Local Partitioning for Directed Graphs Using PageRank

There are several known results concerning how matroids can be induced from given matroids by a bipartite graph and the properties that are inherited in this way. The purpose of this note is to extend some of these results to the situation where the bipartite graph is replaced by an arbitrary directed graph. We show how a directed graph and a matroid can be used to induce a new matroid. If the initial matroid is strongly base orderable, we prove that the induced matroid is also. In particular, a matroid induced from a free matroid by a directed graph is strongly base orderable. A consequence is that the cycle matroid of the complete graph on four nodes cannot be induced from a free matroid by any directed graph.

CHAPTER 4 - Directed Graphs

This paper proposes a new algorithm for detecting and generating simple circuits within a directed graph. Before generating all simple circuits in a directed graph, the algorithm first detects whether there is a simple circuit in directed graph, and the detection can divide the directed graph into trivial graphs or strongly connected components. In the process of detecting the existence of simple circuit, the algorithm repeatedly reduces the number of vertices in vertex set along with edge pruning. Based on depth-first traversal, we can get all simple circuits of the directed graph. Especially, instead of proceeding depth-first traversals based on all vertices, the algorithm starts with a random vertex in a strongly connected component to reduce the number of depth-first traversals, and obtain all simple circuits of this strongly connected component. The difficulty of the algorithm is to detect and delete the repeated circuits in the process of generating all simple circuits, which can reduce both the storage space and time complexity. As a result, the output of the algorithm does not contain any repeated circuits. The algorithm simplifies the edge set of directed graphs and greatly reduces the number of depth-first traversals, thereby reducing the computational complexity and improving the computational efficiency.

An increasing attention has been recently devoted to uncovering community structure in directed graphs which widely exist in real-world complex networks such as social networks, citation networks, World Wide Web, email networks, etc. A two-stage framework for detecting clusters is an effective way for clustering directed graphs while the first stage is to symmetrize the directed graph using some similarity measures. Any state-of-the-art clustering algorithms for undirected graphs can be leveraged in the second stage. Hence, both stages are important to the effectiveness of the clustering result. However, existing symmetrization methods only consider about the direction of edges but ignore the weights of nodes. In this paper, we first attempt to connect link analysis in directed graph clustering. This connection not only takes into consideration the directionality of edges but also uses node ranking scores such as authority and hub score to explicitly capture in-link and out-link similarity. We also demonstrate the generality of our proposed method by showing that existing state-of-the-art symmetrization methods can be derived from our method. Empirical validation shows that our method can find communities effectively in real world networks.

The Maximum Carpool Matching problem is a star packing problem in directed graphs. Formally, given a directed graph G = (V, A), a capacity function c: V -> N, and a weight function w: A -> R^+, a carpool matching is a subset of arcs, M subseteq A, such that every v in V satisfies: (i) d^{in}_M(v) cdot d^{out}_M(v) = 0, (ii) d^{in}_M(v) <= c(v), and (iii) d^{out}_M(v) <= 1. A vertex v for which d^{out}_M(v) = 1 is a passenger, and a vertex for which d^{out}_M(v) = 0 is a driver who has d^{in}_M(v) passengers. In the Maximum Carpool Matching problem the goal is to find a carpool matching M of maximum total weight. The problem arises when designing an online carpool service, such as Zimride, which tries to connect between users based on a similarity function. The problem is known to be NP-hard, even in the unweighted and uncapacitated case. The Maximum Group Carpool Matching problem, is an extension of Maximum Carpool Matching where each vertex represents an unsplittable group of passengers. Formally, each vertex u in V has a size s(u) in N, and the constraint d^{in}_M(v) <= c(v) is replaced with sum_{u:(u,v) in M} s(u) <= c(v). We show that Maximum Carpool Matching can be formulated as an unconstrained submodular maximization problem, thus it admits a 1/2-approximation algorithm. We show that the same formulation does not work for Maximum Group Carpool Matching, nevertheless, we present a local search (1/2 - epsilon)-approximation algorithm for Maximum Group Carpool Matching. For the unweighted variant of both problems when the maximum possible capacity, c_{max}, is bounded by a constant, we provide a local search (1/2 + 1/{2c_{max}} - epsilon)-approximation algorithm. We also show that the problem is APX-hard, even if the maximum degree and c_{max} are at most 3.

Clique, as the most compact cohesive component in a graph, has been employed to identify cohesive subgroups of entities, and explore the sensitive information in the online social network, crowdsourcing network, and cyber-physical network, etc. In this study, we focus on the defense of clique-based attack and target at reducing the risk of entities security/privacy issues in clique structure. Since the ultimate resolution for defending the clique-based attack and risk is wrecking the clique with minimum cost, we establish the problem of clique-destroying in the network from a fundamental algorithm aspect. Interestingly, we notice that the clique-destroying problem in the directed graph is still an unsolved problem, and complexity analysis also does not exist. Therefore, we propose an innovative formal clique-destroying problem and proof the NP-complete problem complexity with solid theoretical analysis, then present effective and efficient algorithms for both undirected and directed graph. Furthermore, we show how to extend our algorithm to data privacy protection applications with controllable parameter k, which could adjust the size of a clique we wish to destroy. By comparing our algorithm with the up-to-date anonymization approaches, the real data experiment demonstrates that our resolution could efficaciously defend the clique-based security and privacy attacks.

Clique, as the most compact cohesive component in a graph, has been employed to identify cohesive subgroups of entities and explore the sensitive information in the online social network, crowdsourcing network, and cyber physical network, etc. In this study, we focus on the defense of clique-based attack and target at reducing the risk of entities security/privacy issues in clique structure. Since the ultimate resolution for defending the clique-based attack and risk is wrecking the clique with minimum cost, we establish the problem of clique-destroying (CD) in the network from a fundamental algorithm aspect. Interestingly, we notice that the clique-destroying problem in the directed graph is still an unsolved problem, and complexity analysis also does not exist. Therefore, we propose an innovative formal clique-destroying problem and proof the NP-complete problem complexity with solid theoretical analysis, then present effective and efficient algorithms for both undirected and directed graph. Furthermore, we show how to extend our algorithm to data privacy protection applications with controllable parameter k, which could adjust the size of a clique we wish to destroy. By comparing our algorithm with the up-to-date anonymization approaches, the real data experiment demonstrates that our resolution could efficaciously defend the clique-based security and privacy attacks.

A heuristic search algorithm is given that determines and resolves the Hamiltonian circuit problem in directed graphs. The heuristic information of each vertex is a set composed of its possible path length values from the starting vertex, which is obtained by the path length extension algorithm. A detailed analysis of algorithm examples shows that the heuristic algorithm can greatly reduce the number of processing nodes compared to the backtracking algorithm without information; meanwhile, the algorithm’s time complexity is related to the number of loops in the directed graph. If a directed graph G(V, E) only contains loops that contain the starting point, then the Hamiltonian problem of G can be determined in polynomial time. The algorithm is implemented using C++ and Python programming, and all of the test graphs are generated by the random graph algorithm. The experiments show that the heuristic algorithm can reduce the number of extended nodes by at least 10% in directed random graphs with no or only a few loops. This heuristic algorithm can save time in applications that perform tasks while extending nodes because it reduce the number of extended nodes.

A directed graphG is a set of vertices V and a subset of V × V called the edges of G. A path in G of length k,is such that (vi, vi+1) is an edge of G for 1 ≦ i ≦ k. A directed graph G is strongly connected if there is a path from every vertex of G to every other vertex. A circuit is a path whose two end vertices are equal. An elementary circuit has no other equal vertices. See (1) for a fuller discussion.Let G be a finite, strongly connected, directed graph (fscdg). The kth power Gk of G is the directed graph with the same vertices as G and edges of the form (i, j) where G has a path of length k from i to j.

Connectivity related concepts are of fundamental interest in graph theory. The area has received extensive attention over four decades, but many problems remain unsolved, especially for directed graphs. A directed graph is 2-edge-connected (resp., 2-vertex-connected) if the removal of any edge (resp., vertex) leaves the graph strongly connected. In this paper we present improved algorithms for computing the maximal 2-edge- and 2-vertex-connected subgraphs of a given directed graph. These problems were first studied more than 35 years ago, with O(mn) time algorithms for graphs with m edges and n vertices being known since the late 1980s. In contrast, the same problems for undirected graphs are known to be solvable in linear time. Henzinger et al. [ICALP 2015] recently introduced O(n2) time algorithms for the directed case, thus improving the running times for dense graphs. Our new algorithms run in time O(m3/2), which further improves the running times for sparse graphs.The notion of 2-connectivity naturally generalizes to k-connectivity for k > 2. For constant values of k, we extend one of our algorithms to compute the maximal k-edge-connected in time O(m3/2 log n), improving again for sparse graphs the best known algorithm by Henzinger et al. [ICALP 2015] that runs in O(n2 log n) time.

Vertex-frequency analysis (VFA) is a useful technique in graph signal processing to extract the correspondence between frequencies and vertices. VFA can be calculated by the windowed graph Fourier transform (WGFT) and the localized graph Fourier transform (LGFT). However, since the inability of the classical graph Fourier transform (GFT) to generate orthogonal bases for directed graphs, the two approaches are confined to undirected graphs. In order to address the absence of VFA on directed graphs, we extend the fundamental concepts of VFA, shifted vertex window function and band-pass transfer filters in spectral domain to directed graphs. We first propose WDGFT based on the shifted window function in vertex domain. The relevant properties, reconstruction formulas, and simulation results of vertex-frequency representation are given. Then, we propose the localized directed graph Fourier transform (LDGFT), where the band-pass transfer filters can capture the localized spectral domain. The transfer functions satisfying the reconstruction condition are discussed. The vertex-frequency representation obtained by LDGFT is provided through numerical examples. Furthermore, we present a polynomial approximation technique to decrease computational cost. The LDGFT can be calculated without any matrix decomposition. Finally, we evaluate the proposed directed VFA framework with two kinds of applications, clustering and malfunction detection. We demonstrate that the proposed VFA framework is a powerful tool for directed graph signal processing.

Given a directed graph $G = (V,E)$ and an integer $k \geq 1$, a $k$-transitive-closure-spanner ($k$-TC-spanner) of $G$ is a directed graph $H = (V, E_H)$ that has (1) the same transitive-closure as $G$ and (2) diameter at most $k$. These spanners were implicitly studied in the context of circuit complexity, data structures, property testing, and access control, and properties of these spanners have been rediscovered over the span of 20 years. We abstract the common task implicitly tackled in these diverse applications as the problem of constructing sparse TC-spanners. We initiate the study of approximability of the size of the sparsest $k$-TC-spanner of a given directed graph. We completely resolve the approximability of $2$-TC-spanners, showing that it is $\Theta(\log n)$ unless $\textsf{P} = \textsf{NP}$. For $k>2$, we present a polynomial time algorithm that finds a $k$-TC-spanner with size within $O((n \log n)^{1-1/k})$ of the optimum. Our techniques also yield algorithms with the first nontrivial approximation ratio for well-studied problems on directed spanners when $k>3$: Directed $k$-Spanner, Client/Server Directed $k$-Spanner, and $k$-Diameter Spanning Subgraph. For constant $k \geq 3$, we show that the size of the sparsest $k$-TC-spanner is hard to approximate within a factor of $2^{\log^{1-\eps} n}$ for any $\eps \in (0,1)$ unless $\NP \subseteq \text{DTIME}(n^{\polylog n})$. Finally, we study the size of the sparsest $k$-TC-spanners for $H$-minor-free graph families. Combining our constructions with our insight that 2-TC-spanners can be used for designing property testers, we obtain a monotonicity tester with $O(\log^2 n /\eps)$ queries for any poset whose transitive reduction, when viewed as an undirected graph, is free of a fixed minor. Previously, the best upper bound on the query complexity for such graphs was $O(\sqrt{n/\eps})$.

In a localized routing algorithm, node A currently holding the message forwards it based on the location of itself, its neighboring nodes and destination. We propose to use depth first search (DFS) method for routing decisions. Each node A, upon receiving the message for the first time, sorts all its neighbors according to a criteria such as their distance to destination and uses that order in DFS algorithm. It is the first localized algorithm that guarantees delivery for (connected) wireless networks modeled by arbitrary graphs, including inaccurate location information. We then propose the first localized QoS routing algorithm for wireless networks. It performs DFS routing algorithm after edges with insufficient bandwidth or insufficient connection time are deleted from the graph, and attempts to minimize hop count. This is also the first paper to apply GPS in QoS routing decisions, and to consider the connection time (estimated lifetime of a link) as a QoS criterion. The average length of measured QoS path in our experiments, obtained by DFS method, was between 1 and 1.34 times longer than the length of QoS path obtained by shortest path algorithm. The overhead is considerably reduced by applying the concept of internal nodes.

Basis functions derived from an undirected graph connecting nearby samples from a Markov decision process (MDP) have proven useful for approximating value functions. The success of this technique is attributed to the smoothness of the basis functions with respect to the state space geometry. This paper explores the properties of bases created from directed graphs which are a more natural fit for expressing state connectivity. Digraphs capture the effect of non-reversible MDPs whose value functions may not be smooth across adjacent states. We provide an analysis using the Dirichlet sum of the directed graph Laplacian to show how the smoothness of the basis functions is affected by the graph's invariant distribution. Experiments in discrete and continuous MDPs with non-reversible actions demonstrate a significant improvement in the policies learned using directed graph bases.

We propose and analyze two algorithms for maintaining approximate Personalized PageRank (PPR) vectors on a dynamic graph, where edges are added or deleted. Our algorithms are natural dynamic versions of two known local variations of power iteration. One, Forward Push, propagates probability mass forwards along edges from a source node, while the other, Reverse Push, propagates local changes backwards along edges from a target. In both variations, we maintain an invariant between two vectors, and when an edge is updated, our algorithm first modifies the vectors to restore the invariant, then performs any needed local push operations to restore accuracy. For Reverse Push, we prove that for an arbitrary directed graph in a random edge model, or for an arbitrary undirected graph, given a uniformly random target node $t$, the cost to maintain a PPR vector to $t$ of additive error $\varepsilon$ as $k$ edges are updated is $O(k + \bar{d} / \varepsilon)$, where $\bar{d}$ is the average degree of the graph. This is $O(1)$ work per update, plus the cost of computing a reverse vector once on a static graph. For Forward Push, we show that on an arbitrary undirected graph, given a uniformly random start node $s$, the cost to maintain a PPR vector from $s$ of degree-normalized error $\varepsilon$ as $k$ edges are updated is $O(k + 1 / \varepsilon)$, which is again $O(1)$ per update plus the cost of computing a PPR vector once on a static graph.