Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · physics/0702077

How does degree heterogeneity affect an order-disorder transition?

R. Lambiotte
View original source ↗Read source document ↗Export metadata ↓

Abstract

We focus on the role played by the node degree distribution on the way collective phenomena emerge on complex networks. To address this question, we focus analytically on a typical model for cooperative behaviour, the Majority Rule, applied to dichotomous networks. The latter are composed of two kinds of nodes, each kind $i$ being characterized by a degree $k_i$. Dichotomous networks are therefore a simple instance of heterogeneous networks, especially adapted in order to reveal the effect of degree heterogeneity. Our main result are that degree heterogeneity affects the location of the order-disorder transition and that the system exhibits non-equipartition of the average opinion between the two kinds of nodes. This effect is observed in the ordered phase and in the disordered phase.

Explore related subjects

physics.soc-ph

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

R. Lambiotte. 2007-04-17. How does degree heterogeneity affect an order-disorder transition?. https://doi.org/10.1209/0295-5075%2F78%2F68002

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Evolutionary dynamics of collective-risk social dilemma game with risk uncertainty and adaptive strategy

The collective-risk dilemma game constitutes the leading metaphor for studying the evolution of cooperation in populations of selfish individuals. Although most theoretical studies often consider the risk level as a fixed parameter, this may not fully reflect the uncertain nature of risk detected in real-world scenarios. Motivated by the observation that individuals can infer their game environment and accordingly adjust their actual behavior, here we investigate this game with uncertain risk. In particular, we introduce a type of adaptive agents who incur an observation cost to infer the risk level during game interactions, and subsequently decide whether to cooperate or defect based on their tolerance thresholds. We first calculate the fixation probabilities of three different strategies in a mutation-free regime. Our results show that adaptive agents are favored by natural selection at low observation costs and moderate tolerance thresholds. Moreover, the fixation of adaptive agents results in higher levels of cooperation when both parameters are relatively low. We then consider behavioral mutation and systematically explore how average cooperation level and group achievement vary with observation costs, tolerance thresholds, and mutation rates. Importantly, the main results remain intact when the mutation is considered. Furthermore, when the tolerance threshold is small, the presence of adaptive agents results in a higher cooperation level and group achievement compared to the traditional setup when only unconditional cooperators and defectors are present. This advantage, however, gradually erodes with an increasing tolerance threshold and these two quantities fall below the levels observed in the classic two-strategy case.

physics.soc-ph↗
arXiv2026

Current frontiers in mobility and well-being

Human mobility has a profound impact on social structure and collective behavior, which allows individuals to reshape group structures in response to strategy information, acting as a double-edged sword for cooperative behavior. It can both disrupt existing cooperative communities and enable individuals to escape adverse environments and seek high-quality interaction partners. This paper surveys current research progress on the impact of mobility on cooperation within the framework of evolutionary game theory, focusing on specific mechanisms of passive escape, dynamic self-organization driven by active choice, and the core trade-off between migration costs and speed. Mobility, an individual's spatial response to local strategic information that turns location into strategic advantage, influences cooperative evolution by altering interactive information and environmental feedback, consequently affecting sustainable social well-being, such as collective benefits. Under specific conditions, this self-interested behavior at the individual level can foster macrosocial harmony, embodying the unity of individual rationality and collective cooperation. Furthermore, we also discusses inter-layer migration and information routing mechanisms in multilayer networks and extends the perspective to human-machine hybrid systems where mobile agents with information regulation capabilities can serve as information supernodes. Our paper aims to provide a unified framework for understanding mobility as a combined physical and informational driving force in the evolution of cooperation.

physics.soc-ph↗
arXiv2026

Heraclitean Dialectical Concept Space

We introduce Heraclitean Dialectical Concept Spaces (HDCS), a topological framework for modelling stagewise structural change in systems of concepts. A stage consists of a nonempty concept set, a feasible family generating an external topology, and a neighbourhood assignment carrying additional relational structure. Neighbourhood overlap induces adjacency and channel ideals. Candidate emergence is represented by the CCER construction: an interaction profile, finite admissible cores, and anchored witnesses determine nonempty open candidate regions in uncovered remainder structure. Selected candidates may be reified as new atomic concepts at a later stage, while partial injective carry maps separately track persistence of existing identities. The resulting history space is obtained as a quotient of the disjoint union of stage spaces with the corresponding final topology. Product concept spaces extend the construction to cross-domain interaction. We derive support characterizations, mediated-interaction components, canonical CCER envelopes, monotonicity and impossibility criteria, product-space factorization, and invariance under concept-space isomorphism. Applications to monetary transformation, zero and representation, mammalian middle-ear evolution, and AI-related labour change illustrate the framework. HDCS is intended as a structural formalism rather than a causal or predictive theory of historical change.

physics.soc-ph↗