Popularity-driven networking

Abstract

We investigate the growth of connectivity in a network. In our model, starting with a set of disjoint nodes, links are added sequentially. Each link connects two nodes, and the connection rate governing this random process is proportional to the degrees of the two nodes. Interestingly, this network exhibits two abrupt transitions, both occurring at finite times. The first is a percolation transition in which a giant component, containing a finite fraction of all nodes, is born. The second is a condensation transition in which the entire system condenses into a single, fully connected, component. We derive the size distribution of connected components as well as the degree distribution, which is purely exponential throughout the evolution. Furthermore, we present a criterion for the emergence of sudden condensation for general homogeneous connection rates.PACS numbers: 89.75, Hc, 02.50.Ey, 05.20.Dd Networks are sets of nodes that are connected by links. Models for the evolution of complex networks fall into two general classes: evolving graphs where the number of nodes is fixed but the number of links grows, and growing networks where the number of nodes and the number of links both grow [1][2][3][4][5][6][7]. The classic evolving random graph model, where pairs of randomly selected nodes are repeatedly connected by links, captures the nucleation of a giant connected component with a macroscopic number of nodes [8,9]. The preferential attachment model of network growth, where newly added nodes connect to existing nodes with probability that is proportional to the degree, yields the broad degree distributions with power-law tails that characterize many real-world complex networks [10,11].In this letter, we study random graphs that evolve according to a preferential attachment mechanism. In our model, the number of nodes is fixed but the number of connections grows. The "popularity" of each node, as measured by the degree, governs the connection process. This fusion between the two seminal network models of random graphs and preferential attachment, is inspired by Facebook, the immense online network of cyber-friends [6]. In Facebook, new connections are formed via friendship requests from one member of the network to another; upon acceptance, the two become friends. Preferential attachment implies that members seek and accept friends according to popularity.In our model for the popularity-driven growth of connectivity in a network, the system consists initially of a set of disjoint nodes. We consider the natural situation where the degree of a node, defined as the number of existing connections, controls the rate by which two nodes connect with each other. Specifically, a node with degree i and a node with degree j connect at a rate that is linear in i and linear in j as well. We find that the degree distribution remains purely exponential throughout the evolution. Thus, the preferential attachment mechanism leads to opposite results in fixed and in growing networks: the tail of the degree distribution is narro...

Preferential attachment without vertex growth: emergence of the giant component

“…A variant of this result for α = 1 was also reported by Ben-Naim and Krapivsky [4]. In fact, Pittel [43] proved much stronger estimates on the size L 1 (m) = L 1 (G α n,m ) of the largest component of G α n,m , in particular near the critical point m c .…”

“…From 2011 onwards Borgs, Chayes, Lovász, Sós, and Vesztergombi [9], Ráth and Szakács [45] and [30,Example 7.9] also studied a natural multigraph variant of G α n,m (see Section 2.1), motivated by the emerging (multi)graph limit theory paradigm. Furthermore, in 2012, Ben-Naim and Krapivsky [4] proposed and studied the model G α n,m by statistical physics methods, motivated by the way connections are formed on Facebook (see also [52] and Appendix A. 2.2).…”

“…Focusing for simplicity on the 'supercritical' phase (where the unique 'giant' component has emerged as the largest component), his result [43, Theorem 1] can be written as follows. 4 If ε = O(1) and ε 4 n → ∞ as n → ∞, then, for a certain function ρ α (ε) with ρ α (ε) = Θ(ε) as ε ց 0, we have 5…”

Unified underpinning of human mobility in the real world and cyberspace

“…In fact, the model in this paper is similar to a preferential attachment model with a fixed number of nodes (short for PA n ) which has been studied in the literature [39]. However, we would like to emphasize that there are some key differences between our model and existing literature on PA n .…”

Empirical Analysis and Modeling of Users' Topic Interests in Online Forums

“…Researchers in this area have attempted to determine the underlying law of complex networks in real society, and to understand their essential evolutionary mechanisms. Thus, theoretical network models have been developed to simulate the actual growth process [4], [5]. These models extract the most important factor of the evolving dynamics, and therefore, they can achieve some statistical characteristics of actual networks, such as power-law degree distribution, and small average path length.…”