Modular Hierarchical Random Networks: Characteristics and Dynamics

Benjamin F. Maier†#,∗ Chen Li†#, Cristián Huepe‡?, and Dirk Brockmann†#†Robert Koch-Institute, Seestraße 10, 13353 Berlin, Germany

#Institute for Theoretical Biology, Humboldt-Universität zu Berlin, Germany‡CHuepe Labs, 954 West 18th Place, Chicago, Illinois 60608, USA and

?Northwestern Institute on Complex Systems, Northwestern University, Evanston, Illinois 60208, USA(Dated: February 25, 2015)

FIG. 1. A generic modular hierarchical random network.

A generic feature of many natural networks is their mod-ular hierarchical structure, nodes naturally fall into groupsof typical scale that can be associated with different lay-ers of the hierarchy. This generic feature is present inmetabolic networks, gene regulatory networks, neural net-works, transportation networks as well as foodwebs [1–4].However, it is still poorly understood why these structuresare so abundant in real world networks. Furthermore, thedegree to which these structures are pronounced in natu-ral networks seems to vary considerably, yet it is unclearhow to define a structural parameter that can express thedegree of modular hierarchical structure in a single orderor structural parameter. A variety of methods have beendevised to extract both hierarchical and modular aspectsfrom natural network data. Interestingly, the developmentof computational recipes for constructing artificial randommodels in which modular hierarchical features can be pre-specified and which can be used as reference models hasreceived surprisingly little attention. Exceptions are meth-ods for generating regular lattice type networks that also

include scale free degree distributions (e.g. [1]). However,a number of natural networks exist that possess modularhierarchical structure with nodes that show little variationin degree and it is important to understand how these net-works behave without the strong impact that degree inho-mogeneities may impose.To clarify some of these questions we designed a minimalmodel for generating random modular hierarchical (MH)networks. The model is based on the simple notion that,from a node’s perspective, the probability of linking to an-other node decreases systematically with the layer depthof a hierarchical template. The method is thus similar torecently introduced network analytical methods and sam-pling techniques [2, 3]. Our model generates random net-works with little degree variation and the strength of mod-ular hierarchical topology is controlled by a single orderparameter without interfering with the network’s degreedistribution (for an interactive visualization of the param-eter’s influence on the generated network’s structure, seehttp://rocs.hu-berlin.de/D3/mhn/). It allows to distinguishbetween strong and weak modular hierarchy and containsordinary random graphs as a limiting case. We propose thismodel as a generic reference model to test the properties ofdifferent dynamical processes, e.g. contagion phenomena,synchronization, etc. as a function of the MH order param-eter.We provide analytical formulas for the degree distributionsof the single layers, as well as for the total degree distribu-tion’s generating function and its first moments. In analogyto [1], we show that even though the degree distribution isnot scale free, the clustering coefficient C(k) may indeedfollow a power law. We discuss a fast algorithm for gener-ating these networks in O(m) time with m being the num-ber of generated edges, allowing the construction of largescale MH networks. This algorithm is of general natureand able to generate networks for arbitrary underlying hi-erarchical trees in O(m) time where m can be computedanalytically for a given tree.Finally, we investigate the spread of generic contagion pro-cesses on such networks and find that the epidemic thresh-old decreases with increasing MH structure. This finding issupported with an approximate analytical formula for theepidemic threshold in such networks.

[1] E. Ravasz, A.-L. Barabási, et al., Science 297, 1551 (2002).[2] M. Sales-Pardo, et al., PNAS 104, 15224 (2007).

[3] A. Clauset, et al., Nature 453, 98 (2008).[4] T. P. Peixoto, Physical Review X 4, 011047 (2014).

∗ bfmaier@physik.hu-berlin.de