We provide a framework for modeling social network formation through conditional multinomial logit models from discrete choice and random utility theory, in which each new edge is viewed as a “choice” made by a node to connect to another node, based on (generic) features of the other nodes available to make a connection. This perspective on network formation unifies existing models such as preferential attachment, triadic closure, and node fitness, which are all special cases, and thereby provides a flexible means for conceptualizing, estimating, and comparing models. The lens of discrete choice theory also provides several new tools for analyzing social network formation; for example, the significance of node features can be evaluated in a statistically rigorous manner, and mixtures of existing models can be estimated by adapting known expectation-maximization algorithms. We demonstrate the flexibility of our framework through examples that analyze a number of synthetic and real-world datasets. For example, we provide rigorous methods for estimating preferential attachment models and show how to separate the effects of preferential attachment and triadic closure. Non-parametric estimates of the importance of degree show a highly linear trend, and we expose the importance of looking carefully at nodes with degree zero. Examining the formation of a large citation graph, we find evidence for an increased role of degree when accounting for age.
There is a long history of parameter estimation in network formation processes @cite_67 @cite_19 . Probably the most studied formation model is preferential attachment @cite_52 , which relates the likelihood of new nodes connecting to a node @math to @math 's degree. More references on PA, since we claim it is the most studied? The original justification of the model was based on the degree distribution of the resulting graph qualitatively matching those of several early large-scale empirical networks. Shortly after, a number of authors proposed non-parametric methods to measure preferential attachment by counting what the degrees of the nodes getting edges are and normalizing them by the relative likelihood of those degrees @cite_41 @cite_7 @cite_73 . There are several generalizations of preferential attachment, where node @math is chosen proportional to some function @math of the degree @math of node @math (here, @math is standard preferential attachment). TODO: Need references if claiming there are several generalizations Much of this research has focused on the case when @math , i.e., attachment is preferential to some (latent) power @math of the degree @cite_17 .
@cite_67 @cite_7 @cite_41 @cite_52 @cite_19 @cite_73 @cite_17
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We consider a growing network, whose growth algorithm is based on the preferential attachment typical for scale-free constructions, but where the long-range bonds are disadvantaged. Thus, the probability of getting connected to a site at distance d is proportional to @math where is a tunable parameter of the model. We show that the properties of the networks grown with @math are close to those of the genuine scale-free construction, while for @math the structure of the network is quite different. Thus, in this regime, the node degree distribution is no longer a power law, and it is well represented by a stretched exponential. On the other hand, the small-world property of the growing networks is preserved at all values of . We give an explicit construction of the weak local limit of a class of preferential attachment graphs. This limit contains all local information and allows several computations that are otherwise hard, for example, joint degree distributions and, more generally, the limiting distribution of subgraphs in balls of any given radius @math around a random vertex in the preferential attachment graph. We also establish the finite-volume corrections which give the approach to the limit. A key ingredient of many current models proposed to capture the topological evolution of complex networks is the hypothesis that highly connected nodes increase their connectivity faster than their less connected peers, a phenomenon called preferential attachment. Measurements on four networks, namely the science citation network, Internet, actor collaboration and science coauthorship network indicate that the rate at which nodes acquire links depends on the node's degree, offering direct quantitative support for the presence of preferential attachment. We find that for the first two systems the attachment rate depends linearly on the node degree, while for the last two the dependence follows a sublinear power law. Our work introduces an approach for estimating the contribution of attachment mechanisms to the formation of growing networks. We present a generic model in which growth is driven by the continuous attachment of new nodes according to random and preferential linkage with a fixed probability. Past approaches apply likelihood analysis to estimate the probability of occurrence of each mechanism at a particular network instance, exploiting the concavity of the likelihood function at each point in time. However, the probability of connecting to existing nodes, and consequently the likelihood function itself, varies as networks grow. We establish conditions under which applying likelihood analysis guarantees the existence of a local maximum of the time-varying likelihood function and prove that a expectation maximization algorithm provides a convergent estimate. Furthermore, the in-degree distributions of the nodes in the growing networks is analytically characterized. Simulations show that, under the proposed conditions, expectation maximization and maximum-likelihood accurately estimate the actual contribution of each mechanism, and in-degree distributions converge to a stationary distributions. We revisit the classic problem of estimating the degree distribution moments of an undirected graph. Consider an undirected graph @math with @math vertices, and define (for @math ) @math . Our aim is to estimate @math within a multiplicative error of @math (for a given approximation parameter @math ) in sublinear time. We consider the sparse graph model that allows access to: uniform random vertices, queries for the degree of any vertex, and queries for a neighbor of any vertex. For the case of @math (the average degree), @math queries suffice for any constant @math (Feige, SICOMP 06 and Goldreich-Ron, RSA 08). Gonen-Ron-Shavitt (SIDMA 11) extended this result to all integral @math , by designing an algorithms that performs @math queries. We design a new, significantly simpler algorithm for this problem. In the worst-case, it exactly matches the bounds of Gonen-Ron-Shavitt, and has a much simpler proof. More importantly, the running time of this algorithm is connected to the degeneracy of @math . This is (essentially) the maximum density of an induced subgraph. For the family of graphs with degeneracy at most @math , it has a query complexity of @math . Thus, for the class of bounded degeneracy graphs (which includes all minor closed families and preferential attachment graphs), we can estimate the average degree in @math queries, and can estimate the variance of the degree distribution in @math queries. This is a major improvement over the previous worst-case bounds. Our key insight is in designing an estimator for @math that has low variance when @math does not have large dense subgraphs. Based on the formation of triad junctions, the mechanism proposed in this paper generates networks that exhibit extended rather than single power law behavior. Triad formation guarantees strong neighborhood clustering and community-level characteristics as the network size grows to infinity. The asymptotic behavior is of interest in the study of directed networks in which (i) the formation of links cannot be described according to the principle of preferential attachment, (ii) the in-degree distribution fits a power law for nodes with a high degree and an exponential form otherwise, (iii) clustering properties emerge at multiple scales and depend on both the number of links that newly added nodes establish and the probability of forming triads, and (iv) groups of nodes form modules that feature fewer links to the rest of the nodes. A detailed understanding of expansion in complex networks can greatly aid in the design and analysis of algorithms for a variety of important network tasks, including routing messages, ranking nodes, and compressing graphs. This has motivated several recent investigations of expansion properties in real-world graphs and also in random models of real-world graphs, like the preferential attachment graph. The results point to a gap between real-world observations and theoretical models. Some real-world graphs are expanders and others are not, but a graph generated by the preferential attachment model is an expander whp. We study a random graph Gn that combines certain aspects of geometric random graphs and preferential attachment graphs. This model yields a graph with power-law degree distribution where the expansion property depends on a tunable parameter of the model. The vertices of Gn are n sequentially generated points x1, x2, ..., xn chosen uniformly at random from the unit sphere in R3. After generating xt, we randomly connect it to m points from those points in x1, x2, ..., xt-1.... We investigate the degree sequences of geometric preferential attachment graphs in general compact metric spaces. We show that, under certain conditions on the attractiveness function, the behaviour of the degree sequence is similar to that of the preferential attachment with multiplicative fitness models investigated by When the metric space is finite, the degree distribution at each point of the space converges to a degree distribution which is an asymptotic power law whose index depends on the chosen point. For infinite metric spaces, we can show that for vertices in a Borel subset of S of positive measure the degree distribution converges to a distribution whose tail is close to that of a power law whose index again depends on the set. The emergence of order in natural systems is a constant source of inspiration for both physical and biological sciences. While the spatial order characterizing for example the crystals has been the basis of many advances in contemporary physics, most complex systems in nature do not offer such high degree of order. Many of these systems form complex networks whose nodes are the elements of the system and edges represent the interactions between them. Traditionally complex networks have been described by the random graph theory founded in 1959 by Paul Erdohs and Alfred Renyi. One of the defining features of random graphs is that they are statistically homogeneous, and their degree distribution (characterizing the spread in the number of edges starting from a node) is a Poisson distribution. In contrast, recent empirical studies, including the work of our group, indicate that the topology of real networks is much richer than that of random graphs. In particular, the degree distribution of real networks is a power-law, indicating a heterogeneous topology in which the majority of the nodes have a small degree, but there is a significant fraction of highly connected nodes that play an important role in the connectivity of the network. The scale-free topology of real networks has very important consequences on their functioning. For example, we have discovered that scale-free networks are extremely resilient to the random disruption of their nodes. On the other hand, the selective removal of the nodes with highest degree induces a rapid breakdown of the network to isolated subparts that cannot communicate with each other. The non-trivial scaling of the degree distribution of real networks is also an indication of their assembly and evolution. Indeed, our modeling studies have shown us that there are general principles governing the evolution of networks. Most networks start from a small seed and grow by the addition of new nodes which attach to the nodes already in the system. This process obeys preferential attachment: the new nodes are more likely to connect to nodes with already high degree. We have proposed a simple model based on these two principles wich was able to reproduce the power-law degree distribution of real networks. Perhaps even more importantly, this model paved the way to a new paradigm of network modeling, trying to capture the evolution of networks, not just their static topology. The linear preferential attachment hypothesis has been shown to be quite successful in explaining the existence of networks with power-law degree distributions. It is then quite important to determine if this mechanism is the consequence of a general principle based on local rules. In this work it is claimed that an effective linear preferential attachment is the natural outcome of growing network models based on local rules. It is also shown that the local models offer an explanation for other properties like the clustering hierarchy and degree correlations recently observed in complex networks. These conclusions are based on both analytical and numerical results for different local rules, including some models already proposed in the literature. Based on the empirical analysis of the dependency network in 18 Java projects, we develop a novel model of network growth which considersboth: an attachment mechanism and the addition of new nodes with a heterogeneous distribution of their initial degree, k0. Empirically we find that the cumulative degree distributions of initial degrees and of the final network, follow power-law behaviors: P(k0) ∝ k01−α, and P(k) ∝ k1−γ, respectively. For the total number of links as a function of the network size, we find empirically K(N) ∝ Nβ, where β is (at the beginning of thenetwork evolution) between 1.25 and 2, while converging to ∼ 1 for large N. This indicates a transition from a growth regime with increasing network density towards a sustainable regime, which revents a collapse because of ever increasing dependencies. Our theoretical framework is able to predict relations between the exponents α, β, γ, which also link issues of software engineering and developer activity. These relations are verified by means of computer simulations and empirical investigations. They indicate that the growth of real Open Source Software networks occurs on the edge between two regimes, which are either dominated by the initial degree distribution of added nodes, or by the preferential attachment mechanism. Hence, the heterogeneous degree distribution of newly added nodes, found empirically, is essential to describe the laws of sustainable growth in networks. We present a detailed study of network evolution by analyzing four large online social networks with full temporal information about node and edge arrivals. For the first time at such a large scale, we study individual node arrival and edge creation processes that collectively lead to macroscopic properties of networks. Using a methodology based on the maximum-likelihood principle, we investigate a wide variety of network formation strategies, and show that edge locality plays a critical role in evolution of networks. Our findings supplement earlier network models based on the inherently non-local preferential attachment. Based on our observations, we develop a complete model of network evolution, where nodes arrive at a prespecified rate and select their lifetimes. Each node then independently initiates edges according to a "gap" process, selecting a destination for each edge according to a simple triangle-closing model free of any parameters. We show analytically that the combination of the gap distribution with the node lifetime leads to a power law out-degree distribution that accurately reflects the true network in all four cases. Finally, we give model parameter settings that allow automatic evolution and generation of realistic synthetic networks of arbitrary scale. We define a class of growing networks in which new nodes are given a spatial position and are connected to existing nodes with a probability mechanism favoring short distances and high degrees. The competition of preferential attachment and spatial clustering gives this model a range of interesting properties. Empirical degree distributions converge to a limit law, which can be a power law with any exponent τ>2 . The average clustering coefficient of the networks converges to a positive limit. Finally, a phase transition occurs in the global clustering coefficients and empirical distribution of edge lengths when the power-law exponent crosses the critical value τ=3 . Our main tool in the proof of these results is a general weak law of large numbers in the spirit of Penrose and Yukich. We investigate the degree sequence of the geometric preferential attachment model of Flaxman, Frieze and Vera (2006), (2007) in the case where the self-loop parameter a is set to 0. We show that, given certain conditions on the attractiveness function F, the degree sequence converges to the same sequence as found for standard preferential attachment in (2001). We also apply our method to the extended model introduced in van der Esker (2008) which allows for an initial attractiveness term, proving similar results. We show that the asynchronous push-pull protocol spreads rumors in preferential attachment graphs (as defined by Barabasi and Albert) in time @math to all but a lower order fraction of the nodes with high probability. This is significantly faster than what synchronized protocols can achieve; an obvious lower bound for these is the average distance, which is known to be Θ(logn loglogn). We study the voter model on heterogeneous graphs. We exploit the nonconservation of the magnetization to characterize how consensus is reached. For a network of @math nodes with an arbitrary but uncorrelated degree distribution, the mean time to reach consensus @math scales as @math , where @math is the @math th moment of the degree distribution. For a power-law degree distribution @math , @math thus scales as @math for @math , as @math for @math , as @math for @math , as @math for @math , and as @math for @math . These results agree with simulation data for networks with both uncorrelated and correlated node degrees. We study simple interacting particle systems on heterogeneous networks, including the voter model and the invasion process. These are both two-state models in which in an update event an individual changes state to agree with a neighbor. For the voter model, an individual imports'' its state from a randomly chosen neighbor. Here the average time @math to reach consensus for a network of @math nodes with an uncorrelated degree distribution scales as @math , where @math is the @math moment of the degree distribution. Quick consensus thus arises on networks with broad degree distributions. We also identify the conservation law that characterizes the route by which consensus is reached. Parallel results are derived for the invasion process, in which the state of an agent is exported'' to a random neighbor. We further generalize to biased dynamics in which one state is favored. The probability for a single fitter mutant located at a node of degree @math to overspread the population the fixation probability is proportional to @math for the voter model and to @math for the invasion process. We consider the problem of diffusing information in networks that contain malicious nodes. We assume that each normal node in the network has no knowledge of the network topology other than an upper bound on the number of malicious nodes in its neighborhood. We introduce a topological property known as r-robustness of a graph, and show that this property provides improved bounds on tolerating malicious behavior, in comparison to traditional concepts such as connectivity and minimum degree. We use this topological property to analyze the canonical problems of distributed consensus and broadcasting, and provide sufficient conditions for these operations to succeed. Finally, we provide a construction for r-robust graphs and show that the common preferential-attachment model for scale-free networks produces a robust graph. In this paper, we introduce the concept of intrinsic strength which is used to describe the node's intrinsic property. Furthermore, we present a single preferential attachment model for the evolution of weighted networks in which the network growth is coupled with dynamical evolution of weights and intrinsic strength. The model yields a nontrivial time evolution of nodes' properties and generalized power law distributions for the weight, strength and degree, as confirmed in many real networks. The numerical simulation results are in good agreement with the analytical expressions. We study a graph-theoretic property known as robustness, which plays a key role in the behavior of certain classes of dynamics on networks (such as resilient consensus and contagion). This property is much stronger than other graph properties such as connectivity and minimum degree, in that one can construct graphs with high connectivity and minimum degree but low robustness. In this paper, we investigate the robustness of common random graph models for complex networks (Erdős-Renyi, geometric random, and preferential attachment graphs). We show that the notions of connectivity and robustness coincide on these random graph models: the properties share the same threshold function in the Erdős-Renyi model, cannot be very different in the geometric random graph model, and are equivalent in the preferential attachment model. This indicates that a variety of purely local diffusion dynamics will be effective at spreading information in such networks.