Networks often exhibit structure at disparate scales. We propose a method for identifying community structure at different scales based on multiresolution modularity and consensus clustering. Our contribution consists of two parts. First, we propose a strategy for sampling the entire range of possible resolutions for the multiresolution modularity quality function. Our approach is directly based on the properties of modularity and, in particular, provides a natural way of avoiding the need to increase the resolution parameter by several orders of magnitude to break a few remaining small communities, necessitating the introduction of ad-hoc limits to the resolution range with standard sampling approaches. Second, we propose a hierarchical consensus clustering procedure, based on a modified modularity, that allows one to construct a hierarchical consensus structure given a set of input partitions. While here we are interested in its application to partitions sampled using multiresolution modularity, this consensus clustering procedure can be applied to the output of any clustering algorithm. As such, we see many potential applications of the individual parts of our multiresolution consensus clustering procedure in addition to using the procedure itself to identify hierarchical structure in networks.

It is common in the study of networks to investigate intermediate-sized (or “meso-scale”) features to try to gain an understanding of network structure and function. For example, numerous algorithms have been developed to try to identify “communities,” which are typically construed as sets of nodes with denser connections internally than with the remainder of a network. In this paper, we adopt a complementary perspective that “communities” are associated with bottlenecks of locally-biased dynamical processes that begin at seed sets of nodes, and we employ several different community-identification procedures (using diffusion-based and geodesic-based dynamics) to investigate community quality as a function of community size. Using several empirical and synthetic networks, we identify several distinct scenarios for “size-resolved community structure” that can arise in real (and realistic) networks: (i) the best small groups of nodes can be better than the best large groups (for a given formulation of the idea of a good community); (ii) the best small groups can have a quality that is comparable to the best medium-sized and large groups; and (iii) the best small groups of nodes can be worse than the best large groups. As we discuss in detail, which of these three cases holds for a given network can make an enormous difference when investigating and making claims about network community structure, and it is important to take this into account to obtain reliable downstream conclusions. Depending on which scenario holds, one may or may not be able to successfully identify “good” communities in a given network (and good communities might not even exist for a given community quality measure), the manner in which different small communities fit together to form meso-scale network structures can be very different, and processes such as viral propagation and information diffusion can exhibit very different dynamics. In addition, our results suggest that, for many large realistic networks, the output of locally-biased methods that focus on communities that are centered around a given seed node might have better conceptual grounding and greater practical utility than the output of global community-detection methods. They also illustrate subtler structural properties that are important to consider in the development of better benchmark networks to test methods for community detection.

Alzheimer's disease is considered a disconnection syndrome, motivating the use of brain network measures to detect changes in whole-brain resting state functional connectivity (FC). We investigated changes in FC within and among resting state networks (RSN) across four different stages in the Alzheimer's disease continuum. FC changes were examined in two independent cohorts of individuals (84 and 58 individuals, respectively) each comprising control, subjective cognitive decline, mild cognitive impairment and Alzheimer's dementia groups. For each participant, FC was computed as a matrix of Pearson correlations between pairs of time series from 278 gray matter brain regions. We determined significant differences in FC modular organization with two distinct approaches, network contingency analysis and multiresolution consensus clustering. Network contingency analysis identified RSN sub-blocks that differed significantly across clinical groups. Multiresolution consensus clustering identified differences in the stability of modules across multiple spatial scales. Significant modules were further tested for statistical association with memory and executive function cognitive domain scores. Across both analytic approaches and in both participant cohorts, the findings converged on a pattern of FC that varied systematically with diagnosis within the frontoparietal network (FP) and between the FP network and default mode network (DMN). Disturbances of modular organization were manifest as greater internal coherence of the FP network and stronger coupling between FP and DMN, resulting in less segregation of these two networks. Our findings suggest that the pattern of interactions within and between specific RSNs offers new insight into the functional disruption that occurs across the Alzheimer's disease spectrum.

The endbrain (telencephalon) is at the rostral end of the central nervous system and is primarily responsible for supporting cognition and affect. Structurally, it consists of right and left cerebral hemispheres, each parceled into multiple cortical and nuclear gray matter regions. The global network organization of axonal macroconnections between the 244 regions forming the endbrain was analyzed with a multiresolution consensus clustering (MRCC) method that provides a hierarchical description of community clustering (modules or subsystems) within the network. Experimental evidence was collated from the neuroanatomical literature for the existence of 10,002 of a possible 59,292 connections within the network, and they cluster into four top-level subsystems and 60 bottom-level subsystems arranged in a 50-level hierarchy. Two top-level subsystems are bihemispheric: One deals with auditory and visual information, and the other corresponds broadly to the default mode network. The other two top-level subsystems are bilaterally symmetrical, and each deals broadly with somatic and visceral information. Because the entire endbrain connection matrix was assembled from multiple subconnectomes, it was easy to show that the status of a region as a connectivity hub is not absolute but, instead, depends on the size and coverage of its anatomical neighborhood. It was also shown numerically that creating an ultradense connection matrix by converting all "absent" connections to a "very weak" connection weight has virtually no effect on the clustering hierarchy. The next logical step in this project is to complete the forebrain connectome by adding the thalamus and hypothalamus (together, the interbrain) to the endbrain analysis.

The analysis of multilayer networks is among the most active areas of network science, and there are now several methods to detect dense "communities" of nodes in multilayer networks. One way to define a community is as a set of nodes that trap a diffusion-like dynamical process (usually a random walk) for a long time. In this view, communities are sets of nodes that create bottlenecks to the spreading of a dynamical process on a network. We analyze the local behavior of different random walks on multiplex networks (which are multilayer networks in which different layers correspond to different types of edges) and show that they have very different bottlenecks that hence correspond to rather different notions of what it means for a set of nodes to be a good community. This has direct implications for the behavior of community-detection methods that are based on these random walks.