Hierarchical Numbering Based Addressing and Stateless Routing Scheme for Wireless Sensor Networks Due to energy and other relevant constraints, addressing of nodes and data routing techniques in sensor networks differ significantly from other networks. In this article, we present an energy-efficient addressing and stateless routing paradigm for wireless sensor networks. We propose a dynamic and globally unique address allocation scheme for sensors in such a way that these addresses can later be used for data routing. We build a tree like organization of sensors rooted by the sink node based on their transmission adjacency and then set labels on each sensor with a number according to the preorder traversal of the tree from the root. In this addressing process, each sensor keeps necessary information so that they can later route data packets to the destination depending on these addresses, without keeping the large routing table and running any no route discovery phase. Moreover, the scheme does not use location information as well (as done by geo-routing) and can be used in the indoor environment. We conduct simulations to measure the soundness of our approach and make a comparison with another similar technique TreeCast. Simulation results reveal that our approach performs better than its counterpart in several important performance metrics like address length and communication energy. A(p): Address of the forwarder node of the query. Initially it is sink's address. sub(p): Subordinate number of the forwarder node of the query packet. aux(p): Auxiliary number of the forwarder node of the query packet. d(p): Depth of the forwarder node. query(p): Query data of interest.Query packet contains neither the sender nor the destination address, rather it contains the address of the intermittent node that passes the packet onward. The sink is the sender and all nodes in the network are the receiver.Below we discuss the routing rules of HN-Routing for the above three types of packets. A wireless sensor network (WSN) is a collection of a large number battery-powered, lowcost sensing devices, commonly known as sensors, which collaborate to perform some distributed sensing task in an environment. These sensors have sensing equipment, a central processing unit, power supply, and a radio transceiver to communicate among themselves. Because of the advancement in microelectronics, fabrication of these small sensors with computation and communication units has become technically and economically feasible. A sensor node senses an event or measures an ambient condition of the environment (e.g., temperature) in its vicinity, and transmits the event data to a remote base-station that ultimately makes the necessary aggregated interpretation of the events.. Though an individual sensor can do very little of its task, a large number of sensors can be coordinated into an organized network to perform a big sensing task. Sensors are normally battery-powered and are by born constrained by energy. To limit energy, they do not transmit the signal to a very long distance to reach the remote sink, rather they form a multihop communication scenario with their short range radio; transmitting data each time to the immediate neighbors and thus fusing it into the entire network. Figure 1 shows a typical sensor network. A good overview of sensor networks is presented in [25] . Sensor networks have been investigated for tremendous applications in home and industries, namely, environment monitoring and forecasting, flood and landslide detection, vehicle tracking and detection, habitat monitoring, and civil or military security and surveillance [32] .. Data routing problem in sensor network has received a great deal of attention in the last few years. Sensor network differs from the conventional network like the Internet in three manners. First, sensor nodes run on feeble hardware that does not allow them to store large routing tables as required by classical routing protocols like distance-vector or link-state routing. Second, sensor nodes are deployed in a large quantity and their identifiers are not topologically ordered. For this, like the traditional IP network the sensor network cannot achieve scalability by address aggregation (by sharing high-order bits as network address) that allows the number of routes to be far fewer than the number of total nodes. Third, sensor networks are highly dynamic-the failure of nodes is quite frequent and the topology changing rate is much higher than that of conventional networks. Distance vector or link state routing requires lots of message exchanges to restore routes in case of node or link failure.. In order to make scalable routing sensor networks, the research community has suggested an array of routing protocols in two major forms: on-demand routing and geographic routing. On-demand routing does not run a periodic exchange of route information, instead it establishes a route whenever required by flooding the route request to the network. On-demand routing reduces the number of route entry per node and can track topology changes by less message exchanges. Initially proposed by Johnson et al. in [13] , the scheme undertook several variations and improvements [14, 15, 24] . The technique works well with a moderate network size, but for a large system route discovery overhead becomes significant [16] . There is an another approach called geographic routing 1 that does not require any route discovery phase. In geo-routing, nodes do not have any routing tables at all, instead they know their locations (in the form of coordinates in Euclidean space, or whatever suitable) and forward data packet based on the geographic location of the destination node. The geo-routing scheme uses the node's location as their addresses, and forwards packets (when possible) in a greedy manner toward the destination. The most widely known proposal is GFG/GPSR [3, 17] . Each node knows its coordinate as well as the coordinates of its neighbors. When a node receives a message containing the coordinate of the destination, it forwards the message to the ''best'' neighbor that is geometrically closest (may be having the minimum Euclidean distance from the destination) to the destination. This greedy approach fails when no better node in the neighborhood with respect to the distance to the destination is obtained. This situation is called ''dead-end,'' and lots of suggestions have been made to handle this. Several geo-routing schemes are found in [7, 8, 18, 20, 21] .. A problem with geographic routing is the availability of location information for all sensors. One proposed solution is to equip all sensors with GPS (Global Positioning System) receivers. However, in comparison to a sensor node, a GPS receiver may be clumsy and expensive, and can be possible for outdoor deployment only. GPS reception might be obstructed by dust or climate condition and for indoor deployment there will be no such reception at all. In [26] , a novel approach is suggested. A small fraction of nodes are designated as perimeter nodes that know their location, whereas the internal nodes compute their position by some localization process (repeatedly taking average of the locations of the neighbors). Another approach in [31] introduces some virtual coordinates based on connectivity or distance information and employ geo-routing on those coordinates. Virtual coordinates do not correspond to real locations of the sensors, rather they describe some relative positions (e.g., hop counts) of the sensors with respect to a set of predefined anchor nodes. In a virtual coordinate scheme, nodes no longer require real location information, but geo-routing is still possible.. Due to the specific application requirements, the routing problem for sensor networks has got a new direction in recent years. In has been widely observed that unlike the traditional network (like the Internet), in sensor application sensor nodes do not pass data toward the specific sensors by their addresses, instead they use some attributes (e.g., type of events, data of events) as the packet destination. This is called data centric routing. Data centric routing is based on the key fact that data itself is more important than the identity of the sender. For example, there might not be any query asking-''get reading from the sensor with Id = 152;'' instead there can be something-''get reading from sensors that read temperature >60 C.'' In data centric routing, nodes express their interests for specific data described by a set of keys (attributes) and the interested/eligible sensors participate in data delivery. Usually the route is discovered at the time of flooding of interests. SPIN (and its variants) [11] , directed diffusion [12] are some data centric routing schemes to name a few.. Geographic routing (location based) can be combined with data centric routing (attribute based) by a technique called geographic hash table (GHT) proposed in [27] . In GHT, a hashing function is devised that maps a key (based on certain attribute) to a geographic location, and data generated with a certain key are replicated in those sensor nodes that are near the mapped geographic location from that key.. In our work, we propose a new type of geo-routing paradigm. Our scheme uses neither location information nor virtual coordinates for geo-routing. In our observation, the virtual coordinate of a node as proposed in [31] in the form of a vector containing the hop counts from a set of designated nodes (called anchors) becomes pretty long for a moderate size network. For example a network of 400 nodes if modeled as a unit disk grid tree requires at least 5 anchors. 2 If the hop distance is expressed by a 4-bit number, the address length becomes 20 bit-quite long for a 400-node network that should require somewhat near 9-10 bits optimally. Moreover, finding anchor nodes for an arbitrary topology of network is another challenge. In our proposed scheme, instead of finding the location (real or virtual), we find an ordering of nodes in the network by assigning some numbers to them such that the geometric routing can be possible based on those assigned numbers. We name the scheme as hierarchical numbering of nodes, and hence the routing scheme is termed as HN-Routing. The basic idea is to organize the nodes in a spanning tree with the sink or base station as the root, and then to assign a number to the nodes according to the order of preorder traversal from the sink. This number forms the pseudo-coordinate of nodes and is used in the routing decision.. HN-Routing uses an address compaction scheme based on the interval routing technique presented in [28] . In contrast to the interval routing, we use intervals for nodes instead of using intervals for each link/edge, since an individual link is hardly significant in a wireless environment. In [28] , a rigorous theoretical study of the interval routing is made for the wired network, but its application in wireless networks was not sought. We improvise the technique for wireless network by introducing hierarchical labels for nodes and make it function in the presence of node failures. Moreover, the unique labels (number) on nodes assigned by HN-Routing can be used as network-layer address (like IP address) if the sensor nodes are connected to the traditional TCP/IP networks. Recently this sort of connectivity is becoming popular [5, 6] .. We organize the article as follows. We give a list of related work in Section 2. We describe our proposed scheme HN-routing in Section 3 followed by Section 4 that explains the routing rules of HN-routing. Then we produce the simulation results in Section 5. Section 6 concludes the article. The routing problem for sensor networks is one of the most prominent research work that has been conducted for the last few decades. Lots of routing techniques are found in the literature. In general, routing protocols in sensor networks can be divided into three major classes: flat-based routing, hierarchical-based routing, and location-based routing. In flat routing all nodes play the same role and they compute routes by exchanging messages among them. Hierarchical routing creates node hierarchy and organizes them into some clusters. Location based routing avoids classical address based routing, and uses location information of nodes for packet routing. In location based routing the destination is not specified by an address but by the location of the destination. Another type of routing that is studied extensively in recent years is query-based or data centric routing. A good number of routing protocols for sensor networks are described in [1] .. A well-known routing technique for sensor network is location-based routing that is usually called geo-routing. In a location-based routing scheme, nodes are addressed by their locations. A few proposals of geo-routing were of pure greed. A source node or any intermediate node selects one of the neighbors according to some criteria. Some notable methods are MFR (Most Forward within Radius), GEDIR (Geographic Distance Routing), and DIR (compass routing). GEDIR [30] algorithm moves the packet to the neighbor of the current node whose distance from the destination is minimum. This simple greedy method works well, but sometimes it fails when a node does not find any neighbor that is nearer to the 2 For a unit disk tree in grid of size n nodes, it requires at least ð ffiffi n p þ9. 12 Þ 2 anchors [31] . destination. This situation is called ''dead-end.'' On the other hand, Compass routing selects the neighbor with the minimum angular distance from the destination. Again, GPSR (Greedy Perimeter Stateless Routing) uses planar graphs to route packet. The packets follow the perimeter of the graph to find their route to the destination. The Greedy Face Routing (GFR) [30] is a technique where packets are forwarded along the faces of the planar graph and is the first one that guarantees success if the source and destination are connected. No ''dead-end'' occurs in this case, but the worst-case cost of GFR is proportional to the number of nodes in the network. The routing schemes GFG [30] and GOAFR (The Greedy Other Adaptive Face Routing) [19] combines greedy and face routing together. When a ''dead-end'' is reached by the greedy method, it runs the best neighbor selection procedure that returns the ''best'' neighbor on the boundary. Whereas GFG does not give competitive worst-case guarantees, GOAFR achieves both worst-case optimality and average-case efficiency.. Usually the location of nodes are determined by GPS-receiver by communicating with satellites. However, it is not always possible to assume that every node can have the location information. There are some technical alternatives. One of them suggested in [26] is to designate some nodes as perimeter nodes and let them know of their locations. Other nodes will compute their locations by some local processing, namely repeatedly taking the average over coordinates of the neighbor nodes. Experimental results [26] show that this scheme with estimated locations works better than that of real coordinates in terms of the packet delivery rate.. Introducing virtual coordinates instead of real positions is another solution to the location problem. The coordinates of the nodes are estimated in [4] by solving a convex linear program, whereas [29] uses multidimensional scaling based on some heuristic. In [22] , authors present an approximation algorithm for computing the virtual coordinates. The work presented in [31] proposes a new type of virtual coordinate for a node u as a vector that is composed by the graph (hop) distances from u to a set of designated nodes in the networks. These virtual coordinates act as the location for the nodes, and routing decisions are taken based on these coordinates. This pseudo-geometric routing is not restricted to planar unit disk graph anymore, but succeeds on general networks.. The HN-Routing presented in this paper is motivated by a functional concept that appears in the interval routing presented in [28] . The goal of interval routing is to find a labeling of the vertices of a graph with the integers 1,2,. . .n such that for every vertex u of the graph, we can assign a disjoint interval [p,q] to every edge e incident to u with the property that if V in [p,q], there exists a route from u to v through e. Figure 2 shows such a labeling. Though the technique drastically reduces the number of routing entries per node and works well for fixed wired network, its application to the dynamic network is questioned, because the entire labels on vertices and intervals on edges are to be recomputed in case of node failure or any topology change.. At this point, we refer to a routing architecture TreeCast [23] which has some resemblance to the current work. TreeCast is a stateless addressing and routing architecture for sensor network that sets labels on nodes and then takes routing decisions based on these addresses. In TreeCast, nodes are organized in a tree structure rooted by the sink and are assigned addresses according to their position in the tree. Child nodes lie within the transmission range of the parent and the parent node assigns addresses to its descendant children. Child nodes randomly generate their own addresses, get the address approved by the parent, and if approved the node builds its own address by appending the parent address with its own address. Figure 3 shows the address allocation by TreeCast.. The addresses assigned by TreeCast not only uniquely identify nodes in the network, but also support data routing depending on addresses. Suppose in Fig. 3

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Discriminative Cue Integration for Medical Image Annotation Abstract. Automatic annotation of medical images is an increasingly important tool for physicians in their daily activity. Hospitals produce nowadays an increasing amount of data. Manual annotation is very costly and prone to human mistakes. This paper proposes a multi-cue approach to automatic medical image annotation. We represent images using global and local features. These cues are then combined together using three alternative approaches, a high-level, a mid-level and a low-level fusion scheme, all based on the Support Vector Machines (SVM) algorithm. We tested our methods on the IRMA database, and with the mid-and high-level integration scheme we did participate to the 2007 ImageCLEFmed benchmark evaluation, in the medical image annotation track. These algorithms ranked first and fifth respectively among all submission. Experiments using the low-level integration scheme also confirm the power of cue integration for this task. The amount of medical image data produced nowadays is constantly growing, with average-sized radiology departments producing several tera-bites of data annually. The cost of manually annotating these images is very high; furthermore, manual classification induces errors in the tag assignment, which means that a part of the available knowledge is not accessible anymore to physicians [Gueld et al. (2002) ]. This calls for automatic annotation algorithms able to perform the task reliably, and benchmark evaluations are thus extremely useful for boosting advances in the field. The ImageCLEFmed annotation task has been established in 2005, and in 2007 it has provided participants with 11000 training and development data, spread across 116 classes. The task consisted in assigning the correct label to 1000 test images. For further informations on the annotation task of ImageCLEF 2007 we refer the reader to ].. An open challenge for automatic annotation of medical images is the intrinsically very high interclass versus intra-class variability. Exemplar images of the same body part, taken from different individuals, might be quite different visually because of differences in age, or because the images were acquired in different hospitals, with slight variations in the image acquisition modality (see Figure 1 for some examples of visual variability within the body region class 'foot'). This problem is known in the literature as the intra-class variability problem, and it calls for classification algorithms able to generalize well while achieving robustness at the same time. Similarly, exemplar images of different classes might share some visual characteristics, so that they actually look similar. This is for instance the case for the classes 'chest unspecified', 'chest expiration', 'chest inspiration' and 'chest supine' (see Figure 2 for some exemplar images): they must be classified differently because of clinical needs, but they present a strong visual similarity because they all contain the body part 'chest'. This problem is known in the literature as the inter-class variability problem, and it calls for classification algorithms able to use the most discriminative information from the available data.. Several authors have tried to address this problem. State of the art approaches use global texture descriptors as features and discriminative algorithms, mainly SVMs, for the classification step. This approach has been adopted successfully by several groups [Mueller et al. (2006) ; W. Hersh (2006) ; Florea et al. (2006) ]. Another popular approach is the use of local features combined to SVMs through ad-hoc kernels; notable examples are in [Mueller et al. (2006) ; Liu et al. (2006) ] and they confirm the suitability of SVMs for the task, while showing the importance of using adhoc kernel functions. Local and global features, and more generally different types of descriptors, have been used separately or combined together in a multi-cue approach. The rationale behind the use of multiple cues is that these are expected to account for more and diverse data characteristics, thus they should lead to better performance. Several authors tried this strategy, proposing different integration schemes and testing various features [Mueller et al. (2006) ; Guld et al. (2006) ]. Performance was lower than expected: the best result obtained by a multi-cue approach ranked 12th among 28 submissions in the ImageCLEF 2006 benchmark evaluation. Still, years of research on visual recognition in other domains have shown clearly that multiple-cue methods outperform single-feature approaches, provided that the features are complementary and that the integration scheme takes advantage from it.. In this paper we follow this route, and we propose to tackle the inter-class versus intra-class variability problem using a discriminative, SVM-based cue integration approach. We extract from images local and global descriptors, so to capture different kinds of information. The two feature types are combined together using three different SVM-based integration schemes. The first is the Discriminative Accumulation Scheme (DAS), a high-level strategy proposed first in [Nilsback and Caputo (2004) ]. For each feature type, an SVM is trained and its output consists of the distance from the separating hyperplane. Then, the decision function is built as a linear combination of the distances, with weighting coefficients determined via cross validation. The second integration scheme is a mid-level strategy. It consists of designing a new Mercer kernel, able to take as input different feature types for each image data. We call it Multi Cue Kernel (MCK); the main advantage of this approach is that features are selected and weighted during the SVM training, thus the final solution is optimal as it minimizes the structural risk. The third integration scheme is a low-level approach. It. Figure 1: Images from the IRMA database. Intra-class variability within the class annotated as 1121-120-914-700, overview image, AP unspecified, foot, muscolosceletal system.. Figure 2: All these images from the IRMA database have as acquisition modality "high beam energy", as body region "chest unspecified", as biological system "unspecified", but they differ for the body orientation: (a) "PA unspecified", (b) "PA expiration" (c) "AP inspiration", (d) "AP supine".. creates a unique feature vector from the original two by concatenating them, and then uses an SVM for classification. We tested our approaches on the IRMA database used for the ImageCLEFmed 2007 benchmark evaluation, in the medical image annotation track. DAS and MCK were submitted to the benchmark evaluation. They achieved respectively a score of 29.9 and 26.85, ranking fifth (DAS) and first (MCK) among all submissions. The low-level approach, developed after the submission deadline, achieved a score of 26.96, which would have corresponded to ranking second among all submissions. These results clearly prove the power of using multiple cues for the task, and underline the impact of the choice of the fusion strategy.. The rest of the paper is organized as follows: section 2 describes the two types of feature descriptors we used at the single cue stage, and gives a brief review of the theory behind SVMs. Section 3 gives details on the three alternative SVM-based cue integration approaches. Section 4 reports the experimental procedure adopted and the results obtained, with a detailed discussion on the performance of each algorithm. The paper concludes with a summary discussion. This paper presented a discriminative multi-cue approach to medical image annotation. We combined global and local information using three alternative fusion strategies: we combined features together in a unique descriptor, we used the discriminative accumulation scheme proposed first by [Nilsback and Caputo (2004) ] and a new class of kernels, the multi cue kernel, able to take as input different cues while keeping them separated during the optimization process. This last method gave the best (a) (b). Figure 7: These images represent the confusion matrices respectively for (a) SIFT oa, (b) Pixel oa, (c) DAS and (d) MCK oa (e) LL oa. We used a colormap corresponding to the number of images varying from zero to five to let the misclassified images stand out. All the position in the matrices containing five or more images appear dark red (better seen in color).. performance in the ImageCLEF 2007 benchmark evaluation, obtaining a score of 26.85, which ranked first among all submissions. All our cue integration schemes achieved very high scored, proving the effectiveness of our approach. This work can be extended in many ways. First, we would like to use various types of local and global descriptors and add shape features as well, so to select the best features for the task and test the performance of the three integration schemes when the number of cues grows. This, combined with a thorough theoretical and algorithmic analysis of the three methods, should make it possible to understand better their strengths and weaknesses, and to evaluate which of them should be preferred for this applications. Finally, our algorithm does not exploit at the moment the natural hierarchical structure of the data, but we believe that this information is crucial for achieving significant improvements in performance. Future work will explore these directions. As we could expect, for the MCK and DAS integration methods the best SIFT weights (see 4 and 5) turn out to be higher than the raw pixel's ones. In the LL integration approach the mixed features are not relatively weighted, nevertheless the obtained results tells that the two features cooperate well to classify images.. The number of support vectors for the MCK run using one-vs-one multiclass SVM extension (MCK oa) is slightly higher than that used by the single cue SIFT oa but lower than that used by PIXEL oa. For the MCK run using one-vs-one multiclass SVM extension (MCK oo) the number of support vectors is even lower than that of both the single cues SIFT oo and PIXEL oo. These results show that combining two features with the MCK algorithm can simplify the classification problem. Comparing the number of support vectors for the LL cue integration approach with that related to the single cue experiments leads to the same conclusions reached for MCK.. For DAS we counted the support vectors summing the ones from SIFT oa and PIXEL oa but considering only once the support vectors associated with the training images that resulted in common between the single cues. The number of support vector for DAS exceeds that obtained for both MCK oa and MCK oo showing a higher complexity of the classification problem. Table 2 shows in details some examples of classification results. The second, third and fourth column contain examples of images misclassified by one of the two cues or by both, but correctly classified by DAS, LL oa and MCK oa. The fifth column shows an example of an image misclassified by both cues and by DAS but correctly classified by LL oa and MCK oa. The sixth column contains an image misclassified by the two cues, by DAS and LL oa but recognized by MCK oa. It is interesting to note that combining local and global features can be useful to recognize images even if they are extremely partial and compromised by the presence of artifacts that for medical images can be prosthesis or reference labels put on the acquisition screen.. The difference between the single-cue approach and the multi-cue integration approach can be better evaluated considering the confusion matrices. They are shown as images in Figure 7 . We used a colormap corresponding to the number of images varying from zero to five to let the misclassified images stand out. It is clear that single-cue and multi-cue methods differ principally for how the wrong images are labeled. The more matrices present sparse values out of the diagonal and far away from it, the worse the method is.. Finally, trying to do a differential analysis on the three cue integration methods used we can say that:. • the high level integration approach is not able to work at its best if the basic features produce very different hypothesis. We can suppose that this kind of behavior is not due to the Discriminative Table 1 : Ranking of our runs, name, best kernel parameters, best SVM-cost parameter C, best feature's weights, number of support vectors, score, gain respect to the best run of other participants and recognition rate. LL stand for low-level cue combination: this approach has been developed after the CLEF competition. The extensions oo and oa refer respectively to the one-vs-one and one-vs-all SVM multiclass approaches. Accumulation Scheme itself but in general it is a problem connected to combining the output of classifiers working on different cues. Some preliminary experiments using the voting scheme confirm this hypothesis;. • the low and the mid level integration methods combining the cues before and during the classification process seem able to better take advantage of the different information associated to the features. The mid-level approach is more refined and complex than the low-level approach thanks to the possibility to weight the features, and although using two cues its results are similar to that obtained by the low-level, MCK would probably gain in performance increasing the number of cues while the LL would suffer for the growth of the integral-feature dimension. Moreover with MCK it is possible to combine cues of an intrinsically different nature simply choosing for each the right kernel, while it could not always be possible to concatenate them in a single vector. The database for the CLEF medical image annotation task was provided by the IRMA group from the University Hospital of Aachen, Germany. It consists of 11000 fully classified anonymous radiographs taken randomly from medical routine and 1000 radiographs for which the classification labels were not available to the ImageCLEF participants. The images' identity is defined through the IRMA code TTTT-DDD-AAA-BBB, a multiaxial classification method based on the acquisition modality (T=technique), the body orientation (D=direction), the body region (A=anatomy) and the biological system examined (B). This code is strictly hierarchical: each sub-code element to the right is connected to only one code element to the left. The way in which the classification performance is judged depends on the level of the hierarchy at which the error is done. Wrong decisions are more penalized if they are at an early stage than at a later stage in the code. The error count produces a score, the lower it is, the better the annotation is. For further details on the database and the ImageCLEF benchmark evaluation for the medical annotation task, we refer the reader to [Deselaers et al. (2008) ]. The original dataset was divided in two parts: training and validation. In order to obtain reliable results, we merged them together and extracted 5 random and disjoint train/test splits of 10000/1000 images. As a preliminary step we run experiments to find the best kernel parameters through cross validation, the obtained results were then used to run our submission experiments on the 1000 unlabeled images of the challenge test set using all the 11000 images of the original dataset as training. We considered as the best parameters the one giving the lower average score on the 5 splits. Note that, due to the score evaluation method, the best score does not correspond necessarily to the best recognition rate.

O B I O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


An algorithm using length-r paths to approximate subgraph isomorphism Abstract The 'LeRP' algorithm approximates subgraph isomorphism for attributed graphs based on counts of Length-R Paths. The algorithm provides a good approximation to the maximal isomorphic subgraph. The basic approach of the LeRP algorithm differs fundamentally from other methods. When comparing structural similarity LeRP uses a neighborhood of nodes that varies in size dynamically. This approach provides sufficient evidence of similarity to permit LeRP to form a node-to-node mapping and can be computed with polynomial effort in the worst-case. Results from over 32,000 simulated cases are reported. We demonstrate that LeRP does not need a high dynamic range of node and edge coloring to perform well. For example, LeRP can input 50-node and 100-node graphs that contain a common 50-node subgraph, and then compute a matching subgraph having 49.74 +/-0.46 nodes (mean +/-one standard deviation). This takes from 0.4 to 0.5 seconds. In this example, 100 trials were evaluated and graphs had discrete coloring for nodes and edges with a dynamic range of four. Test conditions are varied and include strongly regular graphs as well as Model A. , A 2 , A 3 … A R arrays can be stored with O(NDR) memory, provided a sparse representation is used. Also, a tradeoff between speed and memory is possible in LeRP. Precomputation of the best-beta array can be skipped, and the associated values simply recomputed when needed. This reduces the memory requirements to O(NDR).

O O O O O O O B I O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Balanced decomposition of a vertex-colored graph A balanced vertex-coloring of a graph G is a function c fromIn this paper, the balanced decomposition number f (G) of G is introduced; f (G) is the smallest integer s such that for any balanced vertex-coloring c of G, there exists a balanced decompositionBalanced decomposition numbers of some basic families of graphs such as complete graphs, trees, complete bipartite graphs, cycles, 2-connected graphs are studied. Let G be an undirected graph with no multiple edge or loop. Let V (G) and E(G) be the vertex set and edge set of G. A balanced vertex-coloring of G is a pair (R, B) of subsets R, B ⊆ V (G) such that R ∩ B = ∅ and |R| = |B|. We also say that a balanced vertex-coloring (R, B) has order k if |R| = |B| = k. The sets R, B are regarded as a red set and a blue set. If a vertex v is in R ∪ B, v is called colored, and otherwise, v is called uncolored.. Let G be a connected graph. For U ⊆ V (G), the subgraph induced by U is denoted by U G . For a subset M of V (G), we let G − M = V (G) − M G . For subsets L and M of V (G) with L ∩ M = ∅, we let E(L , M) denote the set of edges of G joining a vertex in L and a vertex in M. In this paper, a vertex x is often identified with the set {x}. (Thus if x ∈ V (G), then G − x means G − {x}, and E(x, M) means E({x}, M) for M ⊂ V (G − x).). For subsets L and M of V (G) with L ∩ M = ∅, we let E P (L , M) denote the set of paths of G joining a vertex in L and a vertex in M such that for every path Q = q 1 q 2 . . . q s ∈ E P (L , M), V (Q) ∩ L = {q 1 } and V (Q) ∩ M = {q s }.. Let (R, B) be a balanced vertex-coloring of G. Let U ⊆ V (G). If U G is connected and |U ∩ R| = |U ∩ B|, then U is called a balanced set. A balanced decomposition of a balanced coloring (R, B) of G is a partition of vertices V (G) = V 1 ∪ V 2 ∪ · · · ∪ V r such that all parts V i 's are balanced sets.. Our aim is to get a balanced decomposition in which the number of vertices of each part is as small as possible. The size of a decomposition V (G) = V 1 ∪ · · · ∪ V r is defined as the largest order of V 1 , V 2 , . . ., V r . Let f (k, G) be the smallest integer s such that for any balanced coloring of order k of G, there exists a balanced decomposition of size at most s. The Balanced decomposition number f (G) of G is defined as the largest f (k, G) for all k ≥ 0.. In this paper, we study balanced decomposition numbers of basic families of graphs. In Section 2, basic facts of the balanced decomposition number are studied. In Section 3, the balanced decomposition number of complete bipartite graphs is determined. In Section 4, 2-connected graphs are treated. In Section 5, an application of the balanced decomposition number is discussed, and in Section 6, we mention about the possibility of other applications of the balanced decomposition number. The concept of a balanced decomposition number f (G) seems to have other many applications. Actually, it is related to a kind of linkage problem in graphs. For example, we can regard a balanced decomposition as an analogous decomposition of a k-linked graph (i.e., it has at least 2k vertices and for every sequence s 1 , . . . , s k , t 1 , . . . , t k of distinct vertices there exist disjoint paths P 1 , . . . , P k such that the ends of P i are s i and t i ). (As for a related result, see e.g. [2] .) To see this, for a graph G, assign a red color to all the vertices s 1 , . . . , s k , a blue color to all the vertices t 1 , . . . , t k , respectively. Then, though a balanced set V i does not necessarily form a path which has only s i and t i , it has a good property in that every V i induces a connected graph and its size is at most f (G), and moreover, it contains equal number of red and blue vertices. Also, in view of this, we may be able to regard the concept of a balanced decomposition as an extension of a matching whose endvertices are colored by red and blue.. From Theorem 2, we guess that the connectivity of a graph seems to be important for obtaining a small balanced decomposition number. But it would be interesting for us to consider various sufficient condition for a graph to have a balanced decomposition which has some given properties. The details will be discussed elsewhere.

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Resolution Proof Transformation for Compression and Interpolation Abstract. Verification methods based on SAT, SMT, and Theorem Proving often rely on proofs of unsatisfiability as a powerful tool to extract information in order to reduce the overall effort. For example a proof may be traversed to identify a minimal reason that led to unsatisfiability, for computing abstractions, or for deriving Craig interpolants. In this paper we focus on two important aspects that concern efficient handling of proofs of unsatisfiability: compression and manipulation. First of all, since the proof size can be very large in general (exponential in the size of the input problem), it is indeed beneficial to adopt techniques to compress it for further processing. Secondly, proofs can be manipulated as a flexible preprocessing step in preparation for interpolant computation. Both these techniques are implemented in a framework that makes use of local rewriting rules to transform the proofs. We show that a careful use of the rules, combined with existing algorithms, can result in an effective simplification of the original proofs. We have evaluated several heuristics on a wide range of unsatisfiable problems deriving from SAT and SMT test cases. Symbolic verification methods rely on a representation of the state space as a set of formulae, which are manipulated by formal engines such as SAT-and SMTsolvers. For example Bounded Model Checking [13] represents an execution trace leading to a state violating a property as a propositional formula such that the state is reachable if and only if the formula is satisfiable. When the formula is satisfiable, it is possible to infer a counterexample from the model reported by the solver, showing a path that reaches a violating state. When the formula is unsatisfiable, it is instead possible to extract information that better explains the reason why the violating states are unreachable. For instance this can be useful to derive an abstraction of a set of states as it is done in interpolationbased model checking [49] (to abstract the initial states) or IC3 [17] (to derive a minimal set of clauses to put in a frame).. In this paper we describe a set of techniques that allow efficient manipulation of a propositional proof of unsatisfiability, the by-product of an unsatisfiable run of a state-of-the-art solver that may be used to obtain abstractions in the applications mentioned above. In particular we focus on two important aspects: compression of a proof of unsatisfiability, and rewriting to facilitate the computation of interpolants. These approaches are both realized by means of a set of local rewriting rules that enable proof restructuring and compression. The context of this paper is first order logic. We assume countable sets of individual variables (x, y, z), function (f, g) and predicate (P, Q) symbols. A function symbol of 0-arity is called a constant (a, b, c), while a predicate symbol of 0-arity corresponds to a propositional variable (o, p, q, r). A term is built from function symbols and individual variables (f (c, g(x))); an atom is built from predicate symbols and terms (P (x, f (c))). A literal (s, t) is either an atom (having positive polarity) or its negation (having negative polarity). A formula (φ, ψ) is built from atoms and connectives; we are only interested here in quantifier-free formulae.. A sentence (or ground formula) is a formula without free variables. A clause C is a finite disjunction of literals; a formula in conjunctive normal form (CNF ) is a finite conjunction of clauses. The empty clause, which represents unsatisfiability, is denoted by ⊥. We write clauses as lists of literals and sub-clauses, omitting the "∨" symbol, as for instance pqD (an overline denotes negation).. We use C 1 ⊆ C 2 to indicate that C 1 subsumes C 2 , that is the set of literals C 1 is a subset of the set of literals C 2 . Also we assume that clauses do not contain duplicated literals or both the occurrence of a literal and its negation. Finally, we use v(s) to denote the variable associated with a literal s.. A SAT-solver is a decision procedure that solves the propositional satisfiability problem; most successful state-of-the-art solvers rely on variants of the DPLL algorithm, as the conflict-driven clause-learning (CDCL) [12, 48] , which are based on the resolution inference system [38] . A first order theory T is a collection of sentences; we call SMT(T ) the problem of deciding the satisfiability of a formula w.r.t. a theory T . A theory solver is an algorithm that decides whether a conjunction of ground literals is satisfiable in T . If the conjunction is unsatisfiable in T , then its negation is valid and is called a T -lemma: intuitively, T -lemmata are formulae that encode facts valid in the theory T . An SMT(T )-solver is a procedure to solve SMT(T ); in particular, a lazy solver integrates a theory solver with a CDCL SAT-solver [60] .

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Bipartite multigraphs with expander-like properties A graph with vertex set V and edge set E is called a (d,c)-expander if the maximum degree of a vertex is d and, for every subset W of V that has cardinality at most |V|/2, the number of edges between vertices in W and vertices outside of W is at least c|V|. This note considers a related combinatorial question: "For which integers d and functions f_d does there exist, for every large enough v, a bipartite d-regular multigraph on 2v nodes with node sets V and W having the following property: For every U that is a subset of either V or W, the cardinality of the set of neighbours of U is at least f_d(|U|)?" ::: Graphs with the above property seem to behave well also with respect to other, more complicated, expansion-type properties. We provide results for d in {5,6,7,8} and give a description of a fairly general methodology for devising computer-assisted proofs for a wide class of mathematical claims using so called interval arithmetic. A graph with vertex set V and edge set E is called a (d, c)-expander if the maximum degree of a vertex is d and, for every set W ⊂ V of cardinality at most |V |/2, the inequality |{w ∈ V \ W : {v, w} ∈ E}| ≥ c|V | holds. This note considers a related combinatorial question:. For which integers d and functions f d does there exist, for every large enough v, a bipartite d-regular multigraph on 2v nodes with node sets V and W having the following property: For every U ⊆ V and every U ⊆ W , the cardinality of the set of neighbours of U is at least f d (|U |)?. Graphs with the above property seem to behave well also with respect to other, more complicated, expansion-type properties. Indeed, the author was motivated to study this question by a paper communicated to him in May 2002 (the latest version of the paper is available from URL http://www-math.mit.edu/ ∼ vempala/papers/tspinapprox.ps). In this paper, Papadimitriou and Vempala established approximation hardness of TSP with triangle inequality using as a tool in their construction the fact that for d = 6 and. there exist bipartite multigraphs with the properties described in the above question. In this paper, we prove the following theorem: Theorem 1. For d ∈ {5, 6, 7, 8} and functions f d as described below, there exists, for every large enough v, a bipartite d-regular multigraph on 2v nodes with node sets V and W having the property that for every U ⊆ V and every U ⊆ W the cardinality of the set of neighbours of U is at least f d (|U |).. v/10 ≤ u ≤ 3v/20, 21u/15 + 19v/100 3v/20 ≤ u ≤ 3v/10, u + 31v/100 30v/100 ≤ u ≤ 39v/100, 15u/21 + 59v/140 39v/100 ≤ u ≤ 3v/5, u/2 + 11v/20 3v/5 ≤ u ≤ 7v/10, u/3 + 2v/3 7v/10 ≤ u ≤ v.. The proof of the theorem relies partly on claims that have been verified by a computer. Hence, a second contribution in this paper is the description of a fairly general methodology for devising computer-assisted proofs for a wide class of mathematical claims. The methods described in this paper are fairly general and can be applied to larger values of d without any complications. For smaller d, the analysis of the function P close to the extreme points needs to be adapted.. Our functions f d are all symmetric along the line u+n = v. This property follows from our use of the probabilistic method; the involved probabilities have the same symmetry. One possible direction for future work could be to improve the behaviour of the functions f d in the region where u is close to v. Indeed, for large d, one would expect that the neighbour set of any 0.99v vertices is the entire other side of the bipartite graph.

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


MPC as control strategy for pasta drying processes Process Control and Chemical Engineering), 43 bd du 11 novembre, 69100 Villeurbanne, France ; e-mail of corresponding author: jeroen.detemmerman@biw.kuleuven.be phone number of corresponding author: 32-16-321922 fax number of corresponding author: 32-16-328590 This study focuses on the drying process of pasta in a convectional air drying oven. A model predictive control algorithm is designed, based on a dynamic model describing the mass transfer between pasta and the surrounding air. The multiple input single output control algorithm minimizes a cost function over a prediction horizon which represents the deviation of the process from a desired reference track, thereby incorporating the working limitations of the oven. The performance of the controller and the influence of the prediction horizon are examined in this paper, showing minimal deviation between process and reference track.drying process; model predictive control; nonlinear distributed parameter system; first principle model; moisture content 2Drying is one of the most energy intensive industrial processes with applications in a wide variety of industries including food industry and construction industry (1).Traditional convective drying processes, which are performed in batchwise operations, employ continuous constant air temperature and relative humidity for moisture removal.The rate of migration of the moisture from within the pasta to the evaporation front often controls the overall drying rate for a batch of pasta (2). Thus, for optimal energy consumption and improved quality, it is important to match the energy demand of the product during drying with the external supply of energy. Moreover, optimal control of the moisture content and the efficient application of the energy inputs during drying reduce the drying times and hence the energy consumption. Control of drying operations is therefore helpful, in addition to reduce the cost of production and increase the quality This research is concentrated around the efficient control during pasta drying. Presently, since the moisture content is unknown and uncontrolled during drying, pasta is industrially dried to a moisture level which is much lower than the legal threshold value for microbial safety. The implementation of an efficient controller will enable the process to be followed and to be directed towards a referential track, which is formulated as a function of quality aspects, economical imperatives and consumer concerns. The energy consumption can then be reduced by the efficient tracking of the moisture concentration in pasta, i.e. no additional energy will be spilled by the control actions (air temperature and relative humidity) to remove the extra unnecessary percentages of water out of the pasta with the consequence of a reduction in drying time. In process industry, pasta is namely dried to moisture contents, lower than the critical values for microbial contamination as the process moisture concentrations can not be measured online during production and biological variations in pasta materials influence the final moisture concentration. To reduce the (often unnecessary) last percentages of moisture out of pasta, long drying times are needed, consuming lots of energy. Control structures combined with online process measurements are therefore useful to reduce the drying time, and therefore the energy consumption.In this work, a model predictive control system for pasta drying is built. The idea of model predictive control can be traced back to the 1960s (6,7). However, interest in this field started to surge only in the 1980s after publication of the first papers on IDCOM (8) and dynamic matrix control (DMC) (9) and the first comprehensive exposition of generalized predictive control (GPC) (10,11). DMC was conceived to tackle the A time-varying linear MPC algorithm based on Dufour and Touré (19) will be developed for this research. In order to control the drying of pasta, it is necessary to model the drying process at a fundamental level. The governing transport equation for moisture 6 content is formulated on the basis of a nonlinear partial differential equation (PDE). To Drying is one of the most energy intensive industrial processes with applications in a wide variety of industries including food industry and construction industry (1) .. Traditional convective drying processes, which are performed in batchwise operations, employ continuous constant air temperature and relative humidity for moisture removal.. The rate of migration of the moisture from within the pasta to the evaporation front often controls the overall drying rate for a batch of pasta (2) . Thus, for optimal energy consumption and improved quality, it is important to match the energy demand of the product during drying with the external supply of energy. Moreover, optimal control of the moisture content and the efficient application of the energy inputs during drying reduce the drying times and hence the energy consumption. Control of drying operations is therefore helpful, in addition to reduce the cost of production and increase the quality (3, 4) . Whereas control techniques are widely used in many industries, the number of applications of control in drying is relatively still modest (5) . The use of control tools has started to emerge in drying applications only since 1979. In a second phase started around 1998, new trends based on optimization of a performance function solved by optimal control tools appeared. Since 1992, emerging applications have appeared with few papers in painting, pharmaceuticals, paper and wood applications (5) . However, control structures for drying processes of pasta are not yet studied and applied for industrial use.. Control actions in industrial companies are nowadays manually installed based on the practical process knowledge of the operators.. This research is concentrated around the efficient control during pasta drying. Presently, since the moisture content is unknown and uncontrolled during drying, pasta is industrially dried to a moisture level which is much lower than the legal threshold value for microbial safety. The implementation of an efficient controller will enable the process to be followed and to be directed towards a referential track, which is formulated as a function of quality aspects, economical imperatives and consumer concerns. The energy consumption can then be reduced by the efficient tracking of the moisture concentration in pasta, i.e. no additional energy will be spilled by the control actions (air temperature and relative humidity) to remove the extra unnecessary percentages of water out of the pasta with the consequence of a reduction in drying time. In process industry, pasta is namely dried to moisture contents, lower than the critical values for microbial contamination as the process moisture concentrations can not be measured online during production and biological variations in pasta materials influence the final moisture concentration. To reduce the (often unnecessary) last percentages of moisture out of pasta, long drying times are needed, consuming lots of energy. Control structures combined with online process measurements are therefore useful to reduce the drying time, and therefore the energy consumption.. In this work, a model predictive control system for pasta drying is built. The idea of model predictive control can be traced back to the 1960s (6, 7) . However, interest in this field started to surge only in the 1980s after publication of the first papers on IDCOM (8) and dynamic matrix control (DMC) (9) and the first comprehensive exposition of generalized predictive control (GPC) (10, 11) . DMC was conceived to tackle the multivariable constrained control problems typical for the oil and chemical industries. In DMC, these problems were handled by single loop controllers augmented by various selectors, overrides, decouplers, time-delay compensators, etc. (12) . GPC was intended to offer a new adaptive control alternative. In the tradition of adaptive control input-output (transfer function) models were employed. The GPC approach is not suitable or, at the very least, awkward for multivariable constrained systems which are much more commonly encountered in the oil and chemical industries than situations where adaptive control is needed (12) . Essentially all vendors have adopted a DMC-like approach (13) .. The initial research on MPC is characterized by attempts to understand DMC, which seemed to defy a traditional theoretical analysis because it was formulated in a nonconventional manner. Many MPC approaches have been proposed along the past three decades, most of them based on a receding-horizon strategy, i.e., at each sampling instant k the following actions are taken (14) :. -The plant measurements are updated for use in the feedback/feedforward control loop -the plant model is used to predict the output response to a hypothetical set of future control signals, -a function including the cost of future control actions and future deviations from a reference behavior is optimized to give the best future control sequence, and -the first movement of the optimal control sequence is applied.. These operations are repeated at time k+1. The main advantage of MPC is its ability to address long time delays, inverse responses, significant nonlinearities, multivariable interactions and constraints (14) . The widespread use and success of MPC applications described in the literature attest to the improved performance of MPC compared to the classical control algorithm for control of difficult process dynamics.. However, due to the strong nonlinear character of the equations, a numerical solution technique must be used to solve these equations. The computational effort varies somewhat because some solution methods require only that a feasible (and not necessarily optimal) solution be found or that only an improvement be achieved from time step to time step. Nevertheless the effort is usually formidable when compared to the linear case and stopping with a feasible rather than optimal solution can have unpredictable consequences for the performance. The computational effort can be greatly reduced when the system is linearized first in some manner and then the techniques developed for linear systems are employed online. Nevistic (15) showed excellent simulation results when a linear time varying (LTV) system approximation is used, which is calculated at each time step over the predicted system trajectory (16). Zheng (17, 18) used the MPC formulation in a closed-loop control strategy while reducing the online computational demand. The nonlinear MPC control law was thereby approximated by a linear controller which linearized the nonlinear model and assumed no constraints. The linear controller was then used to compute all future control moves. The online computation effort was significantly reduced in this manner since only the first control move was computed by solving the optimization problem.. A time-varying linear MPC algorithm based on Dufour and Touré (19) will be developed for this research. In order to control the drying of pasta, it is necessary to model the drying process at a fundamental level. The governing transport equation for moisture content is formulated on the basis of a nonlinear partial differential equation (PDE). To provide an insight into the drying process and to elucidate the physics of the transport phenomena that arise during drying, it is necessary to solve this system. The MPC structure which takes into account constraints for the model input and output, is developed afterwards. Experimental results reveal the applicability of the MISO (Multiple Input Single Output) MPC structure in pasta drying companies. In the present study, a distributed parameter model predictive control framework is used

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I B I I I B I I B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O B I O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O


SOFROP: Self-organizing and fair routing protocol for wireless networks with mobile sensors and stationary actors Abstract-Wireless sensor and actor networks (WSAN) have become increasingly popular in recent years. The cooperative operation between sensor and actor nodes results in a major advantage against pure sensor networks and extends the range of possible application scenarios. One emerging application is the Amazon scenario in which stationary actors are deployed at accessible points in a thick forest structure and sensor nodes are thrown in a river flowing through the forest to gather observations from unreachable areas of the forest. This unprecedented and unique setting exposes two important challenges: (a) the dynamics of the river forms a continuously varying topology of sensor nodes requiring a highly adaptive network organization and (b) the inherent features of sensor and actor nodes, combined with rapid changes in the link structure of the network require efficient bandwidth utilization and data transmission.In this paper, we address these challenges by introducing SOFROP, a novel self-organizing and fair routing protocol for WSANs. The extensive simulations that are carried out for evaluation point out two highlights of SOFROP. These are the lightweight and efficient routing that is optimized for fairness and the locally acting adaptive overlay network formation. Wireless sensor and actor networks (WSANs) [1] consist of a large number of tiny sensor nodes, which observe the events in the environment, and limited number of more powerful actors, which collect information from the sensor nodes, process this information and react to the event. In contrast, in most of the pure wireless sensor network (WSN) [2] applications, the network is used only for observation, which is insufficient in many real-world scenarios. For instance, if sensor nodes are able to inform a patrol unit about their battery conditions, this unit should not only collect data but also move closer to the sensor nodes in need of recharging. Thus, WSANs provide functionalities that allow dealing with a wider range of possible application scenarios compared to WSNs.. Coexistence of actor and sensor nodes creates a heterogeneous structure of node resources and communication channels. Hereby, a sensor node has limited data processing capability, transmission rate, energy, and memory. Actors on the other hand possess increased computation capabilities and wider communication ranges. Usually, actors are also equipped with long lasting batteries and larger memory compared to sensor nodes.. In WSANs node mobility is a natural element of many applications such as intelligent transportation, environment monitoring, animal control. There are various communication algorithms in the literature that have been developed for scenarios in which the actors are mobile while sensor nodes are stationary (see [3] [4] ). Although the sensor nodes are homogeneous, the network structure they form is fundamentally determined by the applications, which requires scenariooptimized network organization and routing schemes.. In context of this paper, we consider the scenario of Amazon rain-forest with a river going through it (see Fig. 1 ). The actors are positioned at rare accessible parts of the area and the sensor nodes are thrown in the river to collect information from hard-to-access sides of the area while they float in the river. Equipped with appropriate measurement technologies, sensor nodes are able to gather various kinds of information. For instance Regan et al. [5] deployed such a multi-sensor system in the River Lee Co. Cork, Ireland to monitor water quality parameters such as pH, temperature, conductivity, turbidity and dissolved oxygen. Although nodes move basically in one direction, they suffer from various peculiarities of the scenario such as permanent velocity changes, sudden stops by obstacles. In summary, these circumstances raise the following challenges for the design of an efficient routing protocol: (a) the dynamics of sensor nodes form a continuously varying topology requiring a highly adaptive network organization and (b) rapid changes of the neighborhood and actor association demands an efficient and reliable transmission of data from sensor nodes to the actors.. The limitation regarding the capacities of sensor nodes make the consumption of communication, computation and memory resources a critical constraint for routing protocols in WSANs [6] . An additional constraint arises with the heterogeneous node structure. Therefore the solutions applied neither in traditional wired networks nor in WSNs are suitable for WSANs. In our particular case, the Amazon scenario also comprise challenging and unprecedented characteristics such as the distinct mobility of the sensor nodes and constrained positioning of the actors. For these reasons existing approaches in the field of WSANs are inefficient and cannot be employed to overcome the variety of challenges implied by the differentiating features of the Amazon scenario.. The characteristics of Amazon scenario also create challenges for the quality of service provided to the applications in the network. Quality of service (QoS) in WSANs is characterized according to the employed applications, each of which has various constraints such as reliability, latency and robustness. QoS must be supported to satisfy the requirements of services of WSAN [7] . Throughput, delay, jitter, and packet loss rate are among the most fundamental QoS metrics used to measure the degree of efficiency in these services [8] . QoS performance of a network is improved in terms of these parameters when nodes allocate bandwidth in a fair manner [9] . For instance, even when there are multiple nodes reporting low priority traffic, which causes congestion on a bottleneck node, high priority data traffic must get its share of bandwidth and be transmitted to the relevant actor. All these requirements make fairness a vital part of the algorithm to be used. In the context of this paper, we define QoS as the capability to provide assurance that traffic flows will be treated differently in order to meet the service requirements of the applications.. In this paper, we propose SOFROP (Self-Organizing and Fair ROuting Protocol) to address the described critical issues of the Amazon scenario. There are two main contributions of this work. First, SOFROP uses an overlay network to organize and divide the network into actor areas. We show that SOFROP can provide a dynamic network organization by using localitypreserving communication. Second, SOFROP provides fairness among different types of applications while using the excess bandwidth at sensor nodes with a lightweight algorithm. Additionally, SOFROP uses packet queues to overcome the particular connectivity challenges of the scenario.. The remainder of this paper is organized as follows. Related work is given in Section 2. The system model is described in Section 3. We provide a detailed description of SOFROP in Section 4, which is devided in two phases: network organization and data transmission. We show the simulation results in Section 5 and finally conclude in Section 6. There has been approaches on routing protocols with QoS support in WSANs. The anycast communication paradigm by Hu et al. [10] builds an anycast tree rooted at each event source to reduce latency and energy consumption. The realtime routing framework by Shah et al [11] addresses the coordination of sensor and actor nodes through the delay bound for distributed routing. Another protocol using delay as the main constraint is Ad Hoc On Demand Delay Constrained Distance Vector Routing (AOD 2 V) by Sama and Akkaya [12] , where delay-EDD is used at admission control and EDF is used to determine the departure order of the packets at the intermediate nodes. Boukerche et al. [13] used service differentiation and central processing of routes, aiming low latency and reliable delivery. Paruchi et al. [14] proposed a distributed and randomized communication protocol with a fairness feature regarding power savings of the sensor nodes, which make local decisions on whether to sleep, or be active based on the energy levels of their neighbors. Xia et al. [15] applied feedback control for dynamic bandwidth allocation, which uses deadline miss ratio control to improve QoS in terms of reliability.. Another important asset of SOFROP is the employment of an overlay network scheme for the network organization part by clustering. Clustering algorithms in traditional sensor network, are often used to create a structure of an otherwise flat network topology [16] , [17] . A cluster is a group of interconnected nodes with a dedicated node called clusterhead. A clusterhead is in charge to manage the information flow in the proper cluster, which makes the role and the selection of clusterheads critical for proper network operation. One of the clustering scheme proposed for sensor networks is HEED by Younis and Fahmy [18] , in which authors target QoS by using an efficient energy consumption method.. Network organization behaviors of Amazon scenario requires the actors to be assigned as the pre-determined clusterheads, which are not supposed to change their status throughout the life-time of the network as it occurs in any other existing clustering algorithm for mobile ad hoc networks [19] . Thus the clusterhead election procedure is obsolete. However, the network and the clustering algorithm must be designed in a way that the actor node is always the most attractive clusterhead in its surrounding. Furthermore, the cluster structure permits multi-hop clusters, which is supported by only a few clustering algorithms. Since actors are specially equipped to process data while delivering a long life-time, the number of actors must be minimized. This property reduces the number of clusterheads in the network, which is also important for the case where actors cannot be deployed very close to each other due to restricted access to the environment. A characteristic of the Amazon scenario is that the actors are static while the sensor nodes are mobile. This causes an increased number of re-affiliations for the sensor nodes, since the clusterhead for a node is expected to change frequently. These distinct features of the application scenario require a novel network organization approach. Therefore a locally acting clustering approach, which adopts its basic principles from the multi-hop clustering scheme KHOPCA [20] , is used and it is described in Section 4.. The approaches discussed in this section do not match with the requirements of the Amazon scenario such as simplicity, flexibility and fairness, caused by the dynamics of the application scenario and the characteristics of WSANs. The routing algorithm in this scenario should be able to adapt to the rapid changes in the topology and allocate the bandwidth in a fair manner for QoS support.

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O B I I I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Steady States of Epidemic Spreading in Small-World Networks We consider a standard susceptible-infected-susceptible (SIS) model to study behaviors of steady states of epidemic spreading in smallworld networks. Using analytical methods and large scale simulations, we recover the usual epidemic behavior with a critical threshold λ c below which infectious diseases die out. Whereas for the spreading rate λ far above λ c , it was found that the density of infected individuals ρ as a function of λ has the property ρ ≈ F (K)(ln λ − ln λ c ).Keywords: networks and genealogical trees, diseases, phase transitions.Complex networks have attracted an increasing interest recently. The main reason is that they play an important role in the understanding of complex behaviors in real world networks,[1, 2, 3] including the structure of language, [4, 5]

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I I B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O