Many popular measures used in social network analysis, including centrality, are based on the random walk. The random walk is a model of a stochastic process where a node interacts with one other node at a time. However, the random walk may not be appropriate for modeling social phenomena, including epidemics and information diffusion, in which one node may interact with many others at the same time, for example, by broadcasting the virus or information to its neighbors. To produce meaningful results, social network analysis algorithms have to take into account the nature of interactions between the nodes. In this paper we classify dynamical processes as conservative and non-conservative and relate them to well-known measures of centrality used in network analysis: PageRank and Alpha-Centrality. We demonstrate, by ranking users in online social networks used for broadcasting information, that non-conservative Alpha-Centrality generally leads to a better agreement with an empirical ranking scheme than the conservative PageRank.
The interplay between structural properties of networks and the diffusion processes occurring on them contribute to their complexity. This has been realized by several researchers in the past. For example, @cite_24 @cite_10 emphasized that dynamical processes play an important role in characterizing the structure of complex networks. In @cite_24 they measure the quality of a network partition in terms of the statistical property of the dynamic process taking place in the network. In @cite_10 they study the different equilibrium properties of these processes. However, their works focus on what we call conservative processes: unbiased and biased random walks, discrete and continuous time random walks. In contrast, we also study non-conservative dynamical processes. We also relate these processes to centrality. Although the relationship of PageRank to random walk-type processes is well known, we explain how Alpha-Centrality is related to a type of a non-conservative process. We also carry out an empirical study of different centrality measures, unlike previous works.
@cite_24 @cite_10
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Who are the influential people in an online social network? The answer to this question depends not only on the structure of the network, but also on details of the dynamic processes occurring on it. We classify these processes as conservative and non-conservative. A random walk on a network is an example of a conservative dynamic process, while information spread is non-conservative. The influence models used to rank network nodes can be similarly classified, depending on the dynamic process they implicitly emulate. We claim that in order to correctly rank network nodes, the influence model has to match the details of the dynamic process. We study a real-world network on the social news aggregator Digg, which allows users to post and vote for news stories. We empirically define influence as the number of in-network votes a user's post generates. This influence measure, and the resulting ranking, arises entirely from the dynamics of voting on Digg, which represents non-conservative information flow. We then compare predictions of different influence models with this empirical estimate of influence. The results show that non-conservative models are better able to predict influential users on Digg. We find that normalized alpha-centrality metric turns out to be one of the best predictors of influence. We also present a simple algorithm for computing this metric and the associated mathematical formulation and analytical proofs. motion of a random walker on heterogeneous complex networks. We find that the random walker is attracted toward nodes with larger degree and that the random walk motion is asymmetric. The asymmetry can be quantified with the random walk centrality, the matrix formulation of which is presented. As an interacting system, we consider the zero-range process on complex networks. We find that a structural heterogeneity can lead to a condensation phenomenon. These studies show that structural heterogeneity plays an important role in understanding the properties of dynamical systems. The behavior of complex systems is determined not only by the topological organization of their interconnections but also by the dynamical processes taking place among their constituents. A faithful modeling of the dynamics is essential because different dynamical processes may be affected very differently by network topology. A full characterization of such systems thus requires a formalization that encompasses both aspects simultaneously, rather than relying only on the topological adjacency matrix. To achieve this, we introduce the concept of flow graphs, namely weighted networks where dynamical flows are embedded into the link weights. Flow graphs provide an integrated representation of the structure and dynamics of the system, which can then be analyzed with standard tools from network theory. Conversely, a structural network feature of our choice can also be used as the basis for the construction of a flow graph that will then encompass a dynamics biased by such a feature. We illustrate the ideas by focusing on the mathematical properties of generic linear processes on complex networks that can be represented as biased random walks and their dual consensus dynamics, and show how our framework improves our understanding of these processes. Most methods proposed to uncover communities in complex networks rely on combinatorial graph properties. Usually an edge-counting quality function, such as modularity, is optimized over all partitions of the graph compared against a null random graph model. Here we introduce a systematic dynamical framework to design and analyze a wide variety of quality functions for community detection. The quality of a partition is measured by its Markov Stability, a time-parametrized function defined in terms of the statistical properties of a Markov process taking place on the graph. The Markov process provides a dynamical sweeping across all scales in the graph, and the time scale is an intrinsic parameter that uncovers communities at different resolutions. This dynamic-based community detection leads to a compound optimization, which favours communities of comparable centrality (as defined by the stationary distribution), and provides a unifying framework for spectral algorithms, as well as different heuristics for community detection, including versions of modularity and Potts model. Our dynamic framework creates a systematic link between different stochastic dynamics and their corresponding notions of optimal communities under distinct (node and edge) centralities. We show that the Markov Stability can be computed efficiently to find multi-scale community structure in large networks. Abstract We consider random walks in dynamic random environments given by Markovian dynamics on Z d . We assume that the environment has a stationary distribution μ and satisfies the Poincare inequality w.r.t. μ . The random walk is a perturbation of another random walk (called “unperturbed”). We assume that also the environment viewed from the unperturbed random walk has stationary distribution μ . Both perturbed and unperturbed random walks can depend heavily on the environment and are not assumed to be finite-range. We derive a law of large numbers, an averaged invariance principle for the position of the walker and a series expansion for the asymptotic speed. We also provide a condition for non-degeneracy of the diffusion, and describe in some details equilibrium and convergence properties of the environment seen by the walker. All these results are based on a more general perturbative analysis of operators that we derive in the context of L 2 - bounded perturbations of Markov processes by means of the so-called Dyson–Phillips expansion. Most methods proposed to uncover communities in complex networks rely on their structural properties. Here we introduce the stability of a network partition, a measure of its quality defined in terms of the statistical properties of a dynamical process taking place on the graph. The time-scale of the process acts as an intrinsic parameter that uncovers community structures at different resolutions. The stability extends and unifies standard notions for community detection: modularity and spectral partitioning can be seen as limiting cases of our dynamic measure. Similarly, recently proposed multi-resolution methods correspond to linearisations of the stability at short times. The connection between community detection and Laplacian dynamics enables us to establish dynamically motivated stability measures linked to distinct null models. We apply our method to find multi-scale partitions for different networks and show that the stability can be computed efficiently for large networks with extended versions of current algorithms. There is an ever-increasing interest in investigating dynamics in time-varying graphs (TVGs). Nevertheless, so far, the notion of centrality in TVG scenarios usually refers to metrics that assess the relative importance of nodes along the temporal evolution of the dynamic complex network. For some TVG scenarios, however, more important than identifying the central nodes under a given node centrality definition is identifying the key time instants for taking certain actions. In this paper, we thus introduce and investigate the notion of time centrality in TVGs. Analogously to node centrality, time centrality evaluates the relative importance of time instants in dynamic complex networks. In this context, we present two time centrality metrics related to diffusion processes. We evaluate the two defined metrics using both a real-world dataset representing an in-person contact dynamic network and a synthetically generated randomized TVG. We validate the concept of time centrality showing that diffusion starting at the best ranked time instants (i.e., the most central ones), according to our metrics, can perform a faster and more efficient diffusion process. This paper studies a class of non-Markovian and nonhomogeneous stochastic processes on a finite state space. Relying on a recent paper by Benaim, Hofbauer, and Sorin [SIAM J. Control Optim., 44 (2005), pp. 328-348] it is shown that, under certain assumptions, the asymptotic behavior of occupation measures can be described in terms of a certain set-valued deterministic dynamical system. This provides a unified approach to simulated annealing type processes and permits the study of new models of vertex reinforced random walks and new models of learning in games such as Markovian fictitious play. Abstract Centrality measures play a central role in Complex Networks Theory as much as they provide a tool to rank nodes by their relevance in the processes occurring in a network. In this paper we propose a model for the eigenvector-like centralities of temporal networks that evolve on a continuous time scale. We analytically prove that these centralities can be approximated by the centralities of temporal networks on discrete time scale. We quantify the effectiveness of random walks for searching and construction of unstructured peer-to-peer (P2P) networks. We have identified two cases where the use of random walks for searching achieves better results than flooding: a) when the overlay topology is clustered, and h) when a client re-issues the same query while its horizon does not change much. For construction, we argue that an expander can he maintained dynamically with constant operations per addition. The key technical ingredient of our approach is a deep result of stochastic processes indicating that samples taken from consecutive steps of a random walk can achieve statistical properties similar to independent sampling (if the second eigenvalue of the transition matrix is hounded away from 1, which translates to good expansion of the network; such connectivity is desired, and believed to hold, in every reasonable network and network model). This property has been previously used in complexity theory for construction of pseudorandom number generators. We reveal another facet of this theory and translate savings in random bits to savings in processing overhead. It may not be feasible for sensor networks monitoring nature and inaccessible geographical regions to include powered sinks with Internet connections. We consider the scenario where sinks are not present in large-scale sensor networks, and unreliable sensors have to collectively resort to storing sensed data over time on themselves. At a time of convenience, such cached data from a small subset of live sensors may be collected by a centralized (possibly mobile) collector. In this paper, we propose a decentralized algorithm using fountain codes to guarantee the persistence and reliability of cached data on unreliable sensors. With fountain codes, the collector is able to recover all data as long as a sufficient number of sensors are alive. We use random walks to disseminate data from a sensor to a random subset of sensors in the network. Our algorithms take advantage of the low decoding complexity of fountain codes, as well as the scalability of the dissemination process via random walks. We have proposed two algorithms based on random walks. Our theoretical analysis and simulation-based studies have shown that, the first algorithm maintains the same level of fault tolerance as the original centralized fountain code, while introducing lower overhead than naive random-walk based implementation in the dissemination process. Our second algorithm has lower level of fault tolerance than the original centralized fountain code, but consumes much lower dissemination cost. The concept of centrality is often invoked in social network analysis, and diverse indices have been proposed to measure it. This paper develops a unified framework for the measurement of centrality. All measures of centrality assess a node's involvement in the walk structure of a network. Measures vary along four key dimensions: type of nodal involvement assessed, type of walk considered, property of walk assessed, and choice of summary measure. If we cross-classify measures by type of nodal involvement (radial versus medial) and property of walk assessed (volume versus length), we obtain a four-fold polychotomization with one cell empty which mirrors Freeman's 1979 categorization. At a more substantive level, measures of centrality summarize a node's involvement in or contribution to the cohesiveness of the network. Radial measures in particular are reductions of pair-wise proximities cohesion to attributes of nodes or actors. The usefulness and interpretability of radial measures depend on the fit of the cohesion matrix to the one-dimensional model. In network terms, a network that is fit by a one-dimensional model has a core-periphery structure in which all nodes revolve more or less closely around a single core. This in turn implies that the network does not contain distinct cohesive subgroups. Thus, centrality is shown to be intimately connected with the cohesive subgroup structure of a network. With more than 250 million active users, Facebook (FB) is currently one of the most important online social networks. Our goal in this paper is to obtain a representative (unbiased) sample of Facebook users by crawling its social graph. In this quest, we consider and implement several candidate techniques. Two approaches that are found to perform well are the Metropolis-Hasting random walk (MHRW) and a re-weighted random walk (RWRW). Both have pros and cons, which we demonstrate through a comparison to each other as well as to the "ground-truth" (UNI - obtained through true uniform sampling of FB userIDs). In contrast, the traditional Breadth-First-Search (BFS) and Random Walk (RW) perform quite poorly, producing substantially biased results. In addition to offline performance assessment, we introduce online formal convergence diagnostics to assess sample quality during the data collection process. We show how these can be used to effectively determine when a random walk sample is of adequate size and quality for subsequent use (i.e., when it is safe to cease sampling). Using these methods, we collect the first, to the best of our knowledge, unbiased sample of Facebook. Finally, we use one of our representative datasets, collected through MHRW, to characterize several key properties of Facebook. We investigate random walks on complex networks and derive an exact expression for the mean firstpassage time (MFPT) between two nodes. We introduce for each node the random walk centrality C, which is the ratio between its coordination number and a characteristic relaxation time, and show that it determines essentially the MFPT. The centrality of a node determines the relative speed by which a node can receive and spread information over the network in a random process. Numerical simulations of an ensemble of random walkers moving on paradigmatic network models confirm this analytical prediction. A proof is provided of a strong law of large numbers for a one-dimensional random walk in a dynamic random environment given by a supercritical contact process in equilibrium. The proof is based on a coupling argument that traces the space-time cones containing the infection clusters generated by single infections and uses that the random walk eventually gets trapped inside the union of these cones. For the case where the local drifts of the random walk are smaller than the speed at which infection clusters grow, the random walk eventually gets trapped inside a single cone. This in turn leads to the existence of regeneration times at which the random walk forgets its past. The latter are used to prove a functional central limit theorem and a large deviation principle. The qualitative dependence of the speed, the volatility and the rate function on the infection parameter is investigated, and some open problems are mentioned. Our goal in this paper is to develop a practical framework for obtaining a uniform sample of users in an online social network (OSN) by crawling its social graph. Such a sample allows to estimate any user property and some topological properties as well. To this end, first, we consider and compare several candidate crawling techniques. Two approaches that can produce approximately uniform samples are the Metropolis-Hasting random walk (MHRW) and a re-weighted random walk (RWRW). Both have pros and cons, which we demonstrate through a comparison to each other as well as to the "ground truth." In contrast, using Breadth-First-Search (BFS) or an unadjusted Random Walk (RW) leads to substantially biased results. Second, and in addition to offline performance assessment, we introduce online formal convergence diagnostics to assess sample quality during the data collection process. We show how these diagnostics can be used to effectively determine when a random walk sample is of adequate size and quality. Third, as a case study, we apply the above methods to Facebook and we collect the first, to the best of our knowledge, representative sample of Facebook users. We make it publicly available and employ it to characterize several key properties of Facebook. We consider the problem of detecting communities or modules in networks, groups of vertices with a higher-than-average density of edges connecting them. Previous work indicates that a robust approach to this problem is the maximization of the benefit function known as "modularity" over possible divisions of a network. Here we show that this maximization process can be written in terms of the eigenspectrum of a matrix we call the modularity matrix, which plays a role in community detection similar to that played by the graph Laplacian in graph partitioning calculations. This result leads us to a number of possible algorithms for detecting community structure, as well as several other results, including a spectral measure of bipartite structure in networks and a new centrality measure that identifies those vertices that occupy central positions within the communities to which they belong. The algorithms and measures proposed are illustrated with applications to a variety of real-world complex networks. A complex network can be modeled as a graph representing the ''who knows who'' relationship. In the context of graph theory for social networks, the notion of centrality is used to assess the relative importance of nodes in a given network topology. For example, in a network composed of large dense clusters connected through only a few links, the nodes involved in those links are particularly critical as far as the network survivability is concerned. This may also impact any application running on top of it. Such information can be exploited for various topological maintenance issues to prevent congestion and disruption. This can also be used offline to identify the most important actors in large social interaction graphs. Several forms of centrality have been proposed so far. Yet, they suffer from imperfections: initially designed for small social graphs, they are either of limited use (degree centrality), either incompatible in a distributed setting (e.g. random walk betweenness centrality). In this paper we introduce a novel form of centrality: the second order centrality which can be computed in a distributed manner. This provides locally each node with a value reflecting its relative criticity and relies on a random walk visiting the network in an unbiased fashion. To this end, each node records the time elapsed between visits of that random walk (called return time in the sequel) and computes the standard deviation (or second order moment) of such return times. The key point is that central nodes see regularly the random walk compared to other topology nodes. Both through theoretical analysis and simulation, we show that the standard deviation can be used to accurately identify critical nodes as well as to globally characterize graphs topology in a distributed way. We finally compare our proposal to well-known centralities to assess its competitivity. Social contact networks play an increasingly important role in computational epidemiology. We focus on massive social contact networks that cover an urban region comprising of millions of individuals and billions of time varying interactions. It is impossible to obtain such networks by simply measuring social interactions. As a result, such networks are often synthesized using a diverse set of real-world data. The synthesis method can be viewed as a complex stochastic process that outputs one realization of such a network. The resulting networks are extremely large, dynamic and unstructured. Any meaningful description of such networks is usually done in terms of structural properties. Building on our earlier work, we synthesize a detailed social contact network for the National Capital Territory (NCT) of India. We first synthesize a social visitation network, representing people visiting locations during different time intervals. We then project it to synthesize a people-people contact network. We are not aware of other works on synthesizing the NCT network. Two important questions arise when synthesizing such massive dynamic social contact networks: (i) how does one compare the networks that span the same region and (ii) when is the synthesized network adequate. To address them, we compute a number of network measurements: some of which are classical, while others capture the semantics of social contact networks. These metrics are used to study the similarities and differences between two networks representing the same urban region. For question (ii), we study our ability to understand the dynamics and control of epidemics. Dynamical measures that capture the joint interaction between the local dynamical process and the network structure are presented and used to analyze the NCT networks. Starting from a simple animal-biology example, a general, somewhat counter-intuitive property of diffusion random walks is presented. It is shown that for any (non-homogeneous) purely diffusing system, under any isotropic uniform incidence, the average length of trajectories through the system (the average length of the random walk trajectories from entry point to first exit point) is independent of the characteristics of the diffusion process and therefore depends only on the geometry of the system. This exact invariance property may be seen as a generalization to diffusion of the well-known mean-chord-length property (Case K. M. and Zweifel P. F., Linear Transport Theory (Addison-Wesley) 1967), leading to broad physics and biology applications.