Finding communities in sparse networks

Abstract

Spectral algorithms based on matrix representations of networks are often used to detect communities, but classic spectral methods based on the adjacency matrix and its variants fail in sparse networks. New spectral methods based on non-backtracking random walks have recently been introduced that successfully detect communities in many sparse networks. However, the spectrum of non-backtracking random walks ignores hanging trees in networks that can contain information about their community structure. We introduce the reluctant backtracking operators that explicitly account for hanging trees as they admit a small probability of returning to the immediately previous node, unlike the non-backtracking operators that forbid an immediate return. We show that the reluctant backtracking operators can detect communities in certain sparse networks where the non-backtracking operators cannot, while performing comparably on benchmark stochastic block model networks and real world networks. We also show that the spectrum of the reluctant backtracking operator approximately optimises the standard modularity function. Interestingly, for this family of non- and reluctant-backtracking operators the main determinant of performance on real-world networks is whether or not they are normalised to conserve probability at each node.

A Hybrid Mineral Battery: Energy Storage and Dissolution Behavior of CuFeS2 in a Fixed Bed Flow Cell

“…The rapid decrease in the specific capacitance at high scan rate (Figure 6c)a lso confirmed the slow kinetic response of the electrode materials. [59] This behavior suggested that the current density for charging and discharging of the proposed setup should always be lower than the maximum current density obtained in the CV scan at as low scan rate (0.001 Vs À1 ). In this way,the maximum faradaic response of the electrode materials can be ensuredi nt he FBFC setup.…”

“…However, based on the literature, the quasireversible minor redox peaks centered at approximately 0.55 Vi nb oth the CFe andC FeCu systems represented the pseudocapacitive behavior of the GF owing to the presenceo fs urfacef unctional groups. [51,59] To verify the capacitive behavior of the proposed systems and to estimate the coulombic (h C )a nd energy (h E )e fficiencies of the process, galvanostaticc harge-discharge (GCD) cyclic tests were performed from 0t o1 .05 V. The charging and dischargingw as performed at 200 mA g À1 and 150 mA g À1 ,r espectively,and potential profiles for 500 cycles are provided for comparison in Figure 7. Analogous to the CV scans, the asym-metricalG CD plots reconfirmed the pseudocapacitive behavior of the proposed FBFC systems.…”

Decoupling of timescales reveals sparse convergent CPG network in the adult spinal cord

“…Here, hubs 29 and modules may serve as intermediaries for supraspinal input. Finding modules in sparse networks is generally difficult 49 , but an obvious location to inspect is where the corticospinal projections terminate. The issue of how such few supraspinal fibers can control motor behavior remains to be investigated and understood from a network perspective.…”

Childhood maltreatment is associated with alteration in global network fiber-tract architecture independent of history of depression and anxiety

“…It demonstrates the relative frequencies of the eigenvalues for a normalized Laplacian matrix and illustrates the characteristics of the network structure at the level of the global system as well as the dynamic interactions among the local network structures (Banerjee, 2012; Banerjee and Jost, 2007). Furthermore, it is suitable for describing the community structure of the network (Cheng et al, 2015) and has successfully been applied to real world data including biological networks (A. Singh and Humphries, 2015).…”