Covert networks are social networks that often consist of harmful users. Social Network Analysis (SNA) has played an important role in reducing criminal activities (e.g., counter terrorism) via detecting the influential users in such networks. There are various popular measures to quantify how influential or central any vertex is in a network. As expected, strategic and influential miscreants in covert networks would try to hide herself and her partners (called leaders ) from being detected via these measures by introducing new edges. show that the corresponding computational problem, called Hiding Leader, is NP-Complete for the degree and closeness centrality measures. We study the popular core centrality measure and show that the problem is NP-Complete even when the core centrality of every leader is only @math . On the contrary, we prove that the problem becomes polynomial time solvable for the degree centrality measure if the degree of every leader is bounded above by any constant. We then focus on the optimization version of the problem and show that the Hiding Leader problem admits a @math factor approximation algorithm for the degree centrality measure. We complement it by proving that one cannot hope to have any @math factor approximation algorithm for any constant @math unless there is a @math factor polynomial time algorithm for the Densest @math -Subgraph problem which would be considered a significant breakthrough.
We also provide a few details about previous work on other network modification (design) problems. A set of design problems were introduced in @cite_5 . @cite_60 addressed a shortest path optimization problem via improving edge weights on undirected graphs. The node version of this problem was also studied @cite_66 @cite_1 @cite_61 . @cite_62 proposed approximation algorithms for single-source and all-pair shortest paths minimization. Faster algorithms for some of these problems were also presented in @cite_11 @cite_44 . @cite_63 minimized the diameter of a network by adding shortcut edges. @cite_10 studied the social network effect in the surprise in elections.
@cite_61 @cite_62 @cite_60 @cite_1 @cite_44 @cite_63 @cite_5 @cite_10 @cite_66 @cite_11
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In the network design problem we are given a weighted undirected graph. We wish to find a subgraph which connects all the original vertices and minimizes the sum of the shortest path weights between all vertex pairs, subject to a budget constraint on the sum of its edge weights. In this note we establish NP-completeness for the network design problem, even for the simple case where all edge weights are equal and the budget restricts the choice to spanning trees. This result justifies the development of enumerative optimization methods and of approximation algorithms, such as those described in a recent paper by R. Dionne and M. Florian. This paper studies a multi-facility network synthesis problem, called the Two-level Network Design (TLND) problem, that arises in the topological design of hierarchical communication, transportation, and electric power distribution networks. We are given an undirected network containing two types of nodes---primary and secondary---and fixed costs for installing either a primary or a secondary facility on each edge. Primary nodes require higher grade interconnections than secondary nodes, using the more expensive primary facilities. The TLND problem seeks a minimum cost connected design that spans all the nodes, and connects primary nodes via edges containing primary facilities; the design can use either primary or secondary edges to connect the secondary nodes. The TLND problem generalizes the well-known Steiner network problem and the hierarchical network design problem. In this paper, we study the relationship between alternative model formulations for this problem (e.g., directed and undirected models), and analyze the worst-case performance for a composite TLND heuristic based upon solving embedded subproblems (e.g., minimum spanning tree and either Steiner tree or shortest path subproblems). When the ratio of primary to secondary costs is the same for all edges and when we solve the embedded subproblems optimally, the worst-case performance ratio of the composite TLND heuristic is 4 3. This result applies to the hierarchical network design problem with constant primary-to-secondary cost ratio since its subproblems are shortest path and minimum spanning tree problems. For more general situations, we express the TLND heuristic worst-case ratio in terms of the performance of any heuristic used to solve the embedded Steiner tree subproblem. A companion paper develops and tests a dual ascent procedure that generates tight upper and lower bounds on the optimal value of a multi-level extension of this problem. We study computing all-pairs shortest paths (APSP) on distributed networks (the CONGEST model). The goal is for every node in the (weighted) network to know the distance from every other node using communication. The problem admits (1+o(1))-approximation &#xd5;(n)-time algorithms [2], [3], which are matched with &#x3a9;(n)-time lower bounds [4], [5], , &#xd5; and &#x3a9; hide polylogarithmic factors. Note that the lower bounds also hold even in the unweighted case and in the weighted case with polynomial approximation ratios. . No &#x3a9;(n) lower bound or o(m) upper bound were known for exact computation.In this paper, we present an &#xd5;(n^ 5 4 )-time randomized (Las Vegas) algorithm for exact weighted APSP; this provides the first improvement over the naive O(m)-time algorithm when the network is not so sparse. Our result also holds for the case where edge weights are asymmetric (a.k.a. the directed case where communication is bidirectional). Our techniques also yield an &#xd5;(n^ 3 4 k^ 1 2 +n)-time algorithm for the k-source shortest paths problem where we want every node to know distances from k sources; this improves Elkins recent bound [6] when k= &#x3a9;(n^ 1 4 ).We achieve the above results by developing distributed algorithms on top of the classic scaling technique, which we believe is used for the first time for distributed shortest paths computation. One new algorithm which might be of an independent interest is for the reversed r-sink shortest paths problem, where we want every of r sinks to know its distances from all other nodes, given that every node already knows its distance to every sink. We show an &#xd5;(n&#x221a; r )-time algorithm for this problem. Another new algorithm is called short range extension, where we show that in &#xd5;(n&#x221a; h ) time the knowledge about distances can be extended for additional h hops. For this, we use weight rounding to introduce small additive errors which can be later fixed. We study the following network design problem: Given a communication network, find a minimum cost subset of missing links such that adding these links to the network makes every pair of points within distance at most d from each other. The problem has been studied earlier [17] under the assumption that all link costs as well as link lengths are identical, and was shown to be R(logn)-hard for every d 2 4. We present a novel linear programming based approach to obtain an O(log la log d) approximation algorithm for the case of uniform link lengths and costs. We also extend the Cl(Iogn) hardness to d E Z, 3). On the other hand, if link costs can vary, we show that the prob” ‘-’ n lem is n(Z s )hard for d > 3. This version of our problem can be viewed as a special case of the minimum cost d-spanner problem and thus our hardness result applies there as well. For d = 2, however, we show that the problem continues to be O(logn) approximable by giving an O(log n)-approximation to the more general minimum cost Z-spanner problem. An n(2”s’-’ “)-hardness result also holds when all link costs are identical but link lengths may vary (applies even when all lengths are 1 or 2). Our reduction from the label cower problem [3] also applies to another well-studied network design problem. We show that the directed genemlized steiner network problem [6] is n(2 I’&-’ “)-hard, significantly improving upon the Q(logn) hardness known prior to our work. We also present O(n log d) approximation algorithm for our problem under arbitrary link costs and polynomially bounded link lengths. Same result holds for the minimum cost d-spanner problem. Finally, all our positive results extend to the case where each pair (u,u) of nodes has a distinct distance requirement, say d(u, v). The approximation guarantees above hold provided d is replaced by max,,, d(u, v). All our algorithmic as well as hardness results hold for both undirected and directed versions of the problem. Sanjeev Khanna