--- title: Open Dynamic Multi-Agent Systems url: https://www.emergentmind.com/topics/open-dynamic-multi-agent-systems type: topic --- # Open Dynamic Multi-Agent Systems Open dynamic multi-agent systems are multi-agent systems in which the relevant system constituents change during execution. In the classical open-systems control literature, openness means that membership \(N(t)\) and population size \(n(t)=|N(t)|\) evolve with time, so the state dimension itself changes while interaction proceeds [1709.05142]. In multi-agent reinforcement learning, openness is broadened to include agent openness, task openness, and type openness, so not only the participant set but also the task structure and agent capabilities, preferences, or goals may vary over time [2510.27659]. Across these formulations, openness turns coordination, estimation, planning, optimization, and learning into problems on changing state spaces, changing objective sets, and changing interaction structures. ## 1. Semantic range and disciplinary meanings The term *open* is not used uniformly across the literature. In control, distributed averaging, and consensus, it usually denotes systems with agent arrivals and departures. In normative multi-agent systems, it denotes autonomy, potentially conflicting interests, lack of access to internal states, and possible non-compliance. In recent LLM-agent systems work, it may instead denote open-source availability and architectural pluggability. These meanings overlap only partially, and conflating them obscures both modeling assumptions and guarantees [1709.05142][1005.4815][2604.25602]. | Lens | What changes | Representative formulation | |---|---|---| | Population openness | Agents join, leave, or are replaced | \(N(t)\), \(n(t)\) vary [1709.05142] | | Task/type openness | Tasks appear, disappear, evolve; capabilities or goals change | Agent, task, and type openness [2510.27659] | | Normative openness | Agents are autonomous, externally developed, and may violate rules | Power, permission, obligation, sanction [1005.4815] | | Software openness | Components are pluggable and can be hot-swapped at runtime | Open-source, unified Oxy abstraction [2604.25602] | A recurrent misconception is that open dynamic MAS are merely time-varying graphs. The literature is more specific: graph variation is only one aspect, whereas genuine openness changes the active node set, the dimension of the state, the interpretation of collective objectives, or the institutional rules under which agents interact [2604.13628][2001.00435]. ## 2. State-space models and structural dynamics A common formal pattern is to index the system by switching modes whose dimensions depend on the active population. In topology-identification work, an OMAS is represented by a switched system \[ \dot x(t)=f(x(t))+L^{\mathcal T(k)}x(t)+u(t), \] where the active node set \(\mathcal V_{\mathcal T(k)}\) and its size \(N_{\mathcal T(k)}\) depend on the mode, and switches occur when nodes or edges are added or removed or weights change [2604.13628]. In open dynamic consensus on directed networks, the system is a sequence of graphs \[ \mathcal G_d[k]=(\mathcal V[k],\mathcal E[k]), \] with operational subsets \[ \mathcal R[k]=\mathcal V[k]\cap\mathcal V[k+1],\qquad \mathcal A[k]=\mathcal V[k+1]\setminus\mathcal V[k],\qquad \mathcal D[k]=\mathcal V[k]\setminus\mathcal V[k+1], \] so remaining, arriving, and departing nodes are explicit parts of the model [2508.05895]. Because states at consecutive times may live in different Euclidean spaces, two abstractions recur. One is an *open distance* between vectors on different index sets, \[ d(x,y)= \sqrt{ \sum_{v\in V_1\cap V_2}(x^v-y^v)^2 + \sum_{v\in V_1\setminus V_2}(x^v)^2 + \sum_{v\in V_2\setminus V_1}(y^v)^2 }, \] which compares states after zero-padding on missing coordinates [1906.00890]. The other is an affine switching-time reset, \[ x_{\sigma(t_k^+)}(t_k^+)=\Xi_{\sigma(t_k^+),\sigma(t_k^-)}\,x_{\sigma(t_k^-)}(t_k^-)+\Phi_k, \] which captures both dimension variation and impulsive effects at mode switches [2001.00435]. These devices make it possible to pose stability or consensus questions without pretending that the state space is fixed. Dynamic signed-network models extend the same idea to cooperative and antagonistic interactions. There, switching instants may be caused by node arrivals or departures, edge creation or deletion, sign changes, or direction changes, and the resulting system is analyzed on dynamic signed graphs or signed digraphs rather than unsigned interaction graphs [2504.21443][2602.21738]. ## 3. Gossip, consensus, and statistical steady behavior The most analytically explicit open-MAS model in the literature is randomized gossip with arrivals, departures, and replacements. In the discrete-time model, at each event time \(t\), agents hold scalar states \(x_i(t)\in\mathbb R\), and one of three events occurs: gossip, departure, or arrival. A gossip step picks \(i,j\in N(t)\) uniformly at random and performs \[ x_i(t+1)=x_j(t+1)=\frac{x_i(t)+x_j(t)}{2}, \] while uninvolved agents keep their states; arrivals draw a new initial state from a distribution \(D\) with mean \(0\) and variance \(\sigma^2\); departures remove one uniformly random agent [1709.05142]. Because exact convergence is generally destroyed by membership changes, the analysis shifts to normalized descriptors: \[ \bar x=\frac1n\sum_{i\in N}x_i,\qquad \overline{x^2}=\frac1n\sum_{i\in N}x_i^2,\qquad \mathrm{Var}(x)=\overline{x^2}-\bar x^2, \] and especially to \[ X=\begin{pmatrix}\bar x^2\\ \overline{x^2}\end{pmatrix}. \] For fixed size \(n\), gossip yields \[ E X' = A_g E X,\qquad A_g=\begin{pmatrix} 1 & 0 \\ \frac1n & 1-\frac1n \end{pmatrix}, \] so \[ E\,\mathrm{Var}(x')=\left(1-\frac1n\right)E\,\mathrm{Var}(x). \] Arrival, departure, and replacement have exact affine or linear updates on the same two descriptors, and in the fixed-size replacement model this closes to a \(2\)-dimensional linear recursion with a unique fixed point [1709.05142]. In the continuous-time version, random asynchronous gossip, arrivals, departures, and replacements are modeled as Poisson events, and the expected descriptor dynamics again close on a fixed-size system. For fixed population size with gossip rate \(n\lambda_g\) and replacement rate \(n\lambda_r\), \[ \frac{d}{dt}E X(t) = \begin{pmatrix} -2\lambda_r & \frac{\lambda_r}{n}\\ \lambda_g & -(\lambda_g+\lambda_r) \end{pmatrix}E X(t) +\lambda_r \begin{pmatrix}\frac1n\\1\end{pmatrix}\sigma^2, \] which yields the stationary disagreement level \[ E\operatorname{Var}(x)\big|_{eq} = \frac{1-\frac1n}{1+\rho\left(1-\frac{1}{2n}\right)}\sigma^2, \qquad \rho=\frac{\lambda_g}{\lambda_r}. \] Thus openness does not merely slow consensus; it changes the asymptotic object from a consensus point to a nonzero statistical equilibrium [2009.02970]. Other aggregation tasks require explicit forgetting or handoff mechanisms. For MAX-consensus, the central difficulty is that the departing agent may hold the current maximum, so obsolete maxima must be forgotten. One algorithm uses counters when departing agents can send a last message; another uses age variables and threshold-based resets when departures are silent [1709.05793]. For quantized average consensus on open dynamic directed networks, a departing node must transfer its current mass minus its own original contribution, and successful convergence is characterized by a necessary-and-sufficient condition: \[ |\mathcal N_j^{+}[k]\cap \mathcal R[k]|\ge 1,\qquad \forall v_j\in\mathcal D[k],\ \forall k, \] together with eventual stabilization of the active set and \(T\)-joint strong connectivity afterward [2508.05895]. ## 4. Stability, synchronization, and topology reconstruction When convergence of raw states is not the right objective, stability is often posed relative to a moving equilibrium. In discrete-time OMAS, the *Trajectory of Points of Interest* \(x_k^e\) is defined as the unique fixed point of the frozen-time update at time \(k\), and open stability requires bounded normalized deviation \[ \frac{1}{\sqrt{n_k}}\|x_k-x_k^e\| \] outside a radius determined by three quantities: contraction in the closed dynamics, bounded variation of the point of interest, and bounded error introduced by joining agents [1906.00890]. A broader route models open MAS consensus errors as multi-dimensional switched systems with varying subsystem dimension. In that framework, the disagreement dynamics after suitable transformation form an \(M^3D\) system with state resets \(x^+=\Xi x^-+\Phi\), and slow/fast transition-dependent average dwell time separates stable from unstable modes. The resulting criteria yield global uniform practical stability under non-vanishing switching impulses and global uniform asymptotic stability when the impulses vanish with the state [2001.00435]. This is particularly relevant for open MAS over directed graphs, because disconnected active digraphs induce non-Hurwitz disagreement modes rather than simply invalidating the model. For signed interactions, open systems exhibit richer limiting behaviors than ordinary consensus. On dynamic signed graphs, new agents can join, new edges can appear, and interaction types can switch between cooperative and antagonistic. Using signed edge-based agreement and strict Lyapunov functions for signed edge-Laplacian matrices with multiple zero eigenvalues, the literature establishes global asymptotic stability or practical stability of edge errors under transition-dependent average dwell time. The final behavior depends on structural balance: structurally balanced final graphs yield bipartite consensus, whereas structurally unbalanced final graphs can yield convergence to zero; with dynamic signed digraphs and multiple leader groups, the same framework generalizes to trivial consensus, bipartite consensus, interval consensus, and containment [2504.21443][2602.21738]. Open dynamic MAS also create an identification problem: short dwell times make segment-wise topology estimation unreliable. In fast-switching OMAS, one solution is to estimate local least-squares operators on each interval, \[ \hat L^s=Z_sY_s^\dagger,\qquad P_s=Y_sY_s^\dagger, \] then compare intervals through the projection-based dissimilarity \[ d(s,r)=\|\hat L^sP_r-\hat L^rP_r\|_F+\|\hat L^rP_s-\hat L^sP_s\|_F, \] cluster repeated occurrences of the same mode, and aggregate the sufficient statistics across all intervals in a cluster. This shifts identifiability from a per-interval dwell-time requirement to an aggregate excitation requirement across repeated visits to the same mode [2604.13628]. ## 5. Optimization, planning, and learning under openness In distributed optimization, openness changes both topology and objective composition. Over directed OMAS, departures may fragment the network into multiple strongly connected clusters, each minimizing \[ f_k^q(x)=\sum_{v_j\in\mathcal V_k^q} f_{j,k}(x). \] OPEN-GT addresses this by combining acknowledgement-based detection of active out-neighbors with a max-consensus mechanism that propagates departure information through each cluster. The push-sum and gradient-tracking variables are updated by \[ x_{j,k+1} = \sum_{v_i \in \mathcal{N}_{j,k}^{\mathrm{in}+}} c_{ji,k} (x_{i,k} -\gamma w_{i,k}),\qquad y_{j,k+1} = \sum_{v_i \in \mathcal{N}_{j,k}^{\mathrm{in}+}} c_{ji,k} y_{i,k}, \] \[ z_{j,k+1}=x_{j,k+1}/y_{j,k+1},\qquad w_{j,k+1}=g_{j,k+1}+(1-h_{j,k})\!\left(\sum_{v_i \in \mathcal{N}_{j,k}^{\mathrm{in}+}} c_{ji,k} w_{i,k}-g_{j,k}\right), \] and if the network eventually stops changing, each formed cluster converges to its own cluster-wide optimum for sufficiently small \(\gamma\) [2508.20715]. Decision-theoretic planning faces a different form of openness: an agent must reason not only about others’ actions but about whether they are present at all. In open and typed many-agent systems without communication, one formulation places each other agent’s presence variable inside its mental model rather than in the global state, and uses frame-action anonymity to extrapolate from a small modeled subset of peers to the behavior of large typed populations. The resulting I-POMCP\(_\mathcal O\) planner is designed for temporary absence and reentry of known agents; the paper is explicit that genuinely new unknown agents do not enter this formulation [1911.08642]. In MARL, openness is directly tied to credit assignment. Agent turnover or absence breaks the fixed-agent-set assumption; task turnover or absence breaks stationarity and reward stability; capability, preference, or goal change breaks consistent action-outcome mapping. Empirically, representative temporal and structural algorithms such as DQN and MAPPO exhibit unstable loss functions, slower convergence, degraded coordination, and reduced reward under openness, and the study emphasizes that many current evaluations are still only bounded approximations of openness because they rely on padding and action masking [2510.27659]. Bandit learning in general open systems pushes the same point into regret analysis. With heterogeneous rewards and general arrival patterns, the benchmark arm can change because the active population changes. The literature introduces the pre-training error \[ P_m=\max_{i\in[K]} |\hat\mu_i^m(T_m^a-1)-\mu_m^i|, \qquad D_m=\frac{1}{P_m}, \] the stability indicator \[ S_t=\mathbf 1\{i_t^\star=i_{t-1}^\star\}, \] and global dynamic regret against the current population’s optimal arm. The principal upper bound is \[ R_T \le O\!\left(M_0\log T+\sum_{t=1}^T\sum_{m\in\mathcal A_t}P_m\right) = O\!\left(M_0\log T+\sum_{t=1}^T\frac{|\mathcal A_t|}{D_t}\right), \] with matching lower bounds in hard instances. The result isolates three separate sources of difficulty: turnover, transferability of arriving agents, and persistence of the globally optimal arm [2605.06202]. ## 6. Architectures, graph construction, and institutional adaptation Recent systems work uses the term *open* in a more architectural sense. OxyGent is an open-source framework in which agents, tools, LLMs, and reasoning flows are wrapped as pluggable Oxy nodes with a unified lifecycle, hierarchical data scopes, and permission-driven dynamic planning. The execution trajectory is generated at runtime from authorization relationships rather than from a static DAG, while AOP instrumentation supports non-intrusive monitoring and OxyBank closes the loop from online traces to annotation and joint evolution [2604.25602]. DRAMA likewise targets changing environments through a control-plane/worker-plane architecture, a unified resource abstraction for both agents and tasks, heartbeat-based monitoring, event-triggered reassignment, and explicit takeover of unfinished tasks when agents depart or become unavailable [2508.04332]. These systems are dynamic and open in a software-orchestration sense, but this should not be confused with the classical open-population semantics of arrivals and departures in distributed control. A related infrastructural question is how a usable interaction structure is formed in the first place. DIGCA treats interaction-graph construction as a distributed problem for dynamic DCOPs. New agents discover insertion points through `Announce` and `AnnounceResponse`, choose one candidate by a local selection function, and preserve acyclicity through the ordering rule \[ \phi_i(j)=\mathsf{TRUE}\iff i