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89-614: Routing is the process of selecting a path for traffic in a network or between or across multiple networks. Broadly, routing is performed in many types of networks, including circuit-switched networks , such as the public switched telephone network (PSTN), and computer networks , such as the Internet . In packet switching networks, routing is the higher-level decision making that directs network packets from their source toward their destination through intermediate network nodes by specific packet forwarding mechanisms. Packet forwarding

178-457: A connected acyclic undirected graph. A forest is an undirected graph in which any two vertices are connected by at most one path, or equivalently an acyclic undirected graph, or equivalently a disjoint union of trees. A polytree (or directed tree or oriented tree or singly connected network ) is a directed acyclic graph (DAG) whose underlying undirected graph is a tree. A polyforest (or directed forest or oriented forest )

267-472: A directed graph , or a simple graph to distinguish it from a multigraph ) is a pair G = ( V , E ) , where V is a set whose elements are called vertices (singular: vertex), and E is a set of unordered pairs { v 1 , v 2 } {\displaystyle \{v_{1},v_{2}\}} of vertices, whose elements are called edges (sometimes links or lines ). The vertices u and v of an edge { u , v } are called

356-437: A directed simple graph permitting loops and a directed multigraph permitting loops (or a quiver ) respectively. The edges of a directed simple graph permitting loops G is a homogeneous relation ~ on the vertices of G that is called the adjacency relation of G . Specifically, for each edge ( x , y ) , its endpoints x and y are said to be adjacent to one another, which is denoted x ~ y . A mixed graph

445-441: A graph is a structure consisting of a set of objects where some pairs of the objects are in some sense "related". The objects are represented by abstractions called vertices (also called nodes or points ) and each of the related pairs of vertices is called an edge (also called link or line ). Typically, a graph is depicted in diagrammatic form as a set of dots or circles for the vertices, joined by lines or curves for

534-446: A network is a graph in which a number (the weight) is assigned to each edge. Such weights might represent for example costs, lengths or capacities, depending on the problem at hand. Such graphs arise in many contexts, for example in shortest path problems such as the traveling salesman problem . One definition of an oriented graph is that it is a directed graph in which at most one of ( x , y ) and ( y , x ) may be edges of

623-414: A complete path through them. Such systems generally use next-hop routing. Most systems use a deterministic dynamic routing algorithm. When a device chooses a path to a particular final destination, that device always chooses the same path to that destination until it receives information that makes it think some other path is better. A few routing algorithms do not use a deterministic algorithm to find

712-406: A directed graph is sometimes defined to be an ordered triple G = ( V , E , ϕ ) comprising: To avoid ambiguity, this type of object may be called precisely a directed multigraph . A loop is an edge that joins a vertex to itself. Directed graphs as defined in the two definitions above cannot have loops, because a loop joining a vertex x {\displaystyle x} to itself

801-426: A graph is its number | V | of vertices, usually denoted by n . The size of a graph is its number | E | of edges, typically denoted by m . However, in some contexts, such as for expressing the computational complexity of algorithms, the term size is used for the quantity | V | + | E | (otherwise, a non-empty graph could have size 0). The degree or valency of

890-412: A hub is assortative when it tends to connect to other hubs. A disassortative hub avoids connecting to other hubs. If hubs have connections with the expected random probabilities, they are said to be neutral. There are three methods to quantify degree correlations. The recurrence matrix of a recurrence plot can be considered as the adjacency matrix of an undirected and unweighted network. This allows for

979-413: A path that minimizes their travel time. With such routing, the equilibrium routes can be longer than optimal for all drivers. In particular, Braess's paradox shows that adding a new road can lengthen travel times for all drivers. In a single-agent model used, for example, for routing automated guided vehicles (AGVs) on a terminal, reservations are made for each vehicle to prevent simultaneous use of

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1068-405: A path through a multistage switching fabric . Depending on the application for which path selection is performed, different metrics can be used. For example, for web requests one can use minimum latency paths to minimize web page load time, or for bulk data transfers one can choose the least utilized path to balance load across the network and increase throughput. A popular path selection objective

1157-542: A presence in New York , connected by a fast link with latency 5  ms —and each has a presence in London connected by a 5 ms link. Suppose both ISPs have trans-Atlantic links that connect their two networks, but A 's link has latency 100 ms and B 's has latency 120 ms. When routing a message from a source in A 's London network to a destination in B 's New York network, A may choose to immediately send

1246-415: A record of the routes to various network destinations. Routing tables may be specified by an administrator, learned by observing network traffic or built with the assistance of routing protocols . Routing, in a narrower sense of the term, often refers to IP routing and is contrasted with bridging . IP routing assumes that network addresses are structured and that similar addresses imply proximity within

1335-500: A series of funnels connected by tubes. Here, the amount of water from the original source is infinite. Also, any funnels that have been exposed to the water continue to experience the water even as it passes into successive funnels. The non-conserved model is the most suitable for explaining the transmission of most infectious diseases , neural excitation, information and rumors, etc. The question of how to immunize efficiently scale free networks which represent realistic networks such as

1424-403: A significant use in military intelligence, for uncovering insurgent networks of both hierarchical and leaderless nature. With the recent explosion of publicly available high throughput biological data , the analysis of molecular networks has gained significant interest. The type of analysis in this context is closely related to social network analysis, but often focusing on local patterns in

1513-424: A simple graph cannot start and end at the same vertex. Graphs with self-loops will be characterized by some or all A ii being equal to a positive integer, and multigraphs (with multiple edges between vertices) will be characterized by some or all A ij being equal to a positive integer. Undirected graphs will have a symmetric adjacency matrix (meaning A ij = A ji ). A directed graph or digraph

1602-491: A single central device decides ahead of time the complete path of every packet. In some other small systems, whichever edge device injects a packet into the network decides ahead of time the complete path of that particular packet. In either case, the route-planning device needs to know a lot of information about what devices are connected to the network and how they are connected to each other. Once it has this information, it can use an algorithm such as A* search algorithm to find

1691-438: A standard shortest paths algorithm such as Dijkstra's algorithm . The result is a tree graph rooted at the current node, such that the path through the tree from the root to any other node is the least-cost path to that node. This tree then serves to construct the routing table, which specifies the best next hop to get from the current node to any other node. A link-state routing algorithm optimized for mobile ad hoc networks

1780-410: A value known as the administrative distance to each route, where smaller administrative distances indicate routes learned from a protocol assumed to be more reliable. A local administrator can set up host-specific routes that provide more control over network usage, permits testing, and better overall security. This is useful for debugging network connections or routing tables. In some small systems,

1869-413: A vertex is the number of edges that are incident to it; for graphs with loops, a loop is counted twice. In a graph of order n , the maximum degree of each vertex is n − 1 (or n + 1 if loops are allowed, because a loop contributes 2 to the degree), and the maximum number of edges is n ( n − 1)/2 (or n ( n + 1)/2 if loops are allowed). The edges of a graph define a symmetric relation on

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1958-408: Is a directed acyclic graph whose underlying undirected graph is a forest. More advanced kinds of graphs are: Two edges of a graph are called adjacent if they share a common vertex. Two edges of a directed graph are called consecutive if the head of the first one is the tail of the second one. Similarly, two vertices are called adjacent if they share a common edge ( consecutive if the first one

2047-434: Is a directed graph in which every ordered pair of vertices in the graph is strongly connected. Otherwise, it is called a weakly connected graph if every ordered pair of vertices in the graph is weakly connected. Otherwise it is called a disconnected graph . A k-vertex-connected graph or k-edge-connected graph is a graph in which no set of k − 1 vertices (respectively, edges) exists that, when removed, disconnects

2136-405: Is a graph in which edges have orientations. In one restricted but very common sense of the term, a directed graph is a pair G = ( V , E ) comprising: To avoid ambiguity, this type of object may be called precisely a directed simple graph . In the edge ( x , y ) directed from x to y , the vertices x and y are called the endpoints of the edge, x the tail of the edge and y

2225-400: Is a graph in which some edges may be directed and some may be undirected. It is an ordered triple G = ( V , E , A ) for a mixed simple graph and G = ( V , E , A , ϕ E , ϕ A ) for a mixed multigraph with V , E (the undirected edges), A (the directed edges), ϕ E and ϕ A defined as above. Directed and undirected graphs are special cases. A weighted graph or

2314-569: Is available over the forwarding state, for example, using software-defined networking , routing techniques can be used that aim to optimize global and network-wide performance metrics. This has been used by large internet companies that operate many data centers in different geographical locations attached using private optical links, examples of which include Microsoft's Global WAN, Facebook's Express Backbone, and Google's B4. Global performance metrics to optimize include maximizing network utilization, minimizing traffic flow completion times, maximizing

2403-420: Is called a disconnected graph . In a directed graph, an ordered pair of vertices ( x , y ) is called strongly connected if a directed path leads from x to y . Otherwise, the ordered pair is called weakly connected if an undirected path leads from x to y after replacing all of its directed edges with undirected edges. Otherwise, the ordered pair is called disconnected . A strongly connected graph

2492-451: Is carried in a special path attribute that records the sequence of routing domains through which the reachability information has passed. A route is defined as a pairing between a destination and the attributes of the path to that destination, thus the name, path-vector routing; The routers receive a vector that contains paths to a set of destinations. Path selection involves applying a routing metric to multiple routes to select (or predict)

2581-527: Is directed, because owing money is not necessarily reciprocated. Graphs are the basic subject studied by graph theory. The word "graph" was first used in this sense by J. J. Sylvester in 1878 due to a direct relation between mathematics and chemical structure (what he called a chemico-graphical image). Definitions in graph theory vary. The following are some of the more basic ways of defining graphs and related mathematical structures . A graph (sometimes called an undirected graph to distinguish it from

2670-450: Is due, in part, because two ISPs may be connected through multiple connections. In choosing the single router-level path, it is common practice for each ISP to employ hot-potato routing : sending traffic along the path that minimizes the distance through the ISP's own network—even if that path lengthens the total distance to the destination. For example, consider two ISPs, A and B . Each has

2759-421: Is implied that the graphs discussed are finite. If the graphs are infinite, that is usually specifically stated. In an undirected graph, an unordered pair of vertices { x , y } is called connected if a path leads from x to y . Otherwise, the unordered pair is called disconnected . A connected graph is an undirected graph in which every unordered pair of vertices in the graph is connected. Otherwise, it

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2848-591: Is increasingly employed by banks and insurance agencies in fraud detection, by telecommunication operators in telecommunication network analysis, by medical sector in epidemiology and pharmacology , in law enforcement investigations , by search engines for relevance rating (and conversely by the spammers for spamdexing and by business owners for search engine optimization ), and everywhere else where relationships between many objects have to be analyzed. Links are also derived from similarity of time behavior in both nodes. Examples include climate networks where

2937-451: Is interested in dynamics on networks or the robustness of a network to node/link removal, often the dynamical importance of a node is the most relevant centrality measure. These concepts are used to characterize the linking preferences of hubs in a network. Hubs are nodes which have a large number of links. Some hubs tend to link to other hubs while others avoid connecting to hubs and prefer to connect to nodes with low connectivity. We say

3026-435: Is similar to the distance vector algorithm in the sense that each border router advertises the destinations it can reach to its neighboring router. However, instead of advertising networks in terms of a destination and the distance to that destination, networks are advertised as destination addresses and path descriptions to reach those destinations. The path, expressed in terms of the domains (or confederations) traversed so far,

3115-401: Is subject to instability if there are more than a few hops in the domain. Link state routing needs significant resources to calculate routing tables. It also creates heavy traffic due to flooding. Path-vector routing is used for inter-domain routing. It is similar to distance vector routing. Path-vector routing assumes that one node (there can be many) in each autonomous system acts on behalf of

3204-505: Is the dominant form of message delivery on the Internet. This article focuses on unicast routing algorithms. With static routing , small networks may use manually configured routing tables. Larger networks have complex topologies that can change rapidly, making the manual construction of routing tables unfeasible. Nevertheless, most of the public switched telephone network (PSTN) uses pre-computed routing tables, with fallback routes if

3293-483: Is the edge (for a directed simple graph) or is incident on (for a directed multigraph) ( x , x ) {\displaystyle (x,x)} which is not in { ( x , y ) ∣ ( x , y ) ∈ V 2 and x ≠ y } {\displaystyle \{(x,y)\mid (x,y)\in V^{2}\;{\textrm {and}}\;x\neq y\}} . So to allow loops

3382-688: Is the optimized Link State Routing Protocol (OLSR). OLSR is proactive; it uses Hello and Topology Control (TC) messages to discover and disseminate link-state information through the mobile ad hoc network. Using Hello messages, each node discovers 2-hop neighbor information and elects a set of multipoint relays (MPRs). MPRs distinguish OLSR from other link-state routing protocols. Distance vector and link-state routing are both intra-domain routing protocols. They are used inside an autonomous system , but not between autonomous systems. Both of these routing protocols become intractable in large networks and cannot be used in inter-domain routing. Distance vector routing

3471-429: Is the tail and the second one is the head of an edge), in which case the common edge is said to join the two vertices. An edge and a vertex on that edge are called incident . The graph with only one vertex and no edges is called the trivial graph . A graph with only vertices and no edges is known as an edgeless graph . The graph with no vertices and no edges is sometimes called the null graph or empty graph , but

3560-436: Is the transit of network packets from one network interface to another. Intermediate nodes are typically network hardware devices such as routers , gateways , firewalls , or switches . General-purpose computers also forward packets and perform routing, although they have no specially optimized hardware for the task. The routing process usually directs forwarding on the basis of routing tables . Routing tables maintain

3649-448: Is the union of two disjoint sets, W and X , so that every vertex in W is adjacent to every vertex in X but there are no edges within W or X . A path graph or linear graph of order n ≥ 2 is a graph in which the vertices can be listed in an order v 1 , v 2 , …, v n such that the edges are the { v i , v i +1 } where i = 1, 2, …, n − 1. Path graphs can be characterized as connected graphs in which

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3738-400: Is to reduce the average completion times of traffic flows and the total network bandwidth consumption. Recently, a path selection metric was proposed that computes the total number of bytes scheduled on the edges per path as selection metric. An empirical analysis of several path selection metrics, including this new proposal, has been made available. In some networks, routing is complicated by

3827-477: The Bellman–Ford algorithm . This approach assigns a cost number to each of the links between each node in the network. Nodes send information from point A to point B via the path that results in the lowest total cost (i.e. the sum of the costs of the links between the nodes used). When a node first starts, it only knows of its immediate neighbors and the direct cost involved in reaching them. (This information —

3916-995: The World Wide Web , Internet , gene regulatory networks , metabolic networks, social networks , epistemological networks, etc.; see List of network theory topics for more examples. Euler 's solution of the Seven Bridges of Königsberg problem is considered to be the first true proof in the theory of networks. Network problems that involve finding an optimal way of doing something are studied as combinatorial optimization . Examples include network flow , shortest path problem , transport problem , transshipment problem , location problem , matching problem , assignment problem , packing problem , routing problem , critical path analysis , and program evaluation and review technique . The analysis of electric power systems could be conducted using network theory from two main points of view: Social network analysis examines

4005-628: The diffusion of innovations , news and rumors. Similarly, it has been used to examine the spread of both diseases and health-related behaviors . It has also been applied to the study of markets , where it has been used to examine the role of trust in exchange relationships and of social mechanisms in setting prices. It has been used to study recruitment into political movements , armed groups, and other social organizations. It has also been used to conceptualize scientific disagreements as well as academic prestige. More recently, network analysis (and its close cousin traffic analysis ) has gained

4094-439: The eigenvectors of the adjacency matrix corresponding to a network, to determine nodes that tend to be frequently visited. Formally established measures of centrality are degree centrality , closeness centrality , betweenness centrality , eigenvector centrality , subgraph centrality , and Katz centrality . The purpose or objective of analysis generally determines the type of centrality measure to be used. For example, if one

4183-418: The head of the edge. The edge is said to join x and y and to be incident on x and on y . A vertex may exist in a graph and not belong to an edge. The edge ( y , x ) is called the inverted edge of ( x , y ) . Multiple edges , not allowed under the definition above, are two or more edges with both the same tail and the same head. In one more general sense of the term allowing multiple edges,

4272-424: The Internet and IP networks have become mission critical business tools, there has been increased interest in techniques and methods to monitor the routing posture of networks. Incorrect routing or routing issues cause undesirable performance degradation, flapping or downtime. Monitoring routing in a network is achieved using route analytics tools and techniques. In networks where a logically centralized control

4361-512: The Internet and social networks has been studied extensively. One such strategy is to immunize the largest degree nodes, i.e., targeted (intentional) attacks since for this case p c {\displaystyle pc} is relatively high and fewer nodes are needed to be immunized. However, in most realistic networks the global structure is not available and the largest degree nodes are unknown. Graph (discrete mathematics) In discrete mathematics , particularly in graph theory ,

4450-488: The addresses of suspects and victims, the telephone numbers they have dialed, and financial transactions that they have partaken in during a given timeframe, and the familial relationships between these subjects as a part of police investigation. Link analysis here provides the crucial relationships and associations between very many objects of different types that are not apparent from isolated pieces of information. Computer-assisted or fully automatic computer-based link analysis

4539-487: The analysis of time series by network measures. Applications range from detection of regime changes over characterizing dynamics to synchronization analysis. Many real networks are embedded in space. Examples include, transportation and other infrastructure networks, brain neural networks. Several models for spatial networks have been developed. Content in a complex network can spread via two major methods: conserved spread and non-conserved spread. In conserved spread,

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4628-402: The best link for a packet to get from its original source to its final destination. Instead, to avoid congestion hot spots in packet systems, a few algorithms use a randomized algorithm—Valiant's paradigm—that routes a path to a randomly picked intermediate destination, and from there to its true final destination. In many early telephone switches, a randomizer was often used to select the start of

4717-429: The best path. In high-speed systems, there are so many packets transmitted every second that it is infeasible for a single device to calculate the complete path for each and every packet. Early high-speed systems dealt with this with circuit switching by setting up a path once for the first packet between some source and some destination; later packets between that same source and that same destination continue to follow

4806-504: The best possible routes, while link-state or topological databases may store all other information as well. In case of overlapping or equal routes, algorithms consider the following elements in priority order to decide which routes to install into the routing table: Because a routing metric is specific to a given routing protocol, multi-protocol routers must use some external heuristic to select between routes learned from different routing protocols. Cisco routers, for example, attribute

4895-465: The best route. Most routing algorithms use only one network path at a time. Multipath routing and specifically equal-cost multi-path routing techniques enable the use of multiple alternative paths. In computer networking, the metric is computed by a routing algorithm, and can cover information such as bandwidth , network delay , hop count , path cost, load, maximum transmission unit , reliability, and communication cost. The routing table stores only

4984-399: The connections between nodes, respectively. As the water passes from one funnel into another, the water disappears instantly from the funnel that was previously exposed to the water. In non-conserved spread, the amount of content changes as it enters and passes through a complex network. The model of non-conserved spread can best be represented by a continuously running faucet running through

5073-811: The definitions must be expanded. For directed simple graphs, the definition of E {\displaystyle E} should be modified to E ⊆ { ( x , y ) ∣ ( x , y ) ∈ V 2 } {\displaystyle E\subseteq \{(x,y)\mid (x,y)\in V^{2}\}} . For directed multigraphs, the definition of ϕ {\displaystyle \phi } should be modified to ϕ : E → { ( x , y ) ∣ ( x , y ) ∈ V 2 } {\displaystyle \phi :E\to \{(x,y)\mid (x,y)\in V^{2}\}} . To avoid ambiguity, these types of objects may be called precisely

5162-457: The degree of all but two vertices is 2 and the degree of the two remaining vertices is 1. If a path graph occurs as a subgraph of another graph, it is a path in that graph. A planar graph is a graph whose vertices and edges can be drawn in a plane such that no two of the edges intersect. A cycle graph or circular graph of order n ≥ 3 is a graph in which the vertices can be listed in an order v 1 , v 2 , …, v n such that

5251-505: The development of the field of network medicine . Recent examples of application of network theory in biology include applications to understanding the cell cycle as well as a quantitative framework for developmental processes. The automatic parsing of textual corpora has enabled the extraction of actors and their relational networks on a vast scale. The resulting narrative networks , which can contain thousands of nodes, are then analyzed by using tools from Network theory to identify

5340-424: The down node. When applying link-state algorithms, a graphical map of the network is the fundamental data used for each node. To produce its map, each node floods the entire network with information about the other nodes it can connect to. Each node then independently assembles this information into a map. Using this map, each router independently determines the least-cost path from itself to every other node using

5429-414: The edge's endpoints . The edge is said to join u and v and to be incident on them. A vertex may belong to no edge, in which case it is not joined to any other vertex and is called isolated . When an edge { u , v } {\displaystyle \{u,v\}} exists, the vertices u and v are called adjacent . A multigraph is a generalization that allows multiple edges to have

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5518-425: The edges are the { v i , v i +1 } where i = 1, 2, …, n − 1, plus the edge { v n , v 1 } . Cycle graphs can be characterized as connected graphs in which the degree of all vertices is 2. If a cycle graph occurs as a subgraph of another graph, it is a cycle or circuit in that graph. A tree is an undirected graph in which any two vertices are connected by exactly one path , or equivalently

5607-407: The edges. The edges may be directed or undirected. For example, if the vertices represent people at a party, and there is an edge between two people if they shake hands, then this graph is undirected because any person A can shake hands with a person B only if B also shakes hands with A . In contrast, if an edge from a person A to a person B means that A owes money to B , then this graph

5696-487: The entire autonomous system. This node is called the speaker node. The speaker node creates a routing table and advertises it to neighboring speaker nodes in neighboring autonomous systems. The idea is the same as distance vector routing except that only speaker nodes in each autonomous system can communicate with each other. The speaker node advertises the path, not the metric, of the nodes in its autonomous system or other autonomous systems. The path-vector routing algorithm

5785-407: The fact that no single entity is responsible for selecting paths; instead, multiple entities are involved in selecting paths or even parts of a single path. Complications or inefficiency can result if these entities choose paths to optimize their own objectives, which may conflict with the objectives of other participants. A classic example involves traffic in a road system, in which each driver picks

5874-410: The graph. A k -vertex-connected graph is often called simply a k-connected graph . A bipartite graph is a simple graph in which the vertex set can be partitioned into two sets, W and X , so that no two vertices in W share a common edge and no two vertices in X share a common edge. Alternatively, it is a graph with a chromatic number of 2. In a complete bipartite graph , the vertex set

5963-401: The graph. That is, it is a directed graph that can be formed as an orientation of an undirected (simple) graph. Some authors use "oriented graph" to mean the same as "directed graph". Some authors use "oriented graph" to mean any orientation of a given undirected graph or multigraph. A regular graph is a graph in which each vertex has the same number of neighbours, i.e., every vertex has

6052-493: The key actors, the key communities or parties, and general properties such as robustness or structural stability of the overall network, or centrality of certain nodes. This automates the approach introduced by Quantitative Narrative Analysis, whereby subject-verb-object triplets are identified with pairs of actors linked by an action, or pairs formed by actor-object. Link analysis is a subset of network analysis, exploring associations between objects. An example may be examining

6141-520: The links between two locations (nodes) are determined, for example, by the similarity of the rainfall or temperature fluctuations in both sites. Several Web search ranking algorithms use link-based centrality metrics, including Google 's PageRank , Kleinberg's HITS algorithm , the CheiRank and TrustRank algorithms. Link analysis is also conducted in information science and communication science in order to understand and extract information from

6230-497: The list of destinations, the total cost to each, and the next hop to send data to get there — makes up the routing table, or distance table .) Each node, on a regular basis, sends to each neighbor node its own current assessment of the total cost to get to all the destinations it knows of. The neighboring nodes examine this information and compare it to what they already know; anything that represents an improvement on what they already have, they insert in their own table. Over time, all

6319-484: The message to B in London. This saves A the work of sending it along an expensive trans-Atlantic link, but causes the message to experience latency 125 ms when the other route would have been 20 ms faster. Additionally, a similar routing challenge can be observed in cellular networks, where different packets are destined for various endpoints, and each link exhibits varying spectral efficiency. In this context,

6408-623: The most direct route becomes blocked (see routing in the PSTN ). Dynamic routing attempts to solve this problem by constructing routing tables automatically, based on information carried by routing protocols , allowing the network to act nearly autonomously in avoiding network failures and blockages. Dynamic routing dominates the Internet. Examples of dynamic-routing protocols and algorithms include Routing Information Protocol (RIP), Open Shortest Path First (OSPF) and Enhanced Interior Gateway Routing Protocol (EIGRP). Distance vector algorithms use

6497-484: The network. For example, network motifs are small subgraphs that are over-represented in the network. Similarly, activity motifs are patterns in the attributes of nodes and edges in the network that are over-represented given the network structure. Using networks to analyze patterns in biological systems, such as food-webs, allows us to visualize the nature and strength of interactions between species. The analysis of biological networks with respect to diseases has led to

6586-504: The network. Structured addresses allow a single routing table entry to represent the route to a group of devices. In large networks, structured addressing (routing, in the narrow sense) outperforms unstructured addressing (bridging). Routing has become the dominant form of addressing on the Internet. Bridging is still widely used within local area networks . [REDACTED] [REDACTED] [REDACTED] [REDACTED] Routing schemes differ in how they deliver messages: Unicast

6675-408: The nodes in the network discover the best next hop and total cost for all destinations. When a network node goes down, any nodes that used it as their next hop discard the entry and convey the updated routing information to all adjacent nodes, which in turn repeat the process. Eventually, all the nodes in the network receive the updates and discover new paths to all the destinations that do not involve

6764-514: The numbers of incident edges.) The same remarks apply to edges, so graphs with labeled edges are called edge-labeled . Graphs with labels attached to edges or vertices are more generally designated as labeled . Consequently, graphs in which vertices are indistinguishable and edges are indistinguishable are called unlabeled . (In the literature, the term labeled may apply to other kinds of labeling, besides that which serves only to distinguish different vertices or edges.) The category of all graphs

6853-423: The same degree. A regular graph with vertices of degree k is called a k ‑regular graph or regular graph of degree k . A complete graph is a graph in which each pair of vertices is joined by an edge. A complete graph contains all possible edges. A finite graph is a graph in which the vertex set and the edge set are finite sets . Otherwise, it is called an infinite graph . Most commonly in graph theory it

6942-418: The same pair of endpoints. In some texts, multigraphs are simply called graphs. Sometimes, graphs are allowed to contain loops , which are edges that join a vertex to itself. To allow loops, the pairs of vertices in E must be allowed to have the same node twice. Such generalized graphs are called graphs with loops or simply graphs when it is clear from the context that loops are allowed. Generally,

7031-815: The same part of an infrastructure. This approach is also referred to as context-aware routing. The Internet is partitioned into autonomous systems (ASs) such as internet service providers (ISPs), each of which controls routes involving its network. Routing occurs at multiple levels. First, AS-level paths are selected via the BGP protocol that produces a sequence of ASs through which packets flow. Each AS may have multiple paths, offered by neighboring ASs, from which to choose. These routing decisions often correlate with business relationships with these neighboring ASs, which may be unrelated to path quality or latency. Second, once an AS-level path has been selected, there are often multiple corresponding router-level paths to choose from. This

7120-425: The same path without recalculating until the circuit teardown . Later high-speed systems inject packets into the network without any one device ever calculating a complete path for packets. In large systems, there are so many connections between devices, and those connections change so frequently, that it is infeasible for any one device to even know how all the devices are connected to each other, much less calculate

7209-537: The selection of the optimal path involves considering latency and packet error rate. To address this, multiple independent entities, one for each base station, play a crucial role in path selection while striving to optimize overall network performance. A 2003 measurement study of Internet routes found that, between pairs of neighboring ISPs, more than 30% of paths have inflated latency due to hot-potato routing, with 5% of paths being delayed by at least 12 ms. Inflation due to AS-level path selection, while substantial,

7298-427: The structure of collections of web pages. For example, the analysis might be of the interlinking between politicians' websites or blogs. Another use is for classifying pages according to their mention in other pages. Information about the relative importance of nodes and edges in a graph can be obtained through centrality measures, widely used in disciplines like sociology . For example, eigenvector centrality uses

7387-499: The structure of relationships between social entities. These entities are often persons, but may also be groups , organizations , nation states , web sites , or scholarly publications . Since the 1970s, the empirical study of networks has played a central role in social science, and many of the mathematical and statistical tools used for studying networks have been first developed in sociology . Amongst many other applications, social network analysis has been used to understand

7476-409: The terminology is not consistent and not all mathematicians allow this object. Normally, the vertices of a graph, by their nature as elements of a set, are distinguishable. This kind of graph may be called vertex-labeled . However, for many questions it is better to treat vertices as indistinguishable. (Of course, the vertices may be still distinguishable by the properties of the graph itself, e.g., by

7565-409: The total amount of content that enters a complex network remains constant as it passes through. The model of conserved spread can best be represented by a pitcher containing a fixed amount of water being poured into a series of funnels connected by tubes. Here, the pitcher represents the original source and the water is the content being spread. The funnels and connecting tubing represent the nodes and

7654-777: The traffic delivered prior to specific deadlines and reducing the completion times of flows. Work on the later over private WAN discusses modeling routing as a graph optimization problem by pushing all the queuing to the end-points. The authors also propose a heuristic to solve the problem efficiently while sacrificing negligible performance. Network theory Network theory has applications in many disciplines, including statistical physics , particle physics , computer science, electrical engineering , biology , archaeology , linguistics , economics , finance , operations research , climatology , ecology , public health , sociology , psychology , and neuroscience . Applications of network theory include logistical networks,

7743-435: The vertex set V is taken to be finite (which implies that the edge set E is also finite). Sometimes infinite graphs are considered, but they are usually viewed as a special kind of binary relation , because most results on finite graphs either do not extend to the infinite case or need a rather different proof. An empty graph is a graph that has an empty set of vertices (and thus an empty set of edges). The order of

7832-454: The vertices, called the adjacency relation . Specifically, two vertices x and y are adjacent if { x , y } is an edge. A graph is fully determined by its adjacency matrix A , which is an n × n square matrix, with A ij specifying the number of connections from vertex i to vertex j . For a simple graph, A ij is either 0, indicating disconnection, or 1, indicating connection; moreover A ii = 0 because an edge in

7921-409: Was attributed primarily to BGP's lack of a mechanism to directly optimize for latency, rather than to selfish routing policies. It was also suggested that, were an appropriate mechanism in place, ISPs would be willing to cooperate to reduce latency rather than use hot-potato routing. Such a mechanism was later published by the same authors, first for the case of two ISPs and then for the global case. As

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