What is emergence, after all?

Abstract

The term emergence is increasingly used across scientific disciplines to describe phenomena that arise from interactions among a system's components but cannot be readily inferred by examining those components in isolation. While often invoked to explain higher-level behaviors, such as flocking, synchronization, or collective intelligence, the term is frequently used without precision, sometimes giving rise to ambiguity or even mystique. In this perspective paper, we clarify the scientific meaning of emergence as a measurable and physically grounded phenomenon. Through concrete examples, such as temperature, magnetism, and herd immunity in social networks, we review how collective behavior can arise from local interactions that are constrained by global boundaries. By refining the concept of emergence, we gain a clearer and more grounded understanding of complex systems. Our goal is to show that emergence, when properly framed, offers not mysticism but insight.

Figures (3)

• Figure 1: Usage trajectories of key concepts in complexity discourse, 1900 – 2022. Per-million-word frequency of six terms in the Google Books English corpus, 1900–2022. Seven-year–smoothed curves (smoothing = 3) reveal the rise of "emergence" and "emergent" compared to other famous keywords based on data downloaded via Google Ngram Viewer googleNgramEmergence.
• Figure 2: Coarse-graining, Universality, and RG-flow.(a) Temperature as a coarse-grained variable: under local equilibrium, averaging kinetic energies defines temperature, $\mathcal{F}:(v_1,\ldots,v_N)\!\mapsto\! T$. (b) Local coarse-graining by $2{\times}2$ blocks with majority-vote, $S_I=\operatorname{sgn}(s_i+s_j+s_k+s_l)$ for $s_\ell\in\{\pm1\}$; iterating the map flows toward a uniform block-spin configuration. (c) Liquid–gas criticality shares the Ising universality class. The control parameter is the reduced temperature $\tau=(T-T_c)/T_c$. On a lattice, the lattice-gas mapping $n_i\!\in\!\{0,1\}\mapsto \sigma_i=2n_i-1$ (occupied $\uparrow$, empty $\downarrow$) makes the density deviation $\Delta\rho=\rho-\rho_c$ directly analogous to the magnetization $m$. (d) Increasing temperature drives a continuous transition from an ordered phase to a disordered phase, and the transition point we see scaling behaviors sethna2021statisticalkardar2007statistical. (e) Coarse-graining compresses high-resolution data and induces a renormalization-group (RG) flow in theory space. When the coarse-grained description is related to the microscopic one by parameter redefinitions, the model is said to be renormalizable. A macro-law screens off micro-details but remains autonomous, meaning that macro variables aren’t just convenient summaries but “real enough” to be the subject of genuine laws and explanations—even if many different microstates realize the same macrostate.
• Figure 3: Emergence of herd immunity in social networks.(a–c) Each immune node (blue) deactivates its adjacent edges, forming a local “firewall” that halts transmission to nearby susceptible nodes. The more connections an immune node has, the larger the firewall it establishes. As immunization progresses, these firewalls begin to overlap and coalesce, forming a larger collective barrier that expands nonlinearly—often outpacing the fraction of immunized individuals. (d) Once a critical fraction of nodes (white) is immunized, herd immunity emerges: the remaining susceptible nodes (black) become indirectly protected. This emergent protection is shaped by both the structural and geometric properties of the social network. (e) The thick solid curve shows the remaining susceptible fraction $\pi_\mathcal{S}$ as a function of the immunized fraction $\pi_\mathcal{R}$, constrained by $\pi_\mathcal{S} + \pi_\mathcal{R} \leq 1$. The shaded region quantifies the share of individuals shielded through structural (indirect) immunity. Horizontal intersections of the curve indicate total immunity thresholds, with $\pi^*_\mathcal{R}$ marking the structural herd immunity point. (f) The network in panel (d) can be rearranged to reveal an interface of $\mathcal{SR}$ links separating immune (white) from susceptible (black) nodes. The density of these interface links, $\rho_{\mathcal{SR}}$, serves as a proxy for the potential of epidemic containment and the strength of indirect protection.