# Joint Optimization of Monitor Location and Network Anomaly Detection

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4 flow conservation constraints 3 are, hence, expressed as follows: i j OUT (v) X i j (n,n ) i j IN(v) W (n,n ) iff v = n W (n,n ) iff v = n otherwise X i j (n,n ) = ; v,n,n N (9) 3) The following constraints ensure that the end nodes of paths carrying monitoring flows are selected as monitors: Y n W (n,l) +W (l,n) ; n N, l E (1) 4) Toward preventing looping flows, we define a set of integer variables {H (n,n )(i);n,n,i N}. H (n,n )(i) specifies the number of hops separating node i visited by a flow travelling between the pair of nodes (n,n ) from its originating node n. The idea is to force the flows to travel through nodes in an ascending order of the values of their hop variables, which prevents them from looping. We formulate the looping constraint as follows: H (n,n )(n) = ; (n,n ) N 2 (11) 1 X i j (n,n ) + H (n,n ) ( j) 1 H (n,n ) (i) K 1 X j i (n,n ) + H (n,n ) (i) 1 H (n,n ) ( j) ; K (i, j) E,(n,n ) N 2 (12) H (n,n )(n ) N 1; (n,n ) N 2 (13) Constraints (11) assign the value to the hop variable of the originating node of each path, whereas constraints (13) set the upper bound of the flow lengths to the number of network nodes. Constraints (12) guarantee that flows do not re-visit an already visited node, i.e. a node having a value of hop variable lower than the values of those of visited nodes. C. Extensions The proposed ILP formulations are expressed considering active monitoring systems. The following adaptations should be introduced in order to meet passive monitoring requirements: The set of paths given as input in the path based ILP formulation is replaced by a set of real traffic flows that are candidate to be monitored. 3 The flow that enters a node leaves it except if it is the originating node, in which case the flow only exits; or the terminating node, in which case the flow only enters The link monitoring costs are set to, for passive monitoring techniques proceed by snooping on real traffic flows, and hence do not burden network links. Subsequently, the first components of our cost functions are zero. We might need to constrain some variables to be zero in the second formulation in order to map to real traffic flows that cross the network. For instance, if there are no flows travelling between the pair of nodes (n i, n j ) and crossing link l k, then X lk (n i,n j ) shall be zero. We note that the two formulations can be easily extended to take into account other monitoring constraints. Namely, limiting the monitoring capacities of monitors ( i.e. the maximum number of flows that can be monitored simultaneously by a given monitor), bounding the lenghts of monitored paths, bounding the number of monitored paths, etc. III. EVALUATION In this section, we present our evaluation methodology, metrics, and simulation results. A. Methodology and Metrics We evaluated our ILPs using Cplex11.2 [12] running on a PC equipped with an Intel(R) Core(TM)2 Duo processor, a clock rate of 2, MHz, and 3.9 GB of RAM. All results are the mean over 2 simulations on random topologies generated using the topology generator BRITE (AS level, Waxman model) [11]. Table I depicts a summary of the main characteristics of the topologies considered in our evaluation. We devised and implemented an algorithm that computes the set of paths of an input topology. As we have anticipated owing to the complexity of the problem, we failed to compute the path set for the topologies with 12 nodes and 41 links due to memory failure. We considered an active monitoring scenario where all the network paths are candidate to be monitored and all the nodes are candidate to hold monitors, and we assumed that all the nodes are equidistant from the NOC. The values of Cl l and Cm n are set to 1 l E and n N, respectively. TABLE I SUMMARY OF THE TOPOLOGIES CONSIDERED IN THE EVALUATION Topology # of nodes # of links # of paths TOP(6, 1) TOP(8, 18) TOP(1, 31) TOP(12, 41) * The aim of the evaluation is to compare the performance of the two ILP formulations, and evaluate the trade-off between the monitoring optimization objectives described in section II-B and their impact on the quality of the monitoring solution. For this purpose, we considered the following metrics:

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