Undirected Unweighted Graph Research Articles

$\mathsf {pSCAN}$ : Fast and Exact Structural Graph Clustering

We study the problem of structural graph clustering, a fundamental problem in managing and analyzing graph data. Given an undirected unweighted graph, structural graph clustering is to assign vertices to clusters, and to identify the sets of hub vertices and outlier vertices as well, such that vertices in the same cluster are densely connected to each other while vertices in different clusters are loosely connected. In this paper, we develop a new two-step paradigm for scalable structural graph clustering based on our three observations. Then, we present a $\mathsf {pSCAN}$ approach, within the paradigm, aiming to reduce the number of structural similarity computations, and propose optimization techniques to speed up checking whether two vertices are structure-similar. $\mathsf {pSCAN}$ outputs exactly the same clusters as the existing approaches $\mathsf {SCAN}$ and $\mathsf {SCAN\text{++}}$ , and we prove that $\mathsf {pSCAN}$ is worst-case optimal. Moreover, we propose efficient techniques for updating the clusters when the input graph dynamically changes, and we also extend our techniques to other similarity measures, e.g., Jaccard similarity. Performance studies on large real and synthetic graphs demonstrate the efficiency of our new approach and our dynamic cluster maintenance techniques. Noticeably, for the twitter graph with 1 billion edges, our approach takes 25 minutes while the state-of-the-art approach cannot finish even after 24 hours.

Identifying effective initiators in OSNs: from the spectral radius perspective

AbstractIn this paper, we focus on maximizing the influence of online social networks (OSNs). Particularly, we try to answer how to select proper information initiators such that information can propagate as widely as possible. We stress our attention on the susceptible‐infected model, a type of epidemic models, to describe the process of information diffusion. In general, OSNs can be classified into two categories, Facebook‐like OSNs and Twitter‐like OSNs. The former ones require bidirectional connections, while the latter do not, so we use the undirected unweighted graph and directed unweighted graph to describe them, respectively. We also pay additional attention to the nonidentity of the link probability on information transmission and build the weight graph, which can also cover both the two types of OSNs. In order to determine values of weight graph's weights, we introduce a learning method to obtain useful factors from raw data for assessing the true link probability on information transmission. Based on spectral analysis within the three graphs, our investigations on the information diffusion show that the spectral radius of the graph adjacency matrix can reflect the capability of information propagation, according to which we could determine effective initiators. We conduct our simulations on real OSNs. Experimental results show that our approach could effectively discover the initiators that spread information widely. Copyright © 2016 John Wiley & Sons, Ltd.

New Pairwise Spanners

Let G=(V,E) be an undirected unweighted graph on n vertices. A subgraph H of G is called an (all-pairs) purely additive spanner with stretch β if for every (u,v)∈V×V, d i s t H (u,v)≤d i s t G (u,v) + β. The problem of computing sparse spanners with small stretch β is well-studied. Here we consider the following variant: we are given $\mathcal {P} \subseteq V \times V$ and we seek a sparse subgraph H where d i s t H (u,v)≤d i s t G (u,v) + β for each $(u,v) \in \mathcal {P}$ . That is, distances for pairs outside $\mathcal {P}$ need not be well-approximated in H. Such a subgraph is called a pairwise spanner with additive stretch β and our goal is to construct such subgraphs that are sparser than all-pairs spanners with the same stretch. We show sparse pairwise spanners with additive stretch 4 and with additive stretch 6. We also consider the following special cases: $\mathcal {P} = S \times V$ and $\mathcal {P} = S \times T$ , where S⊆V and T⊆V, and show sparser pairwise spanners for these cases.

Secluded Connectivity Problems

Consider a setting where possibly sensitive information sent over a path in a network is visible to every neighbor of the path, i.e., every neighbor of some node on the path, thus including the nodes on the path itself. The exposure of a path P can be measured as the number of nodes adjacent to it, denoted by N[P]. A path is said to be secluded if its exposure is small. A similar measure can be applied to other connected subgraphs, such as Steiner trees connecting a given set of terminals. Such subgraphs may be relevant due to considerations of privacy, security or revenue maximization. This paper considers problems related to minimum exposure connectivity structures such as paths and Steiner trees. It is shown that on unweighted undirected n-node graphs, the problem of finding the minimum exposure path connecting a given pair of vertices is strongly inapproximable, i.e., hard to approximate within a factor of $$O(2^{\log ^{1-\epsilon }n})$$ for any $$\epsilon >0$$ (under an appropriate complexity assumption), but is approximable with ratio $$\sqrt{\Delta }+3$$ , where $$\Delta $$ is the maximum degree in the graph. One of our main results concerns the class of bounded-degree graphs, which is shown to exhibit the following interesting dichotomy. On the one hand, the minimum exposure path problem is NP-hard on node-weighted or directed bounded-degree graphs (even when the maximum degree is 4). On the other hand, we present a polynomial algorithm (based on a nontrivial dynamic program) for the problem on unweighted undirected bounded-degree graphs. Likewise, the problem is shown to be polynomial also for the class of (weighted or unweighted) bounded-treewidth graphs. Turning to the more general problem of finding a minimum exposure Steiner tree connecting a given set of k terminals, the picture becomes more involved. In undirected unweighted graphs with unbounded degree, we present an approximation algorithm with ratio $$\min \{\Delta , n/k, \sqrt{2n},O(\log k \cdot (k+\sqrt{\Delta }))\}$$ . On unweighted undirected bounded-degree graphs, the problem is still polynomial when the number of terminals is fixed, but if the number of terminals is arbitrary, then the problem becomes NP-hard again.

MIRACLE: A multiple independent random walks community parallel detection algorithm for big graphs

Community detection is a common problem in various types of complex networks. With the emerging of large scale real networks like social networks, community detection meets new technology challenges of extremely large computational cost and lack of prior information. Although several literatures recently try to solve these new challenges, they still have limitations of parallelism and running time. With the scale of data increases sharply, the parallelism is necessary but few codes exist. In this work, we analyze the process of random walking in graphs, and observe that the weight of an edge gotten by processing the vertices visited by the walker could be an indicator to measure the closeness of vertex connection. Based on this finding, we first propose a novel parallel computing community detection algorithm for big unweighted undirected graphs in the true sense. The algorithm consists of three steps, including random walking using multiple independent random walks, weight calculating for edges and community detecting. The time complexity of our algorithm is O(nlogn) without prior information. In order to implement our parallel computing algorithm efficiently, we also propose a novelty graph partition model. Experimental results show that our algorithm is capable of detecting the community structure and the overlapping parts of communities in real-world effectively (reduce the running time by 400 times at least), and handling the challenges of community detection in big graph era.

Maintaining Shortest Paths Under Deletions in Weighted Directed Graphs

We present an improved algorithm for maintaining all-pairs $(1 + \epsilon)$ approximate shortest paths under deletions and weight-increases. The previous state of the art for this problem is total update time $\widetilde{O}(n^2\sqrt{m}/\epsilon)$ over all updates for directed unweighted graphs [S. Baswana, R. Hariharan, and S. Sen, J. Algorithms, 62 (2007), pp. 74--92], and $\widetilde{O}(mn/\epsilon)$ for undirected unweighted graphs [L. Roditty and U. Zwick, in Proceedings of the 45th FOCS, Rome, Italy, 2004, pp. 499--508]. Both algorithms are randomized and have constant query time. Very recently, Henzinger, Krinninger, and Nanongkai presented a deterministic version of the latter algorithm [M. Henzinger, S. Krinninger, and D. Nanongkai, in IEEE FOCS, 2013, pp. 538--547]. Note that $\widetilde{O}(mn)$ is a natural barrier because even with a $(1 + \epsilon)$ approximation, there is no $o(mn)$ combinatorial algorithm for the static all-pairs shortest path problem. Our algorithm works on directed weighte...

Can we determine the filament chirality by the filament footpoint location or the barb-bearing?

We attempt to propose a method for automatically detecting the solar filament chirality and barb bearing. We first introduce the concept of an unweighted undirected graph and adopt the Dijkstra shortest path algorithm to recognize the filament spine. Then, we use the polarity inversion line (PIL) shift method for measuring the polarities on both sides of the filament, and employ the connected components labeling method to identify the barbs and calculate the angle between each barb and the spine to determine the bearing of the barbs, i.e., left or right. We test the automatic detection method with Hα filtergrams from the Big Bear Solar Observatory (BBSO) Hα archive and magnetograms observed with the Helioseismic and Magnetic Imager (HMI) on board the Solar Dynamics Observatory (SDO). Four filaments are automatically detected and illustrated to show the results. The barbs in different parts of a filament may have opposite bearings. The filaments in the southern hemisphere (northern hemisphere) mainly have left-bearing (right-bearing) barbs and positive (negative) magnetic helicity, respectively. The tested results demonstrate that our method is efficient and effective in detecting the bearing of filament barbs. It is demonstrated that the conventionally believed one-to-one correspondence between filament chirality and barb bearing is not valid. The correct detection of the filament axis chirality should be done by combining both imaging morphology and magnetic field observations.

Close to linear space routing schemes

Let $$G=(V,E)$$G=(V,E) be an unweighted undirected graph with n vertices and m edges, and let $$k>2$$k>2 be an integer. We present a routing scheme with a poly-logarithmic header size, that given a source s and a destination t at distance $$\varDelta $$Δ from s, routes a message from s to t on a path whose length is $$O(k\varDelta +m^{1/k})$$O(kΔ+m1/k). The total space used by our routing scheme is $$mn^{O(1/\sqrt{\log n})}$$mnO(1/logn), which is almost linear in the number of edges of the graph. We present also a routing scheme with $$n^{O(1/\sqrt{\log n})}$$nO(1/logn) header size, and the same stretch (up to constant factors). In this routing scheme, the routing table of every$$v\in V$$v?V is at most $$kn^{O(1/\sqrt{\log n})}deg(v)$$knO(1/logn)deg(v), where deg(v) is the degree of v in G. Our results are obtained by combining a general technique of Bernstein (2009), that was presented in the context of dynamic graph algorithms, with several new ideas and observations.

Can we determine the filament chirality by the filament footpoint location or the barb-bearing?or the Barb-bearing?

We attempt to propose a method for automatically detecting the solar filament chirality and barb bearing. We first introduce the concept of an unweighted undirected graph and adopt the Dijkstra shortest path algorithm to recognize the filament spine. Then, we use the polarity inversion line (PIL) shift method for measuring the polarities on both sides of the filament, and employ the connected components labeling method to identify the barbs and calculate the angle between each barb and the spine to determine the bearing of the barbs, i.e., left or right. We test the automatic detection method with Hα filtergrams from the Big Bear Solar Observatory (BBSO) Hα archive and magnetograms observed with the Helioseismic and Magnetic Imager (HMI) on board the Solar Dynamics Observatory (SDO) . Four filaments are automatically detected and illustrated to show the results. The barbs in different parts of a filament may have opposite bearings. The filaments in the southern hemisphere (northern hemisphere) mainly have left-bearing (right-bearing) barbs and positive (negative) magnetic helicity, respectively. The tested results demonstrate that our method is efficient and effective in detecting the bearing of filament barbs. It is demonstrated that the conventionally believed one-to-one correspondence between filament chirality and barb bearing is not valid. The correct detection of the filament axis chirality should be done by combining both imaging morphology and magnetic field observations.

Small Stretch Pairwise Spanners and Approximate $D$-Preservers

Let $G = (V,E)$ be an undirected unweighted graph on $n$ vertices. A subgraph $H$ of $G$ is called a purely additive spanner of $G$ with stretch $\beta$ if for each $(u,v) \in V \times V$, the $u$-$v$ distance in $H$ is at most $\delta_G(u,v) + \beta$. We currently know sparse purely additive spanners with $\beta = O(1)$ only for $\beta = 2,4,6$. When $\beta = 2$, the size of the spanner is $O(n^{3/2})$; when $\beta = 4$, the size of the spanner is $O(n^{1.4}\log^{0.2}n)$; and when $\beta = 6$, the size of the spanner is $O(n^{4/3})$. The following is a natural relaxation of the above problem: we care for only certain distances, these are captured by the set $\mathcal{P} \subseteq V \times V$, and the problem is to construct a sparse subgraph $H$ (also called a $\mathcal{P}$-spanner), where for every $(u,v) \in \mathcal{P}$, the $u$-$v$ distance in $H$ is at most $\delta_G(u,v) + \beta$. In this paper we show algorithms to construct the following for $\beta = 2$: a $\mathcal{P}$-spanner of size $\tilde{O}...

I/O efficient algorithms for the minimum cut problem on unweighted undirected graphs

The problem of finding the minimum cut of an undirected unweighted graph is studied on the external memory model. First, a lower bound of Ω((E/V)Sort(V)) on the number of I/Os is shown for the problem, where V is the number of vertices and E is the number of edges. Then the following are presented, (1) a deterministic minimum cut algorithm that usesO(min⁡{c7(log3+ϵ⁡E)(MST(V,E)),c(log⁡E)(MST(V,E))+(V/M)Sort(V)}) I/Os; here ϵ>0 is a small constant, MST(V,E) is the number of I/Os needed to compute a minimum spanning tree of the graph, and c is the value of the minimum cut. The algorithm performs better on dense graphs than the algorithm of [1], which requires O(E+c2Vlog⁡(V/c)) I/Os, when executed on the external memory model. For a δ-fat graph (for δ>0, the maximum tree packing of the graph is at least (1+δ)c/2), our algorithm computes a minimum cut in O(c(log⁡E)MST(V,E)) I/Os. (2) A randomized algorithm that computes minimum cut with high probability in O(c(log⁡E)⋅MST(V,E)+Sort(E)log2⁡V+VBSort(V)log⁡V) I/Os. (3) A (2+ϵ)-minimum cut algorithm that requires O((E/V)MST(V,E)) I/Os.

Distance Oracles beyond the Thorup--Zwick Bound

We give the first improvement to the space/approximation trade-off of distance oracles since the seminal result of Thorup and Zwick. For unweighted undirected graphs, our distance oracle has size $O(n^{5/3})$ and, when queried about vertices at distance $d$, returns a path of length at most 2d+1. For weighted undirected graphs with $m=n^2/\alpha$ edges, our distance oracle has size $O(n^2 / \sqrt[3]{\alpha})$ and returns a factor 2 approximation. Based on a plausible conjecture about the hardness of set intersection queries, we show that a 2-approximate distance oracle requires space $\widetilde{\Omega}(n^2 / \sqrt{\alpha})$. For unweighted graphs, this implies a $\widetilde{\Omega}(n^{1.5})$ space lower bound to achieve approximation 2d+1.

Preprocess, Set, Query!

Thorup and Zwick (J. ACM 52(1):1–24, 2005 and STOC’01) in their seminal work introduced the notion of distance oracles. Given an n-vertex weighted undirected graph with m edges, they show that for any integer k≥1 it is possible to preprocess the graph in $\tilde {O}(mn^{1/k})$ time and generate a compact data structure of size O(kn 1+1/k ). For each pair of vertices, it is then possible to retrieve an estimated distance with multiplicative stretch 2k−1 in O(k) time. For k=2 this gives an oracle of O(n 1.5) size that produces in constant time estimated distances with stretch 3. Recently, Pǎtrascu and Roditty (In: Proc. of 51st FOCS, 2010) broke the theoretical status-quo in the field of distance oracles and obtained a distance oracle for sparse unweighted graphs of O(n 5/3) size that produces in constant time estimated distances with stretch 2. In this paper we show that it is possible to break the stretch 2 barrier at the price of non-constant query time in unweighted undirected graphs. We present a data structure that produces estimated distances with 1+e stretch. The size of the data structure is O(nm 1−e′) and the query time is $\tilde {O}(m^{1-\varepsilon '})$ . Using it for sparse unweighted graphs we can get a data structure of size O(n 1.87) that can supply in O(n 0.87) time estimated distances with multiplicative stretch 1.75.

Spectral integral variation and unicyclic 3-colored digraphs with second smallest eigenvalue 1

In this article, the spectral integral variation of weighted directed graphs occurring in one place by adding a weighted directed edge is studied. Our results extend those of So (1999) [13] for unweighted undirected graphs and Fan (2003) [6] for mixed graphs. We characterize the unicyclic 3-colored digraphs that have 1 as the second smallest Laplacian eigenvalue. Finally, as an application we characterize the unicyclic 3-colored digraphs for which spectral integral variation occurs in one place by adding an edge of color red and where the changed eigenvalue is the second smallest Laplacian eigenvalue.

Computing the Girth of a Planar Graph in Linear Time

The girth of a graph is the minimum weight of all simple cycles of the graph. We study the problem of determining the girth of an n-node unweighted undirected planar graph. The first non-trivial algorithm for the problem, given by Djidjev, runs in O(n^{5/4} log n) time. Chalermsook, Fakcharoenphol, and Nanongkai reduced the running time to O(n log^2 n). Weimann and Yuster further reduced the running time to O(n log n). In this paper, we solve the problem in O(n) time.

I/O-efficient approximation of graph diameters by parallel cluster growing — a first experimental study

A fundamental step in the analysis of a massive graph is to compute its diameter. In the RAM model, the diameter of a connected undirected unweighted graph can be efficiently 2-approximated using a Breadth-First Search (BFS) traversal from an arbitrary node. However, if the graph is stored on disk, even an external memory BFS traversal is prohibitive, owing to the large number of I/Os it incurs. Meyer [1] proposed a parametrized algorithm to compute an approximation of graph diameter with fewer I/Os than that required for exact BFS traversal of the graph. The approach is based on growing clusters around randomly chosen vertices ‘in parallel’ until their fringes meet. We present an implementation of this algorithm and compare it with some simple heuristics and external-memory BFS in order to determine the trade-off between the approximation ratio and running-time achievable in practice. Our experiments show that with carefully chosen parameters, the new approach is indeed capable to produce surprisingly good diameter approximations in shorter time. We also confirm experimentally, that there are graph-classes where the parametrized approach runs into bad approximation ratios just as the theoretical analysis in [1] suggests.

On computing the diameter of real-world undirected graphs

We propose a new algorithm for the classical problem of computing the diameter of undirected unweighted graphs, namely, the maximum distance among all the pairs of nodes, where the distance of a pair of nodes is the number of edges contained in the shortest path connecting these two nodes. Although its worst-case complexity is O(nm) time, where n is the number of nodes and m is the number of edges of the graph, we experimentally show that our algorithm works in O(m) time in practice, requiring few breadth-first searches to complete its task on almost 200 real-world graphs.

Fully dynamic randomized algorithms for graph spanners

Spanner of an undirected graph G = ( V,E ) is a subgraph that is sparse and yet preserves all-pairs distances approximately. More formally, a spanner with stretch t ∈ ℕ is a subgraph ( V,E S ), E S ⊆ E such that the distance between any two vertices in the subgraph is at most t times their distance in G . Though G is trivially a t -spanner of itself, the research as well as applications of spanners invariably deal with a t -spanner that has as small number of edges as possible. We present fully dynamic algorithms for maintaining spanners in centralized as well as synchronized distributed environments. These algorithms are designed for undirected unweighted graphs and use randomization in a crucial manner. Our algorithms significantly improve the existing fully dynamic algorithms for graph spanners. The expected size (number of edges) of a t -spanner maintained at each stage by our algorithms matches, up to a polylogarithmic factor, the worst case optimal size of a t -spanner. The expected amortized time (or messages communicated in distributed environment) to process a single insertion/deletion of an edge by our algorithms is close to optimal.

All-pairs shortest paths for unweighted undirected graphs in o ( mn ) time

We revisit the all-pairs-shortest-paths problem for an unweighted undirected graph with n vertices and m edges. We present new algorithms with the following running times: { O ( mn /log n ) if m > n log n log log log n O ( mn log log n /log n ) if m > n log log n O ( n 2 log 2 log n /log n ) if m ≤ n log log n . These represent the best time bounds known for the problem for all m ≪ n 1.376 . We also obtain a similar type of result for the diameter problem for unweighted directed graphs.

On non-trivial Nash stable partitions in additive hedonic games with symmetric 0/1-utilities

We consider the problem of computing non-trivial Nash equilibria in additive hedonic games with symmetric 0/1-utilities. Such a game can be represented by an undirected unweighted graph G(V,E) where a non-trivial Nash equilibrium corresponds to a partition of V into at least two sets such that each node has at least as many neighbours in its own set compared to any other set in the partition. We show that computing such an equilibrium is NP-complete. On the other hand, we show that such an equilibrium is computable in polynomial time if (1) G is triangle free, (2) G contains no 4-cycles sharing and edge, and (3) G is not a star. If G is not a star with girth at least 5 we show how to compute a non-trivial equilibrium in O(n)-time.