Shortest Paths and Distances with Differential Privacy

Privacy preserving network publishing has been studied extensively in recent years. Although more works have adopted un-weighted graphs to model network relationships, weighted graph modeling can provide deeper analysis of the degree of relationships. Previous works on weighted graph privacy have concentrated on preserving the shortest path characteristic between pairs of vertices. Two common types of privacy have been proposed. One type of privacy tried to add random noise edge weights to the graph but still maintain the same shortest path. The other privacy, k-shortest path privacy, minimally perturbed edge weights so that there exists k shortest paths. However, the k-shortest path privacy only considers anonymizing same fixed number of shortest paths for all pairs of source and destination vertices. In this work, we present a new concept called [k1, k2]-shortest path privacy to allow different number of shortest paths for different pairs of vertices. A published network graph with [k1, k2]-shortest path privacy has at least k' indistinguishable shortest paths between the source and destination vertices, where k1 l k' l k2. A heuristic algorithm based on modifying only Non-Visited (NV) edges is proposed and experimental results showing the feasibility and characteristics of the proposed approach are presented.

Point-to-point shortest path (PPSP), or s-t connectivity, is a variant of the shortest path problem found in graph theory. In this problem, we are given a graph and pairs of vertices over time, and the output is the shortest path between each pair of vertices. In this paper, we present two algorithms. Our first algorithm approximately solves the PPSP problem on any arbitrary graph. For each pair of vertices queried, it accurately and efficiently estimates the shortest path between the two vertices. Our second algorithm extends the first to work on dynamic graphs. That is, our second algorithm can efficiently account for changes in the graph, such as friend requests on the Facebook network or road closures on road networks. At a high level, both algorithms partition the graph into highly connected communities. To respond to a query q(u, v), they each find the fewest number of partitions between u and v, and the shortest path through each partition. We show that our static graph algorithm can approximate the distance between two vertices with about 20%-35% percent error and anywhere from 80X-70000X faster than a BFS in practice with the right choice of partitions. Additionally, we show that our dynamic graph algorithm can account for updates to the graph anywhere from 20X-20000X faster than rerunning the static graph algorithm for each change to the graph.

Graphs have been widely used to model complex data in many real-world applications. Answering vertex join queries over large graphs is meaningful and interesting, which can benefit friend recommendation in social networks and link prediction, and so on. In this article, we adopt “SimRank” [13] to evaluate the similarity between two vertices in a large graph because of its generality. Note that “Simank” is purely structure dependent, and it does not rely on the domain knowledge. Specifically, we define a S im R ank-based j oin ( SRJ ) query to find all vertex pairs satisfying the threshold from two sets of vertices U and V . To reduce the search space, we propose a shortest-path-distance-based upper bound for SimRank scores to prune unpromising vertex pairs. In the verification, we propose a novel index, called h-go cover + , to efficiently compute the SimRank score of any single vertex pair. Given a graph G , we only materialize the SimRank scores of a small proportion of vertex pairs (i.e., the h-go cover + vertex pairs), based on which the SimRank score of any vertex pair can be computed easily. To find the h-go cover + vertex pairs, we propose an efficient method without building the vertex-pair graph. Hence, large graphs can be dealt with easily. Extensive experiments over both real and synthetic datasets confirm the efficiency of our solution.

We describe cellular-automaton-based algorithms for solving two shortest path problems on arbitrary connected, directed, and weighted graphs with n vertices. The first problem is that of finding the shortest path from a given vertex to another given vertex of the graph. A two-dimensional cellular automaton, shaped as a triangle, with \(O(n^2)\) cells, is used. The algorithm runs in O(n) time. The second problem requires that all shortest paths between pairs of vertices be obtained. An \(n \times n\) cellular automaton solves the problem in \(O(n \log n)\) time.

The growing popularity of online Earth visualization tools and geo-realistic games and the availability of high resolution terrain data have motivated a new class of queries to the interests of the GIS and spatial database community: spatial queries (e.g., kNN) over land surface. However, the fundamental challenges that restrict the applicability of these studies to real world applications are the prohibitive time complexity and storage overhead to precompute the shortest surface paths. In this paper, for the first time, we propose an approximate solution to address both challenges and allow browsing the shortest surface paths in O(log N + √N) time, where N is the size of the terrain. With this method, the time and space requirements for an exhaustive all-pair pre-computation have been reduced from O(N3) to O(N1.5) and O(N) respectively. The substantial savings in both time and storage are gained by taking advantage of the fact that the O(N2) surface paths only deviate from approximate straight lines at O(√N) points, termed rough vertices. As a result, we propose a linear time shortest surface path computation algorithm between two arbitrary vertices and a linear size storage structure, which captures all the shortest surface paths between any pair of vertices. We experimentally verified the applicability and scalability of the proposed methods with large real world and synthetic data sets and showed that accuracy higher than 97% can be obtained in most cases.

Gradient leakage attacks are considered one of the wickedest privacy threats in deep learning as attackers covertly spy gradient updates during iterative training without compromising model training quality, and yet secretly reconstruct sensitive training data using leaked gradients with high attack success rate. Although deep learning with differential privacy is a defacto standard for publishing deep learning models with differential privacy guarantee, we show that differentially private algorithms with fixed privacy parameters are vulnerable against gradient leakage attacks. This paper investigates alternative approaches to gradient leakage resilient deep learning with differential privacy (DP). <i>First</i>, we analyze existing implementation of deep learning with differential privacy, which use fixed noise variance to injects constant noise to the gradients in all layers using fixed privacy parameters. Despite the DP guarantee provided, the method suffers from low accuracy and is vulnerable to gradient leakage attacks. <i>Second</i>, we present a gradient leakage resilient deep learning approach with differential privacy guarantee by using dynamic privacy parameters. Unlike fixed-parameter strategies that result in constant noise variance, different dynamic parameter strategies present alternative techniques to introduce adaptive noise variance and adaptive noise injection which are closely aligned to the trend of gradient updates during differentially private model training. <i>Finally</i>, we describe four complementary metrics to evaluate and compare alternative approaches. Extensive experiments on six benchmark datasets show that differentially private deep learning with dynamic privacy parameters outperforms the deep learning using fixed DP parameters, and existing adaptive clipping approaches in all aspects: compelling accuracy performance, strong differential privacy guarantee, and high attack resilience.

This paper presents the first Learning Automaton-based solution to the dynamic single source shortest path problem. It involves finding the shortest path in a single-source stochastic graph topology where there are continuous probabilistic updates in the edge-weights. The algorithm is significantly more efficient than the existing solutions, and can be used to find the "statistical" shortest path tree in the "average" graph topology. It converges to this solution irrespective of whether there are new changes in edge-weights taking place or not. In such random settings, the proposed learning automata solution converges to the set of shortest paths. On the other hand, the existing algorithms will fail to exhibit such a behavior, and would recalculate the affected shortest paths after each weight-change. The important contribution of the proposed algorithm is that all the edges in a stochastic graph are not probed, and even if they are, they are not all probed equally often. Indeed, the algorithm attempts to almost always probe only those edges that will be included in the shortest path graph, while probing the other edges minimally. This increases the performance of the proposed algorithm. All the algorithms were tested in environments where edge-weights change stochastically, and where the graph topologies undergo multiple simultaneous edge-weight updates. Its superiority in terms of the average number of processed nodes, scanned edges and the time per update operation, when compared with the existing algorithms, was experimentally established. The algorithm can be applicable in domains ranging from ground transportation to aerospace, from civilian applications to military, from spatial database applications to telecommunications networking.

Consider a complete graph Kn with edge weights drawn independently from a uniform distribution U(0, 1). The weight of the shortest (minimum‐weight) path P1 between two given vertices is known to be , asymptotically. Define a second‐shortest path P2 to be the shortest path edge‐disjoint from P1, and consider more generally the shortest path Pk edge‐disjoint from all earlier paths. We show that the cost Xk of Pk converges in probability to uniformly for all k ≤ n − 1. We show analogous results when the edge weights are drawn from an exponential distribution. The same results characterize the collectively cheapest k edge‐disjoint paths, that is, a minimum‐cost k‐flow. We also obtain the expectation of Xk conditioned on the existence of Pk.

For a simple graph [Formula: see text] and for a pair of vertices [Formula: see text], we say that a vertex [Formula: see text] resolves [Formula: see text] and [Formula: see text] if the shortest path from [Formula: see text] to [Formula: see text] is of a different length than the shortest path from [Formula: see text] to [Formula: see text]. A set of vertices [Formula: see text] is a resolving set if for every pair of vertices [Formula: see text] and [Formula: see text] in [Formula: see text], there exists a vertex [Formula: see text] that resolves [Formula: see text] and [Formula: see text]. The minimum weight resolving set problem is to find a resolving set [Formula: see text] for a weighted graph [Formula: see text] such that [Formula: see text] is minimum, where [Formula: see text] is the weight of vertex [Formula: see text]. In this paper, we explore the possible solutions of this problem for grid graphs [Formula: see text] where [Formula: see text]. We give a complete characterization of solutions whose cardinalities are 2 or 3, and show that the maximum cardinality of a solution is [Formula: see text]. We show that the grid has the property that given a landmark set, we only need to investigate whether or not all pairs of vertices that share common neighbors are resolved to determine if the whole graph is resolved. We use this result to provide a characterization of a class of minimals whose cardinalities range from [Formula: see text] to [Formula: see text] and show that the number of such minimals is [Formula: see text].

Graph-based approaches for simulating pedestrian dynamics in building models

Let (V, μ, ρ) be a fuzzy graph. We now provide two popular ways of defining the distance between a pair of vertices. One way is to define the “distance” dis(x,y) between x and y as the length of the shortest strongest path between them. This “distance” is symmetric and is such that dis(x,x) = 0 since by our definition of a fuzzy graph, no path from x to x can have strength greater than μ(x), which is the strength of the path of length 0. However, it does not satisfy the triangle property, as we see from the following example. Let V = {u, v, x, y,z}, ρ(x, u) = ρ(u, v) = ρ(v, z) = 1 and ρ(x, y) = ρ(y, z) = 0.5. Here any path from x to y or from y to z has strength ≤ 1/2 since it must involve either edge (x,y) or edge (y, z). Thus the shortest strongest paths between them have length 1. On the other hand, there is a path from x to z, through u and v, that has length 3 and strength 1. Thus dis(x,z) = 3 > 1 + 1 = dis(x,y) + dis(y, z) in this case.KeywordsHamiltonian PathFuzzy SubsetFuzzy RelationConnected SubgraphFuzzy GraphThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Over the last decade, differential privacy (DP) has emerged as the gold standard of a rigorous and provable privacy framework. However, there are very few practical guidelines on how to apply differential privacy in practice, and a key challenge is how to set an appropriate value for the privacy parameter ɛ. In this work, we employ a statistical tool called hypothesis testing for discovering useful and interpretable guidelines for the state-of-the-art privacy-preserving frameworks. We formalize and implement hypothesis testing in terms of an adversary’s capability to infer mutually exclusive sensitive information about the input data (such as whether an individual has participated or not) from the output of the privacy-preserving mechanism. We quantify the success of the hypothesis testing using the precision- recall-relation, which provides an interpretable and natural guideline for practitioners and researchers on selecting ɛ. Our key results include a quantitative analysis of how hypothesis testing can guide the choice of the privacy parameter ɛ in an interpretable manner for a differentially private mechanism and its variants. Importantly, our findings show that an adversary’s auxiliary information - in the form of prior distribution of the database and correlation across records and time - indeed influences the proper choice of ɛ. Finally, we also show how the perspective of hypothesis testing can provide useful insights on the relationships among a broad range of privacy frameworks including differential privacy, Pufferfish privacy, Blowfish privacy, dependent differential privacy, inferential privacy, membership privacy and mutual-information based differential privacy.

Comparison of weighted and unweighted network analysis in the case of a pig trade network in Northern Germany

Quantitative Information Flow (QIF) and Differential Privacy (DP) are both concerned with the protection of sensitive information, but they are rather different approaches. In particular, QIF considers the expected probability of a successful attack, while DP (in both its standard and local versions) is a max-case measure, in the sense that it is compromised by the existence of a possible attack, regardless of its probability. Comparing systems is a fundamental task in these areas: one wishes to guarantee that replacing a system A by a system B is a safe operation that is the privacy of B is no worse than that of A. In QIF, a refinement order provides strong such guarantees, while, in DP, mechanisms are typically compared w.r.t. the privacy parameter ε in their definition. In this paper, we explore a variety of refinement orders, inspired by the one of QIF, providing precise guarantees for max-case leakage. We study simple structural ways of characterising them, the relation between them, efficient methods for verifying them and their lattice properties. Moreover, we apply these orders in the task of comparing DP mechanisms, raising the question of whether the order based on ε provides strong privacy guarantees. We show that, while it is often the case for mechanisms of the same “family” (geometric, randomised response, etc.), it rarely holds across different families.

The macro nature of social networks has always been a hot topic of scientific research, which is of great help in studying the characteristics of human social behavior. The spectrum is closely related to the nature of the network and determines certain information, including community division, loop, diameter, degree and so on. To ensure the security of data, the differential privacy security framework is used to propose three algorithms for privacy protection of edge weights in spectrum query. Firstly, the basic concepts of edge weight neighbor graph and edge weight difference privacy are given. For the single singular value query, the L1 global sensitivity of the function is proved, and an algorithm satisfying ε-differential privacy is designed by using Laplace mechanism. It can guarantee the privacy of edge weights when we query someone singular value. However, it cannot provide reasonable privacy protection for multiple singular value queries. So the L2 global sensitivity of the query for multiple singular values is further proved, and the Gaussian mechanism is used to design an algorithm that satisfies (ε, δ)-differential privacy. Finally, to combine with multi-singular values query strategy and spectral decomposition, a novel graph data publishing algorithm which can guarantee edge weight (ε, δ)-differential privacy is proposed. To verify the availability of these methods, experimental tests have been carried out in both model networks and actual networks, which shows that the algorithm can better guarantee the availability of data.