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When Does a Digraph Admit a Doubly Stochastic Adjacency Matrix?

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When Does a Digraph Admit a Doubly Stochastic Adjacency Matrix? 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 B|B| 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.
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