Multidimensional network

Last updated

In network theory, multidimensional networks, a special type of multilayer network, are networks with multiple kinds of relations. [1] [2] [3] [4] [5] [6] [7] Increasingly sophisticated attempts to model real-world systems as multidimensional networks have yielded valuable insight in the fields of social network analysis, [3] [4] [8] [9] [10] [11] [12] economics, urban and international transport, [13] [14] [15] ecology, [16] [17] [18] [19] psychology, [20] [21] medicine, biology, [22] commerce, climatology, physics, [23] computational neuroscience, [24] [25] [26] [27] operations management, and finance.

Contents

Terminology

The rapid exploration of complex networks in recent years has been dogged by a lack of standardized naming conventions, as various groups use overlapping and contradictory [28] [29] terminology to describe specific network configurations (e.g., multiplex, multilayer, multilevel, multidimensional, multirelational, interconnected). To fully leverage the dataset information on the directional nature of the communications, some authers consider only direct networks without any labels on vertices, and introduce the definition of edge-labeled multigraphs which can cover many multidimensional situations. [30] The term "fully multidimensional" has also been used to refer to a multipartite edge-labeled multigraph. [31] Multidimensional networks have also recently been reframed as specific instances of multilayer networks. [1] [5] [6] [32] In this case, there are as many layers as there are dimensions, and the links between nodes within each layer are simply all the links for a given dimension.

Definition

Unweighted multilayer networks

In elementary network theory, a network is represented by a graph in which is the set of nodes and the links between nodes, typically represented as a tuple of nodes . While this basic formalization is useful for analyzing many systems, real world networks often have added complexity in the form of multiple types of relations between system elements. An early formalization of this idea came through its application in the field of social network analysis (see, e.g., [33] and papers on relational algebras in social networks) in which multiple forms of social connection between people were represented by multiple types of links. [34]

To accommodate the presence of more than one type of link, a multidimensional network is represented by a triple , where is a set of dimensions (or layers), each member of which is a different type of link, and consists of triples with and . [6]

Note that as in all directed graphs, the links and are distinct.

By convention, the number of links between two nodes in a given dimension is either 0 or 1 in a multidimensional network. However, the total number of links between two nodes across all dimensions is less than or equal to .

Weighted multilayer networks

In the case of a weighted network, this triplet is expanded to a quadruplet , where is the weight on the link between and in the dimension .

The multiplex network of European airports. Each airline denotes a different layer. Visualization made with the muxViz software MuxViz EU Airports.png
The multiplex network of European airports. Each airline denotes a different layer. Visualization made with the muxViz software

Further, as is often useful in social network analysis, link weights may take on positive or negative values. Such signed networks can better reflect relations like amity and enmity in social networks. [31] Alternatively, link signs may be figured as dimensions themselves, [35] e.g. where and This approach has particular value when considering unweighted networks.

This conception of dimensionality can be expanded should attributes in multiple dimensions need specification. In this instance, links are n-tuples . Such an expanded formulation, in which links may exist within multiple dimensions, is uncommon but has been used in the study of multidimensional time-varying networks. [36]

The World Economic Forum map of global risks and global trends, modeled as an interdependent network (also known as network of networks). Muxviz GlobalRisk.png
The World Economic Forum map of global risks and global trends, modeled as an interdependent network (also known as network of networks).

General formulation in terms of tensors

Whereas unidimensional networks have two-dimensional adjacency matrices of size , in a multidimensional network with dimensions, the adjacency matrix becomes a multilayer adjacency tensor, a four-dimensional matrix of size . [3] By using index notation, adjacency matrices can be indicated by , to encode connections between nodes and , whereas multilayer adjacency tensors are indicated by , to encode connections between node in layer and node in layer . As in unidimensional matrices, directed links, signed links, and weights are all easily accommodated by this framework.

In the case of multiplex networks, which are special types of multilayer networks where nodes can not be interconnected with other nodes in other layers, a three-dimensional matrix of size with entries is enough to represent the structure of the system [8] [37] by encoding connections between nodes and in layer .

The multiplex social network of Star Wars saga. Each layer denotes a different episode and two nodes are connected each other if the corresponding characters acted together in one or more scenes. Visualization made with muxViz software Muxviz Star Wars Social Network.png
The multiplex social network of Star Wars saga. Each layer denotes a different episode and two nodes are connected each other if the corresponding characters acted together in one or more scenes. Visualization made with muxViz software

Multidimensional network-specific definitions

Multi-layer neighbors

In a multidimensional network, the neighbors of some node are all nodes connected to across dimensions.

Multi-layer path length

A path between two nodes in a multidimensional network can be represented by a vector r in which the th entry in r is the number of links traversed in the th dimension of . [38] As with overlapping degree, the sum of these elements can be taken as a rough measure of a path length between two nodes.

Network of layers

The existence of multiple layers (or dimensions) allows to introduce the new concept of network of layers, [3] peculiar of multilayer networks. In fact, layers might be interconnected in such a way that their structure can be described by a network, as shown in the figure.

Network of layers in multilayer systems Network of layers in multilayer systems.png
Network of layers in multilayer systems

The network of layers is usually weighted (and might be directed), although, in general, the weights depends on the application of interest. A simple approach is, for each pair of layers, to sum all of the weights in the connections between their nodes to obtain edge weights that can be encoded into a matrix . The rank-2 adjacency tensor, representing the underlying network of layers in the space is given by

where is the canonical matrix with all components equal to zero except for the entry corresponding to row and column , that is equal to one. Using the tensorial notation, it is possible to obtain the (weighted) network of layers from the multilayer adjacency tensor as . [3]

Centrality measures

Degree

In a non-interconnected multidimensional network, where interlayer links are absent, the degree of a node is represented by a vector of length . Here is an alternative way to denote the number of layers in multilayer networks. However, for some computations it may be more useful to simply sum the number of links adjacent to a node across all dimensions. [3] [39] This is the overlapping degree: [4] . As with unidimensional networks, distinction may similarly be drawn between incoming links and outgoing links. If interlayer links are present, the above definition must be adapted to account for them, and the multilayer degree is given by

where the tensors and have all components equal to 1. The heterogeneity in the number of connections of a node across the different layers can be taken into account through the participation coefficient. [4]

Versatility as multilayer centrality

When extended to interconnected multilayer networks, i.e. those systems where nodes are connected across layers, the concept of centrality is better understood in terms of versatility. [10] Nodes that are not central in each layer might be the most important for the multilayer systems in certain scenarios. For instance, this is the case where two layers encode different networks with only one node in common: it is very likely that such a node will have the highest centrality score because it is responsible for the information flow across layers.

Eigenvector versatility

As for unidimensional networks, eigenvector versatility can be defined as the solution of the eigenvalue problem given by , where Einstein summation convention is used for sake of simplicity. Here, gives the multilayer generalization of Bonacich's eigenvector centrality per node per layer. The overall eigenvector versatility is simply obtained by summing up the scores across layers as . [3] [10]

Katz versatility

As for its unidimensional counterpart, the Katz versatility is obtained as the solution of the tensorial equation , where , is a constant smaller than the largest eigenvalue and is another constant generally equal to 1. The overall Katz versatility is simply obtained by summing up the scores across layers as . [10]

HITS versatility

For unidimensional networks, the HITS algorithm has been originally introduced by Jon Kleinberg to rate Web Pages. The basic assumption of the algorithm is that relevant pages, named authorities, are pointed by special Web pages, named hubs. This mechanism can be mathematically described by two coupled equations which reduce to two eigenvalue problems. When the network is undirected, Authority and Hub centrality are equivalent to eigenvector centrality. These properties are preserved by the natural extension of the equations proposed by Kleinberg to the case of interconnected multilayer networks, given by and , where indicates the transpose operator, and indicate hub and authority centrality, respectively. By contracting the hub and authority tensors, one obtains the overall versatilities as and , respectively. [10]

PageRank versatility

PageRank, originally introduced to rank web pages, can also be considered as a measure of centrality for interconnected multilayer networks.

It is worth remarking that PageRank can be seen as the steady-state solution of a special Markov process on the top of the network. Random walkers explore the network according to a special transition matrix and their dynamics is governed by a random walk master equation. It is easy to show that the solution of this equation is equivalent to the leading eigenvector of the transition matrix.

Random walks have been defined also in the case of interconnected multilayer networks [15] and edge-colored multigraphs (also known as multiplex networks). [40] For interconnected multilayer networks, the transition tensor governing the dynamics of the random walkers within and across layers is given by , where is a constant, generally set to 0.85, is the number of nodes and is the number of layers or dimensions. Here, might be named Google tensor and is the rank-4 tensor with all components equal to 1.

As its unidimensional counterpart, PageRank versatility consists of two contributions: one encoding a classical random walk with rate and one encoding teleportation across nodes and layers with rate .

If we indicate by the eigentensor of the Google tensor , denoting the steady-state probability to find the walker in node and layer , the multilayer PageRank is obtained by summing up over layers the eigentensor: [10]

Triadic closure and clustering coefficients

Like many other network statistics, the meaning of a clustering coefficient becomes ambiguous in multidimensional networks, due to the fact that triples may be closed in different dimensions than they originated. [4] [41] [42] Several attempts have been made to define local clustering coefficients, but these attempts have highlighted the fact that the concept must be fundamentally different in higher dimensions: some groups have based their work off of non-standard definitions, [42] while others have experimented with different definitions of random walks and 3-cycles in multidimensional networks. [4] [41]

Community discovery

While cross-dimensional structures have been studied previously, [43] [44] they fail to detect more subtle associations found in some networks. Taking a slightly different take on the definition of "community" in the case of multidimensional networks allows for reliable identification of communities without the requirement that nodes be in direct contact with each other. [3] [8] [9] [45] For instance, two people who never communicate directly yet still browse many of the same websites would be viable candidates for this sort of algorithm.

Modularity maximization

A generalization of the well-known modularity maximization method for community discovery has been originally proposed by Mucha et al. [8] This multiresolution method assumes a three-dimensional tensor representation of the network connectivity within layers, as for edge-colored multigraphs, and a three-dimensional tensor representation of the network connectivity across layers. It depends on the resolution parameter and the weight of interlayer connections. In a more compact notation, making use of the tensorial notation, modularity can be written as , where , is the multilayer adjacency tensor, is the tensor encoding the null model and the value of components of is defined to be 1 when a node in layer belongs to a particular community, labeled by index , and 0 when it does not. [3]

Tensor decomposition

Non-negative matrix factorization has been proposed to extract the community-activity structure of temporal networks. [46] The multilayer network is represented by a three-dimensional tensor , like an edge-colored multigraph, where the order of layers encode the arrow of time. Tensor factorization by means of Kruskal decomposition is thus applied to to assign each node to a community across time.

Statistical inference

Methods based on statistical inference, generalizing existing approaches introduced for unidimensional networks, have been proposed. Stochastic block model is the most used generative model, appropriately generalized to the case of multilayer networks. [47] [48]

As for unidimensional networks, principled methods like minimum description length can be used for model selection in community detection methods based on information flow. [9]

Structural reducibility

Given the higher complexity of multilayer networks with respect to unidimensional networks, an active field of research is devoted to simplify the structure of such systems by employing some kind of dimensionality reduction. [22] [49]

A popular method is based on the calculation of the quantum Jensen-Shannon divergence between all pairs of layers, which is then exploited for its metric properties to build a distance matrix and hierarchically cluster the layers. Layers are successively aggregated according to the resulting hierarchical tree and the aggregation procedure is stopped when the objective function, based on the entropy of the network, gets a global maximum. This greedy approach is necessary because the underlying problem would require to verify all possible layer groups of any size, requiring a huge number of possible combinations (which is given by the Bell number and scales super-exponentially with the number of units). Nevertheless, for multilayer systems with a small number of layers, it has been shown that the method performs optimally in the majority of cases. [22]

Other multilayer network descriptors

Degree correlations

The question of degree correlations in unidimensional networks is fairly straightforward: do networks of similar degree tend to connect to each other? In multidimensional networks, what this question means becomes less clear. When we refer to a node's degree, are we referring to its degree in one dimension, or collapsed over all? When we seek to probe connectivity between nodes, are we comparing the same nodes across dimensions, or different nodes within dimensions, or a combination? [6] What are the consequences of variations in each of these statistics on other network properties? In one study, assortativity was found to decrease robustness in a duplex network. [50]

Path dominance

Given two multidimensional paths, r and s, we say that rdominatess if and only if: and such that . [38]

Shortest path discovery

Among other network statistics, many centrality measures rely on the ability to assess shortest paths from node to node. Extending these analyses to a multidimensional network requires incorporating additional connections between nodes into the algorithms currently used (e.g., Dijkstra's). Current approaches include collapsing multi-link connections between nodes in a preprocessing step before performing variations on a breadth-first search of the network. [28]

Multidimensional distance

One way to assess the distance between two nodes in a multidimensional network is by comparing all the multidimensional paths between them and choosing the subset that we define as shortest via path dominance: let be the set of all paths between and . Then the distance between and is a set of paths such that such that dominates . The length of the elements in the set of shortest paths between two nodes is therefore defined as the multidimensional distance. [38]

Dimension relevance

In a multidimensional network , the relevance of a given dimension (or set of dimensions) for one node can be assessed by the ratio: . [39]

Dimension connectivity

In a multidimensional network in which different dimensions of connection have different real-world values, statistics characterizing the distribution of links to the various classes are of interest. Thus it is useful to consider two metrics that assess this: dimension connectivity and edge-exclusive dimension connectivity. The former is simply the ratio of the total number of links in a given dimension to the total number of links in every dimension: . The latter assesses, for a given dimension, the number of pairs of nodes connected only by a link in that dimension: . [39]

Burst detection

Burstiness is a well-known phenomenon in many real-world networks, e.g. email or other human communication networks. Additional dimensions of communication provide a more faithful representation of reality and may highlight these patterns or diminish them. Therefore, it is of critical importance that our methods for detecting bursty behavior in networks accommodate multidimensional networks. [51]

Diffusion processes on multilayer networks

Illustration of a random walk on the top of a special multilayer system, i.e. a multiplex network Random walks on multilayer networks.png
Illustration of a random walk on the top of a special multilayer system, i.e. a multiplex network

Diffusion processes are widely used in physics to explore physical systems, as well as in other disciplines as social sciences, neuroscience, urban and international transportation or finance. Recently, simple and more complex diffusive processes have been generalized to multilayer networks. [23] [52] One result common to many studies is that diffusion in multiplex networks, a special type of multilayer system, exhibits two regimes: 1) the weight of inter-layer links, connecting layers each other, is not high enough and the multiplex system behaves like two (or more) uncoupled networks; 2) the weight of inter-layer links is high enough that layers are coupled each other, raising unexpected physical phenomena. [23] It has been shown that there is an abrupt transition between these two regimes. [53]

In fact, all network descriptors depending on some diffusive process, from centrality measures to community detection, are affected by the layer-layer coupling. For instance, in the case of community detection, low coupling (where information from each layer separately is more relevant than the overall structure) favors clusters within layers, whereas high coupling (where information from all layer simultaneously is more relevant than the each layer separately) favors cross-layer clusters. [8] [9]

Random walks

As for unidimensional networks, it is possible to define random walks on the top of multilayer systems. However, given the underlying multilayer structure, random walkers are not limited to move from one node to another within the same layer (jump), but are also allowed to move across layers (switch). [15]

Random walks can be used to explore a multilayer system with the ultimate goal to unravel its mesoscale organization, i.e. to partition it in communities, [8] [9] and have been recently used to better understand navigability of multilayer networks and their resilience to random failures, [15] as well as for exploring efficiently this type of topologies. [54]

In the case of interconnected multilayer systems, the probability to move from a node in layer to node in layer can be encoded into the rank-4 transition tensor and the discrete-time walk can be described by the master equation

where indicates the probability of finding the walker in node in layer at time . [3] [15]

There are many different types of walks that can be encoded into the transition tensor , depending on how the walkers are allowed to jump and switch. For instance, the walker might either jump or switch in a single time step without distinguishing between inter- and intra-layer links (classical random walk), or it can choose either to stay in the current layer and jump, or to switch layer and then jump to another node in the same time step (physical random walk). More complicated rules, corresponding to specific problems to solve, can be found in the literature. [23] In some cases, it is possible to find, analytically, the stationary solution of the master equation. [15] [54]

Classical diffusion

The problem of classical diffusion in complex networks is to understand how a quantity will flow through the system and how much time it will take to reach the stationary state. Classical diffusion in multiplex networks has been recently studied by introducing the concept of supra-adjacency matrix, [55] later recognized as a special flattening of the multilayer adjacency tensor. [3] In tensorial notation, the diffusion equation on the top of a general multilayer system can be written, concisely, as

where is the amount of diffusing quantity at time in node in layer . The rank-4 tensor governing the equation is the Laplacian tensor, generalizing the combinatorial Laplacian matrix of unidimensional networks. It is worth remarking that in non-tensorial notation, the equation takes a more complicated form.

Many of the properties of this diffusion process are completely understood in terms of the second smallest eigenvalue of the Laplacian tensor. It is interesting that diffusion in a multiplex system can be faster than diffusion in each layer separately, or in their aggregation, provided that certain spectral properties are satisfied. [55]

Information and epidemics spreading

Recently, how information (or diseases) spread through a multilayer system has been the subject of intense research. [56] [1] [57] [58] [59]

Multilayer network analysis software

Several software programs focusing on the analysis and visualization of multilayer networks have been introduced. Some popular solutions include multinet (C++ / Python / R), MuxViz (R), Pymnet (Python).

Related Research Articles

Tensor–vector–scalar gravity (TeVeS), developed by Jacob Bekenstein in 2004, is a relativistic generalization of Mordehai Milgrom's Modified Newtonian dynamics (MOND) paradigm.

In theoretical physics, massive gravity is a theory of gravity that modifies general relativity by endowing the graviton with a nonzero mass. In the classical theory, this means that gravitational waves obey a massive wave equation and hence travel at speeds below the speed of light.

Scalar–tensor–vector gravity (STVG) is a modified theory of gravity developed by John Moffat, a researcher at the Perimeter Institute for Theoretical Physics in Waterloo, Ontario. The theory is also often referred to by the acronym MOG.

<span class="mw-page-title-main">Post-Newtonian expansion</span> Method of approximation in general relativity

In general relativity, post-Newtonian expansions are used for finding an approximate solution of Einstein field equations for the metric tensor. The approximations are expanded in small parameters that express orders of deviations from Newton's law of universal gravitation. This allows approximations to Einstein's equations to be made in the case of weak fields. Higher-order terms can be added to increase accuracy, but for strong fields sometimes it is preferable to solve the complete equations numerically. This method is a common mark of effective field theories. In the limit, when the small parameters are equal to 0, the post-Newtonian expansion reduces to Newton's law of gravity.

<span class="mw-page-title-main">Assortativity</span> Tendency for similar nodes to be connected

Assortativity, or assortative mixing, is a preference for a network's nodes to attach to others that are similar in some way. Though the specific measure of similarity may vary, network theorists often examine assortativity in terms of a node's degree. The addition of this characteristic to network models more closely approximates the behaviors of many real world networks.

<span class="mw-page-title-main">Network science</span> Academic field

Network science is an academic field which studies complex networks such as telecommunication networks, computer networks, biological networks, cognitive and semantic networks, and social networks, considering distinct elements or actors represented by nodes and the connections between the elements or actors as links. The field draws on theories and methods including graph theory from mathematics, statistical mechanics from physics, data mining and information visualization from computer science, inferential modeling from statistics, and social structure from sociology. The United States National Research Council defines network science as "the study of network representations of physical, biological, and social phenomena leading to predictive models of these phenomena."

<span class="mw-page-title-main">Google matrix</span> Stochastic matrix representing links between entities

A Google matrix is a particular stochastic matrix that is used by Google's PageRank algorithm. The matrix represents a graph with edges representing links between pages. The PageRank of each page can then be generated iteratively from the Google matrix using the power method. However, in order for the power method to converge, the matrix must be stochastic, irreducible and aperiodic.

<span class="mw-page-title-main">SIC-POVM</span> Type of measurement in quantum mechanics

In the context of quantum mechanics and quantum information theory, symmetric, informationally complete, positive operator-valued measures (SIC-POVMs) are a particular type of generalized measurement (POVM). SIC-POVMs are particularly notable thanks to their defining features of (1) being informationally complete; (2)having the minimal number of outcomes compatible with informational completeness, and (3) being highly symmetric. In this context, informational completeness is the property of a POVM of allowing to fully reconstruct input states from measurement data.

In mathematical physics, the concept of quantum spacetime is a generalization of the usual concept of spacetime in which some variables that ordinarily commute are assumed not to commute and form a different Lie algebra. The choice of that algebra still varies from theory to theory. As a result of this change some variables that are usually continuous may become discrete. Often only such discrete variables are called "quantized"; usage varies.

<span class="mw-page-title-main">CheiRank</span>

The CheiRank is an eigenvector with a maximal real eigenvalue of the Google matrix constructed for a directed network with the inverted directions of links. It is similar to the PageRank vector, which ranks the network nodes in average proportionally to a number of incoming links being the maximal eigenvector of the Google matrix with a given initial direction of links. Due to inversion of link directions the CheiRank ranks the network nodes in average proportionally to a number of outgoing links. Since each node belongs both to CheiRank and PageRank vectors the ranking of information flow on a directed network becomes two-dimensional.

In the field of theoretical physics, the Holst action is an equivalent formulation of the Palatini action for General Relativity (GR) in terms of vierbeins by adding a part of a topological term (Nieh-Yan) which does not alter the classical equations of motion as long as there is no torsion,

<span class="mw-page-title-main">Gluon field strength tensor</span> Second rank tensor in quantum chromodynamics

In theoretical particle physics, the gluon field strength tensor is a second order tensor field characterizing the gluon interaction between quarks.

In theoretical physics, the Curtright field (named after Thomas Curtright) is a tensor quantum field of mixed symmetry, whose gauge-invariant dynamics are dual to those of the general relativistic graviton in higher (D>4) spacetime dimensions. Or at least this holds for the linearized theory. For the full nonlinear theory, less is known. Several difficulties arise when interactions of mixed symmetry fields are considered, but at least in situations involving an infinite number of such fields (notably string theory) these difficulties are not insurmountable.

Bimetric gravity or bigravity refers to two different classes of theories. The first class of theories relies on modified mathematical theories of gravity in which two metric tensors are used instead of one. The second metric may be introduced at high energies, with the implication that the speed of light could be energy-dependent, enabling models with a variable speed of light.

Vasiliev equations are formally consistent gauge invariant nonlinear equations whose linearization over a specific vacuum solution describes free massless higher-spin fields on anti-de Sitter space. The Vasiliev equations are classical equations and no Lagrangian is known that starts from canonical two-derivative Frønsdal Lagrangian and is completed by interactions terms. There is a number of variations of Vasiliev equations that work in three, four and arbitrary number of space-time dimensions. Vasiliev's equations admit supersymmetric extensions with any number of super-symmetries and allow for Yang–Mills gaugings. Vasiliev's equations are background independent, the simplest exact solution being anti-de Sitter space. It is important to note that locality is not properly implemented and the equations give a solution of certain formal deformation procedure, which is difficult to map to field theory language. The higher-spin AdS/CFT correspondence is reviewed in Higher-spin theory article.

Infinite derivative gravity is a theory of gravity which attempts to remove cosmological and black hole singularities by adding extra terms to the Einstein–Hilbert action, which weaken gravity at short distances.

<span class="mw-page-title-main">Joos–Weinberg equation</span> Equation for arbitrary spin particles

In relativistic quantum mechanics and quantum field theory, the Joos–Weinberg equation is a relativistic wave equation applicable to free particles of arbitrary spin j, an integer for bosons or half-integer for fermions. The solutions to the equations are wavefunctions, mathematically in the form of multi-component spinor fields. The spin quantum number is usually denoted by s in quantum mechanics, however in this context j is more typical in the literature.

<span class="mw-page-title-main">Dual graviton</span> Hypothetical particle found in supergravity

In theoretical physics, the dual graviton is a hypothetical elementary particle that is a dual of the graviton under electric-magnetic duality, as an S-duality, predicted by some formulations of eleven-dimensional supergravity.

<span class="mw-page-title-main">Dual photon</span> Hypothetical particle dual to the photon

In theoretical physics, the dual photon is a hypothetical elementary particle that is a dual of the photon under electric–magnetic duality which is predicted by some theoretical models, including M-theory.

A graph neural network (GNN) belongs to a class of artificial neural networks for processing data that can be represented as graphs.

References

  1. 1 2 3 De Domenico, Manlio (2023-08-28). "More is different in real-world multilayer networks". Nature Physics. 19 (9). Springer Science and Business Media LLC: 1247–1262. Bibcode:2023NatPh..19.1247D. doi:10.1038/s41567-023-02132-1. ISSN   1745-2473. S2CID   261322676.
  2. Coscia, Michele; Rossetti, Giulio; Pennacchioli, Diego; Ceccarelli, Damiano; Giannotti, Fosca (2013). ""You know because I know": A multidimensional network approach to human resources problem". Proceedings of the 2013 IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining. Advances in Social Network Analysis and Mining (ASONAM). Vol. 2013. pp. 434–441. arXiv: 1305.7146 . doi:10.1145/2492517.2492537. ISBN   9781450322409. S2CID   1810575.
  3. 1 2 3 4 5 6 7 8 9 10 11 De Domenico, M.; Solé-Ribalta, A.; Cozzo, E.; Kivelä, M.; Moreno, Y.; Porter, M.; Gómez, S.; Arenas, A. (2013). "Mathematical Formulation of Multilayer Networks" (PDF). Physical Review X. 3 (4): 041022. arXiv: 1307.4977 . Bibcode:2013PhRvX...3d1022D. doi:10.1103/PhysRevX.3.041022. S2CID   16611157. Archived from the original (PDF) on 2014-02-25. Retrieved 2016-02-13.
  4. 1 2 3 4 5 6 Battiston, F.; Nicosia, V.; Latora, V. (2014). "Structural measures for multiplex networks". Physical Review E. 89 (3): 032804. arXiv: 1308.3182 . Bibcode:2014PhRvE..89c2804B. doi:10.1103/PhysRevE.89.032804. PMID   24730896. S2CID   13931603.
  5. 1 2 Kivela, M.; Arenas, A.; Barthelemy, M.; Gleeson, J. P.; Moreno, Y.; Porter, M. A. (2014). "Multilayer networks". Journal of Complex Networks. 2 (3): 203–271. arXiv: 1309.7233 . doi:10.1093/comnet/cnu016. S2CID   11390956.
  6. 1 2 3 4 Boccaletti, S.; Bianconi, G.; Criado, R.; del Genio, C. I.; Gómez-Gardeñes, J.; Romance, M.; Sendiña-Nadal, I.; Wang, Z.; Zanin, M. (2014). "The structure and dynamics of multilayer networks". Physics Reports. 544 (1): 1–122. arXiv: 1407.0742 . Bibcode:2014PhR...544....1B. doi:10.1016/j.physrep.2014.07.001. PMC   7332224 . PMID   32834429.
  7. Battiston, Federico; Nicosia, Vincenzo; Latora, Vito (2017-02-01). "The new challenges of multiplex networks: Measures and models". The European Physical Journal Special Topics. 226 (3): 401–416. arXiv: 1606.09221 . Bibcode:2017EPJST.226..401B. doi:10.1140/epjst/e2016-60274-8. ISSN   1951-6355. S2CID   7205907.
  8. 1 2 3 4 5 6 Mucha, P.; et al. (2010). "Community structure in time-dependent, multiscale, and multiplex networks" (PDF). Science. 328 (5980): 876–878. arXiv: 0911.1824 . Bibcode:2010Sci...328..876M. CiteSeerX   10.1.1.749.3504 . doi:10.1126/science.1184819. PMID   20466926. S2CID   10920772.
  9. 1 2 3 4 5 De Domenico, M.; Lancichinetti, A.; Arenas, A.; Rosvall, M. (2015). "Identifying Modular Flows on Multilayer Networks Reveals Highly Overlapping Organization in Interconnected Systems". Physical Review X. 5 (1): 011027. arXiv: 1408.2925 . Bibcode:2015PhRvX...5a1027D. doi:10.1103/PhysRevX.5.011027. S2CID   6364922.
  10. 1 2 3 4 5 6 De Domenico, M.; Sole-Ribalta, A.; Omodei, E.; Gomez, S.; Arenas, A. (2015). "Ranking in interconnected multilayer networks reveals versatile nodes". Nature Communications. 6: 6868. arXiv: 1311.2906 . Bibcode:2015NatCo...6.6868D. doi: 10.1038/ncomms7868 . PMID   25904405.
  11. Battiston, Federico; Iacovacci, Jacopo; Nicosia, Vincenzo; Bianconi, Ginestra; Latora, Vito (2016-01-27). "Emergence of Multiplex Communities in Collaboration Networks". PLOS ONE. 11 (1): e0147451. arXiv: 1506.01280 . Bibcode:2016PLoSO..1147451B. doi: 10.1371/journal.pone.0147451 . ISSN   1932-6203. PMC   4731389 . PMID   26815700.
  12. Rossi, Luca; Dickison, Mark E.; Magnani, Matteo (July 18, 2016). Multilayer Social Networks (1st ed.). Cambridge University Press.
  13. Cardillo, A.; et al. (2013). "Emergence of network features from multiplexity". Scientific Reports. 3: 1344. arXiv: 1212.2153 . Bibcode:2013NatSR...3E1344C. doi:10.1038/srep01344. PMC   3583169 . PMID   23446838.
  14. Gallotti, R.; Barthelemy, M. (2014). "Anatomy and efficiency of urban multimodal mobility". Scientific Reports. 4: 6911. arXiv: 1411.1274 . Bibcode:2014NatSR...4E6911G. doi:10.1038/srep06911. PMC   4220282 . PMID   25371238.
  15. 1 2 3 4 5 6 De Domenico, M.; Sole-Ribalta, A.; Gomez, S.; Arenas, A. (2014). "Navigability of interconnected networks under random failures". PNAS. 111 (23): 8351–8356. Bibcode:2014PNAS..111.8351D. doi: 10.1073/pnas.1318469111 . PMC   4060702 . PMID   24912174.
  16. Stella, M.; Andreazzi, C.S.; Selakovic, S.; Goudarzi, A.; Antonioni, A. (2016). "Parasite spreading in spatial ecological multiplex networks". Journal of Complex Networks. 5 (3): 486–511. arXiv: 1602.06785 . doi:10.1093/comnet/cnw028. S2CID   14398574.
  17. Pilosof, S.; Porter, M.A.; Pascual, M.; Kefi, S. (2017). "The Multilayer Nature of Ecological Networks". Nature Ecology & Evolution. 1 (4): 0101. arXiv: 1511.04453 . Bibcode:2017NatEE...1..101P. doi:10.1038/s41559-017-0101. PMID   28812678. S2CID   11387365.
  18. Timóteo, S.; Correia, M.; Rodríguez-Echeverría, S.; Freitas, H.; Heleno, R. (2018). "Multilayer networks reveal the spatial structure of seed-dispersal interactions across the Great Rift landscapes". Nature Communications. 9 (1): 140. Bibcode:2018NatCo...9..140T. doi:10.1038/s41467-017-02658-y. PMC   5762785 . PMID   29321529.
  19. Costa, J.M.; Ramos, J.A.; Timóteo, S.; da Silva, L.P.; Ceia, R.C.; Heleno, R. (2018). "Species activity promote the stability of fruit-frugivore interactions across a five-year multilayer network". bioRxiv   10.1101/421941 .
  20. Fiori, K. L.; Smith, J; Antonucci, T. C. (2007). "Social network types among older adults: A multidimensional approach". The Journals of Gerontology Series B: Psychological Sciences and Social Sciences. 62 (6): P322–30. doi: 10.1093/geronb/62.6.p322 . PMID   18079416.
  21. Stella, M.; Beckage, N. M.; Brede, M. (2017). "Multiplex lexical networks reveal patterns in early word acquisition in children". Scientific Reports. 21 (7): 619–23. arXiv: 1609.03207 . Bibcode:2017NatSR...746730S. doi:10.1038/srep46730. PMID   5402256. S2CID   13561769.
  22. 1 2 3 De Domenico, M.; Nicosia, V.; Arenas, A.; Latora, V. (2015). "Structural reducibility of multilayer networks". Nature Communications. 6: 6864. arXiv: 1405.0425 . Bibcode:2015NatCo...6.6864D. doi: 10.1038/ncomms7864 . PMID   25904309.
  23. 1 2 3 4 De Domenico, M.; Granell, C.; Porter, Mason A.; Arenas, A. (7 April 2016). "The physics of spreading processes in multilayer networks". Nature Physics. 12 (10): 901–906. arXiv: 1604.02021 . Bibcode:2016NatPh..12..901D. doi:10.1038/nphys3865. S2CID   5063264.
  24. Timme, N.; Ito, S.; Myroshnychenko, M.; Yeh, F.C.; Hiolski, E.; Hottowy, P.; Beggs, J.M. (2014). "Multiplex Networks of Cortical and Hippocampal Neurons Revealed at Different Timescales". PLOS ONE. 9 (12): e115764. Bibcode:2014PLoSO...9k5764T. doi: 10.1371/journal.pone.0115764 . PMC   4275261 . PMID   25536059.
  25. De Domenico, M.; Sasai, S.; Arenas, A. (2016). "Mapping multiplex hubs in human functional brain networks". Frontiers in Neuroscience. 10: 326. arXiv: 1603.05897 . doi: 10.3389/fnins.2016.00326 . PMC   4945645 . PMID   27471443.
  26. Battiston, F.; Nicosia, V.; Chavez, M.; Latora, V. (2017). "Multilayer motif analysis of brain networks". Chaos: An Interdisciplinary Journal of Nonlinear Science. 27 (4): 047404. arXiv: 1606.09115 . Bibcode:2017Chaos..27d7404B. doi:10.1063/1.4979282. PMID   28456158. S2CID   5206551.
  27. De Domenico, M. (2017). "Multilayer modeling and analysis of human brain networks". GigaScience. 6 (5): 1–8. doi:10.1093/gigascience/gix004. PMC   5437946 . PMID   28327916.
  28. 1 2 Bródka, P.; Stawiak, P.; Kazienko, P. (2011). "Shortest Path Discovery in the Multi-layered Social Network". 2011 International Conference on Advances in Social Networks Analysis and Mining. pp. 497–501. arXiv: 1210.5180 . doi:10.1109/ASONAM.2011.67. ISBN   978-1-61284-758-0. S2CID   8279639.
  29. Barrett, L.; Henzi, S. P.; Lusseau, D. (2012). "Taking sociality seriously: The structure of multi-dimensional social networks as a source of information for individuals". Philosophical Transactions of the Royal Society B. 367 (1599): 2108–18. doi:10.1098/rstb.2012.0113. PMC   3385678 . PMID   22734054.
  30. Zignani, Matteo; Quadri, Christian; Gaitto, Sabrina; Gian Paolo Rossi (2014). "Exploiting all phone media? A multidimensional network analysis of phone users' sociality". arXiv: 1401.3126 [cs.SI]. Ch. 4: "Here we introduce the definition of edge-labeled multigraph which can cover many multidimensional situations. To fully leverage the dataset information on the directional nature of the communications, we consider only direct networks without any labels on vertices".
  31. 1 2 Contractor, Noshir; Monge, Peter; Leonardi, Paul M. (2011). "Network Theory: Multidimensional Networks and the Dynamics of Sociomateriality: Bringing Technology Inside the Network". International Journal of Communication. 5: 39.
  32. Magnani, M.; Rossi, L. (2011). "The ML-Model for Multi-layer Social Networks". 2011 International Conference on Advances in Social Networks Analysis and Mining. p. 5. doi:10.1109/ASONAM.2011.114. ISBN   978-1-61284-758-0. S2CID   18528564.
  33. Goffman (1986). Frame analysis: an essay on the organization of experience . Northeastern University Press. ISBN   9780930350918.
  34. Wasserman, Stanley (1994-11-25). Social Network Analysis: Methods and Applications. Cambridge: Cambridge University Press. ISBN   9780521387071.
  35. Leskovec, Jure; Huttenlocher, Daniel; Kleinberg, Jon (2010). "Predicting positive and negative links in online social networks" (PDF). Proceedings of the 19th international conference on World wide web. Vol. 2010. pp. 641–650. arXiv: 1003.2429 . CiteSeerX   10.1.1.154.3679 . doi:10.1145/1772690.1772756. ISBN   9781605587998. S2CID   7119014.
  36. Kazienko, P. A.; Musial, K.; Kukla, E. B.; Kajdanowicz, T.; Bródka, P. (2011). "Multidimensional Social Network: Model and Analysis". Computational Collective Intelligence. Technologies and Applications. Lecture Notes in Computer Science. Vol. 6922. pp. 378–387. doi:10.1007/978-3-642-23935-9_37. ISBN   978-3-642-23934-2.
  37. Nicosia, V.; Bianconi, G.; Nicosia, V.; Barthelemy, M. (2013). "Growing multiplex networks". Physical Review Letters. 111 (5): 058701. arXiv: 1302.7126 . Bibcode:2013PhRvL.111e8701N. doi:10.1103/PhysRevLett.111.058701. PMID   23952453. S2CID   18564513.
  38. 1 2 3 M. Magnani, A. Monreale, G. Rossetti, F. Giannotti: "On multidimensional network measures", SEBD 2013, Rocella Jonica, Italy
  39. 1 2 3 Berlingerio, M.; Coscia, M.; Giannotti, F.; Monreale, A.; Pedreschi, D. (2011). "Foundations of Multidimensional Network Analysis" (PDF). 2011 International Conference on Advances in Social Networks Analysis and Mining. p. 485. CiteSeerX   10.1.1.717.5985 . doi:10.1109/ASONAM.2011.103. ISBN   978-1-61284-758-0. S2CID   14134143.
  40. Battiston, F.; Nicosia, V.; Latora, V. (2016). "Efficient exploration of multiplex networks". New Journal of Physics. 18 (4): 043035. arXiv: 1505.01378 . Bibcode:2016NJPh...18d3035B. doi:10.1088/1367-2630/18/4/043035. S2CID   17690611.
  41. 1 2 Cozzo, Emanuele; Kivelä, Mikko; Manlio De Domenico; Solé, Albert; Arenas, Alex; Gómez, Sergio; Porter, Mason A.; Moreno, Yamir (2015). "Structure of triadic relations in multiplex networks" (PDF). New Journal of Physics. 17 (7): 073029. arXiv: 1307.6780 . Bibcode:2015NJPh...17g3029C. doi:10.1088/1367-2630/17/7/073029. S2CID   2321303.
  42. 1 2 Bródka, Piotr; Kazienko, Przemysław; Musiał, Katarzyna; Skibicki, Krzysztof (2012). "Analysis of Neighbourhoods in Multi-layered Dynamic Social Networks". International Journal of Computational Intelligence Systems. 5 (3): 582–596. arXiv: 1207.4293 . doi:10.1080/18756891.2012.696922. S2CID   1373823.
  43. Jianyong Wang; Zhiping Zeng; Lizhu Zhou (2006). "CLAN: An Algorithm for Mining Closed Cliques from Large Dense Graph Databases" (PDF). 22nd International Conference on Data Engineering (ICDE'06). p. 73. doi:10.1109/ICDE.2006.34. ISBN   978-0-7695-2570-9. S2CID   5474939.
  44. Cai, D.; Shao, Z.; He, X.; Yan, X.; Han, J. (2005). "Community Mining from Multi-relational Networks". Knowledge Discovery in Databases: PKDD 2005. Lecture Notes in Computer Science. Vol. 3721. p. 445. doi:10.1007/11564126_44. ISBN   978-3-540-29244-9.
  45. Berlingerio, M.; Pinelli, F.; Calabrese, F. (2013). "ABACUS: Frequent p Attern mining-BAsed Community discovery in m Ultidimensional networkS". Data Mining and Knowledge Discovery. 27 (3): 294–320. arXiv: 1303.2025 . doi:10.1007/s10618-013-0331-0. S2CID   17320129.
  46. Gauvin, L.; Panisson, A.; Cattuto, C. (2014). "Detecting the community structure and activity patterns of temporal networks: a non-negative tensor factorization approach". PLOS ONE. 9 (1): e86028. arXiv: 1308.0723 . Bibcode:2014PLoSO...986028G. doi: 10.1371/journal.pone.0086028 . PMC   3908891 . PMID   24497935.
  47. Peixoto, T.P. (2015). "Inferring the mesoscale structure of layered, edge-valued, and time-varying networks". Physical Review E. 92 (4): 042807. arXiv: 1504.02381 . Bibcode:2015PhRvE..92d2807P. doi:10.1103/PhysRevE.92.042807. PMID   26565289. S2CID   19407001.
  48. Valles-Català, T.; Massucci, F.; Guimerà, R.; Sales-Pardo, M. (2016). "Multilayer stochastic block models reveal the multilayer structure of complex networks". Physical Review X. 6 (1): 011036. arXiv: 1411.1098 . Bibcode:2016PhRvX...6a1036V. doi: 10.1103/PhysRevX.6.011036 .
  49. Sánchez-García, R.J.; Cozzo, E.; Moreno, Y. (2014). "Dimensionality reduction and spectral properties of multilayer networks". Physical Review E. 89 (5): 052815. arXiv: 1311.1759 . Bibcode:2014PhRvE..89e2815S. doi:10.1103/PhysRevE.89.052815. PMID   25353852. S2CID   15447580.
  50. Zhou, D.; Stanley, H. E.; d’Agostino, G.; Scala, A. (2012). "Assortativity decreases the robustness of interdependent networks". Physical Review E. 86 (6): 066103. arXiv: 1203.0029 . Bibcode:2012PhRvE..86f6103Z. doi:10.1103/PhysRevE.86.066103. PMID   23368000. S2CID   13273722.
  51. Quadri, C.; Zignani, M.; Capra, L.; Gaito, S.; Rossi, G. P. (2014). "Multidimensional Human Dynamics in Mobile Phone Communications". PLOS ONE. 9 (7): e103183. Bibcode:2014PLoSO...9j3183Q. doi: 10.1371/journal.pone.0103183 . PMC   4113357 . PMID   25068479.
  52. Salehi, M.; et al. (2015). "Spreading Processes in Multilayer Networks". IEEE Transactions on Network Science and Engineering. 2 (2): 65–83. arXiv: 1405.4329 . doi:10.1109/TNSE.2015.2425961. S2CID   3197397.
  53. Radicchi, F.; Arenas, A. (2013). "Spreading Processes in Multilayer Networks". Nature Physics. 9 (11): 717–720. arXiv: 1307.4544 . Bibcode:2013NatPh...9..717R. doi:10.1038/nphys2761. S2CID   717835.
  54. 1 2 Battiston, F.; Nicosia, V.; Latora, V. (2016). "Efficient exploration of multiplex networks". New Journal of Physics. 18 (4): 043035. arXiv: 1505.01378 . Bibcode:2016NJPh...18d3035B. doi:10.1088/1367-2630/18/4/043035. S2CID   17690611.
  55. 1 2 Gomez, S.; et al. (2013). "Diffusion dynamics on multiplex networks". Physical Review Letters. 110 (2): 028701. arXiv: 1207.2788 . Bibcode:2013PhRvL.110b8701G. doi:10.1103/PhysRevLett.110.028701. PMID   23383947. S2CID   16280230.
  56. De Domenico, Manlio; Granell, Clara; Porter, Mason A.; Arenas, Alex (2016-08-22). "The physics of spreading processes in multilayer networks". Nature Physics. 12 (10). Springer Science and Business Media LLC: 901–906. arXiv: 1604.02021 . Bibcode:2016NatPh..12..901D. doi:10.1038/nphys3865. ISSN   1745-2473. S2CID   5063264.
  57. Granell, Clara; Gómez, Sergio; Arenas, Alex (2013-09-17). "Dynamical Interplay between Awareness and Epidemic Spreading in Multiplex Networks". Physical Review Letters. 111 (12): 128701. arXiv: 1306.4136 . Bibcode:2013PhRvL.111l8701G. doi:10.1103/PhysRevLett.111.128701. PMID   24093306. S2CID   11083463.
  58. Battiston, Federico; Cairoli, Andrea; Nicosia, Vincenzo; Baule, Adrian; Latora, Vito (2016-06-01). "Interplay between consensus and coherence in a model of interacting opinions". Physica D: Nonlinear Phenomena. Nonlinear Dynamics on Interconnected Networks. 323–324: 12–19. arXiv: 1506.04544 . Bibcode:2016PhyD..323...12B. doi:10.1016/j.physd.2015.10.013. S2CID   16442344.
  59. Battiston, Federico; Nicosia, Vincenzo; Latora, Vito; Miguel, Maxi San (2016-06-17). "Robust multiculturality emerges from layered social influence". arXiv: 1606.05641 [physics.soc-ph].