Sphere and Dot Product Representations of Graphs

Kang, Ross J.; Müller, Tobias

doi:10.1007/s00454-012-9394-8

Cited by 51 publications

(48 citation statements)

References 24 publications

(29 reference statements)

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“…Although for social networks the necessary data to obtain a representation should be readily available, this is not true in general. Given the difficulty in building a representation from the graph (see [21]), such representation-less algorithms could prove useful, and would be interesting from a theoretical point of view. We note that our algorithms for Clique on 2-dot product graphs and for Independent Set on d 0 -dot product graphs and on d + -dot product graphs already do not require a representation to be given.…”

Section: Discussionmentioning

confidence: 99%

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Algorithms for diversity and clustering in social networks through dot product graphs

2015

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. (2015) 'Algorithms for diversity and clustering in social networks through dot product graphs.', Social networks., 41 . pp. 48-55. Further information on publisher's website:http://dx.doi.org/10.1016/j.socnet.2015.01.001Publisher's copyright statement: NOTICE: this is the author's version of a work that was accepted for publication in Social Networks. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reected in this document. Changes may have been made to this work since it was submitted for publication. A denitive version was subsequently published in Social Networks, 41, May 2015, 10.1016/j.socnet.2015.01.001. Additional information:Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. We consider diversity and clustering in social networks by using a d-dot product graph model for the network. Diversity is considered through the size of the largest independent set of the graph, and clustering through the size of the largest clique. We present both positive and negative results on the potential of this model. We obtain a tight result for the diversity problem, namely that it is polynomial-time solvable for d = 2, but NP-hard for d ≥ 3. We show that the clustering problem is polynomialtime solvable for d = 2. To our knowledge, these results are also the first on the computational complexity of combinatorial optimization problems on dot product graphs. We also give new insights into the structure of dot product graphs. We also consider the situation when two individuals u and v are connected if and only if their preferences are not antithetical, that is, if and only if a u · a v ≥ 0, and the situation when two individuals u and v are connected if and only if their preferences are neither antithetical nor "orthogonal", that is, if and only if a u · a v > 0. For these two cases we prove that the diversity problem is polynomial-time solvable for any fixed d and that the clustering problem is polynomial-time solvable for d ≤ 3.

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Section: Discussionmentioning

confidence: 99%

“…However, Kang and Müller [21] showed the problem of deciding whether a graph has dot product dimension d is NP-hard for all fixed d ≥ 2 (membership in NP is still open). They also proved that an exponential number of bits is sufficient and can be necessary to store a d-dot product representation of a dot product graph.…”

Section: Introductionmentioning

confidence: 99%

Algorithms for diversity and clustering in social networks through dot product graphs

2015

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“…In the case M = R m , the problem of describing the set G (1) (R m ) is equivalent to asking which graphs are realizable as R m -geometric graphs in a given dimension m. There is a vast literature about this problem and, commonly, geometric graphs realizable in dimension m are called m-sphere graphs while the minimal dimension m such that a given graph is an m-sphere graph is called its sphericity. In [35] it is proven that every graph has finite sphericity; in [32] the authors prove that the problem of deciding, given a graph , whether is an m-sphere is NP-hard for all m > 1. We can also observe that, for each m > 0, there are graphs that are not m-sphere graphs.…”

Section: Riemannian Casementioning

confidence: 99%

Random Geometric Complexes and Graphs on Riemannian Manifolds in the Thermodynamic Limit

Lerario

Mulas

2020

Discrete Comput Geom

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We investigate some topological properties of random geometric complexes and random geometric graphs on Riemannian manifolds in the thermodynamic limit. In particular, for random geometric complexes we prove that the normalized counting measure of connected components, counted according to isotopy type, converges in probability to a deterministic measure. More generally, we also prove similar convergence results for the counting measure of types of components of each k-skeleton of a random geometric complex. As a consequence, in the case of the 1-skeleton (i.e., for random geometric graphs) we show that the empirical spectral measure associated to the normalized Laplace operator converges to a deterministic measure.

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“…where norm( RCap k ) and norm( ACap k ) are the normalised vector of the capabilities requirements of tasks and capabilities of agents allocated to G k , respectively; and mc r k is the indicator of the r th capability, which describes that to what extant G k misses the r th capability, if mc r k ≤0, mc r k = 0, otherwise, mc r k = mc r k . Finally, the similarity value between the vector of the missed capabilities requirements of tasks allocated to G k and the normalised vector of capabilities of an unallocated agent A u (i.e., norm( Cap u )=(nc 1 u , nc 2 u , ..., nc R u )) can be calculated by the dot product of two vectors [24], which can be described as follows.…”

Section: Agent Allocationmentioning

confidence: 99%

Dynamic Task Allocation for Heterogeneous Agents in Disaster Environments Under Time, Space and Communication Constraints

Zhang

Bai

2015

The Computer Journal

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Task allocation for heterogeneous agents in disaster environments under time, space and communication constraints is a challenging issue in both theory and practice. This paper presents a dynamic task allocation approach for such situations. The proposed approach consists of an information collection mechanism, a group task allocation mechanism and a group coordination mechanism. Initially, the information collection mechanism is applied to help agents in communication networks to reduce their communication connections and select one agent in each network as the network leader in a decentralised manner so as to facilitate the collection of information for task allocation under communication constraints. Then, the group task allocation mechanism is employed by each network leader to allocate tasks and agents in its network to groups with suitable space ranges by considering time, space and communication constraints as well as the differing capabilities of agents. During task execution, due to the dynamics of disaster environments, the original allocation (by the group task allocation mechanism) of tasks and agents in groups may be unsuitable. In order to achieve continuous coordination of the heterogeneous agents among groups under communication constraints, the group coordination mechanism is employed. Experimental results demonstrate that the proposed approach can have better performance than many existing approaches in terms of information collection and dynamic task allocation in disaster environments under time, space and communication constraints.Keywords: Dynamic task allocation; Disaster environments; Time, space and communication constraints a task, it first needs to move to the location of the task [7,6,8]; 3) Communication constraints. In disaster environments, due to the destruction of local communication facilities, the amount of information transferred among agents (i.e., the constraint of communication capacities) and the communication distances of agents (i.e., the constraint of communication ranges) are limited [9, 10]. 4) Dynamics of environments. In disaster environments, agents can enter and leave the environments and tasks can be continuously discovered and finished in the environments [11,12]; 5) Differing capabilities of agents. In disaster environments, each agent has its own capabilities, which determine what kind of tasks it can conduct [7]. 6) Local views of agents. Due to the communication constraints of disaster environments, each agent can only acquire the information about tasks close to its location and the information of agents with which it can directly communicate [13];Various approaches have been proposed to achieve efficient task allocation in disaster environments [14,11,15]. Some approaches deal with task allocation through a central point [7,6,8]. In such approaches, the central controller (i.e., the agent in charge of task allocation) can create an optimal solution for task allocation by considering time and space constraints as well as the differing capabilities of agents ...

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Sphere and Dot Product Representations of Graphs

Cited by 51 publications

References 24 publications

Algorithms for diversity and clustering in social networks through dot product graphs

Algorithms for diversity and clustering in social networks through dot product graphs

Random Geometric Complexes and Graphs on Riemannian Manifolds in the Thermodynamic Limit

Dynamic Task Allocation for Heterogeneous Agents in Disaster Environments Under Time, Space and Communication Constraints

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