Entropies of complex networks with hierarchically constrained topologies

Bianconi, Ginestra; Coolen, Anthony C. C.; Vicente, C. J. Pérez

doi:10.1103/physreve.78.016114

Cited by 61 publications

(120 citation statements)

References 44 publications

(64 reference statements)

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“…(64). Using a similar derivation as the one reported in [33,35] it is possible to prove that for sparse networks Ω is given by…”

Section: Multiplex Ensemble With Given Multidegree Sequencementioning

confidence: 95%

“…(6) and the expression for P C ( G) given by Eq. (33) it is easy to show that the entropy of the canonical multiplex ensemble S, that we call Shannon entropy, is given by…”

Section: Multiplex Ensemble With Given Expected Degree Sequence In Eamentioning

confidence: 99%

“…For single networks an equilibrium statistical mechanics framework has been recently formulated [30][31][32][33][34][35][36][37][38][39][40][41][42] in order to characterize network ensembles. A network ensemble is defined as a set of networks that satisfy a given number of structural constraints, i.e.…”

Section: Introductionmentioning

confidence: 99%

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Statistical mechanics of multiplex networks: Entropy and overlap

2013

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There is growing interest in multiplex networks where individual nodes take part in several layers of networks simultaneously. This is the case for example in social networks where each individual node has different kind of social ties or transportation systems where each location is connected to another location by different types of transport. Many of these multiplex are characterized by a significant overlap of the links in different layers. In this paper we introduce a statistical mechanics framework to describe multiplex ensembles. A multiplex is a system formed by N nodes and M layers of interactions where each node belongs to the M layers at the same time. Each layer α is formed by a network G α . Here we introduce the concept of correlated multiplex ensembles in which the existence of a link in one layer is correlated with the existence of a link in another layer. This implies that a typical multiplex of the ensemble can have a significant overlap of the links in the different layers. Moreover we characterize microcanonical and canonical multiplex ensembles satisfying respectively hard and soft constraints and we discuss how to construct multiplex in these ensembles. Finally we provide the expression for the entropy of these ensembles that can be useful to address different inference problems involving multiplexes.

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“…(64). Using a similar derivation as the one reported in [33,35] it is possible to prove that for sparse networks Ω is given by…”

Section: Multiplex Ensemble With Given Multidegree Sequencementioning

confidence: 95%

“…(6) and the expression for P C ( G) given by Eq. (33) it is easy to show that the entropy of the canonical multiplex ensemble S, that we call Shannon entropy, is given by…”

Section: Multiplex Ensemble With Given Expected Degree Sequence In Eamentioning

confidence: 99%

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Statistical mechanics of multiplex networks: Entropy and overlap

2013

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“…In extending this theory to evolving graphs we are motivated by the necessity to bridge an existing gap between the static and the dynamical treatment of graphs. While many statistical aspects of random graph topology are now understood (like the influence of topology on processes occurring on graphs, percolation and critical phenomena, loop statistics, or the entropies of different topologies in various random graph ensembles [3][4][5][6][7][8][9][10]), much less work has been invested in the mathematical study of the dynamics of graphical structures (see [11][12][13] for recent examples). Besides their mathematical interest, dynamical problems are prominent in application areas where the issue of sampling uniformly the space of graphs with certain prescribed macroscopic properties is vital.…”

Section: Introductionmentioning

confidence: 99%

Constrained Markovian Dynamics of Random Graphs

2009

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We introduce a statistical mechanics formalism for the study of constrained graph evolution as a Markovian stochastic process, in analogy with that available for spin systems, deriving its basic properties and highlighting the role of the 'mobility' (the number of allowed moves for any given graph). As an application of the general theory we analyze the properties of degree-preserving Markov chains based on elementary edge switchings. We give an exact yet simple formula for the mobility in terms of the graph's adjacency matrix and its spectrum. This formula allows us to define acceptance probabilities for edge switchings, such that the Markov chains become controlled Glauber-type detailed balance processes, designed to evolve to any required invariant measure (representing the asymptotic frequencies with which the allowed graphs are visited during the process). As a corollary we also derive a condition in terms of simple degree statistics, sufficient to guarantee that, in the limit where the number of nodes diverges, even for state-independent acceptance probabilities of proposed moves the invariant measure of the process will be uniform. We test our theory on synthetic graphs and on realistic larger graphs as studied in cellular biology.

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“…Many different entropy measures have been developed in the context of complex networks [15][16][17][18][19][20][21][22][23]. For example, Bianconi [17] proposes entropy as an indicator to assess the role of each structural feature in a given real network, and observes that the ensembles with fixed scale-free degree distribution have smaller entropy than the ensembles with homogeneous degree distribution, indicating a higher level of order in scale-free networks.…”

Section: Introductionmentioning

confidence: 99%

Network Entropies of the Chinese Financial Market

2016

Entropy

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Based on the data from the Chinese financial market, this paper focuses on analyzing three types of network entropies of the financial market, namely, Shannon, Renyi and Tsallis entropies. The findings suggest that Shannon entropy can reflect the volatility of the financial market, that Renyi and Tsallis entropies also have this function when their parameter has a positive value, and that Renyi and Tsallis entropies can reflect the extreme case of the financial market when their parameter has a negative value.

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Entropies of complex networks with hierarchically constrained topologies

Cited by 61 publications

References 44 publications

Statistical mechanics of multiplex networks: Entropy and overlap

Statistical mechanics of multiplex networks: Entropy and overlap

Constrained Markovian Dynamics of Random Graphs

Network Entropies of the Chinese Financial Market

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