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L.A.N. Amaral, A. Scala, M. Barthelemy, and H.E. Stanley, 2000. "Classes of small–world networks," Proceedings, National Academy of Sciences of the United States, volume 97, number 21, pp. 1149–1152.

M.A. Jovanovic, F.S. Annexstein, and K.A. Berman, 2001. "Modeling peer–to–peer networks topologies through "small world" models and power laws," paper presented at the IX Telecommunications Forum Telfor 2001 (Belgrade).

It is expected that Internet of Things (IoT) revolution will enable new solutions and business for consumers and entrepreneurs by connecting billions of physical world devices with varying capabilities. However, for successful realization of IoT, challenges such as heterogeneous connectivity, ubiquitous coverage, reduced network and device complexity, enhanced power savings, and enhanced resource management have to be solved. All these challenges are heavily impacted by the IoT network topology supported by massive number of connected devices. Small-world networks and scale-free networks are important complex network models with massive number of nodes and have been actively used to study the network topology of brain networks, social networks, and wireless networks. These models, also, have been applied to IoT networks to enhance synchronization, error tolerance, and more. However, due to interdisciplinary nature of the network science, with heavy emphasis on graph theory, it is not easy to study the various tools provided by complex network models. Therefore, in this paper, we attempt to introduce basic concepts of graph theory, including small-world networks and scale-free networks, and provide system models that can be easily implemented to be used as a powerful tool in solving various research problems related to IoT.

Due to interdisciplinary nature of the network science, with heavy emphasis on graph theory, it is not easy to study the various tools provided by complex network models. There are numerous small-world and scale-free network introductory research papers from statistical mechanics branch, but there are very few introductory research papers from information and communications technology branch that can be used for wireless network optimization. Therefore, in this paper, we attempt to introduce basic concepts of graph theory, including small-world networks and scale-free networks, and provide system models with wireless channel characteristics that can be easily implemented to be used as a powerful tool in solving various research problems related to IoT. Furthermore, we evaluate the proposed system models of small-world networks and scale-free networks based on various complex network metrics, such as average path length and clustering coefficient.

The remainder of the paper is organized as follows. Section 2 presents recent research activities related to applications of small-world and scale-free concepts to IoT networks. Section 3 provides an overview on basic concepts related to complex networks. In Section 4, we describe the proposed system model for small-world networks and system model for scale-free network in Section 5. In Section 6, we present various network characteristics that are needed to be added to the conventional complex network models for IoT modeling. The numerical results are presented in Section 7. Finally, we conclude in Section 8.

In this section, we discuss the design and implementation issues related to modeling small-world networks through algorithm description, system model presentation, and metric calculation module overview.

The small-world model has been actively applied to the communications networks research due to resulting network topology with features such as smaller average transmission delay and more robust network connectivity. The small-world network is constructed by randomly rewiring the edges of a ring lattice with nodes. The following procedure describes the basic steps of the small-world network construction. By varying the rewiring probability , one can analyze the transition of the network from a lattice structure to a random structure with .

Figure 4 shows the system model for implementing a small-world network. The system contains five major blocks: node initialization block, node connection block, rewiring block, average path length calculation block, and clustering coefficient calculation block. Parameters , , and correspond to total number of nodes, initial degree of all the nodes, and rewiring probability, respectively. Furthermore, the node connection matrix or adjacency matrix gives information about all the node connections after the completion of the rewiring process. The node connection matrix shown in equation (6) describes the node connections for the graph example in Figure 1. Note that the small-world network is modeled by a relational graph where the distance is based on edges or hops rather than the absolute distance used in spatial graphs.

Major features of scale-free networks that are different compared to random networks and small-world networks are dynamic addition of new nodes and preferential attachment to existing nodes with rich connections. Due to these features, in contrast to random networks and small-world networks with Poisson distribution, scale-free networks have degree distribution following power-law nature, resulting in higher probability of finding nodes with a large number of links. The following algorithm shows the steps towards construction of a scale-free network.

In average path length calculation block, after completion of a scale-free network construction, the node connection matrix that contains the link information between all the nodes is used for average path length calculation. The steps used in the average path length calculation block for small-world network are also used in the scale-free network system model. Furthermore, the BFS algorithm is recommended for average path length calculation of scale-free networks with a large number of nodes and edges.

In contrast to the relational graphs used in general complex network models, spatial graphs are more appropriate models for wireless sensor networks (WSNs). This is because in WSN, due to practical constraints, such as energy capacity and radio transmission range, the links are restricted by the distance between nodes, rather than relational factors as in small-world networks and scale-free networks. Thus, a new incoming node will have a limited number of candidate target nodes to be connected subject to required power consumption to communicate between nodes and that can be simply defined as , where is the path loss exponent and is the minimum power for acceptable reception quality. Furthermore, in heterogeneous WSN, the nodes will have different communication and energy capabilities. Possible WSN nodes are a sink node, a small number of cluster head nodes with high hardware capabilities and energy capacity, and a large number of low cost sensor nodes that are continuously added to the network over time.

Important optimization criteria of WSN are energy efficiency, average path length based on geographical distance, and network tolerance. Based on these WSN optimization criteria, the rewiring scheme in the conventional small-world network algorithm is modified to include performance metric such as energy efficiency as shown below.

In this section, we study the behavior of the small-world network implemented based on the system architecture with metric calculation modules described in Section 4. We initially assumed a regular ring lattice model with = 24 nodes and initial degree = 4 for all the nodes. The small-world network was created according to the system architecture described previously with various rewiring probability ranging from 0 to 1. Figure 7 shows the average path length of the implemented small-world network. One could observe that the average path length is around 4.2 for = 0 (without rewiring) and decreases to 2.6 for high rewiring probability (random network). Even with small number of random rewiring, there is a drastic decrease in average path length. Note that the theoretical average path length value for random network ( = 1) can be calculated as . Figure 8 shows the clustering coefficient of the implemented small-world network. One could observe that the clustering coefficient remains relatively constant with value around 0.5. However, there is rapid drop in the clustering coefficient for rewiring probability greater than 0.1. Thus, we can observe that the small-world network remains highly clustered like regular lattice for less than 0.1. From Figures 7 and 8, we can conclude that the behavior of the small-world network was fully confirmed, having highly clustered behavior as the regular lattice and small average path length as the random graphs, based on the proposed system architecture.

In this section, we study the behavior of the scale-free network implemented based on the system architecture described in Section 5. Figure 9 compares the average path length for random networks, denoted as RN in the plot, and scale-free networks, denoted as SFN in the plot, for different number of nodes in the network with = 4. The random network was constructed based on the algorithm described in Section 3. As for the scale-free network, the initial number of nodes was set to = 2 and the number of target nodes for preferential attachment by the incoming node was set to . To calculate the average path length, the BFS algorithm was utilized. As shown in Figure 9, with increase in network size , substantial decrease in average path length in scale-free network is observed compared to a random network. Figure 10 shows the degree distribution of a scale-free network with total number of nodes in the network equal to = 500 for different number of initial nodes and target nodes set as , 5, and 7. The degree distribution was calculated using the degree distribution calculation block described in Section 5. As seen in Figure 10, the degree distribution generated by the proposed system model follows the power-law distribution for all different values of and proving that the generated network evolves into a scale-free network. Note that the noise in the tail occurs due to limited number of data to average out the noise. One of the advantages of the scale-free network is the error and attack tolerance. Figure 11 shows the error tolerance performance as a function number of nodes removed from the network. To study the error tolerance performance of the generated scale-free network, out of = 1000 nodes, randomly selected nodes were removed with removal of all the connected edges to that node. Average path length metric was used to study the disruption effect to the scale-free network and random network due to removal of nodes. We can see that when 20% of the nodes in the network are removed, the average path length of the network increases around 16% and 12% in random network and scale-free network, respectively. Note that a peak point can be observed from the figure. This point is called a critical point where the network breaks into numerous isolated clusters, resulting in rapid drop in average path length. From the small average path length, power-law degree distribution, and error tolerance performance shown in Figures 9, 10, and 11, we can conclude that the generated network fully satisfies the scale-free characteristics. 041b061a72


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