### Citations

3931 | Emergence of scaling in random networks
- Barabási, Albert
- 1999
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Citation Context ...d variable k with w to avoid confusion with the variable k commonly used in degree distribution function Edge Union of Networks on the Same Vertex Set 4 2.3. Barabási-Albert Barabási-Albert network =-=[32]-=- is denoted by Bm where m is the number of new edges at each iteration. The network construction begins with some arbitrary small number of vertices connected randomly. At each iteration, one new vert... |

3323 |
Collective dynamics of smallworld’networks
- Watts, Stogatz
- 1998
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Citation Context ...|E|/(|V | 2 ) . The slight difference is that Erdős-Rényi has precisely |E| edges while Gilbert graph has approximately |E| edges with high probability. 2.2. Watts-Strogatz A Watts-Strogatz network =-=[31]-=-, Ww,q is parameterised by w and q for mean degree‡ and probability of rewiring respectively. The construction begins with a regular ring lattice where each vertex connects to w/2 neighbours on each s... |

3062 | On random graphs
- ERDÖS, RÉNYI
- 1959
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Citation Context ... vertex set is implied to be n for all the networks in the rest of the paper. Lastly, let V and E be vertex set and edge set respectively. 2.1. Erdős-Rényi A realisation of an Erdős-Rényi network =-=[29]-=- is selected with equal probability from the set of all possible graphs with n = |V | vertices and |E| edges. However to generate a huge random Erdős-Rényi is difficult. To circumvent this problem o... |

887 | The evolution of social and economic networks
- Jackson, Watts
- 2002
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Citation Context ...́nyi, Watts-Strogatz and Barabási-Albert models. These models generate networks with only one type of edges and hence may be thought of as relating to agents interacting under a homogeneous relation =-=[1, 2, 3]-=-. However recently research has started to address the limitations of homogeneous networks. In the real world, agents may be related through more than one form of relations. E.g. think of social inter... |

860 |
Networks: an introduction
- Newman
- 2009
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Citation Context ...́nyi, Watts-Strogatz and Barabási-Albert models. These models generate networks with only one type of edges and hence may be thought of as relating to agents interacting under a homogeneous relation =-=[1, 2, 3]-=-. However recently research has started to address the limitations of homogeneous networks. In the real world, agents may be related through more than one form of relations. E.g. think of social inter... |

377 | An information flow model for conflict and fission in small groups
- Zachary
- 1977
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Citation Context ...ies of the union of the different types of edges. Combining edges of different types is known as the edge union of networks on the same vertex set. We note that the famous Zachary Karate Club Network =-=[26]-=- is an example of an edge union of 8 networks on the same vertex set. Any two members (vertices) of the karate club are connected if they have consistent interactions outside classes and club meetings... |

348 |
Markov graphs
- Frank, Strauss
- 1986
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Citation Context ...properties for such cases. In addition, the methods provides a framework for future studies on combinations of other families of network generators like Kronecker Graph [42], Exponential Random Graph =-=[43]-=- and Multifractal Network Generator [44]. Moreover the present investigation can be further developed to study a closely related model, which considers the percolation across one network layer of an e... |

251 | Random graph models of social networks.
- Newman, Watts, et al.
- 2002
(Show Context)
Citation Context ...an be estimated like any ER network. Edge Union of Networks on the Same Vertex Set 15 However the “clustering coefficient” of ER is often measured by an alternative similar metric, Transitivity Ratio =-=[37]-=-. It is the ratio of the number of triangles to the number of connected triples: Transitivity Ratio of Gcp = ( n 3 ) p3( n 3 ) p2 = p. (26) 5.4. Clustering coefficient of other combinations In the nex... |

154 | Winners Don't Take All: Characterizing the Competition for Links on the Web,
- Pennock, Flake, et al.
- 2002
(Show Context)
Citation Context ...rdős-Rényi with Barabási-Albert It has been noticed that the degree distribution of the BA model often does not represent natural occurring networks. This has lead to various generalised BA models =-=[33, 34, 35]-=- in which preferential and random uniform attachment are combined. This construction is similar to the union of ER with BA, where ER adds some uniform attachment to BA. To compute the degree distribut... |

144 |
Structure of growing networks with preferential linking
- Dorogovtsev, Mendes, et al.
- 2000
(Show Context)
Citation Context ...rdős-Rényi with Barabási-Albert It has been noticed that the degree distribution of the BA model often does not represent natural occurring networks. This has lead to various generalised BA models =-=[33, 34, 35]-=- in which preferential and random uniform attachment are combined. This construction is similar to the union of ER with BA, where ER adds some uniform attachment to BA. To compute the degree distribut... |

142 | Random graphs,”
- Gilbert
- 1959
(Show Context)
Citation Context ...ertices and |E| edges. However to generate a huge random Erdős-Rényi is difficult. To circumvent this problem one may instead let the number of edges fluctuate slightly and consider a Gilbert graph =-=[30]-=- Gp in which every vertex pair is connected with probability p where p ≈ |E|/(|V | 2 ) . The slight difference is that Erdős-Rényi has precisely |E| edges while Gilbert graph has approximately |E| e... |

106 | Realistic, mathematically tractable graph generation and evolution, using kronecker multiplication
- LESKOVEC, CHAKRABARTI, et al.
- 2005
(Show Context)
Citation Context ... to understand the statistical properties for such cases. In addition, the methods provides a framework for future studies on combinations of other families of network generators like Kronecker Graph =-=[42]-=-, Exponential Random Graph [43] and Multifractal Network Generator [44]. Moreover the present investigation can be further developed to study a closely related model, which considers the percolation a... |

95 |
On the properties of small-world network models,”
- Barrat, Weigt
- 2000
(Show Context)
Citation Context ... number of triangles for any vertex in WS given q. Thus: Clustering Coefficient of C ≥ 1 n n∑ i=0 T (vi,ws)( deg(vi,c) 2 ) (21) ≈ α n n∑ i=0 1( deg(vi,c) 2 ) (22) = α n∑ k=0 P C(k) 1( k 2 ) (23) From =-=[36]-=-, the clustering coefficient of WS decreases at the rate (1 − q)3 and α ≈ (w/2)2 for q = 0. This is also equal to the rate of decrease to the number of triangles since (1− q)3 is the probability that ... |

75 |
Multirelational organization of large-scale social networks in an online world.
- Szell, Lambiotte, et al.
- 2010
(Show Context)
Citation Context ...al relations and so forth. There are many names for networks with multiple types of connections, different researchers use different names: Multigraph [4], Multi-Layered [5, 6, 7, 8], MultiRelational =-=[9, 10, 11, 12, 13]-=-, Multi-Dimensional [14, 15, 16, 17, 18, 19], Multiplex [11, 20, 21, 22, 23], Multi-Modal [24] or graphs on the same vertex set [25]. We will here use the term multi-dimensional. We may think of these... |

66 |
Markets: Reasoning about a Highly Connected World
- Networks
- 2010
(Show Context)
Citation Context ...́nyi, Watts-Strogatz and Barabási-Albert models. These models generate networks with only one type of edges and hence may be thought of as relating to agents interacting under a homogeneous relation =-=[1, 2, 3]-=-. However recently research has started to address the limitations of homogeneous networks. In the real world, agents may be related through more than one form of relations. E.g. think of social inter... |

46 | Multiplicative attribute graph model of real-world networks
- Kim, Leskovec
- 2012
(Show Context)
Citation Context ...classes at the university. The 8 different relationships are then combine as edges (of different types) on the same vertex set. Other real-world applications of these kind of networks can be found in =-=[27]-=- and [28]. As another example of the relevance of multi-dimensional networks think of the network of email exchanges between people. Assume that the preferential attachment of the Barabási-Albert mod... |

43 | Community mining from multirelational networks.
- Cai, Shao, et al.
- 2005
(Show Context)
Citation Context ...al relations and so forth. There are many names for networks with multiple types of connections, different researchers use different names: Multigraph [4], Multi-Layered [5, 6, 7, 8], MultiRelational =-=[9, 10, 11, 12, 13]-=-, Multi-Dimensional [14, 15, 16, 17, 18, 19], Multiplex [11, 20, 21, 22, 23], Multi-Modal [24] or graphs on the same vertex set [25]. We will here use the term multi-dimensional. We may think of these... |

43 |
Growing scale-free networks with small world behavior,”
- Klemm, Eguíluz
- 2002
(Show Context)
Citation Context ... compute the degree distribution, we can use the method from section 4.2: P C(k) ∼ k∑ k∑ j Pce(|j, k + − j)PG(j)PB(k + − j) (11) where the approximate degree distribution of BA is PB(k) ∼ 2m2k−3 =-=[41]-=-. However the hypergeometric function, Pce(|j, k + − j) is computationally expensive. It is difficult to calculate the sums in Eq. (11) for large values of k and thereby assess how the additional u... |

22 |
Anomalous percolation properties of growing networks
- Dorogovtsev, Mendes, et al.
(Show Context)
Citation Context ...rdős-Rényi with Barabási-Albert It has been noticed that the degree distribution of the BA model often does not represent natural occurring networks. This has lead to various generalised BA models =-=[33, 34, 35]-=- in which preferential and random uniform attachment are combined. This construction is similar to the union of ER with BA, where ER adds some uniform attachment to BA. To compute the degree distribut... |

14 |
Emergence of network features from multiplexity. Scientific reports 3.
- Cardillo, Gómez-Gardeñes, et al.
- 2013
(Show Context)
Citation Context ...types of connections, different researchers use different names: Multigraph [4], Multi-Layered [5, 6, 7, 8], MultiRelational [9, 10, 11, 12, 13], Multi-Dimensional [14, 15, 16, 17, 18, 19], Multiplex =-=[11, 20, 21, 22, 23]-=-, Multi-Modal [24] or graphs on the same vertex set [25]. We will here use the term multi-dimensional. We may think of these networks as compose of layers of different relations (dimensions). Each lay... |

13 |
Exposing multi-relational networks to single-relational network analysis algorithms,”
- Rodriguez, Shinavier
- 2010
(Show Context)
Citation Context ...al relations and so forth. There are many names for networks with multiple types of connections, different researchers use different names: Multigraph [4], Multi-Layered [5, 6, 7, 8], MultiRelational =-=[9, 10, 11, 12, 13]-=-, Multi-Dimensional [14, 15, 16, 17, 18, 19], Multiplex [11, 20, 21, 22, 23], Multi-Modal [24] or graphs on the same vertex set [25]. We will here use the term multi-dimensional. We may think of these... |

12 |
Analysis of complex contagions in random multiplex networks,” Physical Review E,
- Yagan, Gligor
- 2012
(Show Context)
Citation Context ...types of connections, different researchers use different names: Multigraph [4], Multi-Layered [5, 6, 7, 8], MultiRelational [9, 10, 11, 12, 13], Multi-Dimensional [14, 15, 16, 17, 18, 19], Multiplex =-=[11, 20, 21, 22, 23]-=-, Multi-Modal [24] or graphs on the same vertex set [25]. We will here use the term multi-dimensional. We may think of these networks as compose of layers of different relations (dimensions). Each lay... |

11 | Social Decision Making with Multi-Relational Networks and GrammarBased Particle Swarms
- Rodriguez
- 2007
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Citation Context |

9 | Probability models for degree distributions of protein interaction networks
- Stumpf, Ingram
- 2007
(Show Context)
Citation Context ...t the university. The 8 different relationships are then combine as edges (of different types) on the same vertex set. Other real-world applications of these kind of networks can be found in [27] and =-=[28]-=-. As another example of the relevance of multi-dimensional networks think of the network of email exchanges between people. Assume that the preferential attachment of the Barabási-Albert model is tru... |

7 |
Scalable Link Prediction on Multidimensional Networks," Data Mining Workshops (ICDMW),
- Rossetti, Berlingerio, et al.
- 2011
(Show Context)
Citation Context ...ny names for networks with multiple types of connections, different researchers use different names: Multigraph [4], Multi-Layered [5, 6, 7, 8], MultiRelational [9, 10, 11, 12, 13], Multi-Dimensional =-=[14, 15, 16, 17, 18, 19]-=-, Multiplex [11, 20, 21, 22, 23], Multi-Modal [24] or graphs on the same vertex set [25]. We will here use the term multi-dimensional. We may think of these networks as compose of layers of different ... |

7 | Temporal scale of processes in dynamic networks
- Caceres, Berger-Wolf, et al.
(Show Context)
Citation Context ...rtex set [39]. The edge union of the network snapshots is analogous to extending the time interval in the sampling, i.e. aggregation of a temporal stream of edges over a fixed number of time distance =-=[40]-=-. Aggregation helps to identify the optimal size of the time interval at which to resolve a temporal network. For example assume that vertex vi and vj are connected over time T with probability 1− exp... |

6 | Finding and characterizing communities in multidimensional networks,”
- Berlingerio, Coscia, et al.
- 2011
(Show Context)
Citation Context ...ny names for networks with multiple types of connections, different researchers use different names: Multigraph [4], Multi-Layered [5, 6, 7, 8], MultiRelational [9, 10, 11, 12, 13], Multi-Dimensional =-=[14, 15, 16, 17, 18, 19]-=-, Multiplex [11, 20, 21, 22, 23], Multi-Modal [24] or graphs on the same vertex set [25]. We will here use the term multi-dimensional. We may think of these networks as compose of layers of different ... |

2 |
Athina Markopoulou. Multigraph sampling of online social networks
- Gjoka, Butts, et al.
- 2011
(Show Context)
Citation Context ...rests, such as chess, a third could be professional relations and so forth. There are many names for networks with multiple types of connections, different researchers use different names: Multigraph =-=[4]-=-, Multi-Layered [5, 6, 7, 8], MultiRelational [9, 10, 11, 12, 13], Multi-Dimensional [14, 15, 16, 17, 18, 19], Multiplex [11, 20, 21, 22, 23], Multi-Modal [24] or graphs on the same vertex set [25]. W... |

2 |
Thiran P, Layered complex networks
- Kurant
- 2006
(Show Context)
Citation Context ...s, a third could be professional relations and so forth. There are many names for networks with multiple types of connections, different researchers use different names: Multigraph [4], Multi-Layered =-=[5, 6, 7, 8]-=-, MultiRelational [9, 10, 11, 12, 13], Multi-Dimensional [14, 15, 16, 17, 18, 19], Multiplex [11, 20, 21, 22, 23], Multi-Modal [24] or graphs on the same vertex set [25]. We will here use the term mul... |

2 |
and Jari Saramki. Temporal networks
- Holme
- 2012
(Show Context)
Citation Context ...s common for real-world networks to have similar “layered dimensions”. In temporal networks where connections vary across time, each layer is a time snapshot of the connections of the same vertex set =-=[39]-=-. The edge union of the network snapshots is analogous to extending the time interval in the sampling, i.e. aggregation of a temporal stream of edges over a fixed number of time distance [40]. Aggrega... |

1 |
Nefedov Multiple-Membership Communities Detection and Its Applications for Mobile Networks
- Nikolai
- 2011
(Show Context)
Citation Context ...s, a third could be professional relations and so forth. There are many names for networks with multiple types of connections, different researchers use different names: Multigraph [4], Multi-Layered =-=[5, 6, 7, 8]-=-, MultiRelational [9, 10, 11, 12, 13], Multi-Dimensional [14, 15, 16, 17, 18, 19], Multiplex [11, 20, 21, 22, 23], Multi-Modal [24] or graphs on the same vertex set [25]. We will here use the term mul... |

1 |
Chun-Chi Liu and Xianghong Jasmine Zhou. Identifying Multi-Layer Gene Regulatory Modules from Multi-Dimensional Genomic Data
- Li, Zhang
- 2012
(Show Context)
Citation Context ...s, a third could be professional relations and so forth. There are many names for networks with multiple types of connections, different researchers use different names: Multigraph [4], Multi-Layered =-=[5, 6, 7, 8]-=-, MultiRelational [9, 10, 11, 12, 13], Multi-Dimensional [14, 15, 16, 17, 18, 19], Multiplex [11, 20, 21, 22, 23], Multi-Modal [24] or graphs on the same vertex set [25]. We will here use the term mul... |

1 |
Nomikos and Thrasyvoulos Spyropoulos and Franck Legendre. Collection and analysis of multi-dimensional network data for opportunistic networking research
- Hossmann, George
- 2012
(Show Context)
Citation Context ...ny names for networks with multiple types of connections, different researchers use different names: Multigraph [4], Multi-Layered [5, 6, 7, 8], MultiRelational [9, 10, 11, 12, 13], Multi-Dimensional =-=[14, 15, 16, 17, 18, 19]-=-, Multiplex [11, 20, 21, 22, 23], Multi-Modal [24] or graphs on the same vertex set [25]. We will here use the term multi-dimensional. We may think of these networks as compose of layers of different ... |

1 |
and Dorothea Wagner Approximating Clustering Coefficient and Transitivity
- Schank
(Show Context)
Citation Context ...ies of networks. The clustering coefficient of WS is measured by Eq. 17 and the “clustering coefficient” (transitivity ratio) of ER is measured by Eq. 26. Since these metrics are different in general =-=[38]-=-, we are only able to deduce a weak estimate of the union (Fig. 7). Firstly in the composite network, the maximum number of triangles at vertex vi,c is ( deg(vi,c) 2 ) . Among them, T (vi,ws) triangle... |

1 |
Lszl Lovszb, Tams Vicseka, Multifractal network generator PNAS
- Palla
(Show Context)
Citation Context ...the methods provides a framework for future studies on combinations of other families of network generators like Kronecker Graph [42], Exponential Random Graph [43] and Multifractal Network Generator =-=[44]-=-. Moreover the present investigation can be further developed to study a closely related model, which considers the percolation across one network layer of an effect induced by an adjacent layer. Such... |

1 |
Stanley and Shlomo Havlin Networks formed from interdependent networks Nature Physics 8
- Gao, Buldyrev, et al.
- 2012
(Show Context)
Citation Context ...ther developed to study a closely related model, which considers the percolation across one network layer of an effect induced by an adjacent layer. Such networks are known as interdependent networks =-=[45, 46]-=-. For example, two networks G1(V1, E1) and G2(V2, E2) form an interdependent network if there exists an edge between a vertex in V1 and V2. Let the set of such edges between G1 and G2 be E3. Hence an ... |

1 |
Shlomo Havlin Cascading Failures in Interdependent Lattice Networks: The Critical Role of the Length of Dependency
- Li, Bashan, et al.
(Show Context)
Citation Context ...ther developed to study a closely related model, which considers the percolation across one network layer of an effect induced by an adjacent layer. Such networks are known as interdependent networks =-=[45, 46]-=-. For example, two networks G1(V1, E1) and G2(V2, E2) form an interdependent network if there exists an edge between a vertex in V1 and V2. Let the set of such edges between G1 and G2 be E3. Hence an ... |