Dependency
Dependency theory is a socioeconomic framework originating in Latin America during the 1960s that attributes the persistent underdevelopment of peripheral nations to their structural subordination within the global capitalist system, where economic surplus is systematically extracted by core industrialized countries through unequal exchange and trade relations.[1][2] Developed as a critique of modernization theory, which viewed underdevelopment as an internal, transitional phase solvable via Western-style industrialization, dependency theory instead emphasizes external exploitation—such as deteriorating terms of trade and reliance on primary commodity exports—as the primary causal mechanism perpetuating poverty and inequality.[3] Key proponents, including Raúl Prebisch and André Gunder Frank, argued that integration into the world economy hinders autonomous growth, advocating delinking or import-substitution strategies to foster self-reliance.[4] The theory gained prominence amid post-colonial disillusionment with imported development models, influencing policy debates in the Global South and contributing to structuralist economics at institutions like the United Nations Economic Commission for Latin America.[5] However, it has faced substantial empirical challenges, including the rapid industrialization of East Asian economies like South Korea and Taiwan through export-oriented policies within the global system, which contradicted predictions of inevitable dependency traps.[6] Critics contend that the framework overemphasizes external determinism while downplaying internal factors such as governance, institutions, and policy choices, leading to its decline in academic favor since the 1980s amid evidence of successful market reforms in formerly peripheral states.[7][8] Despite these shortcomings, elements of dependency analysis persist in discussions of global value chains and neocolonial dynamics, though causal realism favors hybrid approaches integrating domestic agency with international constraints.[9]Philosophy
Ontological Dependency
Ontological dependence denotes a metaphysical relation in which the existence, identity, or essential properties of one entity are metaphysically contingent upon those of another, such that the dependent entity could not obtain without the independent one serving as its ground or bearer. This relation contrasts with ontologically independent substances, as articulated in Aristotle's Categories, where primary substances—such as individual animals or plants—exist per se and support accidents like qualities or quantities, which cannot exist separately but inhere in substances as their substrate.[10] Aristotle's framework posits that non-substances derive their reality from substances, establishing a foundational asymmetry in being.[11] Medieval scholasticism, particularly in Thomas Aquinas, extended this through the real distinction between essence (what a thing is) and existence (that it is), applicable to all finite beings. In creatures, essence limits existence but does not produce it; instead, esse is an act participated from God as the subsistent act of being itself, rendering all contingent entities dependent on the divine for their actuality.[12] This dependency forms a vertical chain from God to creation, where lower beings receive causal and existential support from higher ones, avoiding any egalitarian ontology of mutual independence.[13] In modern analytic metaphysics, Kit Fine refines ontological dependence by distinguishing rigid dependencies, as in his theory of embodiments, where an entity's parts or grounds are non-substitutable—such as letters forming a specific word token that rigidly constitutes its identity without variable replacement.[14] Fine's approach emphasizes non-modal, essence-based necessities in dependence, countering purely possible or causal reductions by prioritizing what metaphysically necessitates existence or identity.[15] Such relations imply a stratified reality, with independent bases (substances or ultimate grounds) hierarchically sustaining dependents, aligning with causal realism by revealing directed existential chains that explain derivativeness without reducing to temporal causation alone.[16] This hierarchy counters views of ontological flatness, underscoring how dependencies delineate real orders of priority in the structure of being.[11]Mathematics and Logic
Linear Dependence
In a vector space over a field, a finite set of vectors \{ \mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_k \} is linearly dependent if there exist scalars c_1, c_2, \dots, c_k, not all zero, such that c_1 \mathbf{v}_1 + c_2 \mathbf{v}_2 + \dots + c_k \mathbf{v}_k = \mathbf{0}.[17] This condition implies that at least one vector in the set can be expressed as a linear combination of the others, indicating redundancy in the spanning capabilities of the set.[18] Equivalently, the set is linearly dependent if its span has dimension less than k, or if the matrix formed by the vectors as columns has rank strictly less than k.[19] For example, in \mathbb{R}^2, the vectors \mathbf{v}_1 = (1, 0) and \mathbf{v}_2 = (2, 0) are linearly dependent because $2\mathbf{v}_1 - \mathbf{v}_2 = \mathbf{0}, reflecting their collinearity along the x-axis.[18] In contrast, including the zero vector in any set guarantees dependence, as $1 \cdot \mathbf{0} + 0 \cdot \mathbf{v}_i = \mathbf{0} for other vectors \mathbf{v}_i.[20] A key property is that subsets inherit dependence: if a set S_1 \subseteq S_2 and S_1 is dependent, then S_2 is dependent, as the nontrivial relation in S_1 extends trivially to S_2.[20] Moreover, in a vector space of dimension m, any set exceeding m vectors must be dependent.[19] The concept traces to 19th-century foundations of linear algebra, particularly Hermann Grassmann's 1844 Die Lineale Ausdehnungslehre, which introduced linear combinations and independence in a manner akin to modern treatments, distinguishing formal algebraic structure from geometric interpretations.[21] Grassmann's work emphasized extensors and their decompositions, prefiguring bases and dependence relations, while William Rowan Hamilton's contributions on quaternions and vector methods around 1843 influenced related ideas but focused less on abstract dependence.[22] Linear dependence underpins solving linear systems A\mathbf{x} = \mathbf{b}, where dependence among columns of A implies either no solution or infinitely many, as the homogeneous equation A\mathbf{x} = \mathbf{0} admits nontrivial solutions./02:_Vectors_matrices_and_linear_combinations/2.04:_Linear_independence) In eigenvalue problems, eigenvectors for distinct eigenvalues are linearly independent, enabling diagonalization of matrices with full sets of such eigenvectors and simplifying computations in dynamical systems.[23] For dimensionality reduction, techniques like principal component analysis identify orthogonal directions of variance, effectively projecting onto a basis that discards dependent components to minimize information loss while reducing feature count.[24] To verify dependence computationally, form the matrix with vectors as columns and apply Gaussian elimination to compute its rank; dependence holds if rank < number of vectors.[25] For square matrices, a zero determinant confirms dependence via singularity.[25] These methods, rooted in row reduction, scale via algorithms like QR decomposition for large sparse matrices, ensuring numerical stability in practice.[26]Functional and Relational Dependencies
In relational databases, a functional dependency (FD) is a constraint where the value of one attribute (or set of attributes), known as the determinant, uniquely determines the value of another attribute (or set) within a relation, ensuring data consistency and integrity.[27] This concept emerged as part of Edgar F. Codd's relational model, introduced in 1970, which structures data into tables (relations) with rows and columns to minimize redundancy through declarative constraints rather than procedural code.[28] For instance, in an employee relation with attributes {EmployeeID, Name, Department, Salary}, the FD EmployeeID → Name means each EmployeeID maps to exactly one Name, preventing duplicate or conflicting entries for the same identifier.[27] William W. Armstrong formalized inference rules for FDs in 1974, known as Armstrong's axioms, which provide a sound and complete set for deriving all implied dependencies from a given set: reflexivity (if Y ⊆ X, then X → Y), augmentation (if X → Y, then XZ → YZ for any Z), and transitivity (if X → Y and Y → Z, then X → Z).[29] These axioms enable reasoning about dependency closures, essential for database design processes like normalization, where relations are decomposed into higher normal forms (e.g., 3NF or BCNF) to eliminate anomalies such as insertion (inability to add data without nulls), deletion (loss of unrelated data), and update (inconsistent changes across duplicates).[30] From first principles, normalization enforces causal constraints mirroring real-world determinism—e.g., a unique key should causally imply dependent facts—reducing redundancy that empirically leads to storage savings of up to 50% in denormalized systems and query performance gains through optimized joins, as observed in relational schema evaluations.[31] Beyond basic FDs, relational dependencies encompass broader constraints in formal relational systems, including multivalued dependencies (MVDs, where X →→ Y means X determines a set of Y values independently of other attributes) and join dependencies, which generalize decomposition properties to preserve data equivalence under projections and natural joins.[32] In logical terms, these align with implication in first-order predicate logic underlying the relational model, where dependencies express universal quantifications (∀ tuples, if X matches, then Y follows), distinct from probabilistic or non-monotonic logics.[33] In graph-theoretic representations, relational dependencies can be modeled as directed acyclic graphs (DAGs), with nodes as attributes and edges indicating determination paths, facilitating visualization of inference chains and anomaly detection in complex schemas.[34] Empirical studies on normalized versus denormalized databases show that enforcing such dependencies reduces update anomalies by 70-90% in transactional workloads, enhancing overall system reliability without sacrificing query speed when indexes are applied.[35]Computer Science
Software Package Dependencies
Software package dependencies refer to the reliance of a software module or application on external libraries, frameworks, or components provided by third-party packages, which are managed through systems like package managers to enable reuse and modularity in development.[36] These dependencies can be direct, explicitly declared by the developer, or transitive, automatically included via the dependencies of other packages, forming a dependency graph that must be resolved during build or runtime processes.[37] A major challenge arises in the form of "dependency hell," a situation where conflicting versions of the same package—often introduced through transitive dependencies—prevent successful resolution, installation, or execution of software.[38] For instance, in Maven, which was first released in July 2004, transitive dependencies are fetched automatically, but version conflicts are resolved using a "nearest wins" strategy, where the version closest in the dependency tree prevails, potentially leading to suboptimal or vulnerable selections if not explicitly overridden.[37] Similarly, npm, introduced in 2010 for Node.js, employs a flattened dependency tree and semantic versioning to mitigate issues, yet real-world examples include conflicts where a direct dependency requires version 1.x of a library while a transitive one demands 2.x, forcing manual exclusions or overrides that complicate maintenance.[39] Such conflicts have been documented in projects using Gradle and Maven, where debugging involves tools like dependency trees to identify and exclude problematic paths.[40] To address these issues, package managers incorporate resolution algorithms, such as Maven's explicit dependency management inpom.xml files to enforce versions or exclusions, and npm's package-lock.json for reproducible installs that lock transitive versions. Auditing tools further aid in verification; OWASP Dependency-Check, an open-source software composition analysis utility, scans project dependencies against known vulnerability databases like the National Vulnerability Database (NVD) to identify outdated or insecure components.[41]
Security risks from dependencies are acute, as evidenced by supply chain attacks where malicious code is injected into trusted packages or updates. The 2020 SolarWinds Orion compromise involved Russian state actors inserting malware into software updates distributed to approximately 18,000 customers, exploiting the trust in vendor-supplied dependencies to enable persistent access to networks, including U.S. government agencies.[42] This incident highlighted how transitive dependencies amplify attack surfaces, prompting recommendations for integrity checks and minimal dependency footprints.
Despite these challenges, modularity via package dependencies promotes scalability by enabling independent development, testing, and deployment of components, which reduces coupling and allows teams to update isolated parts without full rebuilds.[43] In large projects, this decomposition facilitates evolvability, as modular designs support phased adaptations and reuse, empirically linked to faster design evolution in empirical studies of software architectures.[44] Causal analysis shows that by limiting recompilation to affected modules, build times decrease proportionally to project size, enhancing efficiency in scalable systems.[45]