Deterministic Gossiping ARO SWARMS MURI Deterministic Gossiping A. S. Morse Yale University February 23, 2010 University of Pennsylvania

Consensus Process Consider a group of n agents labeled 1 to n The groups’ neighbor graph N is an undirected, connected graph with vertices labeled 1,2,...,n. 7 4 1 3 5 2 6 The neighbors of agent i , other than itself, correspond to those vertices which are adjacent to vertex i Each agent i controls a real-valued scalar quantity xi called a consensus variable. The goal of a consensus process is for all n agents to ultimately reach a consensus by adjusting their individual consensus variables to a common value. This is to be accomplished over time by sharing information among neighbors in a distributed manner.

Consensus Process A consensus process is a recursive process which evolves with respect to a discrete time scale. In a standard consensus process, agent i sets the value of its consensus variable at time t +1 equal to the average of the values of its neighbors’ consensus variables at time t, assuming agent i is a neighbor of itself. Average at time t of values of consensus variables of neighbors of agent i. Ni = set of indices of agent i0s neighbors. ni = number of indices in Ni A time-varying consensus process is one in which the neighbor graph N depends on t.

Gossip Process A gossip process is a consensus process in which at each clock time, each agent is allowed to average its consensus variable with the consensus variable of at most one of its neighbors. The index of the neighbor of agent i which agent i gossips with at time t. This is called a gossip and is denoted by (i, j). In the most commonly studied version of gossiping, the specific sequence of gossips which occurs during a gossiping process is determined probabilistically. In a deterministic gossiping process, the sequence of gossips which occurs is determined by a pre-specified protocol. If more than one pair of agents gossip at a given time, the event is called a multi-gossip.

A finite sequence of multi-gossips determines a subgraph M of the groups’ neighbor graph N where (i, j) is an edge in M iff (i, j) is a gossip in the sequence. A finite sequence of multi-gossips is complete if the sub-graph it determines is a spanning tree of N. An infinite multi-gossip sequence is periodic with period T if the finite sequence of multi-gossips which occur in any given period repeats itself on each successive period of length T. An infinite periodic multi-gossip sequence with period T is periodically complete if the finite sequence of multi-gossips which occur in any given period is complete. A uniformly aperiodic multi-gossip sequence is composed of an infinite sequence of successive complete subsequences, each of at length at most T.

State Space Models For a time-varying consensus process each value of M(t) is a “stochastic matrix” with positive diagonals. For a gossip process each value of M(t) is a “doubly stochastic matrix” with positive diagonals. A square matrix S is stochastic if it has only nonnegative entries and if its row sums all equal 1. A square matrix S is doubly stochastic if it has only nonnegative entries and if its row and column sums all equal 1. Stochastic S1 = 1 Doubly Stochastic S1 = 1 and S01 =1

State Space Models Reaching a consensus: Convergability A compact subset of matrices M is convergable if for each infinite sequence of matrices S1, S2, .... from M, the matrix product Si Si-1S1 ! 1c. What are conditions for convergability?

The graph of a nonnegative square matrix M, written °(M), is a directed graph on n vertices with an arc from i to j iff mji  0. Strong connectivity: There is a directed path from each vertex to each other vertex. A compact subset of matrices M is convergable if for each infinite sequence of matrices S1, S2, .... from M, the matrix product Si Si-1S1 ! 1c. Weak connectivity: There is an undirected path between each pair of vertices Rooted: For at least one vertex v, there is a directed path from v to each other vertex. Let S = set of stochastic matrices whose graphs have self arcs at all vertices. A compact subset of S is convergable iff the graphs of its matrices are all rooted. Let D = set of doubly stochastic matrices whose graphs have self arcs at all vertices. A compact subset of D is convergable iff the graphs of its matrices are all weakly connected. What about convergence rates?

Periodically Complete Multi-Gossip Sequences The convergence rate of a periodically complete multi-gossip sequence is no slower than ¸ = second largest eigenvalue of the stochastic matrix determined by a sequence of multi-gossips in a period T. For a given spanning tree T ½ N, how does ¸ depend on the order in which the gossips denoted by the edges of T are carried out? 2. For a given spanning tree T ½ N, what is the minimal value of T assuming multi-gossiping? 1. ¸ is independent of the order in which the gossips are carried out. 2. The minimal value of T is the chromatic index of T which for a tree is the degree of the tree.

Uniformly Aperiodically Complete Multi-Gossip Sequences Coefficient of Ergodicity ²(S) <1 iff S is a “scrambling matrix” A stochastic matrix is a scrambling matrix if no two rows are orthogonal. Any compact set C of scrambling matrices is convergable Convergence rate:

A compact subset of the set of stochastic matrices whose graphs have self arcs at all vertices is convergable iff the graphs of its matrices are all rooted. A compact subset of the set of doubly stochastic matrices whose graphs have self arcs at all vertices is convergable iff the graphs of its matrices are all weakly connected. Are there analogs of the coefficient of ergodicity for stochastic matrices with rooted graphs or for doubly stochastic matrices with weakly connected graphs? Any compact set C of scrambling matrices is convergable

Semi-norms of A 2 Rm£n For any p define Satisfies the triangle inequality and is thus a semi-norm Is sub-multiplicative for the set of all A satisfying A1 =1 For any doubly stochastic matrix Sn£n |S|p · 1, p=1,2 Let q be the integer quotient of n divided by 2. Then |S|1 <1 iff the number of nonzero entries in each column of S exceeds q. |S|2 = the second largest singular value of S.

A finite sequence of multi-gossips determines a subgraph M of the groups’ neighbor graph N where (i, j) is an edge in M iff (i, j) is a gossip in the sequence. A finite sequence of multi-gossips is complete if the sub-graph it determines is a spanning tree of N. A finite sequence of multi-gossips is complete if and only if the graph of the stochastic matrix it determines is weakly connected. A finite sequence of multi-gossips is complete if and only if the stochastic matrix S it determines, satisfies |S|2 < 1. |S|2 = the second largest singular value of S.

Uniformly Aperiodically Complete Multi-Gossip Sequences Assume that: Each agent gossips with at most one neighbor at one time. At any time t, there are no “uncommitted” agent pairs. An agent which has gossiped with neighbor i, gossips with all of its other neighbors exactly once before it again gossips with neighbor i If N is a graph of degree d, then a complete gossip sequence is achieved in at most 2d - 1 steps.

Uniformly Aperiodically Complete Multi-Gossip Sequences Assume N is a graph of degree d. The convergence rate of a uniformly aperiodically complete multi-gossip sequence is no slower than T = 2d -1 where: C = set of all complete multi-gossip sequences of length at most T ¸¾ = the second largest singular values of the stochastic matrix determined by multi-gossip sequence ¾2C