Abstract:
Lower-cost shortest path tree is a commonly-used multicast tree type, which can minimize end-to-end delay and at the same time reduce bandwidth as possible. This article presents an algorithm for lower cost shortest path tree. The algorithm adjusts the nodes' minimum cost to the current shortest path tree dynamically, and gradually gets shortest path tree with low total cost by selecting the node with minimum cost to current shortest path tree in turn. The algorithm has better performance and lower complexity than Destination-Driven Shortest Path tree (DDSP) algorithm so that is a very fine shortest path tree algorithm by algorithm analysis and simulation.

Abstract:
Low-Cost Shortest Path Tree is a commonly-used multicast tree type, which can minimize the end-to-end delay and at the same time reduce the bandwidth requirement if possible. Based on the low-cost shortest path tree algorithm DDSP (destination-driven shortest path) and through improving on the search procedure, a Fast Low-cost Shortest Path Tree (FLSPT) algorithm is presented in this paper. The Shortest Path Tree constructed by the FLSPT algorithm is the same as that constructed by the DDSP algorithm, but its computation complexity is lower than that of the DDSP algorithm. The simulation results with random network models show that FLSPT algorithm is more effective.

Abstract:
We consider an intuitionistic fuzzy shortest path problem (IFSPP) in a directed graph where the weights of the links are intuitionistic fuzzy numbers. We develop a method to search for an intuitionistic fuzzy shortest path from a source node to a destination node. We coin the concept of classical Dijkstra’s algorithm which is applicable to graphs with crisp weights and then extend this concept to graphs where the weights of the arcs are intuitionistic fuzzy numbers. It is claimed that the method may play a major role in many application areas of computer science, communication network, transportation systems, and so forth. in particular to those networks for which the link weights (costs) are ill defined. 1. Introduction Graphs [1–4] are a very important model of networks. There are many real-life problems of network of transportation, communication, circuit systems, and so forth, which are modeled into graphs and hence solved. Graph theory has wide varieties of applications in several branches of engineering, science, social science, medical science, economics, and so forth, to list a few only out of many. Many real-life situations of communication network, transportation network, and so forth cannot be modeled into crisp graphs because of the reason that few or all of the arcs/links have the cost/weight which is ill defined. The weights of the arcs are not always crisp but intuitionistic fuzzy (or fuzzy). One of the first studies on fuzzy shortest path problem (FSPP) in graphs was done by Dubois and Prade [5] and then by Klein [6]. However, few more solutions to FSPP proposed in [7–10] are also interesting. Though the work of Dubois and Prade [5] was a major breakthrough, that paper lacked any practical interpretation even if fuzzy shortest path is found, but still this may not actually be any of the path in the corresponding network for which it was found. There are very few works reported in the literature on finding an intuitionistic fuzzy shortest path in a graph. Mukherjee [11] used a heuristic methodology for solving the IF shortest path problem using the intuitionistic fuzzy hybrid geometric (IFHG) operator, with the philosophy of Dijkstra’s algorithm. In [12], Karunambigai et al. in a team work with Atanassov, present a model based on dynamic programming to find the shortest paths in intuitionistic fuzzy graphs. Nagoor Gani and Mohammed Jabarulla in [13] also developed a method on searching intuitionistic fuzzy shortest path in a network. But all these algorithms have both merits and demerits (none is absolutely the best), as all these are

Abstract:
We consider the problem of classifying graphs using graph kernels. We define a new graph kernel, called the generalized shortest path kernel, based on the number and length of shortest paths between nodes. For our example classification problem, we consider the task of classifying random graphs from two well-known families, by the number of clusters they contain. We verify empirically that the generalized shortest path kernel outperforms the original shortest path kernel on a number of datasets. We give a theoretical analysis for explaining our experimental results. In particular, we estimate distributions of the expected feature vectors for the shortest path kernel and the generalized shortest path kernel, and we show some evidence explaining why our graph kernel outperforms the shortest path kernel for our graph classification problem.

Abstract:
In integrated network,a crucial aspect of Quality of Service(QoS)is to find multiple shortest feasible paths that meet end to end delay and cost constraints.This paper expands Breadth First Search(BFS) algorithm,and proposes a multiple feasible paths algorithm under delay constraints(KDCP).The KDCP algorithm is then extended as KEDCP to get paths under delay and cost constraints.Simulation result shows the two algorithms have well performance compared with other mimetic algorithms.

Abstract:
In two-stage robust optimization the solution to a problem is built in two stages: In the first stage a partial, not necessarily feasible, solution is exhibited. Then the adversary chooses the "worst" scenario from a predefined set of scenarios. In the second stage, the first-stage solution is extended to become feasible for the chosen scenario. The costs at the second stage are larger than at the first one, and the objective is to minimize the total cost paid in the two stages. We give a 2-approximation algorithm for the robust mincut problem and a ({\gamma}+2)-approximation for the robust shortest path problem, where {\gamma} is the approximation ratio for the Steiner tree. This improves the factors (1+\sqrt2) and 2({\gamma}+2) from [Golovin, Goyal and Ravi. Pay today for a rainy day: Improved approximation algorithms for demand-robust min-cut and shortest path problems. STACS 2006]. In addition, our solution for robust shortest path is simpler and more efficient than the earlier ones; this is achieved by a more direct algorithm and analysis, not using some of the standard demand-robust optimization techniques.

Abstract:
Low-cost shortest path tree is a commonly-used multicast tree type. On the foundation of the Fast Low-cost Shortest Path Tree (FLSPT) algorithm, the ordinal circularly linked list was selected as the calculating and saving center of sequence Q of nodes which were waiting for development. The fast low-cost shortest path tree algorithm based on ordinal circularly double linked list named Fast Low-cost Shortest Path Tree (DKFLSPT) was put forward. The shortest path tree constructed by DKFLSPT algorithm is the same as that constructed by FLSPT algorithm, making use of the part principle of ordinal circularly double linked list to improve the search procedure which can get the shortest path of nodes. The imitated experiment of random network indicates that the efficiency of DKFLSPT algorithm can be raised by 19%.

Abstract:
The shortest path problem is among the most fundamental combinatorial optimization problems to answer reachability queries. It is hard to deter-mine which vertices or edges are visited during shortest path traversals. In this paper, we provide an empirical analysis on how traversal algorithms behave on social networks. First, we compute the shortest paths between set of vertices. Each shortest path is considered as one transaction. Second, we utilize the pat-tern mining approach to identify the frequency of occurrence of the vertices. We evaluate the results in terms of network properties, i.e. degree distribution, clustering coefficient.

Abstract:
A hypergraph is a set V of vertices and a set of non-empty subsets of V, called hyperedges. Unlike graphs, hypergraphs can capture higher-order interactions in social and communication networks that go beyond a simple union of pairwise relationships. In this paper, we consider the shortest path problem in hypergraphs. We develop two algorithms for finding and maintaining the shortest hyperpaths in a dynamic network with both weight and topological changes. These two algorithms are the first to address the fully dynamic shortest path problem in a general hypergraph. They complement each other by partitioning the application space based on the nature of the change dynamics and the type of the hypergraph.

Abstract:
1. The birth of network science. 2. What are random networks? 3. Adjacency matrix. 4. Degree distribution. 5. What are simple networks? Classical random graphs. 6. Birth of the giant component. 7. Topology of the Web. 8.Uncorrelated networks. 9. What are small worlds? 10. Real networks are mesoscopic objects. 11. What are complex networks? 12. The configuration model. 13. The absence of degree--degree correlations. 14.Networks with correlated degrees.15.Clustering. 16. What are small-world networks? 17. `Small worlds' is not the same as `small-world networks'. 18. Fat-tailed degree distributions. 19.Reasons for the fat-tailed degree distributions. 20. Preferential linking. 21. Condensation of edges. 22. Cut-offs of degree distributions. 23. Reasons for correlations in networks. 24. Classical random graphs cannot be used for comparison with real networks. 25. How to measure degree--degree correlations. 26. Assortative and disassortative mixing. 27. Disassortative mixing does not mean that vertices of high degrees rarely connect to each other. 28. Reciprocal links in directed nets. 29. Ultra-small-world effect. 30. Tree ansatz. 31.Ultraresilience against random failures. 32. When correlated nets are ultraresilient. 33. Vulnerability of complex networks. 34. The absence of an epidemic threshold. 35. Search based on local information. 36.Ultraresilience disappears in finite nets. 37.Critical behavior of cooperative models on networks. 38. Berezinskii-Kosterlitz-Thouless phase transitions in networks. 39.Cascading failures. 40.Cliques & communities. 41. Betweenness. 42.Extracting communities. 43. Optimal paths. 44.Distributions of the shortest-path length & of the loop's length are narrow. 45. Diffusion on networks. 46. What is modularity? 47.Hierarchical organization of networks. 48. Convincing modelling of real-world networks:Is it possible? 49. The small Web..