# Investigating Mesoscale Structures of Networks via Transport Properties

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Investigating Mesoscale Structures of Networks via Transport Properties

seminar at National Institute for Mathematical Sciences (NIMS), December 10, 2014.

SHL and P. Holme, Pathlength scaling in graphs with incomplete navigational information, Physica A 390, 3996 (2011); Exploring Maps with Greedy Navigators, Phys. Rev. Lett. 108, 128701 (2012). SHL, M. Cucuringu, and M. A. Porter, Density-based and transport-based core-periphery structures in networks, Phys. Rev. E 89, 032810 (2014); SHL, M. D. Fricker, and M. A. Porter, Mesoscale Analyses of Fungal Networks, e-print arXiv:1406.5855; M. Cucuringu, M. P. Rombach, SHL, and M. A. Porter, Detection of Core-Periphery Structure in Networks Using Spectral Methods and Geodesic Paths, e-print arXiv:1410.6572; SHL, D. Kim, and H. Jeong, Is Nestedness in Networks Generalized Core-Periphery Structures?, in preparation. SHL, M. Farazmand, G. Haller, and M. A. Porter, Finding Lagrangian Coherent Structures Using Community Detection, in preparation.

Sang Hoon Lee Department of Energy Science, Sungkyunkwan University

http://sites.google.com/site/lshlj82

statistical physics: micro interactions macro

regular/random networks (interactions)

magnet gas

microscale structure

macroscale properties

microscale structure

macroscale properties

irregular, or complex (partially random) networks

How about this? Something new but ubiquitous topology

ref) M. E. J. Newman, Phys. Rev. E 74, 036104 (2006).

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Collaborations Between Network Scientists

This figure shows a network of collaborationsbetween scientists working on networks. Itwas compiled from the bibliographies of tworeview articles, by M. Newman (SIAM Review2003) and by S. Boccaletti et al. (Physics Re-ports 2006). Vertices represent scientists whosenames appear as authors of papers in those bib-liographies and an edge joins any two whosenames appear on the samepaper. A small num-ber of other references were added by handto bring the network up to date. This figureshows the largest component of the resultingnetwork, which contains 379 individuals. Sizesof vertices are proportional to their so-calledcommunity centrality. Colors represent ver-tex degrees with redder vertices having higherdegree.

a snapshot of network of network scientists

They are everywhere, indeed.

the Internet biochemical network

brain

ad infinitum

The most complicated system in the universe known to itself

microscale structure: neuron

macroscopic structure: brain or cognition

Volume 12, Number 6, 2006 THE NEUROSCIENTIST 521

with its growth by creation of new nodes, which preferen-tially form connections to existing hubs. One fMRI studyhas reported a power law degree distribution for a func-tional network of activated voxels (Eguluz and others2005). But the degree distribution of whole-brain fMRInetworks of cortical regions has also been described asan exponentially truncated power law (Achard and oth-ers 2006), meaning broadly that the probability of veryhighly connected hubs is less in the brain than in the

WWW, but there is more probability of a hub in thebrain than in a random graph. The hubs of this networkwere predominantly regions of the heteromodal and uni-modal association cortex.

Truncated power law degree distributions are wide-spread in complex systems that are physically embeddedor constrained, such as transport or infrastructural net-works, and in systems in which nodes have a finite life span,such as the social network of collaborating Hollywood

Fig. 6. Small-world functional brain networks (Achard and others 2006). Anatomical map of a small-world human brainfunctional network created by thresholding the scale 4 wavelet correlation matrix representing functional connectivity inthe frequency interval 0.03 to 0.06 Hz. A, Four hundred five undirected edges, ~10% of the 4005 possible interregionalconnections, are shown in a sagittal view of the right side of the brain. Nodes are located according to the y and z coor-dinates of the regional centroids in Talairach space. Edges representing connections between nodes separated by aEuclidean distance 7.5 cm are blue. B, Degree distribution of a small-world brain functional network. Plot of the log of the cumulative prob-ability of degree, log(P(ki)), versus log of degree, log(ki). The plus sign indicates observed data, the solid line is the best-fitting exponentially truncated power law, the dotted line is an exponential, and the dashed line is a power law. C, Resilience of the human brain functional network (right column) compared with random (left column) and scale-free(middle column) networks. Size of the largest connected cluster in the network (scaled to maximum; y axis) versus theproportion of tota

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