Localization in random bipartite graphs: Numerical and empirical study

Slanina, František

doi:10.1103/physreve.95.052149

Cited by 8 publications

(6 citation statements)

References 106 publications

(173 reference statements)

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“…The very different nature of localization behavior in modular and anti-modular net-works is reflected in the localization-delocalization transition seen for the principal eigenmode (associated with the largest eigenvalue λ N −1 of L ) as r is varied [Fig 4 (k)]. Thus, as the mesoscopic nature of the network changes from modular to anti-modular, we observe that the eigenmode becomes completely delocalized (IP R → 1/N as r diverges), irrespective of the extent of heterogeneity in module sizes (similar to transitions seen in the spectral behavior of network adjacency matrices [31]).…”

mentioning

confidence: 58%

Anti-modular nature of partially bipartite networks makes them infra small-world

Singh¹,

Ashraf²,

Sinha³

2019

Preprint

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Strong inter-dependence in complex systems can manifest as partially bipartite networks characterized by interactions occurring primarily between distinct groups of nodes (identified as modules). In this paper, we show that the anti-modular character of such networks, e.g., those defined by the adjacent occurrence of alphabetic characters in corpora of natural language texts, can result in striking structural properties which place them outside the well-known regular/small-world/random network paradigm. Using an ensemble of model networks whose modularity can be tuned, we demonstrate that strong module size heterogeneity in anti-modular random networks imparts them with higher communication efficiency and lower clustering than their randomized counterparts, making them infra small-world. Passage to anti-modularity is associated with characteristic changes in spectral properties of the network, including a delocalization transition exhibited by the principal eigenvector (PEV) of the normalized Laplacian. This is accompanied by the emergence of prominent bimodality in the distribution of PEV components, which can function as a signature for identifying anti-modular organization in empirical networks.

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confidence: 58%

Anti-modular nature of partially bipartite networks makes them infra small-world

Singh¹,

Ashraf²,

Sinha³

2019

Preprint

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“…However, we know that many real-world networks follow power-law degree distributions and thus contain several large degree nodes, naturally forming imperfect wheel graph (i.e., star, friendship, etc.). Our study offers a platform to understand PEV localization behaviors of real-world systems, as well as to relate them with the network's structural properties by providing fundamental insight to localization and delocalization behavior of eigenvectors of networks [42,43]. Furthermore, since eigenvectors and eigenvalues provide information [44] for energy controllability and synchronization of complex networks [45,46], the investigation carried out here for PEV of adjacency matrix can be extended for finding localization of eigenvector for other matrix representations of networks.…”

Section: Resultsmentioning

confidence: 94%

Principal eigenvector localization and centrality in networks: Revisited

Pradhan

Angeliya

Jalan

2020

Physica A: Statistical Mechanics and its Applications

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Complex networks or graphs provide a powerful framework to understand importance of individuals and their interactions in real-world complex systems. Several graph theoretical measures have been introduced to access importance of the individual in systems represented by networks. Particularly, eigenvector centrality (EC) measure has been very popular due to its ability in measuring importance of the nodes based on not only number of interactions they acquire but also particular structural positions they have in the networks. Furthermore, the presence of certain structural features, such as the existence of high degree nodes in a network is recognized to induce localization transition of the principal eigenvector (PEV) of the network's adjacency matrix. Localization of PEV has been shown to cause difficulties in assigning centrality weights to the nodes based on the EC. We revisit PEV localization and its relation with failure of EC problem, and by using simple model networks demonstrate that in addition to the localization of the PEV, the delocalization of PEV may also create difficulties for using EC as a measure to rank the nodes. Our investigation while providing fundamental insight to the relation between PEV localization and centrality of nodes in networks, suggests that for the networks having delocalized PEVs, it is better to use degree centrality measure to rank the nodes.

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“…Constructing networks whose edges are determined or weighted by inter-particle similarities may be particularly useful for achieving a better understanding of mesoscale physics in polydisperse packings, which are thought to depend on the spatial distributions of particles of different types [301]. A perhaps nonintuitive choice is to use a bipartite representation of a granular network, such as the approach used in [302].…”

Section: Definitions Of Nodes and Edgesmentioning

confidence: 99%

Network Analysis of Particles and Grains

Papadopoulos,

Porter,

Daniels

et al. 2017

Preprint

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The arrangements of particles and forces in granular materials have a complex organization on multiple spatial scales that ranges from local structures to mesoscale and system-wide ones. This multiscale organization can affect how a material responds or reconfigures when exposed to external perturbations or loading. The theoretical study of particle-level, force-chain, domain, and bulk properties requires the development and application of appropriate physical, mathematical, statistical, and computational frameworks. Traditionally, granular materials have been investigated using particulate or continuum models, each of which tends to be implicitly agnostic to multiscale organization. Recently, tools from network science have emerged as powerful approaches for probing and characterizing heterogeneous architectures across different scales in complex systems, and a diverse set of methods have yielded fascinating insights into granular materials. In this paper, we review work on network-based approaches to studying granular matter and explore the potential of such frameworks to provide a useful description of these systems and to enhance understanding of their underlying physics. We also outline a few open questions and highlight particularly promising future directions in the analysis and design of granular matter and other kinds of material networks.

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Localization in random bipartite graphs: Numerical and empirical study

Cited by 8 publications

References 106 publications

Anti-modular nature of partially bipartite networks makes them infra small-world

Anti-modular nature of partially bipartite networks makes them infra small-world

Principal eigenvector localization and centrality in networks: Revisited

Network Analysis of Particles and Grains

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