node k ∈ SC?,

P(M(∞) ∈ k|(M(∞) ∈ SC?|M(0) = j)) =

1

|SC?|.

Unfortunately, the formulation given above does not simplify our design problem, since constructing transition

probability matrices for complex chains with given stationary distributions is an open problem in the literature.

However, we believe that pointing out the equivalence relation is necessary as it brings a different perspective to

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S1 D1 S2D2

Fig. 3.

the AVT problem. Moreover, we will make use of the equivalence in the proof of the following lemma:

Line network topology. S and D denotes source nodes and destinations nodes respectively.

Lemma 7. Consider a network F(E) and the sets SSand SD. Partition the network into two disjoint sets (P,Pc)

such that there exists at least one source class-destination pair in both sides of the network. For a given partition,

we denote the links going from one set to the other set as cut edges, i.e, an edge (i,j) is a cut edge if i ∈ P and

j ∈ Pc, or j ∈ P and i ∈ Pc. If there exists a feasible stochastic code, then there exits at least two (cut) edges

between P and Pcfor all such partitions.

Proof: Proof of the lemma is given in Appendix F.

Unlike the results we have proposed in the previous sections, Lemma 7 gives us a topology based method to detect

infeasible cases. We note that the condition given in the lemma has to be satisfied by a network where stochastic

AVT codes are feasible, besides the connectivity constraint that we imposed. Therefore, one can conclude that the

connectivity assumption is not sufficient for the existence of a solution under stochastic codes. At this point, we

remind our readers that the connectivity assumption is a sufficient condition for the existence of an ACG solution [5].

Thus, we conclude that demands on stochastic AVT codes are stricter than the ones for the consensus problems.

We would like to note that, as stated in the hypothesis, the conditions in Lemma 7 is valid for the cases where there

are at least two source classes in the network. For the scenarios where there is only one source class, Lemma 7 is

not valid.

In the following, we will give examples of infeasible network topologies for CAR under stochastic AVT codes. We

will be considering nontrivial CAR instances with more than one source and destination nodes.

1) Line network topology: A line network topology is given in Fig. 3. The vertical line shows a cut where

both partitions have one source-destination pair. By Lemma 7, such a network does not have a solution for

CAR, since the cut shown in the figure has a single cut edge.

2) Mesh network topology with a bottleneck link: The network is shown in Fig. 4. It does not have a solution,

since there is a single cut edge for the given partition.

VI. CONSTRUCTION OF PARTIALLY DIRECTED AVT SOLUTIONS

As we have discussed in the preceding sections, it is difficult to construct feasible AVT codes due to the non-linearity

of the constraints in Theorem 1. For this reason, we have proposed several simplifications, which do not affect

the feasibility of an AVT code in Sections IV-A and IV-B. While these simplifications are valuable in determining

whether AVT codes are feasible or infeasible, they did not lead to a constructive way for designing AVT solutions.

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S1

D1

S2

D2

Fig. 4.

In the following, we will formulate an integer programming problem whose solution will be utilized to construct so

Mesh network topology. S and D denotes source nodes and destinations nodes respectively.

called partially directed AVT solutions. These solutions belong to a subset of the AVT solutions given in Theorem 1.

We first introduce the definition of a partially directed AVT solution:

Definition 5. Consider a network F(E) and the sets SSand SD. A code W is a partially directed AVT solution,

if it satisfies Theorem 1 and each link on the network, except the links among the source nodes, can be utilized

only in one direction, i.e., WijWji= 0, ∀ i,j ?∈ SS.

It should be clear from Definition 5 why these codes are called partially directed solutions, i.e., communication is

directed only among the non-source nodes. Before proposing our construction, we will introduce some necessary

definitions.

Since the communications among source nodes are bidirectional, we can employ Reduction 2 in Section IV-B

and assume that each source class has already converged to the average value of its members. We enumerate the

source classes (arbitrarily ordered) and we define the set of source classes as SC. We also define E?⊂ E which

contains all edges except the edges among the source nodes. For any given U ⊂ S, we define a set of edges

E(U) = {(i,j) ∈ E?|i,j ∈ U}. In other words, E(U) is the set of edges whose end points belong to the set U.

In the following lemma, we introduce our method to construct partially directed AVT codes:

Lemma 8. Consider a network F(E) and the sets SSand SD. For k ∈ SCand l ∈ SD, we define a variable bkl

i

as:

bkl

i=

1,

if i = k,

−1,

if i = l,

0,

otherwise.

Consider the following integer programming formulation:

miminize

max

k∈SC,l∈SD

?

(i,j)∈E?

zkl

ij,

(10)

subject to

?

(i,j) ∈ E?, k ∈ SC, l ∈ SD,

j|(i,j)∈E?

zkl

ij−

?

j|(j,i)∈E?

zkl

ji= bkl

i

i ∈ {SC∪ {M + 1,...,N}},k ∈ SC,l ∈ SD,

(11)

zkl

ij≤ uij,

(12)