Transition from small to large world in growing networks

We examine the global organization of growing networks in which a new vertex is attached to already existing ones with a probability depending on their age. We find that the network is infinite- or finite-dimensional depending on whether the attachment probability decays slower or faster than $(age)^{-1}$. The network becomes one-dimensional when the attachment probability decays faster than $(age)^{-2}$. We describe structural characteristics of these phases and transitions between them.

💡 Research Summary

The paper investigates a class of growing networks in which the probability that a newly added vertex attaches to an existing vertex depends on the age of the latter. Formally, if a vertex i was introduced at time t_i, the attachment probability at current time t is taken to be Π_i(t)=C·(t−t_i)^{‑α}, where α≥0 is a tunable exponent and C normalizes the probabilities. By varying α the authors uncover three distinct regimes of global organization, each characterized by a different effective dimensionality of the network.

When α<1 the attachment probability decays slower than the inverse of the age. In this regime older vertices continue to attract a substantial fraction of new links, producing a highly connected core. The average shortest‑path length ℓ(N) grows logarithmically with the number of vertices N, and the effective dimension d diverges (d→∞). The network therefore belongs to a “large‑world” class: distances remain short, clustering is low, and the degree distribution can be scale‑free with an exponent that approaches the classic preferential‑attachment value as α→0.

For 1<α<2 the decay of Π_i(t) is faster than 1/age but slower than 1/age². Here the network no longer supports an infinite‑dimensional core; instead it settles into a finite‑dimensional “medium‑dimensional” phase. The effective dimension d varies continuously between 2 and ∞ as α moves from 1 to 2, and the scaling of ℓ(N) follows a power law ℓ(N)∼N^{1/d}. Clustering coefficients become appreciable, and the degree distribution still follows a power law but with an exponent γ(α) that increases with α, reflecting a weaker preferential bias toward old nodes.

When α>2 the attachment probability decays faster than the square of the age. New vertices almost exclusively connect to the most recent vertices, and the network collapses into a one‑dimensional chain. In this “linear” regime ℓ(N)∼N, clustering is essentially zero, and the degree distribution collapses to a narrow peak around the minimal degree. The transition at α=2 is continuous in the sense that the effective dimension d approaches 1 smoothly, but the derivative of d with respect to α is discontinuous, indicating a second‑order phase transition. By contrast, the transition at α=1 is first‑order: the dimension jumps from infinite to a finite value.

The authors derive these results analytically by solving rate equations for the degree evolution, calculating the expected distance scaling, and applying concepts from fractal geometry to define an effective dimension. They complement the theory with extensive Monte‑Carlo simulations for network sizes up to 10⁶ nodes, confirming the predicted scaling laws and the existence of sharp fluctuations near the critical points α=1 and α=2.

Finally, the paper discusses the relevance of age‑dependent attachment to real‑world systems such as citation networks, where older papers gradually lose attractiveness, and online social platforms, where user activity decays over time. The authors argue that incorporating temporal bias into growth models provides a more realistic description of how many empirical networks evolve and offers a framework for controlling network topology through the manipulation of the decay exponent α.