Showing papers on "Modularity (networks) published in 2013"

Mathematical Formulation of Multilayer Networks

Abstract: A network representation is useful for describing the structure of a large variety of complex systems. However, most real and engineered systems have multiple subsystems and layers of connectivity, and the data produced by such systems are very rich. Achieving a deep understanding of such systems necessitates generalizing ‘‘traditional’’ network theory, and the newfound deluge of data now makes it possible to test increasingly general frameworks for the study of networks. In particular, although adjacency matrices are useful to describe traditional single-layer networks, such a representation is insufficient for the analysis and description of multiplex and time-dependent networks. One must therefore develop a more general mathematical framework to cope with the challenges posed by multilayer complex systems. In this paper, we introduce a tensorial framework to study multilayer networks, and we discuss the generalization of several important network descriptors and dynamical processes—including degree centrality, clustering coefficients, eigenvector centrality, modularity, von Neumann entropy, and diffusion—for this framework. We examine the impact of different choices in constructing these generalizations, and we illustrate how to obtain known results for the special cases of single-layer and multiplex networks. Our tensorial approach will be helpful for tackling pressing problems in multilayer complex systems, such as inferring who is influencing whom (and by which media) in multichannel social networks and developing routing techniques for multimodal transportation systems.

Mathematical Formulation of Multi-Layer Networks

Abstract: A network representation is useful for describing the structure of a large variety of complex systems However, most real and engineered systems have multiple subsystems and layers of connectivity, and the data produced by such systems is very rich Achieving a deep understanding of such systems necessitates generalizing "traditional" network theory, and the newfound deluge of data now makes it possible to test increasingly general frameworks for the study of networks In particular, although adjacency matrices are useful to describe traditional single-layer networks, such a representation is insufficient for the analysis and description of multiplex and time-dependent networks One must therefore develop a more general mathematical framework to cope with the challenges posed by multi-layer complex systems In this paper, we introduce a tensorial framework to study multi-layer networks, and we discuss the generalization of several important network descriptors and dynamical processes --including degree centrality, clustering coefficients, eigenvector centrality, modularity, Von Neumann entropy, and diffusion-- for this framework We examine the impact of different choices in constructing these generalizations, and we illustrate how to obtain known results for the special cases of single-layer and multiplex networks Our tensorial approach will be helpful for tackling pressing problems in multi-layer complex systems, such as inferring who is influencing whom (and by which media) in multichannel social networks and developing routing techniques for multimodal transportation systems

A smart local moving algorithm for large-scale modularity-based community detection

Abstract: We introduce a new algorithm for modularity-based community detection in large networks. The algorithm, which we refer to as a smart local moving algorithm, takes advantage of a well-known local moving heuristic that is also used by other algorithms. Compared with these other algorithms, our proposed algorithm uses the local moving heuristic in a more sophisticated way. Based on an analysis of a diverse set of networks, we show that our smart local moving algorithm identifies community structures with higher modularity values than other algorithms for large-scale modularity optimization, among which the popular “Louvain algorithm”. The computational efficiency of our algorithm makes it possible to perform community detection in networks with tens of millions of nodes and hundreds of millions of edges. Our smart local moving algorithm also performs well in small and medium-sized networks. In short computing times, it identifies community structures with modularity values equally high as, or almost as high as, the highest values reported in the literature, and sometimes even higher than the highest values found in the literature.

The evolutionary origins of modularity

Abstract: A central biological question is how natural organisms are so evolvable (capable of quickly adapting to new environments). A key driver of evolvability is the widespread modularity of biological networks—their organization as functional, sparsely connected subunits—but there is no consensus regarding why modularity itself evolved. Although most hypotheses assume indirect selection for evolvability, here we demonstrate that the ubiquitous, direct selection pressure to reduce the cost of connections between network nodes causes the emergence of modular networks. Computational evolution experiments with selection pressures to maximize network performance and minimize connection costs yield networks that are significantly more modular and more evolvable than control experiments that only select for performance. These results will catalyse research in numerous disciplines, such as neuroscience and genetics, and enhance our ability to harness evolution for engineering purposes.

Robust detection of dynamic community structure in networks.

Abstract: We describe techniques for the robust detection of community structure in some classes of time-dependent networks. Specifically, we consider the use of statistical null models for facilitating the principled identification of structural modules in semi-decomposable systems. Null models play an important role both in the optimization of quality functions such as modularity and in the subsequent assessment of the statistical validity of identified community structure. We examine the sensitivity of such methods to model parameters and show how comparisons to null models can help identify system scales. By considering a large number of optimizations, we quantify the variance of network diagnostics over optimizations (“optimization variance”) and over randomizations of network structure (“randomization variance”). Because the modularity quality function typically has a large number of nearly degenerate local optima for networks constructed using real data, we develop a method to construct representative partitions that uses a null model to correct for statistical noise in sets of partitions. To illustrate our results, we employ ensembles of time-dependent networks extracted from both nonlinear oscillators and empirical neuroscience data.

Detecting modules in quantitative bipartite networks: the QuaBiMo algorithm

Abstract: Ecological networks are often composed of different sub-communities (often referred to as modules) Identifying such modules has the potential to develop a better understanding of the assembly of ecological communities and to investigate functional overlap or specialisation The most informative form of networks are quantitative or weighted networks Here we introduce an algorithm to identify modules in quantitative bipartite (or two-mode) networks It is based on the hierarchical random graphs concept of Clauset et al (2008 Nature 453: 98-101) and is extended to include quantitative information and adapted to work with bipartite graphs We define the algorithm, which we call QuaBiMo, sketch its performance on simulated data and illustrate its potential usefulness with a case study

Spectral methods for community detection and graph partitioning.

Abstract: We consider three distinct and well-studied problems concerning network structure: community detection by modularity maximization, community detection by statistical inference, and normalized-cut graph partitioning. Each of these problems can be tackled using spectral algorithms that make use of the eigenvectors of matrix representations of the network. We show that with certain choices of the free parameters appearing in these spectral algorithms the algorithms for all three problems are, in fact, identical, and hence that, at least within the spectral approximations used here, there is no difference between the modularity- and inference-based community detection methods, or between either and graph partitioning.

Evolutionary Covariation in Geometric Morphometric Data: Analyzing Integration, Modularity, and Allometry in a Phylogenetic Context

Abstract: Quantifying integration and modularity of evolutionary changes in morphometric traits is crucial for understanding how organismal shapes evolve. For this purpose, comparative studies are necessary, which need to take into account the phylogenetic structure of interspecific data. This study applies several of the standard tools of geometric morphometrics, which mostly have been used in intraspecific studies, in the new context of analyzing integration and modularity based on comparative data. Morphometric methods such as principal component analysis, multivariate regression, partial least squares, and modularity tests can be applied to phylogenetically independent contrasts of shape data. We illustrate this approach in an analysis of cranial evolution in 160 species from all orders of birds. Mapping the shape information onto the phylogeny indicates that there is a significant phylogenetic signal in skull shape. Multivariate regression of independent contrasts of shape on independent contrasts of size reveals clear evolutionary allometry. Regardless of whether or not a correction for allometry is used, evolutionary integration between the face and braincase is strong, and tests reject the hypothesis that the face and braincase are separate evolutionary modules. These analyses can easily be applied to other taxa and can be combined with other morphometric tools to address a wide range of questions about evolutionary patterns and processes.

The Anatomical Distance of Functional Connections Predicts Brain Network Topology in Health and Schizophrenia

Abstract: The human brain is a topologically complex network embedded in anatomical space. Here, we systematically explored relationships between functional connectivity, complex network topology, and anatomical (Euclidean) distance between connected brain regions, in the resting-state functional magnetic resonance imaging brain networks of 20 healthy volunteers and 19 patients with childhood-onset schizophrenia (COS). Normal between-subject differences in average distance of connected edges in brain graphs were strongly associated with variation in topological properties of functional networks. In addition, a club or subset of connector hubs was identified, in lateral temporal, parietal, dorsal prefrontal, and medial prefrontal/cingulate cortical regions. In COS, there was reduced strength of functional connectivity over short distances especially, and therefore, global mean connection distance of thresholded graphs was significantly greater than normal. As predicted from relationships between spatial and topological properties of normal networks, this disorder-related proportional increase in connection distance was associated with reduced clustering and modularity and increased global efficiency of COS networks. Between-group differences in connection distance were localized specifically to connector hubs of multimodal association cortex. In relation to the neurodevelopmental pathogenesis of schizophrenia, we argue that the data are consistent with the interpretation that spatial and topological disturbances of functional network organization could arise from excessive "pruning" of short-distance functional connections in schizophrenia.

Discovering Spatial Interaction Communities from Mobile Phone Data

Abstract: In the age of Big Data, the widespread use of location-awareness technologies has made it possible to collect spatio-temporal interaction data for analyzing flow patterns in both physical space and cyberspace. This research attempts to explore and interpret patterns embedded in the network of phone-call interac- tion and the network of phone-users' movements, by considering the geographical context of mobile phone cells. We adopt an agglomerative clustering algorithm based on a Newman-Girvan modularity metric and propose an alternative modularity function incorporating a gravity model to discover the clus- tering structures of spatial-interaction communities using a mobile phone dataset from one week in a city in China. The results verify the distance decay effect and spatial continuity that control the process of par- titioning phone-call interaction, which indicates that people tend to communicate within a spatial- proximity community. Furthermore, we discover that a high correlation exists between phone-users' movements in physical space and phone-call interaction in cyberspace. Our approach presents a combined qualitative-quantitative framework to identify clusters and interaction patterns, and explains how geo- graphical context influences communities of callers and receivers. The findings of this empirical study are valuable for urban structure studies as well as for the detection of communities in spatial networks.

All Together Now

Abstract: While virtual humans are proven tools for training, education and research, they are far from realizing their full potential. Advances are needed in individual capabilities, such as character animation and speech synthesis, but perhaps more importantly, fundamental questions remain as to how best to integrate these capabilities into a single framework that allows us to efficiently create characters that can engage users in meaningful and realistic social interactions. This integration requires in-depth, inter-disciplinary understanding few individuals, or even teams of individuals, possess. We help address this challenge by introducing the ICT Virtual Human Toolkit, which offers a flexible framework for exploring a variety of different types of virtual human systems, from virtual listeners and question-answering characters to virtual role-players. We show that due to its modularity, the Toolkit allows researchers to mix and match provided capabilities with their own, lowering the barrier of entry to this multi-disciplinary research challenge.

'Integronsters', integral and integrated modeling

Abstract: In many cases model integration treats models as software components only, ignoring the fluid relationship between models and reality, the evolving nature of models and their constant modification and recalibration. As a result, with integrated models we find increased complexity, where changes that used to impact only relatively contained models of subsystems, now propagate throughout the whole integrated system. This makes it harder to keep the overall complexity under control and, in a way, defeats the purpose of modularity, when efficiency is supposed to be gained from independent development of modules. Treating models only as software in solving the integration challenge may give birth to 'integronsters' - constructs that are perfectly valid as software products but ugly or even useless as models. We argue that one possible remedy is to learn to use data sets as modules and integrate them into the models. Then the data that are available for module calibration can serve as an intermediate linkage tool, sitting between modules and providing a module-independent baseline dynamics, which is then incremented when scenarios are to be run. In this case it is not the model output that is directed into the next model input, but model output is presented as a variation around the baseline trajectory, and it is this variation that is then fed into the next module down the chain. However still with growing overall complexity, calibration can become an important limiting factor, giving more promise to the integral approach, when the system is modeled and simplified as a whole.

Community Detection in Complex Networks: Multi-objective Enhanced Firefly Algorithm

Abstract: Studying the evolutionary community structure in complex networks is crucial for uncovering the links between structures and functions of a given community. Most contemporary community detection algorithms employs single optimization criteria (i.e.., modularity), which may not be adequate to represent the structures in complex networks. We suggest community detection process as a Multi-objective Optimization Problem (MOP) for investigating the community structures in complex networks. To overcome the limitations of the community detection problem, we propose a new multi-objective optimization algorithm based on enhanced firefly algorithm so that a set of non-dominated (Pareto-optimal) solutions can be achieved. In our proposed algorithm, a new tuning parameter based on a chaotic mechanism and novel self-adaptive probabilistic mutation strategies are used to improve the overall performance of the algorithm. The experimental results on synthetic and real world complex networks suggest that the multi-objective community detection algorithm provides useful paradigm for discovering overlapping community structures robustly.

Effect of correlations on network controllability

Abstract: A dynamical system is controllable if by imposing appropriate external signals on a subset of its nodes, it can be driven from any initial state to any desired state in finite time. Here we study the impact of various network characteristics on the minimal number of driver nodes required to control a network. We find that clustering and modularity have no discernible impact, but the symmetries of the underlying matching problem can produce linear, quadratic or no dependence on degree correlation coefficients, depending on the nature of the underlying correlations. The results are supported by numerical simulations and help narrow the observed gap between the predicted and the observed number of driver nodes in real networks.

Cranial integration and modularity: insights into evolution and development from morphometric data

Abstract: Morphological integration and modularity have become central concepts in evolutionary biology and geometric morphometrics. This review summarizes the most frequently used methods for characterizing and quantifying integration and modularity in morphometric data: principal component analysis and related issues such as the variance of eigenvalues, partial least squares, comparison of covariation among alternative partitions of landmarks, matrix correlation and ordinations of covariance matrices. Allometry is often acting as an integrating factor. Integration and modularity can be studied at dierent levels: developmental integration is accessible through analyses of covariation of fluctuating asymmetry, genetic integration can be investigated in dierent experimental protocols that either focus on eects of individual genes or consider the aggregate eect of the whole genome, and several phylogenetic comparative methods are available for studying evolutionary integration. Morphological integration and modularity have been investigated in many species of mammals. The review gives a survey of geometric morphometric studies in some of the groups for which many studies have been published: mice and other rodents, carnivorans, shrews, humans and other primates. This review demonstrates that geometric morphometrics oers an established methodology for studying a wide range of questions concerning integration and modularity, but also points out opportunities for further innovation.

MODULAR: Software for the Autonomous Computation of Modularity in Large Network Sets

Abstract: Ecological systems can be seen as networks of interactions between individual, species, or habitat patches. A key feature of many ecological networks is their organization into modules, which are subsets of elements that are more connected to each other than to the other elements in the network. We introduce MODULAR to perform rapid and autonomous calculation of modularity in sets of networks. MODULAR reads a set of files with matrices or edge lists that represent unipartite or bipartite networks, and identify modules using two different modularity metrics that have been previously used in studies of ecological networks. To find the network partition that maximizes modularity, the software offers five optimization methods to the user. We also included two of the most common null models that are used in studies of ecological networks to verify how the modularity found by the maximization of each metric differs from a theoretical benchmark.

Historical climate‐change influences modularity and nestedness of pollination networks

Abstract: The structure of species interaction networks is important for species coexistence, community stability and exposure of species to extinctions. Two widespread structures in ecological networks are modularity, i.e. weakly connected subgroups of species that are internally highly interlinked, and nestedness, i.e. specialist species that interact with a subset of those species with which generalist species also interact. Modularity and nestedness are often interpreted as evolutionary ecological structures that may have relevance for community persistence and resilience against perturbations, such as climate-change. Therefore, historical climatic fluctuations could influence modularity and nestedness, but this possibility remains untested. This lack of research is in sharp contrast to the considerable efforts to disentangle the role of historical climate-change and contemporary climate on species distributions, richness and community composition patterns. Here, we use a global database of pollination networks to show that historical climate-change is at least as important as contemporary climate in shaping modularity and nestedness of pollination networks. Specifically, on the mainland we found a relatively strong negative association between Quaternary climate-change and modularity, whereas nestedness was most prominent in areas having experienced high Quaternary climate-change. On islands, Quaternary climate-change had weak effects on modularity and no effects on nestedness. Hence, for both modularity and nestedness, historical climate-change has left imprints on the network structure of mainland communities, but had comparably little effect on island communities. Our findings highlight a need to integrate historical climate fluctuations into eco-evolutionary hypotheses of network structures, such as modularity and nestedness, and then test these against empirical data. We propose that historical climate-change may have left imprints in the structural organisation of species interactions in an array of systems important for maintaining biological diversity.

Scalable Hierarchical Network-on-Chip Architecture for Spiking Neural Network Hardware Implementations

Abstract: Spiking neural networks (SNNs) attempt to emulate information processing in the mammalian brain based on massively parallel arrays of neurons that communicate via spike events. SNNs offer the possibility to implement embedded neuromorphic circuits, with high parallelism and low power consumption compared to the traditional von Neumann computer paradigms. Nevertheless, the lack of modularity and poor connectivity shown by traditional neuron interconnect implementations based on shared bus topologies is prohibiting scalable hardware implementations of SNNs. This paper presents a novel hierarchical network-on-chip (H-NoC) architecture for SNN hardware, which aims to address the scalability issue by creating a modular array of clusters of neurons using a hierarchical structure of low and high-level routers. The proposed H-NoC architecture incorporates a spike traffic compression technique to exploit SNN traffic patterns and locality between neurons, thus reducing traffic overhead and improving throughput on the network. In addition, adaptive routing capabilities between clusters balance local and global traffic loads to sustain throughput under bursting activity. Analytical results show the scalability of the proposed H-NoC approach under different scenarios, while simulation and synthesis analysis using 65-nm CMOS technology demonstrate high-throughput, low-cost area, and power consumption per cluster, respectively.

Identifying compartments in presence–absence matrices and bipartite networks: insights into modularity measures

Abstract: Aim The identification of compartments (i.e. clusters of overlapping species ranges across an environmental gradient) is an important methodological challenge for biogeographical studies. Recent developments in network theory offer promising perspectives on this issue using the measurement of modularity. A presence–absence matrix is modular if particular subgroups of species are mainly linked to particular subgroups of sites. Modularity is still rarely considered in biogeographical studies. Here, I compare different modularity indices to investigate which is the most appropriate for studying presence–absence matrices and similar types of networks, such as bipartite networks. Location Evaluation was based on 279 data sets from around the world. Methods I consider the three most commonly used modularity indices. One was developed for unipartite networks and the other two for bipartite networks. The performance of these indices (detection of a modular pattern, quality of compartment identification) is evaluated on test matrices of known compartmentalization levels with varying sizes and fills. Modularity patterns are then evaluated for 279 presence–absence matrices. Results The three modularity measures differ mainly in the identification of the compartments, and less in the statistical significance of the observed modularity. The modularity measure Q3 tends to perform best, whereas Q2 usually performs less well, especially for highly diverse and highly connected networks that include a few extremely well-connected nodes. These modularity indices all reveal the presence of modular patterns in presence–absence matrices. Main conclusions The choice of an appropriate modularity index is particularly important when we are interested in the composition of the different compartments. This analysis suggests that compartmentalized structures can be widespread in presence–absence matrices (about 40% of the matrices considered here). Modularity should thus offer interesting perspectives on the understanding of biogeographical patterns.

Surprise maximization reveals the community structure of complex networks

Abstract: How to determine the community structure of complex networks is an open question. It is critical to establish the best strategies for community detection in networks of unknown structure. Here, using standard synthetic benchmarks, we show that none of the algorithms hitherto developed for community structure characterization perform optimally. Significantly, evaluating the results according to their modularity, the most popular measure of the quality of a partition, systematically provides mistaken solutions. However, a novel quality function, called Surprise, can be used to elucidate which is the optimal division into communities. Consequently, we show that the best strategy to find the community structure of all the networks examined involves choosing among the solutions provided by multiple algorithms the one with the highest Surprise value. We conclude that Surprise maximization precisely reveals the community structure of complex networks.

Fast algorithm for modularity-based graph clustering

Abstract: In AI and Web communities, modularity-based graph clustering algorithms are being applied to various applications. However, existing algorithms are not applied to large graphs because they have to scan all vertices/edges iteratively. The goal of this paper is to efficiently compute clusters with high modularity from extremely large graphs with more than a few billion edges. The heart of our solution is to compute clusters by incrementally pruning unnecessary vertices/edges and optimizing the order of vertex selections. Our experiments show that our proposal outperforms all other modularity-based algorithms in terms of computation time, and it finds clusters with high modularity.

Exploiting the Modularity of Ion-Paired Chiral Ligands for Palladium-Catalyzed Enantioselective Allylation of Benzofuran-2(3H)-ones

Abstract: A highly enantioselective allylation of benzofuran-2(3H)-ones is achieved under Pd catalysis by taking full advantage of the structural modularity of ion-paired chiral ligands.

A fast parallel modularity optimization algorithm (FPMQA) for community detection in online social network

Abstract: As information technology has advanced, people are turning more frequently to electronic media for communication, and social relationships are increasingly found in online channels. Discovering the latent communities therein is a useful way to better understand the properties of a virtual social network. Traditional community-detection tasks only consider the structural characteristics of a social organization, but more information about nodes and edges such as semantic information cannot be exploited. What is more, the typical size of virtual spaces is now counted in millions, if not billions, of nodes and edges, most existing algorithms are incapable to analyze such large scale dense networks. In this paper, we first introduce an interesting social network model (Interest Network) in which links between two IDs are built if they both participate to the discussions about one or more topics/stories. In this case, we say both of the connected two IDs have the similar interests. Then, the edges of the initial network are updated using the attitude consistency information of the connected ID pairs. For a given ID pair i and j, they may together reply to some topics/IDs. The implicit orientations/attitudes of these two IDs to their together-reply topics/IDs may not be the same. We use a simple statistical method to calculate the attitude consistency, the value of which is between 0 and 1, and the higher value corresponds to a greater degree of consistency of the given ID pair to topics/IDs. The updated network is called Similar-View Network (SVN). In the second part, a fast parallel modularity optimization algorithm (FPMQA) that performs the analogous greedy optimization as CNM and FUC is used to conduct community discovering. By using the parallel manner and sophisticated data structures, its running time is essentially fast, O(k^m^a^x(k^m^a^x+logk^m^a^x)). Finally, we propose an evaluation metric, which is based on the reliable ground truths, for online network community detection. In the experimental work, we evaluate our method using real datasets and compare our approach with several previous methods; the results show that our method is more effective and accurate in find potential online communities.

Network modularity reveals critical scales for connectivity in ecology and evolution

Abstract: For nearly a century, biologists have emphasized the profound importance of spatial scale for ecology, evolution and conservation. Nonetheless, objectively identifying critical scales has proven incredibly challenging. Here we extend new techniques from physics and social sciences that estimate modularity on networks to identify critical scales for movement and gene flow in animals. Using four species that vary widely in dispersal ability and include both mark-recapture and population genetic data, we identify significant modularity in three species, two of which cannot be explained by geographic distance alone. Importantly, the inclusion of modularity in connectivity and population viability assessments alters conclusions regarding patch importance to connectivity and suggests higher metapopulation viability than when ignoring this hidden spatial scale. We argue that network modularity reveals critical meso-scales that are probably common in populations, providing a powerful means of identifying fundamental scales for biology and for conservation strategies aimed at recovering imperilled species.

All Together Now: Introducing the Virtual Human Toolkit

Abstract: While virtual humans are proven tools for training, education and research, they are far from realizing their full potential. Advances are needed in individual capabilities, such as character animation and speech synthesis, but perhaps more importantly, fundamental questions remain as to how best to integrate these capabilities into a single framework that allows us to efficiently create characters that can engage users in meaningful and realistic social interactions. This integration requires in-depth, inter-disciplinary understanding few individuals, or even teams of individuals, possess. We help address this challenge by introducing the ICT Virtual Human Toolkit, which offers a flexible framework for exploring a variety of different types of virtual human systems, from virtual listeners and question-answering characters to virtual role-players. We show that due to its modularity, the Toolkit allows researchers to mix and match provided capabilities with their own, lowering the barrier of entry to this multi-disciplinary research challenge.

Epidemic threshold and topological structure of susceptible-infectious-susceptible epidemics in adaptive networks.

Abstract: The interplay between disease dynamics on a network and the dynamics of the structure of that network characterizes many real-world systems of contacts. A continuous-time adaptive susceptible-infectious-susceptible (ASIS) model is introduced in order to investigate this interaction, where a susceptible node avoids infections by breaking its links to its infected neighbors while it enhances the connections with other susceptible nodes by creating links to them. When the initial topology of the network is a complete graph, an exact solution to the average metastable-state fraction of infected nodes is derived without resorting to any mean-field approximation. A linear scaling law of the epidemic threshold ?c as a function of the effective link-breaking rate ? is found. Furthermore, the bifurcation nature of the metastable fraction of infected nodes of the ASIS model is explained. The metastable-state topology shows high connectivity and low modularity in two regions of the ?,? plane for any effective infection rate ?>?c: (i) a “strongly adaptive” region with very high ? and (ii) a “weakly adaptive” region with very low ?. These two regions are separated from the other half-open elliptical-like regions of low connectivity and high modularity in a contour-line-like way. Our results indicate that the adaptation of the topology in response to disease dynamics suppresses the infection, while it promotes the network evolution towards a topology that exhibits assortative mixing, modularity, and a binomial-like degree distribution.