Choosing to Grow a Graph: Modeling Network Formation as Discrete Choice

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.Comment: 12 pages, 5 figures, 4 table

Scale-free behavior of networks with the copresence of preferential and uniform attachment rules

Complex networks in different areas exhibit degree distributions with heavy upper tail. A preferential attachment mechanism in a growth process produces a graph with this feature. We herein investigate a variant of the simple preferential attachment model, whose modifications are interesting for two main reasons: to analyze more realistic models and to study the robustness of the scale free behavior of the degree distribution. We introduce and study a model which takes into account two different attachment rules: a preferential attachment mechanism (with probability 1-p) that stresses the rich get richer system, and a uniform choice (with probability p) for the most recent nodes. The latter highlights a trend to select one of the last added nodes when no information is available. The recent nodes can be either a given fixed number or a proportion (\alpha n) of the total number of existing nodes. In the first case, we prove that this model exhibits an asymptotically power-law degree distribution. The same result is then illustrated through simulations in the second case. When the window of recent nodes has constant size, we herein prove that the presence of the uniform rule delays the starting time from which the asymptotic regime starts to hold. The mean number of nodes of degree k and the asymptotic degree distribution are also determined analytically. Finally, a sensitivity analysis on the parameters of the model is performed

Influence of the seed in affine preferential attachment trees

We study randomly growing trees governed by the affine preferential attachment rule. Starting with a seed tree $S$, vertices are attached one by one, each linked by an edge to a random vertex of the current tree, chosen with a probability proportional to an affine function of its degree. This yields a one-parameter family of preferential attachment trees $(T_n^S)_{n\geq|S|}$, of which the linear model is a particular case. Depending on the choice of the parameter, the power-laws governing the degrees in $T_n^S$ have different exponents. We study the problem of the asymptotic influence of the seed $S$ on the law of $T_n^S$. We show that, for any two distinct seeds $S$ and $S'$, the laws of $T_n^S$ and $T_n^{S'}$ remain at uniformly positive total-variation distance as $n$ increases. This is a continuation of Curien et al. (2015), which in turn was inspired by a conjecture of Bubeck et al. (2015). The technique developed here is more robust than previous ones and is likely to help in the study of more general attachment mechanisms.Comment: 31 pages, 1 figur

A Directed Preferential Attachment Model with Poisson Measurement

When modeling a directed social network, one choice is to use the traditional preferential attachment model, which generates power-law tail distributions. In a traditional directed preferential attachment, every new edge is added sequentially into the network. However, for real datasets, it is common to only have coarse timestamps available, which means several new edges are created at the same timestamp. Previous analyses on the evolution of social networks reveal that after reaching a stable phase, the growth of edge counts in a network follows a non-homogeneous Poisson process with a constant rate across the day but varying rates from day to day. Taking such empirical observations into account, we propose a modified preferential attachment model with Poisson measurement, and study its asymptotic behavior. This modified model is then fitted to real datasets, and we see it provides a better fit than the traditional one.Comment: 35 pages, 7 figure

The NN-stars network evolution model

A new network evolution model is introduced in this paper. The model is based on co-operations of $N$ units. The units are the nodes of the network and the co-operations are indicated by directed links. At each evolution step $N$ units co-operate which formally means that they form a directed $N$-star subgraph. At each step either a new unit joins to the network and it co-operates with $N-1$ old units or $N$ old units co-operate. During the evolution both preferential attachment and uniform choice are applied. Asymptotic power law distributions are obtained both for the in-degrees and the out-degrees.Comment: 28 pages, 5 figure

The power of 2 choices over preferential attachment

We introduce a new type of preferential attachment tree that includes choices in its evolution, like with Achlioptas processes. At each step in the growth of the graph, a new vertex is introduced. Two possible neighbor vertices are selected independently and with probability proportional to degree. Between the two, the vertex with smaller degree is chosen, and a new edge is created. We determine with high probability the largest degree of this graph up to some additive error term

A Preferential Attachment Paradox: How Preferential Attachment Combines with Growth to Produce Networks with Log-normal In-degree Distributions

Every network scientist knows that preferential attachment combines with growth to produce networks with power-law in-degree distributions. How, then, is it possible for the network of American Physical Society journal collection citations to enjoy a log-normal citation distribution when it was found to have grown in accordance with preferential attachment? This anomalous result, which we exalt as the preferential attachment paradox, has remained unexplained since the physicist Sidney Redner first made light of it over a decade ago. Here we propose a resolution. The chief source of the mischief, we contend, lies in Redner having relied on a measurement procedure bereft of the accuracy required to distinguish preferential attachment from another form of attachment that is consistent with a log-normal in-degree distribution. There was a high-accuracy measurement procedure in use at the time, but it would have have been difficult to use it to shed light on the paradox, due to the presence of a systematic error inducing design flaw. In recent years the design flaw had been recognised and corrected. We show that the bringing of the newly corrected measurement procedure to bear on the data leads to a resolution of the paradox.Comment: 13 pages, 4 figures, 2 table, 1 supplementary notes file; fixed broken figure and table reference

Toward Universal Testing of Dynamic Network Models

Numerous networks in the real world change over time, in the sense that nodes and edges enter and leave the networks. Various dynamic random graph models have been proposed to explain the macroscopic properties of these systems and to provide a foundation for statistical inferences and predictions. It is of interest to have a rigorous way to determine how well these models match observed networks. We thus ask the following goodness of fit question: given a sequence of observations/snapshots of a growing random graph, along with a candidate model M, can we determine whether the snapshots came from M or from some arbitrary alternative model that is well-separated from M in some natural metric? We formulate this problem precisely and boil it down to goodness of fit testing for graph-valued, infinite-state Markov processes and exhibit and analyze a universal test based on non-stationary sampling for a natural class of models.Comment: Accepted to the 31st International Conference on Algorithmic Learning Theor

Asymmetry and structural information in preferential attachment graphs

Graph symmetries intervene in diverse applications, from enumeration, to graph structure compression, to the discovery of graph dynamics (e.g., node arrival order inference). Whereas Erd\H{o}s-R\'enyi graphs are typically asymmetric, real networks are highly symmetric. So a natural question is whether preferential attachment graphs, where in each step a new node with $m$ edges is added, exhibit any symmetry. In recent work it was proved that preferential attachment graphs are symmetric for $m=1$, and there is some non-negligible probability of symmetry for $m=2$. It was conjectured that these graphs are asymmetric when $m \geq 3$. We settle this conjecture in the affirmative, then use it to estimate the structural entropy of the model. To do this, we also give bounds on the number of ways that the given graph structure could have arisen by preferential attachment. These results have further implications for information theoretic problems of interest on preferential attachment graphs.Comment: 24 pages; to appear in Random Structures & Algorithm

Distances in scale free networks at criticality

Scale-free networks with moderate edge dependence experience a phase transition between ultrasmall and small world behaviour when the power law exponent passes the critical value of three. Moreover, there are laws of large numbers for the graph distance of two randomly chosen vertices in the giant component. When the degree distribution follows a pure power law these show the same asymptotic distances of $\frac{\log N}{\log\log N}$ at the critical value three, but in the ultrasmall regime reveal a difference of a factor two between the most-studied rank-one and preferential attachment model classes. In this paper we identify the critical window where this factor emerges. We look at models from both classes when the asymptotic proportion of vertices with degree at least~$k$ scales like $k^{-2} (\log k)^{2\alpha + o(1)}$ and show that for preferential attachment networks the typical distance is $\big(\frac{1}{1+\alpha}+o(1)\big)\frac{\log N}{\log\log N}$ in probability as the number~$N$ of vertices goes to infinity. By contrast the typical distance in a rank one model with the same asymptotic degree sequence is $\big(\frac{1}{1+2\alpha}+o(1)\big)\frac{\log N}{\log\log N}.$ As $\alpha\to\infty$ we see the emergence of a factor two between the length of shortest paths as we approach the ultrasmall regime.Comment: 38 page