When does a digraph admit a doubly stochastic adjacency matrix?

Bahman Gharesifard and Jorge Cortes´

Abstract— Digraphs with doubly stochastic adjacency matri- of weight-balanced digraphs. To the authors’ knowledge, the ces play an essential role in a variety of cooperative control establishment of this relationship is also a novel contribution. problems including distributed averaging, optimization, and The paper is organized as follows. Section II presents some gossiping. In this paper, we fully characterize the class of digraphs that admit an edge weight assignment that makes the mathematical preliminaries from graph theory. Section III digraph adjacency matrix doubly stochastic. As a by-product introduces the problem statement. Section IV examines the of our approach, we also unveil the connection between weight- connection between weight-balanced and doubly stochastic balanced and doubly stochastic adjacency matrices. Several adjacency matrices. Section V gives necessary and sufficient examples illustrate our results. conditions for the existence of a doubly stochastic adja- cency matrix assignment for a given digraph. Section VI I. INTRODUCTION studies the properties of the topological character associated A digraph is doubly stochastic if, at each vertex, the sum with strongly connected doubly stochasticable digraphs. We of the weights of the incoming edges as well as the sum of gather our conclusions and ideas for future work in Sec- the weights of the outgoing edges are equal to one. Doubly tion VII. stochastic digraphs play a key role in networked control II. MATHEMATICAL PRELIMINARIES problems. Examples include distributed averaging [1], [2], We adopt some basic notions from [1], [16], [17]. A [3], [4], distributed convex optimization [5], [6], [7], and directed graph, or simply digraph, is a pair G V, E , gossip algorithms [8], [9]. Because of the numerous algo- = ( ) where V is a finite set called the vertex set and E V V rithms available in the literature that use doubly stochastic ⊆ × is the edge set. If V n, i.e., the cardinality of V is interaction topologies, it is an important research question | | = n Z , we say that G is of order n. We say that an edge to characterize when a digraph can be given a nonzero edge ∈ >0 u,v E is incident away from u and incident toward v, weight assignment that makes it doubly stochastic. This is ( ) ∈ and we call u an in-neighbor of v and v an out-neighbor the question we investigate in this paper. of u. The in-degree and out-degree of v, denoted d (v) and We refer to a digraph as doubly stochasticable if it admits in d (v), are the number of in-neighbors and out-neighbors of a doubly stochastic adjacency matrix. In studying this class out v, respectively. We call a vertex v isolated if it has zero in- of digraphs, we unveil their close relationship with a special and out-degrees. class of weight-balanced digraphs. A digraph is weight- An undirected graph, or simply graph, is a pair G = balanced if, at each node, the sum of the weights of the (V, E), where V is a finite set called the vertex set and incoming edges equals the sum of the weights of the outgo- the edge set E consists of unordered pairs of vertices. In ing edges. The notion of weight-balanced digraph is key in a graph, neighboring relationships are always bidirectional, establishing convergence results of distributed algorithms for and hence we simply use neighbor, degree, etc. for the average-consensus [10], [2] and consensus on general func- notions introduced above. A graph is regular if each vertex tions [11] via Lyapunov stability analysis. Weight-balanced has the same number of neighbors. The union G ∪ G of digraphs also appear in the design of leader-follower strate- 1 2 digraphs G = (V , E ) and G = (V , E ) is defined by gies under time delays [12], virtual leader strategies under 1 1 1 2 2 2 G ∪ G = (V ∪ V , E ∪ E ). The intersection of two asymmetric interactions [13] and stable flocking algorithms 1 2 1 2 1 2 digraphs can be defined similarly. A digraph G is generated for agents with significant inertial effects [14]. We call by a set of digraphs G ,...,G if G = G ∪ · · · ∪ G . We a digraph weight-balanceable if it admits an edge weight 1 m 1 m let E− ⊆ V × V denote the set obtained by changing the assignment that makes it weight-balanced. A characterization order of the elements of E, i.e., (v,u) ∈ E− if (u,v) ∈ E. of weight-balanceable digraphs was presented in [15]. The digraph G = (V, E ∪ E−) is the mirror of G. In this paper, we provide a constructive a necessary and A weighted digraph is a triplet G = (V,E,A), where sufficient condition for a digraph to be doubly stochasticable. (V, E) is a digraph and A ∈ Rn×n is the adjacency matrix. This condition involves a very particular structure that the ≥0 We denote the entries of A by a , where i, j ∈ {1,...,n}. cycles of the digraph must enjoy. As a by-product of our ij The adjacency matrix has the property that the entry a > 0 characterization, we unveil the connection between the struc- ij if (v ,v ) ∈ E and a = 0, otherwise. If a matrix A ture of doubly stochasticable digraphs and a special class i j ij satisfies this property, we say that A can be assigned to the digraph G = (V, E). Note that any digraph can be Bahman Gharesifard and Jorge Cortes´ are with the Department of Mechanical and Aerospace Engineering, University of California San Diego, trivially seen as a weighted digraph by assigning weight 1 {bgharesifard,cortes}@ucsd.edu to each one of its edges. We will find it useful to extend the definition of union of digraphs to weighted digraphs. The v1 / v2 O `B union G ∪ G of weighted digraphs G = (V , E , A ) and BB || 1 2 1 1 1 1 BB|| G = (V , E , A ) is defined by G ∪ G = (V ∪ V , E ∪ || BB 2 2 2 2 1 2 1 2 1 ~|| B E2, A), where v3 / v4

A|V1∩V2 = A1|V1∩V2 + A2|V1∩V2 , Fig. 1. A weight-balanceable digraph for which their exists no doubly stochastic adjacency assignment. A|V1\V2 = A1, A|V2\V1 = A2. For a weighted digraph, the weighted out-degree and in- degree are, respectively, Theorem 2.1: A digraph G = (V, E) is weight- balanceable if and only if the edge set E can be decomposed n n w w into k subsets E1,...,Ek such that d (vi) = aij , d (vi) = aji. out in (i) E = E ∪ E ∪ . . . ∪ E and Xj=1 Xj=1 1 2 k (ii) every subgraph G = (V, Ei), for i = {1,...,k}, is a A. Graph connectivity notions weight-balanceable digraph. A directed path in a digraph, or in short path, is an ordered Theorem 2.2: Let G = (V, E) be a directed digraph. The sequence of vertices so that any two consecutive vertices in following statements are equivalent. the sequence are an edge of the digraph. A cycle in a digraph (i) Every element of E lies in a cycle. is a directed path that starts and ends at the same vertex and (ii) G is weight-balanceable. has no other repeated vertex. Two cycles are disjoint if they (iii) G is strongly semiconnected. do not have any vertex in common. III. PROBLEM STATEMENT A digraph is strongly connected if there is a path between each pair of distinct vertices and is strongly semiconnected Our main goal in this paper is to obtain necessary and if the existence of a path from v to w implies the existence sufficient conditions that characterize when a digraph is of a path from w to v, for all v, w ∈ V . Clearly, strongly doubly stochasticable. Note that strongly semiconnected- connectedness implies strongly semiconnectedness, but the ness is a necessary, and sufficient, condition for a digraph converse is not true. The strongly connected components to be weight-balanceable. All doubly stochastic digraphs of a directed graph G are its maximal strongly connected are weight-balanced; thus a necessary condition for a di- subdigraphs. graph to be doubly stochasticable is strongly semiconnected- ness. Moreover, weight-balanceable digraphs that are doubly B. Basic notions from linear algebra stochasticable do not have any isolated vertex. However, Rn×n n none of these conditions is sufficient. A simple example A matrix A ∈ ≥0 is weight-balanced if j=1 aij = n n×n illustrates this. Consider the digraph G shown in Figure 1. aji, for all i ∈ {1,...,n}. A matrix AP∈ R is j=1 ≥0 Note that this digraph is strongly connected; thus there exists row-stochastic if each of its rows sums 1. One can similarly P a set of positive weights which makes the digraph weight- define a column-stochastic matrix. We denote the set of all Rn×n Rn×n balanced. However, there exists no set of nonzero weights row-stochastic matrices on ≥0 by RStoc( ≥0 ). A non- Rn×n that makes this digraph doubly stochastic. Suppose zero matrix A ∈ ≥0 is doubly stochastic if it is both row- n×n stochastic and column-stochastic. A matrix A ∈ {0, 1} is 0 α1 0 0 a permutation matrix, where n ∈ Z , if A has exactly one  0 0 α2 0  ≥1 A = , entry 1 in each row and each column. A matrix A ∈ Rn×n is α 0 0 α ≥0  3 4  irreducible if, for any nontrivial partition J ∪K of the index  α 0 0 0   5  set {1,...,n}, there exist j ∈ J and k ∈ K such that a 6= jk where α ∈ R , for all i ∈ {1,..., 5}, is a doubly 0. We denote by Irr(Rn×n) the set all irreducible matrices on i >0 ≥0 stochastic adjacency assignment for this digraph. Then a Rn×n. Note that a weighted digraph G is strongly connected ≥0 simple computation shows that the only solution that makes if and only if its adjacency matrix is irreducible [17]. the digraph doubly stochastic is by choosing α3 = 0, which is not possible by assumption. Thus this digraph is not doubly C. Weight-balanced and doubly stochastic digraphs stochasticable. A weighted digraph G is weight-balanced (resp. doubly The following result will simplify our analysis by allowing stochastic) if its adjacency matrix is weight-balanced (resp. us to restrict our attention to strongly connected digraphs. doubly-stochastic). Note that G is weight-balanced if and Lemma 3.1: A strongly semiconnected digraph is doubly w w only if dout(v) = din(v), for all v ∈ V . A digraph is stochasticable if and only if all of its strongly connected called weight-balanceable (resp. doubly stochasticable) if it components are doubly stochasticable. admits a weight-balanced (resp. doubly stochastic) adjacency Proof: Let G1 and G2 be two strongly connected com- matrix. The following two results establish a constructive ponents of the digraph. Note that there can be no edges from centralized approach for determining whether a digraph is v1 ∈ G1 to v2 ∈ G2 (or vice versa). If this were the case, then weight-balanceable, see [15]. the strongly semiconnectedness of the digraph would imply that there is a path from v2 to v1 in the digraph, and hence for all j ∈ {1,...,n}. Thus G1 ∪ G2 would be strongly connected, contradicting the fact a1j a2j anj that G and G are maximal. Therefore, the adjacency matrix + + · · · + = 1, (2) 1 2 Cj Cj Cj of the digraph is a block-diagonal matrix, where each block corresponds to the adjacency matrix of a strongly connected From Equations (1) and (2), we have component, and the result follows. 1 1 1 1 As a result of Lemma 3.1, we are interested in charac- a1j − + · · · + anj − = 0, (3) C1 Cj  Cn Cj  terizing the class of strongly connected digraphs which are doubly stochasticable. for all j ∈ {1,...,n}. Suppose that, up to rearranging,

C1 = min{Ck | k ∈ {1,...,n}}, IV. THE RELATIONSHIP BETWEEN WEIGHT-BALANCED k AND DOUBLY STOCHASTIC ADJACENCY MATRICES and, 0 < C1 < Ci, for all i ∈ {2,...,n}. Then (3) gives As an intermediate step of the characterization of doubly 1 1 1 1 stochasticable digraphs, we will find it useful to study the a − + · · · + an − = 0; 21 C C  1 C C  relationship between weight-balanced and doubly stochas- 2 1 n 1 tic digraphs. The example in Figure 1 underscores the thus aj1 = 0, for all j ∈ {2,...,n}, which contradicts the n importance of characterizing the set of weight-balanceable irreducibility assumption. If the set {Ck}k=1 has more than digraphs that are also doubly-stochasticable. one element giving the minimum, the proof follows a similar We start by introducing the row-stochastic normalization argument. Suppose Rn×n Rn×n map φ : Irr( ≥0 ) → RStoc( ≥0 ) defined by C1 = C2 = min{Ck | k ∈ {1,...,n}}, k aij φ : aij 7→ n . and suppose that 0 < C1 = C2 < Ci, for all i ∈ {3,...,n}. l=1 ail P Then we have Note that, for A ∈ Irr(Rn×n), φ(A) is doubly stochastic if ≥0 1 1 1 1 and only if a31 − + · · · + an1 − = 0, n C3 C1  Cn C1  aij n = 1, and ail Xi=1 l=1 P 1 1 1 1 for all j ∈ {1,...,n}. The following result characterizes a32 − + · · · + an2 − = 0, C3 C2  Cn C2  when the digraph associated with an irreducible weight- balanced adjacency matrix is doubly stochasticable. and thus aj1 = 0 = aj2, for all j ∈ {3,...,n}, which Rn×n contradicts the irreducibility assumption. The same argument Theorem 4.1: Let A ∈ Irr( ≥0 ) be an adjacency ma- trix associated to a weight-balanced digraph. Then φ(A) holds for an arbitrary number of minima. n Corollary 4.2: Any strongly connected digraph is doubly is doubly stochastic if and only if l=1 ail = C, for all R stochasticable after adding enough number of self-loops. i ∈ {1,...,n}, for some C ∈ >0. P Proof: The implication from right to left is immediate. Proof: Any strongly connected digraph is weight- Suppose then that A is associated to a strongly connected balanceable. The result follows from noting that, for any weight-balanced digraph. Then we need to show that if A weight-balanced matrix, it is enough to add self-loops with satisfies the following set of equations appropriate weights to the vertices of the digraph to make the conditions of Theorem 4.1 hold. n n Regular undirected graphs trivially satisfy the conditions a = a , jl lj of Theorem 4.1 and hence the following result. Xl=1 Xl=1 n Corollary 4.3: All undirected regular graphs are doubly aij stochasticable. n = 1, ail Xi=1 l=1 P V. NECESSARY AND SUFFICIENT CONDITIONS FOR for all j ∈ {1,...,n}, there exists C ∈ R>0 such that DOUBLY STOCHASTICABILITY n n l=1 ail = C, for all i ∈ {1,...,n}. Let Ck = l=1 akl, In this section, we provide a characterization of the kP∈ {1,...,n}. Then the doubly stochastic conditionsP can structure of digraphs that are doubly stochasticable. The main be written as contributions are Theorem 5.4 and Corollary 5.5. a a a 1j + 2j + · · · + nj = 1, (1) Let G = (V, E) be a strongly semiconnected digraph. Let C1 C2 Cn Gcyc denote a union of some disjoint cycles of G (note that for all j ∈ {1,...,n}. Note that C 6= 0, for all k ∈ Gcyc can be just one cycle). One can extend the adjacency k Rn×n {1,...,n}, since A is irreducible. By the weight-balanced matrix associated to Gcyc to a matrix Acyc ∈ , by adding assumption, we have zero rows and columns for the vertices of G that are not included in Gcyc. Note that the matrix Acyc is the adjacency a1j + a2j + · · · + anj = Cj, matrix for a subdigraph of G. We call Acyc the extended adjacency matrix associated to Gcyc. We have the following Proof: Since one can assign a doubly stochastic ad- result. jacency matrix to G, the digraph cannot have any isolated Lemma 5.1: The extended adjacency matrix associated to vertex. Let A be a doubly stochastic matrix associated to G. Gcyc is a permutation matrix if and only if Gcyc contains all By the Birkhoff–von Neumann theorem [18], a square matrix the vertices of G. is doubly stochastic if and only if it is a convex combination Proof: It is clear that if Gcyc is a union of some of permutation matrices. Therefore, disjoint cycles and contains all the vertices of G, then the n! adjacency matrix associated to Gcyc is a permutation matrix. i A = λ¯iA , Conversely, suppose that G does not contain one of the perm cyc Xi=1 vertices of G. Then the adjacency matrix associated to Gcyc ¯ R n! ¯ i has a zero row and thus is not a permutation matrix. where λi ∈ ≥0, i=1 λi = 1, and Aperm is a permutation By Theorem 2.2, any strongly semiconnected digraph can matrix for each i ∈P {1,...,n!}. By Lemma 5.1, for all λ¯i > i be generated by the cycles contained in it. Thus it makes 0, one can associate to the corresponding Aperm a union of i sense to define a minimal set of such cycles that can generate disjoint cycles that contains all the vertices. Thus each Aperm the digraph. That is what we define next. is an extended adjacency matrix associated to an element of ¯ Definition 5.2: Let G = (V, E) be a strongly semicon- C(G). Let us rename all the nonzero coefficients λi > 0 by nected digraph. Let C(G) denote the set of all subdigraphs λi. In order to complete the proof, we need to show that of G that are either isolated vertices, cycles of G, or a union at least p(G) of the λi’s are nonzero. Suppose otherwise. i of disjoint cycles of G. P (G) ⊆ C(G) is a principal cycle Since each Aperm with nonzero coefficient is associated to set of G if its elements generate G, and there is no subset an element of C(G), this means that the digraph G can be of C(G) with strictly smaller cardinality that satisfies this generated by fewer elements than p(G), which contradicts property. Definition 5.2. Note that there might exist more than one principal set, The following result fully characterizes the set of strongly however, by definition, the cardinalities of all principal cycle connected doubly stochasticable digraphs. sets are the same. We denote this cardinality by p(G). Corollary 5.5: A strongly connected digraph G is doubly i ξ Principal cycle sets give rise to weight-balanced assignments. stochasticable if and only if there exists a set {Gcyc}i=1 ⊆ i Proposition 5.3: Let G = (V, E) be a strongly semicon- C(G), where ξ ≥ p(G), that generates G and such that Gcyc nected digraph. Then, the union of the elements of a principal contains all the vertices of G, for each i ∈ {1,...,ξ}. cycle set of G, considered as subdigraphs with trivial weight Proof: Suppose G is doubly stochasticable. By The- assignment, gives a set of positive integer weights which orem 5.4, A can be written as the union of at least p(G) make the digraph weight-balanced. elements of C(G) which contain all the vertices and generate Proof: Since each element of a principal cycle P (G) is G. This proves the implication from left to right. Suppose ξ i i either an isolated vertex, a cycle, or union of disjoint cycles, G = ∪i=1Gcyc, where Gcyc ∈C(G) contain all the vertices, it is weight-balanced. By definition, G can be written as for all i ∈ {1,...,ξ}. Consider the adjacency matrix the union of the elements of P (G). Thus by Theorem 2.1, ξ the weighted union of the elements of P (G) gives a set of A = Ai , weights makes the digraph G weight-balanced. cyc Xi=1 Note that, in general, the assignment in Proposition 5.3 i uses fewer number of cycles than the ones used in Theo- where Acyc is the extended adjacency matrix associated i rem 2.2. Note that a cycle, or a union of disjoint cycles, that to Gcyc. Note that A is weight-balanced and satisfies the contains all the vertices has the maximum number of edges conditions of Theorem 4.1 (the sum of each row is equal to that an element of C(G) can have. Thus these elements are ξ). Thus G is doubly stochasticable. the obvious candidates for constructing a principal cycle set. Corollary 5.5 suggests the definition of the following Next, we state a necessary condition for a digraph to be notion. Given a strongly connected doubly stochasticable doubly stochasticable. digraph G, DS(G) ⊆ C(G) is a DS-cycle set of G if all Theorem 5.4: Let G be a strongly semiconnected digraph. its elements contain all the vertices of G, they generate G, Suppose that one can assign a doubly stochastic adjacency and there is no subset of C(G) with strictly smaller cardi- matrix A to G. Then nality that satisfies these properties. Corollary 5.5 implies that DS-cycle sets exist for any strongly connected doubly ξ stochasticable digraph. The cardinality of any DS-cycle set A = λ Ai , i cyc of G is the DS-character of G, denoted ds(G). If a doubly Xi=1 stochastic digraph is not strongly connected, one can use this where notion on each strongly connected component. ξ R ξ • {λi}i=1 ⊂ >0, i=1 λi = 1, and ξ ≥ p(G). Example 5.6: (Weight-balanceable, not doubly stochasti- i • Acyc, i ∈ {1,...,ξP}, is the extended adjacency matrix cable digraph): Consider the digraph G shown in Figure 2(a). associated to an element of C(G) that contains all the It is shown in [19] that there exists a set of weights vertices. which makes this digraph weight-balanced. We show that the v1 / v2 (a) v2 o v1 (b) v1 / v2 X1 1B > F O X1 1 1 BB || 1 11 11 BB || 11 1 1 BB || 1 11  v 1 B || 1 1 v3 5 1 v v 1 v 1 `BB 11 3 5 o 11 3 1 | B 1 |> O `BB 1 | 1 || BB ||1 BB 1 || 11 || BB | 1 B 1 | 1 || B || 11 BB11 || ~| v ||  B  ~|| v4 4 v5 o v4 v4 v1 / v2 Fig. 2. The digraph of Examples 5.6 and 5.7 are shown in plots (a) and O (b), respectively.  v5 o v3 v v v v 1 / 2 1 / 2 G O X11 Fig. 4. Cycles of the digraph of Example 5.7 which are not in the 1 principal cycle set. 11  11  v5 v3 v5 o 1 v3 `B 1 BB || 11 BB || 1 not all the weight-balanced adjacency assignments become BB || 11 B ~||  doubly stochastic under the row-stochastic normalization v4 v4 map. An example is given by the adjacency matrix

Fig. 3. The only principal cycle set for the digraph of Example 5.7 contains 03000 the above cycles.  00300  A = 00021 . •    20002  digraph is not doubly stochasticable. The edge (v2,v3) only    10020  appears in the cycle Gcyc = {v1,v2,v3}. Thus this cycle   appears in any set of elements of C(G) that generates G. An alternative question to the one considered above would Since Gcyc does not include all the vertices, by Corollary 5.5, be to find a set of edge weights (some possibly zero) there exists no doubly stochastic adjacency assignment for that make the digraph doubly stochastic. Such assignments this digraph. One can verify this by trying to find such exist for the digraph in Figure 1. However, such weight assignment explicitly, i.e., by seeking αi ∈ R>0, where assignments are not guaranteed, in general, to preserve the i ∈ {1,..., 8}, such that connectivity of the digraph. The following result gives a sufficient condition for the existence of such an edge weight 0 α1 0 0 0 assignment.  0 0 α2 α3 0  Proposition 5.8: A strongly connected digraph G admits A = α 0 0 0 0  4  an edge weight assignment (where some entries might be  α 0 α 0 α   5 6 7  zero) such that the resulting weighted digraph is strongly  0 0 α 0 0   8  connected and doubly stochastic if there exists a cycle is doubly stochastic. A simple computation shows that such containing all the vertices of G. an assignment is not possible unless α2 = α5 = α6 = 0, Regarding Proposition 5.8, note that even if connectivity which is a contradiction. • is preserved, fewer edges lead to smaller algebraic con- Example 5.7 (Doubly stochasticable digraph): Consider nectivity [20], which in turn affects negatively the rate the digraph G shown in Figure 2(b). One can observe that of convergence of the consensus, optimization, and gossip the only principal cycle set of G contains the two cycles algorithms executed over doubly stochastic digraphs, see shown in Figure 3. Both of these cycles pass through all the e.g., [8], [9], [21], [22]. vertices of the digraph and thus, using Corollary 5.5, this digraph is doubly stochasticable. Note that this digraph has VI. PROPERTIES OF THE TOPOLOGICAL CHARACTER OF another three cycles, shown in Figure 4, none of which is DOUBLY STOCHASTICABLE DIGRAPHS in the principal cycle set. The adjacency matrix assignment In this section, we investigate the properties of DS- 02000 cycle sets and of their cardinality ds(G). We start with the  00200  following definition. A = 00011   Definition 6.1: Let G = (V, E) be a strongly connected  10001    digraph and let A be a weight-balanced adjacency matrix  10010  C R   which satisfies the conditions of Theorem 4.1 with ∈ >0. obtained by the sum of the elements of the principal cycle Then we call the weighted digraph G = (V,E,A) a C- set, is weight-balanced and satisfies the conditions of The- regular digraph. orem 4.1 and thus is doubly stochasticable. Also note that We have the following result. Theorem 6.2: Let G be a strongly connected doubly variety of distributed algorithms for consensus and optimiza- stochasticable digraph of order n ∈ Z>0 with DS-character tion. Future work will investigate the design of distributed ds(G). Then the following statements hold, algorithms that a network of agents can run in order to obtain Zn×n the set of weights that makes the adjacency matrix doubly • There exists a weight assignment Awb ∈ ≥0 that makes G a C-regular digraph with C ≥ ds(G). stochastic. n×n • There exist no integer weight assignment A Z wb ∈ ≥0 ACKNOWLEDGMENTS that makes G a C-regular digraph with C < ds(G). Proof: By Corollary 5.5, it is clear that one can generate This research was partially supported by NSF Awards Zn×n CCF-0917166 and CMMI-0908508. Awb ∈ ≥0 that makes G weight-balanced and also satisfies the conditions of Theorem 4.1 for C = ds(G), just by taking REFERENCES the weighted union of the members of a DS-cycle set. Let [1] F. Bullo, J. Cortes,´ and S. Mart´ınez, Distributed Control of Robotic C > ds(G). Choose a set of integer numbers λi ∈ Z>0, for Networks, ser. Applied Mathematics Series. Princeton University ds(G) i ∈ {1,..., ds(G)}, such that λi = C. Consider the Press, 2009, electronically available at http://coordinationbook.info. i=1 [2] R. Olfati-Saber and R. M. Murray, “Consensus problems in networks adjacency matrix P of agents with switching topology and time-delays,” IEEE Transac- ds(G) tions on Automatic Control, vol. 49, no. 9, pp. 1520–1533, 2004. i [3] W. Ren and R. W. Beard, Distributed Consensus in Multi-vehicle A = λiAcyc, Cooperative Control, ser. Communications and Control Engineering. Xi=1 Springer, 2008. [4] L. Xiao and S. Boyd, “Fast linear iterations for distributed averaging,” i where Acyc is the extended adjacency matrix associated Systems & Control Letters, vol. 53, pp. 65–78, 2004. to the ith element of the DS-cycle set. The matrix A is [5] B. Johansson, M. Rabi, and M. Johansson, “A randomized incremental subgradient method for distributed optimization in networked sys- weight-balanced adjacency and satisfies the conditions of tems,” SIAM Journal on Control and Optimization, 2009, submitted. Theorem 4.1. This proves the first part of the theorem. [6] A. Nedic and A. Ozdaglar, “Distributed subgradient methods for multi- Now assume that there exists a weight-balanced adjacency agent optimization,” IEEE Transactions on Automatic Control, vol. 54, Zn×n no. 1, pp. 48–61, 2009. matrix A ∈ ≥0 that makes G a C-regular digraph with [7] M. Zhu and S. Mart´ınez, “On distributed convex optimization under 1 inequality and equality constraints via primal-dual subgradient meth- C < ds(G). Then C A is a doubly stochastic adjacency matrix for G. Thus using Theorem 5.4 ods,” IEEE Transactions on Automatic Control, 2009, submitted. [8] S. Boyd, A. Ghosh, B. Prabhakar, and D. Shah, “Randomized gossip C algorithms,” IEEE Transactions on Information Theory, vol. 52, no. 6, 1 pp. 2508–2530, 2006. A = λ Ai , C i cyc [9] J. Liu, A. S. Morse, B. D. O. Anderson, and C. Yu, “The contraction Xi=1 coefficient of a complete gossip sequence,” Proceedings of the IEEE, i 2009, submitted. where Acyc, i ∈ {1,...,C}, is the extended adjacency matrix [10] R. Olfati-Saber, J. A. Fax, and R. M. Murray, “Consensus and associated to an element of C(G) which contains all the cooperation in networked multi-agent systems,” Proceedings of the vertices. But this contradicts the minimality in the definition IEEE, vol. 95, no. 1, pp. 215–233, 2007. [11] J. Cortes,´ “Distributed algorithms for reaching consensus on general of a DS-cycle set. functions,” Automatica, vol. 44, no. 3, pp. 726–737, 2008. We finish this section by bounding the DS-character of a [12] J. Hu and Y. Hong, “Leader-following coordination of multi-agent digraph. systems with coupling time delays,” Physica A, vol. 374, no. 2, pp. 853–863, 2007. Lemma 6.3: Let G = (V, E) be a strongly connected [13] H. Shi, L. Wang, and T. Chu, “Virtual leader approach to coordinated doubly stochasticable digraph. Then control of multiple mobile agents with asymmetric interactions,” Physica D, vol. 213, no. 1, pp. 51–65, 2006. [14] D. Lee and M. W. Spong, “Stable flocking of multiple inertial agents max dout(v) ≤ ds(G) ≤ |E| − |V | + 1. v∈V on balanced graphs,” IEEE Transactions on Automatic Control, vol. 52, Proof: The first inequality follows from the fact that no. 8, pp. 1469–1475, 2007. none of the out-edges of the vertex v with maximum out- [15] L. Hooi-Tong, “On a class of directed graphs - with an application to traffic-flow problems,” Operations Research, vol. 18, no. 1, pp. 87–94, degree are contained in the same element of any DS-cycle 1970. set DS(G). To show the second inequality, take any element [16] R. Diestel, Graph Theory, 2nd ed., ser. Graduate Texts in Mathematics. of DS(G). This element must contain |V | edges. The rest Springer, 2005, vol. 173. [17] N. Biggs, Algebraic Graph Theory, 2nd ed. Cambridge University of the edges of the digraph can be represented by at most Press, 1994. |E| − |V | elements of C(G), and hence the bound follows. [18] G. Birkhoff, “Tres observaciones sobre el algebra lineal,” Universidad Nacional de Tucuman,´ Revista A, vol. 5, pp. 147–151, 1946. [19] B. Gharesifard and J. Cortes,´ “Distributed strategies for making a digraph weight-balanced,” in Allerton Conf. on Communications, VII. CONCLUSIONS Control and Computing, Monticello, IL, Oct. 2009. We have provided necessary and sufficient conditions for [20] C. W. Wu, “Algebraic connectivity of directed graphs,” Linear and Multilinear Algebra, vol. 53, no. 3, pp. 203–223, 2005. the existence of an edge weight assignment that makes [21] M. Cao and C. W. Wu, “Topology design for fast convergence the adjacency matrix of a given digraph doubly stochastic. of network consensus algorithms,” in The 2007 IEEE International We have unveiled the particular connection of this class of Symposium on Circuits and Systems (ISCAS), New Orleans, LA, USA, 2007, pp. 1029–1032. digraphs with a special subset of weight-balanced digraphs. [22] A. Ghosh and S. Boyd, “Upper bounds on algebraic connectivity via The characterization provided here enlarges the range of convex optimization,” Linear Algebra and its Applications, vol. 418, network interconnection topologies for which one can run a pp. 693–707, 2006.