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Influence propagation in social networks is a central problem in modern social network analysis, with important societal applications in politics and advertising. A large body of work has focused on cascading models, viral marketing, and finite-horizon diffusion. There is, however, a need for more developed, mathematically principled \emph{adversarial models}, in which multiple, opposed actors strategically select nodes whose influence will maximally sway the crowd to their point of view. In the present work, we develop and analyze such a model based on harmonic functions and linear diffusion. We prove that our general problem is NP-hard and that the objective function is monotone and submodular; consequently, we can greedily approximate the solution within a constant factor. Introducing and analyzing a convex relaxation, we show that the problem can be approximately solved using smooth optimization methods. We illustrate the effectiveness of our approach on a variety of example networks., Comment: 23 pages, 5 figures, comments welcome!!

Real-world network datasets are typically obtained in ways that fail to capture all edges. The patterns of missing data are often non-uniform as they reflect biases and other shortcomings of different data collection methods. Nevertheless, uniform missing data is a common assumption made when no additional information is available about the underlying missing-edge pattern, and link prediction methods are frequently tested against uniformly missing edges. To investigate the impact of different missing-edge patterns on link prediction accuracy, we employ 9 link prediction algorithms from 4 different families to analyze 20 different missing-edge patterns that we categorize into 5 groups. Our comparative simulation study, spanning 250 real-world network datasets from 6 different domains, provides a detailed picture of the significant variations in the performance of different link prediction algorithms in these different settings. With this study, we aim to provide a guide for future researchers to help them select a link prediction algorithm that is well suited to their sampled network data, considering the data collection process and application domain., Comment: Submitted to PLOS ONE

Many empirical networks originate from correlational data, arising in domains as diverse as psychology, neuroscience, genomics, microbiology, finance, and climate science. Specialized algorithms and theory have been developed in different application domains for working with such networks, as well as in statistics, network science, and computer science, often with limited communication between practitioners in different fields. This leaves significant room for cross-pollination across disciplines. A central challenge is that it is not always clear how to best transform correlation matrix data into networks for the application at hand, and probably the most widespread method, i.e., thresholding on the correlation value to create either unweighted or weighted networks, suffers from multiple problems. In this article, we review various methods of constructing and analyzing correlation networks, ranging from thresholding and its improvements to weighted networks, regularization, dynamic correlation networks, threshold-free approaches, and more. Finally, we propose and discuss a variety of key open questions currently confronting this field., Comment: 1 figure

We provide a rearrangement based algorithm for fast detection of subgraphs of $k$ vertices with long escape times for directed or undirected networks. Complementing other notions of densest subgraphs and graph cuts, our method is based on the mean hitting time required for a random walker to leave a designated set and hit the complement. We provide a new relaxation of this notion of hitting time on a given subgraph and use that relaxation to construct a fast subgraph detection algorithm and a generalization to $K$-partitioning schemes. Using a modification of the subgraph detector on each component, we propose a graph partitioner that identifies regions where random walks live for comparably large times. Importantly, our method implicitly respects the directed nature of the data for directed graphs while also being applicable to undirected graphs. We apply the partitioning method for community detection to a large class of model and real-world data sets., Comment: 22 pages, 10 figures, 1 table, comments welcome!!

Despite the numerous ways now available to quantify which parts or subsystems of a network are most important, there remains a lack of centrality measures that are related to the complexity of information flows and are derived directly from entropy measures. Here, we introduce a ranking of edges based on how each edge's removal would change a system's von Neumann entropy (VNE), which is a spectral-entropy measure that has been adapted from quantum information theory to quantify the complexity of information dynamics over networks. We show that a direct calculation of such rankings is computationally inefficient (or unfeasible) for large networks: e.g.\ the scaling is $\mathcal{O}(N^3)$ per edge for networks with $N$ nodes. To overcome this limitation, we employ spectral perturbation theory to estimate VNE perturbations and derive an approximate edge-ranking algorithm that is accurate and fast to compute, scaling as $\mathcal{O}(N)$ per edge. Focusing on a form of VNE that is associated with a transport operator $e^{-\beta{ L}}$, where ${ L}$ is a graph Laplacian matrix and $\beta>0$ is a diffusion timescale parameter, we apply this approach to diverse applications including a network encoding polarized voting patterns of the 117th U.S. Senate, a multimodal transportation system including roads and metro lines in London, and a multiplex brain network encoding correlated human brain activity. Our experiments highlight situations where the edges that are considered to be most important for information diffusion complexity can dramatically change as one considers short, intermediate and long timescales $\beta$ for diffusion., Comment: 27 pages, 9 figures. Revision based on submission to SIAM MMS journal

Network classification aims to group networks (or graphs) into distinct categories based on their structure. We study the connection between classification of a network and of its constituent nodes, and whether nodes from networks in different groups are distinguishable based on structural node characteristics such as centrality and clustering coefficient. We demonstrate, using various network datasets and random network models, that a classifier can be trained to accurately predict the network category of a given node (without seeing the whole network), implying that complex networks display distinct structural patterns even at the node level. Finally, we discuss two applications of node-level network classification: (i) whole-network classification from small samples of nodes, and (ii) network bootstrapping., Comment: 8 pages, 5 figures

Populations of oscillators are present throughout nature. Very often synchronization is observed in such populations if they are allowed to interact. A paradigmatic model for the study of such phenomena has been the Kuramoto model. However, considering real oscillations are rarely isochronous as a function of energy, it is natural to extend the model by allowing the natural frequencies to vary as a function of some dynamical resource supply. Beyond just accounting for a dynamical supply of resources, however, competition over a \emph{shared} resource supply is important in a variety of biological systems. In neuronal systems, for example, resource competition enables the study of neural activity via fMRI. It is reasonable to expect that this dynamical resource allocation should have consequences for the synchronization behavior of the brain. This paper presents a modified Kuramoto dynamics which includes additional dynamical terms that provide a relatively simple model of resource competition among populations of Kuramoto oscillators. We design a mutlilayer system which highlights the impact of the competition dynamics, and we show that in this designed system, correlations can arise between the synchronization states of two populations of oscillators which share no phase-coupling edges. These correlations are interesting in light of the often observed variance between functional and structural connectivity measures in real systems. The model presented here then suggests that some of the observed discrepancy may be explained by the way in which the brain dynamically allocates resources to different regions according to demand. If true, models such as this one provide a theoretical framework for analyzing the differences between structural and functional measures, and possibly implicate dynamical resource allocation as an integral part of the neural computation process., Comment: 12 pages, 2 figures

In composite materials composed of soft polymer matrix and stiff, high-aspect-ratio particles, the composite undergoes a transition in mechanical strength when the inclusion phase surpasses a critical density. This phenomenon (rheological or mechanical percolation) is well-known to occur in many composites at a critical density that exceeds the conductivity percolation threshold. Conductivity percolation occurs as a consequence of contact percolation, which refers to the conducting particles' formation of a connected component that spans the composite. Rheological percolation, however, has evaded a complete theoretical explanation and predictive description. A natural hypothesis is that rheological percolation arises due to rigidity percolation, whereby a rigid component of inclusions spans the composite. We model composites as random isotropic dispersions of soft-core rods, and study rigidity percolation in such systems. Building on previous results for two-dimensional systems, we develop an approximate algorithm that identifies spanning rigid components through iteratively identifying and compressing provably rigid motifs -- equivalently, decomposing giant rigid components into rigid assemblies of successively smaller rigid components. We apply this algorithm to random rod systems to estimate a rigidity percolation threshold and explore its dependence on rod aspect ratio. We show that this transition point, like the contact percolation transition point, scales inversely with the average (aspect ratio-dependent) rod excluded volume. However, the scaling of the rigidity percolation threshold, unlike the contact percolation scaling, is valid for relatively low aspect ratio. Moreover, the critical rod contact number is constant for aspect ratio above some relatively low value; and lies below the prediction from Maxwell's isostatic condition.

To design control strategies that predictably manipulate a system's behavior, it is first necessary to understand how the system's structure relates to its response. Many complex systems can be represented as multilayer networks whose response and control can be studied in the framework of network control theory. It remains unknown how a system's layered architecture dictates its control properties, particularly when control signals can only access the system through a single input layer. Here, we use the framework of linear control theory to probe the control properties of a duplex network with directed interlayer links. We determine the manner in which the structural properties of layers and relative interlayer arrangement together dictate the system's response. For this purpose, we calculate the exact expression of optimal control energy in terms of layer spectra and the relative alignment between the eigenmodes of the input layer and the deeper target layer. For a range of numerically constructed duplex networks, we then calculate the control properties of the two layers as a function of target-layer densities and layer topology. The alignment of layer eigenmodes emerges as an important parameter that sets the cost and routing of the optimal energy for control. We understand these results in a simplified limit of a single-mode approximation, and we build metrics to characterize the routing of optimal control energy through the eigenmodes of each layer. Our analytical and numerical results together provide insights into the relationships between structure and control in duplex networks. More generally, they serve as a platform for future work designing optimal \emph{interlayer} control strategies., Comment: 14 pages, 5 figures, 19 pages supplementary material, 10 supplementary figures

The shortest-path, commute time, and diffusion distances on undirected graphs have been widely employed in applications such as dimensionality reduction, link prediction, and trip planning. Increasingly, there is interest in using asymmetric structure of data derived from Markov chains and directed graphs, but few metrics are specifically adapted to this task. We introduce a metric on the state space of any ergodic, finite-state, time-homogeneous Markov chain and, in particular, on any Markov chain derived from a directed graph. Our construction is based on hitting probabilities, with nearness in the metric space related to the transfer of random walkers from one node to another at stationarity. Notably, our metric is insensitive to shortest and average walk distances, thus giving new information compared to existing metrics. We use possible degeneracies in the metric to develop an interesting structural theory of directed graphs and explore a related quotienting procedure. Our metric can be computed in $O(n^3)$ time, where $n$ is the number of states, and in examples we scale up to $n=10,000$ nodes and $\approx 38M$ edges on a desktop computer. In several examples, we explore the nature of the metric, compare it to alternative methods, and demonstrate its utility for weak recovery of community structure in dense graphs, visualization, structure recovering, dynamics exploration, and multiscale cluster detection., Comment: 26 pages, 9 figures, for associated code, visit https://github.com/zboyd2/hitting_probabilities_metric, accepted at SIAM J. Math. Data Sci

A fundamental understanding of synchronized behavior in multi-agent systems can be acquired by studying analytically tractable Kuramoto models. However, such models typically diverge from many real systems whose dynamics evolve under non-negligible resource constraints. Here we construct a system of coupled Kuramoto oscillators that consume or produce resources as a function of their oscillation frequency. At high coupling, we observe strongly synchronized dynamics, whereas at low coupling we observe independent oscillator dynamics, as expected from standard Kuramoto models. For intermediate coupling, which typically induces a partially synchronized state, we empirically observe that (and theoretically explain why) the system can exist in either (i) a state in which the order parameter oscillates in time, or (ii) a state in which multiple synchronization states are simultaneously stable. Whether (i) or (ii) occurs depends upon whether the oscillators consume or produce resources, respectively. Relevant for systems as varied as coupled neurons and social groups, our study lays important groundwork for future efforts to develop quantitative predictions of synchronized dynamics for systems embedded in environments marked by sparse resources., Comment: 8 pages, 5 figure, submitted to Physical Review E

Network properties govern the rate and extent of spreading processes on networks, from simple contagions to complex cascades. Recent advances have extended the study of spreading processes from static networks to temporal networks, where nodes and links appear and disappear. We review previous studies on the effects of temporal connectivity for understanding the spreading rate and outbreak size of model infection processes. We focus on the effects of "accessibility", whether there is a temporally consistent path from one node to another, and "reachability", the density of the corresponding "accessibility graph" representation of the temporal network. We study reachability in terms of the overall level of temporal concurrency between edges, quantifying the overlap of edges in time. We explore the role of temporal resolution of contacts by calculating reachability with the full temporal information as well as with a simplified interval representation approximation that demands less computation. We demonstrate the extent to which the computed reachability changes due to this simplified interval representation., Comment: To appear in Holme and Saramaki (Editors). "Temporal Network Theory". Springer- Nature, New York. 2019

Modularity based community detection encompasses a number of widely used, efficient heuristics for identification of structure in networks. Recently, a belief propagation approach to modularity optimization provided a useful guide for identifying non-trivial structure in single-layer networks in a way that other optimization heuristics have not. In this paper, we extend modularity belief propagation to multilayer networks. As part of this development, we also directly incorporate a resolution parameter. We show that adjusting the resolution parameter affects the convergence properties of the algorithm and yields different community structures than the baseline. We compare our approach with a widely used community detection tool, GenLouvain, across a range of synthetic, multilayer benchmark networks, demonstrating that our method performs comparably to the state of the art. Finally, we demonstrate the practical advantages of the additional information provided by our tool by way of two real-world network examples. We show how the convergence properties of the algorithm can be used in selecting the appropriate resolution and coupling parameters and how the node-level marginals provide an interpretation for the strength of attachment to the identified communities. We have released our tool as a Python package for convenient use.

We describe centralities in temporal networks using a supracentrality framework to study centrality trajectories, which characterize how the importances of nodes change in time. We study supracentrality generalizations of eigenvector-based centralities, a family of centrality measures for time-independent networks that includes PageRank, hub and authority scores, and eigenvector centrality. We start with a sequence of adjacency matrices, each of which represents a time layer of a network at a different point or interval of time. Coupling centrality matrices across time layers with weighted interlayer edges yields a \emph{supracentrality matrix} $\mathbb{C}(\omega)$, where $\omega$ controls the extent to which centrality trajectories change over time. We can flexibly tune the weight and topology of the interlayer coupling to cater to different scientific applications. The entries of the dominant eigenvector of $\mathbb{C}(\omega)$ represent \emph{joint centralities}, which simultaneously quantify the importance of every node in every time layer. Inspired by probability theory, we also compute \emph{marginal} and \emph{conditional centralities}. We illustrate how to adjust the coupling between time layers to tune the extent to which nodes' centrality trajectories are influenced by the oldest and newest time layers. We support our findings by analysis in the limits of small and large $\omega$., Comment: 20 pages, 5 figures

Network properties govern the rate and extent of various spreading processes, from simple contagions to complex cascades. Recently, the analysis of spreading processes has been extended from static networks to temporal networks, where nodes and links appear and disappear. We focus on the effects of "accessibility", whether there is a temporally consistent path from one node to another, and "reachability", the density of the corresponding "accessibility graph" representation of the temporal network. The level of reachability thus inherently limits the possible extent of any spreading process on the temporal network. We study reachability in terms of the overall levels of temporal concurrency between edges and the structural cohesion of the network agglomerating over all edges. We use simulation results and develop heterogeneous mean field model predictions for random networks to better quantify how the properties of the underlying temporal network regulate reachability.

Characterizing the importances (i.e., centralities) of nodes in social, biological, and technological networks is a core topic in both network science and data science. We present a linear-algebraic framework that generalizes eigenvector-based centralities, including PageRank and hub/authority scores, to provide a common framework for two popular classes of multilayer networks: multiplex networks (which have layers that encode different types of relationships) and temporal networks (in which the relationships change over time). Our approach involves the study of joint, marginal, and conditional "supracentralities" that one can calculate from the dominant eigenvector of a supracentrality matrix [Taylor et al., 2017], which couples centrality matrices that are associated with individual network layers. We extend this prior work (which was restricted to temporal networks with layers that are coupled by adjacent-in-time coupling) by allowing the layers to be coupled through a (possibly asymmetric) interlayer-adjacency matrix $\tilde{{\bf A}}$, where the entry $\tilde{A}_{tt'} \geq 0$ encodes the coupling between layers $t$ and $t'$. Our framework provides a unifying foundation for centrality analysis of multiplex and temporal networks; it also illustrates a complicated dependency of the supracentralities on the topology and weights of interlayer coupling. By scaling $\tilde{{\bf A}}$ by an interlayer-coupling strength $\omega\ge0$ and developing a singular perturbation theory for the limits of weak ($\omega\to0^+$) and strong coupling ($\omega\to\infty$), we also reveal an interesting dependence of supracentralities on the dominant left and right eigenvectors of $\tilde{{\bf A}}$., Comment: 35 pages with 7 figures, followed by 5 pages of supporting material

Adaptive voter models (AVMs) are simple mechanistic systems that model the emergence of mesoscopic structure from local networked processes driven by conflict and homophily. AVMs display rich behavior, including a phase transition from a fully-fragmented regime of "echo-chambers" to a regime of persistent disagreement governed by low-dimensional quasistable manifolds. Many extant methods for approximating the behavior of AVMs are either restricted in scope, expensive in computation, or inaccurate in predicting important statistics. In this work, we develop a novel, second-order moment closure approximation method for binary-state rewire-to-random and rewire-to-same model variants. We incorporate a small amount of noise via a random mutation term, which renders the system ergodic. Using ergodicity, we then approximate the voting process, which is non-Markovian in the second moments of the system, with a Markovian term near the phase transition. This approximation exploits an asymmetry between different classes of voting events. The resulting scheme enables us to predict the location of the phase transition and the active edge density in the regime of persistent disagreement, across the entire space of parameters and opinion densities. Numerically, our results are nearly exact for the rewire-to-random model, and competitive with other current approaches for the rewire-to-same model. Moreover, our computations display constant scaling in the mean degree, enabling approximations for denser systems than previously possible. We conclude with suggestions for model refinements and extensions., Comment: 20 Pages, 4 figures. Final revised copy

Much of the community detection literature studies structural communities, communities defined solely by the connectivity patterns of the network. Often, networks contain additional metadata which can inform community detection such as the grade and gender of students in a high school social network. In this work, we introduce a tuning parameter to the content map equation that allows users of the Infomap community detection algorithm to control the metadata's relative importance for identifying network structure. On synthetic networks, we show that our algorithm can overcome the structural detectability limit when the metadata is well-aligned with community structure. On real-world networks, we show how our algorithm can achieve greater mutual information with the metadata at a cost in the traditional map equation. Our tuning parameter, like the focusing knob of a microscope, allows users to "zoom in" and "zoom out" on communities with varying levels of focus on the metadata.

Models of contagion dynamics, originally developed for infectious diseases, have proven relevant to the study of information, news, and political opinions in online social systems. Modelling diffusion processes and predicting viral information cascades are important problems in network science. Yet, many studies of information cascades neglect the variation in infectivity across different pieces of information. Here, we employ early-time observations of online cascades to estimate the infectivity of distinct pieces of information. Using simulations and data from real-world Twitter retweets, we demonstrate that these estimated infectivities can be used to improve predictions about the virality of an information cascade. Developing our simulations to mimic the real-world data, we consider the effect of the limited effective time for transmission of a cascade and demonstrate that a simple model for slow but non-negligible decay of the infectivity captures the essential properties of retweet distributions. These results demonstrate the interplay between the intrinsic infectivity of a tweet and the complex network environment within which it diffuses, strongly influencing the likelihood of becoming a viral cascade., Comment: 16 pages, 10 figures

Attributed network data is becoming increasingly common across fields, as we are often equipped with information about nodes in addition to their pairwise connectivity patterns. This extra information can manifest as a classification, or as a multidimensional vector of features. Recently developed methods that seek to extend community detection approaches to attributed networks have explored how to most effectively combine connectivity and attribute information to identify quality communities. These methods often rely on some assumption of the dependency relationships between attributes and connectivity. In this work, we seek to develop a statistical test to assess whether node attributes align with network connectivity. The objective is to quantitatively evaluate whether nodes with similar connectivity patterns also have similar attributes. To address this problem, we use a node sampling and label propagation approach. We apply our method to several synthetic examples that explore how network structure and attribute characteristics affect the empirical p-value computed by our method. Finally, we apply the test to a network generated from a single cell mass cytometry (CyTOF) dataset and show that our test can identify markers associated with distinct sub populations of single cells.

The spreading of epidemics is very much determined by the structure of the contact network, which may be impacted by the mobility dynamics of the individuals themselves. In confined scenarios where a small, closed population spends most of its time in localized environments and has easily identifiable mobility patterns --such as workplaces, university campuses or schools-- it is of critical importance to identify the factors controlling the rate of disease spread. Here we present a discrete-time, metapopulation-based model to describe the transmission of SIS-like diseases that take place in confined scenarios where the mobilities of the individuals are not random but, rather, follow clear recurrent travel patterns. This model allows analytical determination of the onset of epidemics, as well as the ability to discern which contact structures are most suited to prevent the infection to spread. It thereby determines whether common prevention mechanisms, as isolation, are worth implementing in such a scenario and their expected impact.

The stochastic block model (SBM) is a probabilistic model for community structure in networks. Typically, only the adjacency matrix is used to perform SBM parameter inference. In this paper, we consider circumstances in which nodes have an associated vector of continuous attributes that are also used to learn the node-to-community assignments and corresponding SBM parameters. While this assumption is not realistic for every application, our model assumes that the attributes associated with the nodes in a network's community can be described by a common multivariate Gaussian model. In this augmented, attributed SBM, the objective is to simultaneously learn the SBM connectivity probabilities with the multivariate Gaussian parameters describing each community. While there are recent examples in the literature that combine connectivity and attribute information to inform community detection, our model is the first augmented stochastic block model to handle multiple continuous attributes. This provides the flexibility in biological data to, for example, augment connectivity information with continuous measurements from multiple experimental modalities. Because the lack of labeled network data often makes community detection results difficult to validate, we highlight the usefulness of our model for two network prediction tasks: link prediction and collaborative filtering. As a result of fitting this attributed stochastic block model, one can predict the attribute vector or connectivity patterns for a new node in the event of the complementary source of information (connectivity or attributes, respectively). We also highlight two biological examples where the attributed stochastic block model provides satisfactory performance in the link prediction and collaborative filtering tasks.

Using particle-scale models to accurately describe property enhancements and phase transitions in macroscopic behavior is a major engineering challenge in composite materials science. To address some of these challenges, we use the graph theoretic property of rigidity to model me- chanical reinforcement in composites with stiff rod-like particles. We develop an efficient algorithmic approach called rigid graph compression (RGC) to describe the transition from floppy to rigid in disordered fiber networks ('rod-hinge systems'), which form the reinforcing phase in many composite systems. To establish RGC on a firm theoretical foundation, we adapt rigidity matroid theory to identify primitive topological network motifs that serve as rules for composing interacting rigid par- ticles into larger rigid components. This approach is computationally efficient and stable, because RGC requires only topological information about rod interactions (encoded by a sparse unweighted network) rather than geometrical details such as rod locations or pairwise distances (as required in rigidity matroid theory). We conduct numerical experiments on simulated two-dimensional rod-hinge systems to demonstrate that RGC closely approximates the rigidity percolation threshold for such systems, through comparison with the pebble game algorithm (which is exact in two dimensions). Importantly, whereas the pebble game is derived from Laman's condition and is only valid in two dimensions, the RGC approach naturally extends to higher dimensions.

Even though transitivity is a central structural feature of social networks, its influence on epidemic spread on coevolving networks has remained relatively unexplored. Here we introduce and study an adaptive SIS epidemic model wherein the infection and network coevolve with non-trivial probability to close triangles during edge rewiring, leading to substantial reinforcement of network transitivity. This new model provides a unique opportunity to study the role of transitivity in altering the SIS dynamics on a coevolving network. Using numerical simulations and Approximate Master Equations (AME), we identify and examine a rich set of dynamical features in the new model. In many cases, the AME including transitivity reinforcement provides accurate predictions of stationary-state disease prevalences and network degree distributions. Furthermore, for some parameter settings, the AME accurately trace the temporal evolution of the system. We show that higher transitivity reinforcement in the model leads to lower levels of infective individuals in the population, when closing a triangle is the dominant rewiring mechanism. These methods and results may be useful in developing ideas and modeling strategies for controlling SIS type epidemics., Comment: 16 pages, 21 figures

Community detection is a commonly used technique for identifying groups in a network based on similarities in connectivity patterns. To facilitate community detection in large networks, we recast the network to be partitioned into a smaller network of 'super nodes', each super node comprising one or more nodes in the original network. To define the seeds of our super nodes, we apply the 'CoreHD' ranking from dismantling and decycling. We test our approach through the analysis of two common methods for community detection: modularity maximization with the Louvain algorithm and maximum likelihood optimization for fitting a stochastic block model. Our results highlight that applying community detection to the compressed network of super nodes is significantly faster while successfully producing partitions that are more aligned with the local network connectivity, more stable across multiple (stochastic) runs within and between community detection algorithms, and overlap well with the results obtained using the full network.

We introduce the Convex Hull of Admissible Modularity Partitions (CHAMP) algorithm to prune and prioritize different network community structures identified across multiple runs of possibly various computational heuristics. Given a set of partitions, CHAMP identifies the domain of modularity optimization for each partition ---i.e., the parameter-space domain where it has the largest modularity relative to the input set---discarding partitions with empty domains to obtain the subset of partitions that are "admissible" candidate community structures that remain potentially optimal over indicated parameter domains. Importantly, CHAMP can be used for multi-dimensional parameter spaces, such as those for multilayer networks where one includes a resolution parameter and interlayer coupling. Using the results from CHAMP, a user can more appropriately select robust community structures by observing the sizes of domains of optimization and the pairwise comparisons between partitions in the admissible subset. We demonstrate the utility of CHAMP with several example networks. In these examples, CHAMP focuses attention onto pruned subsets of admissible partitions that are 20-to-1785 times smaller than the sets of unique partitions obtained by community detection heuristics that were input into CHAMP., Comment: http://www.mdpi.com/1999-4893/10/3/93

Because of increasing global urbanization and its immediate consequences, including changes in patterns of food demand, circulation and land use, the next century will witness a major increase in the extent of paved roads built worldwide. To model the effects of this increase, it is crucial to understand whether possible self-organized patterns are inherent in the global road network structure. Here, we use the largest updated database comprising all major roads on Earth, together with global urban and cropland inventories, to suggest that road length distributions within croplands are indistinguishable from urban ones, once rescaled to account for the difference in mean road length. Such similarity extends to road length distributions within urban or agricultural domains of given area. We find two distinct regimes for the scaling of the mean road length with the associated area, holding in general at small and at large values of the latter. In suitably large urban and cropland domains, we find that mean and total road lengths increase linearly with their domain area, differently from earlier suggestions. Scaling regimes suggest that simple and universal mechanisms regulate urban and cropland road expansion at the global scale. As such, our findings bear implications for global road infrastructure growth based on land-use change and for planning policies sustaining urban expansions.

Community structure describes the organization of a network into subgraphs that contain a prevalence of edges within each subgraph and relatively few edges across boundaries between subgraphs. The development of community-detection methods has occurred across disciplines, with numerous and varied algorithms proposed to find communities. As we present in this Chapter via several case studies, community detection is not just an "end game" unto itself, but rather a step in the analysis of network data which is then useful for furthering research in the disciplinary domain of interest. These case-study examples arise from diverse applications, ranging from social and political science to neuroscience and genetics, and we have chosen them to demonstrate key aspects of community detection and to highlight that community detection, in practice, should be directed by the application at hand., Comment: 21 pages, 5 figures

Assessing whether a given network is typical or atypical for a random-network ensemble (i.e., network-ensemble comparison) has widespread applications ranging from null-model selection and hypothesis testing to clustering and classifying networks. We develop a framework for network-ensemble comparison by subjecting the network to stochastic rewiring. We study two rewiring processes, uniform and degree-preserved rewiring, which yield random-network ensembles that converge to the Erdos-Renyi and configuration-model ensembles, respectively. We study convergence through von Neumann entropy (VNE), a network summary statistic measuring information content based on the spectra of a Laplacian matrix, and develop a perturbation analysis for the expected effect of rewiring on VNE. Our analysis yields an estimate for how many rewires are required for a given network to resemble a typical network from an ensemble, offering a computationally efficient quantity for network-ensemble comparison that does not require simulation of the corresponding rewiring process., Comment: 24 pages, 6 figures

We study a model for switching strategies in the Prisoner's Dilemma game on adaptive networks of player pairings that coevolve as players attempt to maximize their return. We use a node-based strategy model wherein each player follows one strategy at a time (cooperate or defect) across all of its neighbors, changing that strategy and possibly changing partners in response to local changes in the network of player pairing and in the strategies used by connected partners. We compare and contrast numerical simulations with existing pair approximation differential equations for describing this system, as well as more accurate equations developed here using the framework of approximate master equations. We explore the parameter space of the model, demonstrating the relatively high accuracy of the approximate master equations for describing the system observations made from simulations. We study two variations of this partner-switching model to investigate the system evolution, predict stationary states, and compare the total utilities and other qualitative differences between these two model variants., Comment: 23 pages, 12 figures

Network representations of systems from various scientific and societal domains are neither completely random nor fully regular, but instead appear to contain recurring structural building blocks. These features tend to be shared by networks belonging to the same broad class, such as the class of social networks or the class of biological networks. At a finer scale of classification within each such class, networks describing more similar systems tend to have more similar features. This occurs presumably because networks representing similar purposes or constructions would be expected to be generated by a shared set of domain specific mechanisms, and it should therefore be possible to classify these networks into categories based on their features at various structural levels. Here we describe and demonstrate a new, hybrid approach that combines manual selection of features of potential interest with existing automated classification methods. In particular, selecting well-known and well-studied features that have been used throughout social network analysis and network science and then classifying with methods such as random forests that are of special utility in the presence of feature collinearity, we find that we achieve higher accuracy, in shorter computation time, with greater interpretability of the network classification results., Comment: 14 pages including 4 figures and a table. Methods and supplementary material included to the end

Applied network science often involves preprocessing network data before applying a network-analysis method, and there is typically a theoretical disconnect between these steps. For example, it is common to aggregate time-varying network data into windows prior to analysis, and the tradeoffs of this preprocessing are not well understood. Focusing on the problem of detecting small communities in multilayer networks, we study the effects of layer aggregation by developing random-matrix theory for modularity matrices associated with layer-aggregated networks with $N$ nodes and $L$ layers, which are drawn from an ensemble of Erd\H{o}s-R\'enyi networks. We study phase transitions in which eigenvectors localize onto communities (allowing their detection) and which occur for a given community provided its size surpasses a detectability limit $K^*$. When layers are aggregated via a summation, we obtain $K^*\varpropto \mathcal{O}(\sqrt{NL}/T)$, where $T$ is the number of layers across which the community persists. Interestingly, if $T$ is allowed to vary with $L$ then summation-based layer aggregation enhances small-community detection even if the community persists across a vanishing fraction of layers, provided that $T/L$ decays more slowly than $ \mathcal{O}(L^{-1/2})$. Moreover, we find that thresholding the summation can in some cases cause $K^*$ to decay exponentially, decreasing by orders of magnitude in a phenomenon we call super-resolution community detection. That is, layer aggregation with thresholding is a nonlinear data filter enabling detection of communities that are otherwise too small to detect. Importantly, different thresholds generally enhance the detectability of communities having different properties, illustrating that community detection can be obscured if one analyzes network data using a single threshold., Comment: 11 pages, 8 figures

One of the fundamental structural properties of many networks is triangle closure. Whereas the influence of this transitivity on a variety of contagion dynamics has been previously explored, existing models of coevolving or adaptive network systems use rewiring rules that randomize away this important property. In contrast, we study here a modified coevolving voter model dynamics that explicitly reinforces and maintains such clustering. Employing extensive numerical simulations, we establish that the transitions and dynamical states observed in coevolving voter model networks without clustering are altered by reinforcing transitivity in the model. We then use a semi-analytical framework in terms of approximate master equations to predict the dynamical behaviors of the model for a variety of parameter settings.

Many systems are naturally represented by a multilayer network in which edges exist in multiple layers that encode different, but potentially related, types of interactions, and it is important to understand limitations on the detectability of community structure in these networks. Using random matrix theory, we analyze detectability limitations for multilayer (specifically, multiplex) stochastic block models (SBMs) in which L layers are derived from a common SBM. We study the effect of layer aggregation on detectability for several aggregation methods, including summation of the layers' adjacency matrices for which we show the detectability limit vanishes as O(L^{-1/2}) with increasing number of layers, L. Importantly, we find a similar scaling behavior when the summation is thresholded at an optimal value, providing insight into the common - but not well understood - practice of thresholding pairwise-interaction data to obtain sparse network representations., Comment: 7 pages, 4 figures

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., Comment: 20 pages, 5 figures (some with multiple parts)