Vertex-neighbor-integrity of magnifiers, expanders, and hypercubes

AbstractA set of vertices S is subverted from a graph G by removing the closed neighborhood N[S] from G. We denote the survival subgraph of the vertex subversion strategy S by G/S. The vertex-neighbor-integrity of G is defined to be VNI(G)=minS⊆V(G){|S|+ω(G/S)}, where ω(H) is the order of the largest connected component in the graph H. The graph parameter VNI was introduced by Cozzens and Wu [3] to measure the vulnerability of a spy network. Cozzens and Wu showed that the VNI of paths, cycles, trees and powers of paths on n vertices are all on the order of n. Here we prove that the VNI of any member of a family of magnifier graphs is linear in the order of the graph. We also find upper and lower bounds on the VNI of hypercubes. Finally, we show that the decision problem corresponding to computing the vertex-neighbor-integrity of a graph is NP-complete

Domination Cover Pebbling: Structural Results

This paper continues the results of "Domination Cover Pebbling: Graph Families." An almost sharp bound for the domination cover pebbling (DCP) number for graphs G with specified diameter has been computed. For graphs of diameter two, a bound for the ratio between the cover pebbling number of G and the DCP number of G has been computed. A variant of domination cover pebbling, called subversion DCP is introducted, and preliminary results are discussed.Comment: 15 page

X-Vine: Secure and Pseudonymous Routing Using Social Networks

Distributed hash tables suffer from several security and privacy vulnerabilities, including the problem of Sybil attacks. Existing social network-based solutions to mitigate the Sybil attacks in DHT routing have a high state requirement and do not provide an adequate level of privacy. For instance, such techniques require a user to reveal their social network contacts. We design X-Vine, a protection mechanism for distributed hash tables that operates entirely by communicating over social network links. As with traditional peer-to-peer systems, X-Vine provides robustness, scalability, and a platform for innovation. The use of social network links for communication helps protect participant privacy and adds a new dimension of trust absent from previous designs. X-Vine is resilient to denial of service via Sybil attacks, and in fact is the first Sybil defense that requires only a logarithmic amount of state per node, making it suitable for large-scale and dynamic settings. X-Vine also helps protect the privacy of users social network contacts and keeps their IP addresses hidden from those outside of their social circle, providing a basis for pseudonymous communication. We first evaluate our design with analysis and simulations, using several real world large-scale social networking topologies. We show that the constraints of X-Vine allow the insertion of only a logarithmic number of Sybil identities per attack edge; we show this mitigates the impact of malicious attacks while not affecting the performance of honest nodes. Moreover, our algorithms are efficient, maintain low stretch, and avoid hot spots in the network. We validate our design with a PlanetLab implementation and a Facebook plugin.Comment: 15 page

Approximate Consensus in Highly Dynamic Networks: The Role of Averaging Algorithms

In this paper, we investigate the approximate consensus problem in highly dynamic networks in which topology may change continually and unpredictably. We prove that in both synchronous and partially synchronous systems, approximate consensus is solvable if and only if the communication graph in each round has a rooted spanning tree, i.e., there is a coordinator at each time. The striking point in this result is that the coordinator is not required to be unique and can change arbitrarily from round to round. Interestingly, the class of averaging algorithms, which are memoryless and require no process identifiers, entirely captures the solvability issue of approximate consensus in that the problem is solvable if and only if it can be solved using any averaging algorithm. Concerning the time complexity of averaging algorithms, we show that approximate consensus can be achieved with precision of $\varepsilon$ in a coordinated network model in $O(n^{n+1} \log\frac{1}{\varepsilon})$ synchronous rounds, and in $O(\Delta n^{n\Delta+1} \log\frac{1}{\varepsilon})$ rounds when the maximum round delay for a message to be delivered is $\Delta$. While in general, an upper bound on the time complexity of averaging algorithms has to be exponential, we investigate various network models in which this exponential bound in the number of nodes reduces to a polynomial bound. We apply our results to networked systems with a fixed topology and classical benign fault models, and deduce both known and new results for approximate consensus in these systems. In particular, we show that for solving approximate consensus, a complete network can tolerate up to 2n-3 arbitrarily located link faults at every round, in contrast with the impossibility result established by Santoro and Widmayer (STACS '89) showing that exact consensus is not solvable with n-1 link faults per round originating from the same node

On the Computational Complexity of Vertex Integrity and Component Order Connectivity

The Weighted Vertex Integrity (wVI) problem takes as input an $n$-vertex graph $G$, a weight function $w:V(G)\to\mathbb{N}$, and an integer $p$. The task is to decide if there exists a set $X\subseteq V(G)$ such that the weight of $X$ plus the weight of a heaviest component of $G-X$ is at most $p$. Among other results, we prove that: (1) wVI is NP-complete on co-comparability graphs, even if each vertex has weight $1$; (2) wVI can be solved in $O(p^{p+1}n)$ time; (3) wVI admits a kernel with at most $p^3$ vertices. Result (1) refutes a conjecture by Ray and Deogun and answers an open question by Ray et al. It also complements a result by Kratsch et al., stating that the unweighted version of the problem can be solved in polynomial time on co-comparability graphs of bounded dimension, provided that an intersection model of the input graph is given as part of the input. An instance of the Weighted Component Order Connectivity (wCOC) problem consists of an $n$-vertex graph $G$, a weight function $w:V(G)\to \mathbb{N}$, and two integers $k$ and $l$, and the task is to decide if there exists a set $X\subseteq V(G)$ such that the weight of $X$ is at most $k$ and the weight of a heaviest component of $G-X$ is at most $l$. In some sense, the wCOC problem can be seen as a refined version of the wVI problem. We prove, among other results, that: (4) wCOC can be solved in $O(\min\{k,l\}\cdot n^3)$ time on interval graphs, while the unweighted version can be solved in $O(n^2)$ time on this graph class; (5) wCOC is W[1]-hard on split graphs when parameterized by $k$ or by $l$; (6) wCOC can be solved in $2^{O(k\log l)} n$ time; (7) wCOC admits a kernel with at most $kl(k+l)+k$ vertices. We also show that result (6) is essentially tight by proving that wCOC cannot be solved in $2^{o(k \log l)}n^{O(1)}$ time, unless the ETH fails.Comment: A preliminary version of this paper already appeared in the conference proceedings of ISAAC 201

MANCaLog: A Logic for Multi-Attribute Network Cascades (Technical Report)

The modeling of cascade processes in multi-agent systems in the form of complex networks has in recent years become an important topic of study due to its many applications: the adoption of commercial products, spread of disease, the diffusion of an idea, etc. In this paper, we begin by identifying a desiderata of seven properties that a framework for modeling such processes should satisfy: the ability to represent attributes of both nodes and edges, an explicit representation of time, the ability to represent non-Markovian temporal relationships, representation of uncertain information, the ability to represent competing cascades, allowance of non-monotonic diffusion, and computational tractability. We then present the MANCaLog language, a formalism based on logic programming that satisfies all these desiderata, and focus on algorithms for finding minimal models (from which the outcome of cascades can be obtained) as well as how this formalism can be applied in real world scenarios. We are not aware of any other formalism in the literature that meets all of the above requirements