2 instead of choosing a random group to fragment we al- ways choose the largest group. At each time step with probability p a pair of units is selected and joined to- gether, whereas with probability 1 − p the largest cluster is found and fragmented. As in the original model of market dynamics [5, 6], in the system of N units any randomly selected pair can join each other in the aggre- gation event, whereas all the links between pairs within the cluster are removed when the cluster is destroyed. In terms of topology (see later) this situation is represented by a random graph with an annealed link structure. The effects of a fixed and evolving network underlying the aggregation processes will be studied in section III. the number of casualties inflicted by a force of strength s is proportional to s . In these models the parameter p is an external fixed parameter, however, in the real dynam- ics this probability might emerge from another stochastic process, inherently related to the main dynamics. One would expect that the stochastic processes would be quite different in the market and conflict dynamical systems. The empirical studies of war and terrorist actions demonstrate [7–10] that the power-law behaviour is non- universal , with the exponent τ depending on the type of conflict (insurgence, guerrilla, terrorist, etc...) and on the geographical location and period of time consid- ered. In contrast the original model with the coagulation- fragmentation of random clusters always leads to a uni- versal exponent τ = 5 / 2. Thus the EZ model is not able to capture the underlying mechanisms which lead to the variety of observed power-laws. Moreover, the full fragmentation of clusters, which is suitable for herding in market dynamics, might not be appropriate for the conflicts. In this paper we study new models of conflicts with the cluster dynamics on networks when only the largest cluster can be involved in an action. This mimics the sit- uations in which an attack is planned by a central body, and the largest attack force is used, and where the units interact within a social or technological network. Follow- ing the action, the network is either reset back to the original structure (fixed network model), or evolves by adding new links between units involved in the action (emergent network model). The evolving network model provides a mechanism for understanding the power-law exponent dependence on the evolution time, similar to the empirical data. Furthermore, in our simulations we analyse the temporal behavior of each unit, which is be- hind the emergent power-laws. This allows us to unravel the changing nature of the group actions and monitor emergent structure of their connections. For comparison, we also study the cluster dynamics with maximal cluster fragmentation on an annealed random graph structure, which is closer to the original model and the situation in real market dynamics. In Section II we introduce the microscopic rules of the model and present the simulations on the annealed random graph, the situation which parallels the origi- nal coagulation-fragmentation model of market crashes. We also define and compute the temporal quantities that can be monitored in real data. In section III we present simulation results corresponding to the cluster dynamics on a network with fixed and evolving network topology. Finally, in section IV we give a brief summary of the re- sults and a discussion, in particular on the mechanisms revealed within our model, which might drive the evolu- tion of modern conflicts. In each case, the size of the largest (fragmented) cluster starts playing a dominant role over a threshold s$_{0}$ (max- imum of the fragmented size distribution), which moves upward with reduced fragmentation probability. The slope τ of the cluster-size distribution over the threshold is fully determined by the distribution of the fragmented cluster size and it is continuously varying with the proba- bility p . With the reduced fragmentation probability (in- creasing aggregation probability p ) the slope of the size distribution reduces and converges towards the universal curve obtained in the random cluster fragmentation (top line in Fig. 1d). Intuitively, this is understandable since, for vanishingly small but finite probability of crush, a very large clusters may occur in the system increasing the probability to be selected by random picking unit. However, the nature of correlations that the system de- velops over time reveals that these are different processes. The time fluctuations in the number of clusters and their In this section we consider a minimal extension of the original aggregation–fragmentation model [5, 6], in which Starting with N = 1000 units, we apply the rules of random aggregation and fragmentation events, as de- scribed above. If we always select the largest cluster for fragmentation, instead of a randomly selected cluster, it affects the system dynamics in a different ways, depend- ing on potential size that the cluster can grow, which, on the other hand is determined by the fragmentation probability 1 − p . For a sizable probability of fragmenta- tion, the system has no large clusters, which causes the probability distribution of cluster sizes to decay rapidly. The distribution of all sizes of clusters and the size of the fragmented cluster is shown in Fig. 1d for several values of the probability 1 − p = 10%, 5%, and 2%. It exhibits a power-law tail for sizes above a threshold value s > s$_{0}$ with the scaling exponent τ defined by the expression P ( s ) ∼ s − $^{τ}$, s > s$_{0}$ . (2) A. Cluster size statistics and temporal structure II. CLUSTER-AGGREGATION AND LARGEST CLUSTER FRAGMENTATION WITH AN ANNEALED RANDOM GRAPH TOPOLOGY