Critical Node Detection with Connectivity Based on Bounded Path Lengths

Node cooperation approaches improve physical layer security in wireless networks without applying data encryption. Selecting the appropriate nodes to cooperate as relays or friendly jammers, to degrade eavesdropper's link, is a challenging problem which has attracted a lot of attention in recent years. Various approaches, based on conventional exhaustive search, have been suggested for optimal relay-jammer selection and power allocation in the literature. The complexity of these methods is proportional to the number of intermediate nodes. As the number of intermediate nodes exceeds a specific bound, conventional exhaustive search appears infeasible due to high computational complexity. In this paper, we propose a novel approach based on genetic algorithm (GA), for relay-jammer selection, power allocation and computational complexity reduction. Our system model consists of a source-destination pair, one eavesdropper and a number of intermediate nodes which cooperate as relays or friendly jammers. Considering the achievable secrecy rate as the cost function for GA optimization, the proposed algorithm is performed in two steps. First, we determine the desired channel gains that maximize the secrecy rate using GA. In the second step, nodes for which the corresponding channel gains have the smallest deviation from the obtained values in the first step, will be chosen as relays or jammers. We compare the proposed method with two kinds of conventional exhaustive search techniques one with fixed and the other with adaptive power allocated to jammers. Simulation results show that the proposed method reduces computational complexity while attaining a secrecy rate close to conventional techniques. Also the power which is allocated to jammers during each phase of transmission will be declined using GA optimization. Despite of conventional exhaustive search, the complexity of the proposed algorithm is almost the same as the number of cooperative nodes increases. So, the proposed method will outperform the previous works in terms of simplicity, efficiency and security.

We consider the shortest paths between all pairs of nodes in a directed or undirected complete graph with edge lengths which are uniformly and independently distributed in [0, 1]. We show that die longest of these paths is bounded by c log n/n almost surely, where c is a constant and n is the number of nodes. Our bound is the best possible up to a constant. We apply this result to some well-known problems and obtain several algorithmic improvements over existing results. Our results hold with obvious modifications to random (as opposed to complete) graphs and to any distribution of weights whose density is positive and bounded from below at a neighborhood of zero. As a corollary of our proof we get a new result concerning the diameter of random graphs.

With the development of emerging social networks, such as Facebook and MySpace, security and privacy threats arising from social network analysis bring a risk of disclosure of confidential knowledge when the social network data is shared or made public. In addition to the current social network anonymity de-identification techniques, we study a situation, such as in a business transaction network, in which weights are attached to network edges that are considered to be confidential (e.g., transactions). We consider perturbing the weights of some edges to preserve data privacy when the network is published, while retaining the shortest path and the approximate cost of the path between some pairs of nodes in the original network. We develop two privacy-preserving strategies for this application. The first strategy is based on a Gaussian randomization multiplication, the second one is a greedy perturbation algorithm based on graph theory. In particular, the second strategy not only yields an approximate length of the shortest path while maintaining the shortest path between selected pairs of nodes, but also maximizes privacy preservation of the original weights. We present experimental results to support our mathematical analysis.

Finding shortest path in the presence of barriers: An alternate approach

A pyramid organization is a hierarchical structure based on the principle of unity of command (Koontz, 1980) that every member except the top in the organization should have a single immediate superior. On the other hand an organization characterized by System 4 (Likert, 1976) has a structure in which relations between members of the same section are added to the pyramid organization structure. Members of middle layers of System 4 which are both members of the upper units and chiefs of the lower units are called linking pins, and this type of organization is called a linking pin organization. In the linking pin organization there exist relations between each superior and his direct subordinates and those between members which have the same immediate superior. The linking pin organization structure can be expressed as a structure where every pair of siblings which are nodes which have the same parent in a rooted tree is adjacent, if we let nodes and edges in the structure correspond to members and relations between members in the organization respectively. Then the linking pin organization structure is characterized by the number of subordinates of each member, that is, the number of children of each node and the number of levels in the organization, that is, the height of the rooted tree, and so on (Robbins, 2003; Takahara & Mesarovic, 2003). Moreover, the path between a pair of nodes in the structure is equivalent to the route of communication of information between a pair of members in the organization, and adding edges to the structure is equivalent to forming additional relations other than those between each superior and his direct subordinates and between members which have the same direct subordinate. The purpose of our study is to obtain an optimal set of additional relations to the linking pin organization such that the communication of information between every member in the organization becomes the most efficient. This means that we obtain a set of additional edges to the structure minimizing the sum of lengths of shortest paths between every pair of all nodes. We have obtained an optimal depth for a model of adding edges between every pair of nodes with the same depth to a complete K-ary linking pin structure of height H(H = 2, 3, ...) where every pair of siblings in a complete K-ary tree of height H is adjacent (Sawada, 2008). A complete K-ary tree is a rooted tree in which all leaves have the same depth and all internal nodes have K(K = 2, 3, ...) children (Cormen et al., 2001). Figure 1 shows an example

Despite the numerous works related to the problem of finding the shortest paths (SP), attention to development of speed-efficient algorithms for constructing SP does not decrease. This is primarily due to the fact that in the overwhelming majority of cases such algorithms are often used to solve individual subtasks in many applications in various fields of natural science, and the time to solve the general optimization problem is largely determined by the time of constructing SP. Three groups of single-criterion algorithms are considered: network combinatorial algorithms; algebraic or matrix algorithms; algorithms based on methods for solving linear programming problems (LP-methods). The article presents an overview, analysis and classification of methods and algorithms for constructing the shortest paths on networks and graphs between given subsets of the network nodes (Single Source Shortest Path, SSSP) and between all pairs of nodes (Shortest Path Tree, SPT and All Pairs Shortest Paths, APSP). Estimates of the time complexity of the best known algorithms for solving SSST and APSP problems by combinatorial, matrix and LP methods for networks with non-negative arcs lengths and networks with negative arcs lengths and cycles of negative lengths are given. It is noted that for solving individual SSSP problems there are "almost optimal" algorithms in theory and practice, at the same time, for solving a wider class of problems, including APSP problem, there are prerequisites for improving existing algorithms. In recent years the evolution of methods for solving the problems of finding SP has been associated with development and further improvement of effective structures of abstract data types for representing objects of the problem and the creation of parallel algorithms for multiprocessor solving the problem. The main directions of further research on the development of effective methods and algorithms for solving the problems of finding the shortest paths are determined.

Little is known about geographic properties of large-scale social networks. In this paper, we examine the geospatial attributes of a planetary-scale social network of 240 million people and 1.3 billion edges. We study the interplay among topological, geographical, and algorithmically generated paths connecting pairs of nodes in a social network. Starting in the realm of cyberspace, we find that topologically shortest paths of average length of 6.6 exist between pairs of nodes in the network and that the average degree of separation among nodes is robust to removal of hub nodes. Moving to the realm of locations and distances in geographic space, we find that topologically shortest paths in the social graph grow with increasing geographic distance between path's endpoint nodes. We discover that shortest topological paths are geographically inefficient, but that geography provides an important cue for local algorithmic policies for navigating between source and target nodes. Local algorithmic strategies for navigating the larger network structure in the absence of global navigation procedures have varying success. At the early stages of the navigation, navigating to a hub node helps, while in the middle stage, geography provides the most important clue. While local algorithms for navigating have trouble reaching the target node, they are successful in reaching nodes that are geographically close to the target. Taken together, our results demonstrate a complex interplay between topological and geographical properties of social networks and explain the success of local strategies for navigating such networks.

Centrality, which quantifies the importance of individual nodes, is among the most essential concepts in modern network theory. As there are many ways in which a node can be important, many different centrality measures are in use. Here, we concentrate on versions of the common betweenness and closeness centralities. The former measures the fraction of paths between pairs of nodes that go through a given node, while the latter measures an average inverse distance between a particular node and all other nodes. Both centralities only consider shortest paths (i.e., geodesics) between pairs of nodes. Here we develop a method, based on absorbing Markov chains, that enables us to continuously interpolate both of these centrality measures away from the geodesic limit and toward a limit where no restriction is placed on the length of the paths the walkers can explore. At this second limit, the interpolated betweenness and closeness centralities reduce, respectively, to the well-known current-betweenness and resistance-closeness (information) centralities. The method is tested numerically on four real networks, revealing complex changes in node centrality rankings with respect to the value of the interpolation parameter. Nonmonotonic betweenness behaviors are found to characterize nodes that lie close to intercommunity boundaries in the studied networks.

This study aims to optimize the number of nodes and arcs in a constructed network employing a given criteria. For this, we subject the constructed network using the methods like Shortest Path (SP), Critical Path (CP) and Max-Flow (MF) between any pair of nodes. A set of iterative algorithms have been generated using matrix operations to meet the above requirement, which is named as Pandit’s algorithm. The equivalent algorithms for SP, CP and MF are hereby represented as Pandit’s Mini-additive, Maxi-additive and Max-min algorithms respectively. The generated algorithms take less number of iterations when compared to the established methods. The proposed new algorithms are shown with suitable numerical examples and discuss their merits over the established methods. These new algorithms take less computational time and hence suggested for solving the bulk networks such as transportation GIS, telecommunication networks, information systems, project scheduling etc.

This paper presents a new solution to the dynamic all-pairs shortest path routing problem, using a pursuit estimator learning approach. It involves finding the shortest path in a stochastic network, where there are continuous probabilistically-based updates in link-costs. In this paper we present the details of the algorithm and also provide an example to illustrate how the algorithm would function. The experimental results of the algorithm show that the algorithm is few orders of magnitude superior to the algorithms available in the literature. It can be used to find the shortest path (between all pairs of nodes in a network) within the statistical average network, which converges irrespective of whether there are new changes in link-costs or not. On the other hand, the existing algorithms will fail to exhibit such a behavior and would recalculate the affected shortest paths after each link-cost update.

This paper presents a new solution to the dynamic all-pairs shortest path routing problem, using a linear reinforcement learning scheme. It involves finding the shortest path in a stochastic network, where there are continuous probabilistically-based updates in link-costs. In this paper we present the details of the algorithm and also provide an example to illustrate how the algorithm would function. The initial experimental results of the algorithm show that the algorithm is few orders of magnitude superior to the algorithms available in the literature. It can be used to find the shortest path (between all pairs of nodes in a network) within the "statistical" average network, which converges irrespective of whether there are new changes in link-costs or not. On the other hand, the existing algorithms fails to exhibit such a behavior and would recalculate the affected shortest paths after each link-cost update.

Estimating the distances of the shortest path between given pairs of nodes in a graph is a basic operation in a wide variety of applications including social network analysis, web retrieval, etc. Such applications require a response on the order of a few milliseconds, but exact algorithms to compute the distance of the shortest path exactly do not work on real-world large-scale networks, because of their infeasible time complexities. The landmark-based methods approximate distances by using a few nodes as landmarks, and can accurately estimate shortest-path distances with feasible time complexities. However, they fail at estimating small distances, as it is difficult for a few selected landmarks to cover the shortest paths of many close node pairs. To tackle this problem, we present a novel method EigenSP, that estimates the shortest-path distance by using an adjacency matrix approximated by a few eigenvalues and eigenvectors. The average relative error rate of EigenSP is lower than that of the landmark-based methods on large graphs with many short distances. Empirical results suggest that EigenSP estimates small distances better than the landmark-based methods.

EIA-CNDP: An exact iterative algorithm for critical node detection problem

Wireless full-duplex (FD) communication allows a node to send and receive data frames simultaneously within the same frequency channel. Generally, FD transmission is effective when a pair of FD transmission nodes have the same data-transmission duration. The difference in the time duration of the frames transmitted by a pair of the nodes wastes the channel resources where more data could be transmitted. The wasted time duration decreases the system throughput in networks. To adjust the data-frame transmission duration in a pair of FD nodes, previous work has proposed the application of frame aggregation technique to FD communication. However, the work has assumed the network is in saturated condition where each node has enough number of data-frames in its queue. In real networks, each node does not always have enough data frames. For applying the frame aggregation to FD networks, therefore, it is necessary to consider the networks under non-saturated condition. This paper proposes throughput and delay analysis for FD WLANs with frame aggregation. As the contributions, this paper provides the analytical expressions of FD networks under non-saturated condition. From the proposed model, the appropriate aggregation size for both the network throughput and the delay can be derived for each offered load, packet payload size, and the number of nodes in the networks.

Both access and core optical networks require connectivity between a given set of pairs of nodes under path-length constraints. In the context of Long Reach Passive Optical Networks (LR-PON) there is a limit on the length of the fibre between a local-exchange (LE) site and its metro-core (MC) node, and in the transparent optical core network the length of fibre between each pair of MC nodes must be within a given threshold. A failure in the network could affect hundreds of thousands of customers. Therefore, it is often important ensuring multiple connectivity between a given set of pair of nodes through at least two node-disjoint paths while respecting the path-length constraints. Node-disjointness allows switching to an alternative path whenever a link or a node fails. Minimising the cost associated with the fibre while maximising the disjointness for a given network is a very challenging task. In this paper, we develop and present a general approach that is scalable for solving very large instances of this problem.