Network network, a collection of distinct points together with a set of connections joining certain pairs of those points, may be regarded as the most elementary abstraction of organised relation. In the language of mathematics such a structure is called a graph; the points are vertices and the connections are edges. The conception of a network therefore rests upon the simple notion that a system may be reduced to a set of entities whose mutual interactions are represented by links. From this elementary definition a great variety of phenomena—electrical circuits, transport routes, molecular assemblages, and logical arrangements—may be examined within a single formal framework. Historical origins. The first recognised problem of this kind appears in the eighteenth‑century work of Leonhard Euler on the Seven Bridges of Königsberg. Euler’s argument that no walk could cross each bridge exactly once without repetition introduced the idea of a graph as a tool for reasoning about connectivity. Shortly thereafter, Gustav Kirchhoff applied similar ideas to electrical networks, formulating laws that related the currents in a circuit to the topology of the underlying graph. These early investigations established that the pattern of connections, independent of the physical nature of the elements, determines the behaviour of the whole. The abstract formulation proceeds by assigning to each vertex a degree, the number of incident edges, and by considering paths, sequences of edges linking successive vertices. A network is said to be connected when a path exists between any two vertices; otherwise it separates into components. The notion of a cycle, a closed path returning to its origin without retracing any edge, proves essential in distinguishing tree‑like structures from those containing redundant connections. Such elementary concepts already afford a means of classifying networks according to their structural complexity. Beyond mere connectivity, many networks are endowed with a notion of flow. In a hydraulic or electrical system, each edge may be assigned a capacity, and a flow is a function assigning to each edge a quantity respecting the capacities and conserving the amount at each interior vertex. The classical problem of determining the greatest possible flow from a source vertex to a sink vertex, together with the complementary principle that the minimal capacity of a cut separating source from sink bounds this flow, constitutes a fundamental theorem of network theory. This result, though later expressed in more general terms, was already implicit in the work of Kirchhoff and in the later combinatorial studies of the early twentieth century. The relevance of networks to the theory of computation follows directly from the observation that a computing device may be represented as a configuration of logical elements linked by wires. The machine conceived by Alan Turing, though described as a tape traversed by a head, can be interpreted as a linear network in which each cell of the tape is a vertex and the head’s movement corresponds to a transition along an edge. More generally, switching circuits, as studied by Claude Shannon in the mid‑twentieth century, are precisely Boolean networks: each vertex performs a logical operation on the signals received from incident edges, and the overall behaviour of the circuit is determined by the topology of the network. Thus the study of networks supplies a natural language for expressing the architecture of any mechanical or electromechanical computer. In the realm of logic, networks have been employed to model the interdependence of propositions. A logical network consists of vertices representing statements and edges indicating logical implication or equivalence. By analysing the connectivity of such a network, one may deduce the consequences of a given set of premises, a method that anticipates modern proof‑theoretic techniques. The correspondence between logical deduction and traversal of a graph underscores the deep unity between reasoning and the combinatorial properties of networks. The biological sciences have long recognised that many processes are organised as networks. In the study of morphogenesis, for example, the diffusion of chemical substances across a tissue may be represented as a network of interacting compartments, each vertex corresponding to a region of the organism and each edge to a pathway for diffusion. The equations governing such systems are linear combinations of the concentrations at neighbouring vertices, a formulation that parallels the analysis of electrical networks. Theoretical work on pattern formation thus draws upon the same combinatorial foundations that underlie circuit theory. A further aspect of network theory concerns reliability. In engineering, a network is often required to continue functioning despite the failure of some of its components. The redundancy introduced by multiple paths between vertices enhances fault tolerance; the degree to which a network can sustain the loss of edges without becoming disconnected is measured by its connectivity. Early investigations into the probability of network failure employed combinatorial enumeration of spanning subgraphs, leading to results that remain central to modern reliability analysis. The enumeration of particular classes of networks constitutes a rich field of combinatorial mathematics. Kirchhoff’s theorem, which expresses the number of spanning trees of a connected graph as a determinant derived from the network’s incidence structure, provides a powerful tool for counting. Such results illuminate the relationship between algebraic properties of a network and its combinatorial structure, a theme that recurs throughout the theory. Algorithmic treatment of networks began with elementary procedures for traversing a graph. Depth‑first and breadth‑first searches, methods for visiting vertices systematically, allow the determination of connectivity, the detection of cycles, and the construction of spanning trees. The problem of finding a shortest path between two vertices, solved by the method of successive approximations, illustrates how quantitative information may be incorporated into the purely topological framework. These algorithms, though simple in conception, form the basis of more elaborate procedures employed in optimisation and decision problems. When networks are allowed to be infinite, questions of decidability arise. The reachability problem—whether a given vertex can be reached from another by a finite sequence of edges—may be undecidable in certain classes of infinite graphs, a circumstance that mirrors the limitations discovered in the theory of computable functions. The existence of such pathological examples demonstrates that the combinatorial simplicity of a network does not guarantee algorithmic tractability. Boolean networks, composed of vertices each performing a binary operation on the states of incident vertices, provide a rudimentary model of neural activity. Early work on such systems anticipated the later development of artificial neural networks, showing that the dynamics of a network can exhibit complex behaviour even when each component follows a simple rule. The study of fixed points and cycles within these networks yields insight into the stability of logical circuits and, by extension, into the operation of mechanical calculators. Complexity considerations reveal that many natural questions about networks are computationally demanding. Determining a minimal set of edges whose removal disconnects a graph, or finding a maximal matching of vertices, are problems known to require effort beyond that of simple traversal. The classification of such problems as difficult in the sense of unsolvable by a universal machine underscores the intimate connection between network theory and the limits of computation. Beyond abstract mathematics, networks have found concrete application in the design of transport and communication systems. The layout of railway lines, the routing of telegraph cables, and the organisation of early telephone exchanges are all instances of engineering a network to satisfy constraints of cost, capacity, and reliability. The planning of such systems relies upon the same principles of connectivity, flow, and redundancy that appear in the theoretical treatment. Social structures may likewise be represented as networks of acquaintance or kinship. By modelling individuals as vertices and social ties as edges, one may analyse the spread of information, the formation of cliques, and the robustness of a community to the loss of members. Though the data are of a different nature, the mathematical tools applied remain those developed for more physical networks, illustrating the universality of the graph concept. In summary, the notion of a network furnishes a unifying abstraction that bridges the disparate realms of electricity, mechanics, biology, logic, and society. By reducing a system to vertices and edges, one isolates the pattern of interrelation that governs its behaviour, enabling rigorous analysis through combinatorial, algebraic, and algorithmic methods. The study of networks, from Euler’s bridges to modern computational models, thus occupies a central place in the mathematical description of organised complexity. [role=marginalia, type=clarification, author="a.darwin", status="adjunct", year="2026", length="38", targets="entry:network", scope="local"] The notion of a network mirrors the inter‑connectedness observed in living systems: each vertex may be likened to an individual organism, each edge to the functional relationship—whether competitive, symbiotic, or reproductive—by which natural selection operates upon the whole. [role=marginalia, type=objection, author="a.simon", status="adjunct", year="2026", length="37", targets="entry:network", scope="local"] One must caution that the reduction of any organised relation to mere vertices and edges omits higher‑order interactions; many physical and chemical systems require hyperedges or simplicial complexes to capture multi‑body constraints, a fact the entry overlooks. [role=marginalia, type=objection, author="Reviewer", status="adjunct", year="2026", length="42", targets="entry:network", scope="local"] Network thinking can obscure hierarchy and power: who controls the nodes and the protocols is often more decisive than the topology of the graph. See Also See "Machine" See "Automaton"