Modelling conflicts with cluster dynamics on networks Bosiljka Tadi´c 1 and G.J.Rodgers 2 $^{1}$Department for Theoretical Physics; Joˇzef Stefan Institute; P.O. Box 3000; SI-1001 Ljubljana; Slovenia, $^{2}$Department of Mathematical Sciences; Brunel University; Uxbridge Middlesex UB8 3PH United Kingdom We introduce cluster dynamical models of conflicts in which only the largest cluster can be involved in an action. This mimics the situations in which an attack is planned by a central body, and the largest attack force is used. We study the model in its annealed random graph version, on a fixed network, and on a network evolving through the actions. The sizes of actions are distributed with a power-law tail, however, the exponent is non-universal and depends on the frequency of actions and sparseness of the available connections between units. Allowing the network reconstruction over time in a self-organized manner, e.g., by adding the links based on previous liaisons between units, we find that the power-law exponent depends on the evolution time of the network. Its lower limit is given by the universal value 5/2, derived analytically for the case of random fragmentation processes. In the temporal patterns behind the size of actions we find long-range correlations in the time series of number of clusters and non-trivial distribution of time that a unit waits between two actions. In the case of an evolving network the distribution develops a power-law tail, indicating that through the repeated actions, the system develops internal structure which is not just more effective in terms of the size of events, but also has a full hierarchy of units. PACS numbers: 89.75.-k; 05.65.+b; 89.20.-a The study of coagulation-fragmentation models in physics have a long history, and have been used to ex- plain a variety of physical phenomena, including the for- mation of aerosols, colloidal aggregates, polymers and celestial bodies [1]. More recently the basic theory, orig- inally developed by Smoluchowski [2] and Becker-Doring [3] has been adapted to model group behaviour in human- dynamics. The first microscopic model of herding in fi- nancial markets consisted of randomly connected agents [4]. In this model agents are connected with probability a and disconnected with probability 1 − a . Agents that are in the same group share information and make the same decisions in the market. The parameter a was tuned to the percolation threshold to obtain a power-law distribu- tion of group sizes and hence a power-law distribution of returns. Inspired by this, in 2000 Eguiluz and Zimmermann [5] (EZ) introduced a coagulation-fragmentation model of herding in financial markets. The EZ model is a ki- netic version of the model in [4], in which the group sizes emerge more naturally in the limit of large time. At each time step with probability p an edge is introduced be- tween two randomly selected agents or with probability 1 − p the group of a randomly selected agent is frag- mented. In [6] Dhulst and Rodgers solved this model exactly and showed that in the limit t → ∞ the system evolves to a stationary cluster size distribution with the number of clusters of size s , n$_{s}$ , is given by limit p → 1 this result can be expanded for large s to give a power-law with n$_{s}$ ∼ s − τ and τ = 5 / 2. This basic model has been used [7] to explain the ap- parent ubiquity of the exponent τ = 5 / 2 in the data from modern insurgent warfare. Many authors have studied wars and conflicts empirically, and many have concluded that the distribution of daily casualties, or of casualties per attack, has a power-law distribution [8, 9]. A wide range of exponents have been reported. In [10] an ex- ponent of τ = 1 . 8 is reported for the intensity of 119 old wars between 1816 and 1980. Casualty numbers in global terrorist attacks, since 1968, have τ = 1 . 7 for G7 countries and τ = 2 . 5 for non-G7 countries [11]. More recently, in [7] the daily data from killings and injuries in Columbia and civilian casualties in Iraq were examined. The total data sets gave good power-law distributions with τ = 3 for Columbia and τ = 2 for Iraq. By reducing the time window within these data sets, and then sliding the time window forward, the authors of [7] were able to calculate a series for τ as a function of time. This revealed that as time window is shifted towards recent time the exponent τ in both the Columbian and Iraqi datasets was tending to the value τ = 5 / 2. This caused the authors of [7] to develop a new model of modern in- surgent warfare based on a dynamical model of herding introduced in [5, 6]. In financial markets, the return is equal to the differ- ence in the number of buyers and sellers at a particular time, or alternatively to the supply and demand balance at a particular time [12]. Hence in the models of finan- cial markets the probability of having a return of size s is proportional to sn$_{s}$ . In models of wars and conflicts the units can be thought of as troops, weapons or equip- ment and it is an assumption in these models [7] that n$_{s}$ = p s − $^{1}$(2 s − 2)! ( p + 1) 2 s − $^{1}$s ! $_{2}$N (1) where N is the number of agents in the system. In the arXiv:1001.3012v1 [physics.soc-ph] 18 Jan 2010 I. INTRODUCTION