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Distributed linear consensus
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Optimal network topology for responsive collective behavior

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Let us consider a group of N + 1 identical agents performing a distributed consensus protocol on their scalar state-variable xi(t). The dynamics of the system is determined by the state vector X(t) = {xi(t); i = 0, …, N} and the adjacency matrix of the underlying graph A = {aij; i, j = 0, …, N}, where aij = 1 if agent i is connected to j and 0 otherwise. Given a certain connectivity graph, the system evolves according to$dxidt=ω0ki∑j=0Naij(xj(t)−xi(t))=∑j=0Nwijxj(t)$(2)where ω0 is the natural response frequency of our identical agents, and $ki=∑j=0Naij$ is the degree (or number of neighbors) of agent i. The quantity wij = ω0(aij/ki − δij)—where δij is a Kronecker delta—is introduced for the sake of a compact notation. Note that, by definition, wii = −ω0 and ∑jwij = 0 for all i.

To model the response of the system, we consider a leader-follower consensus scenario where one agent—for example, agent i = 0, the leader—does not abide by the dynamics of Eq. 2 but instead follows an arbitrary trajectory x0(t) = u(t). In the presence of this single leader, Eq. 2 can be recast as$dxidt=∑j=1Nwijxj(t)+wi0u(t)$(3)for i = 1, …, N. The solution of Eq. 3 (up to an integration constant) can be written compactly in matrix notation on the frequency domain as$X(ω)=(iωI−WF)−1WLu(ω)$(4)where I is the identity matrix of dimension N, WF = {wij} is the N × N consensus protocol matrix between the follower agents (also known as state matrix A in LTI systems), and WL = {wi0} is the N × 1 consensus protocol matrix between the followers and the leader (also known as input matrix B).

The response function or susceptibility measures the capacity of the multi-agent system to follow the leader’s trajectory, u(t), and can be expressed in the frequency domain (33) as$H(ω)=(δXδu)(ω)=(iωI−WF)−1WL$(5)

The entries of the vector H = {hi}i = 1,…,N correspond to the frequency response of each individual agent, with |hi(ω)| ≤ 1 for all i and ω (33). As is clear from Eq. 5, the response functions have a nontrivial dependency on the topology of the agents’ connectivity through the entries of WF and WL. The collective response of the system can be characterized by performing a singular value decomposition of H, giving a single singular value σ2 = ∑i|hi|2 = H2. Throughout this work, we will use H2(ω) (or its normalized form, H2/N) as a measure of the collective frequency response of multi-agent systems. From Eq. 4, one can see that if u(t) is taken to be a white-noise stochastic perturbation, the fluctuations in the collective will be correlated as 〈XX〉 ∝ H2.

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