Network science

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Network science is an academic field which studies complex networks such as telecommunication networks, computer networks, biological networks, cognitive and semantic networks, and social networks, considering distinct elements or actors represented by nodes (or vertices) and the connections between the elements or actors as links (or edges). The field draws on theories and methods including graph theory from mathematics, statistical mechanics from physics, data mining and information visualization from computer science, inferential modeling from statistics, and social structure from sociology. The United States National Research Council defines network science as "the study of network representations of physical, biological, and social phenomena leading to predictive models of these phenomena." [1]

Background and history

The study of networks has emerged in diverse disciplines as a means of analyzing complex relational data. The earliest known paper in this field is the famous Seven Bridges of Königsberg written by Leonhard Euler in 1736. Euler's mathematical description of vertices and edges was the foundation of graph theory, a branch of mathematics that studies the properties of pairwise relations in a network structure. The field of graph theory continued to develop and found applications in chemistry (Sylvester, 1878).

Dénes Kőnig, a Hungarian mathematician and professor, wrote the first book in Graph Theory, entitled "Theory of finite and infinite graphs", in 1936. [2]

Moreno's sociogram of a 1st grade class. Moreno Sociogram 1st Grade.png
Moreno's sociogram of a 1st grade class.

In the 1930s Jacob Moreno, a psychologist in the Gestalt tradition, arrived in the United States. He developed the sociogram and presented it to the public in April 1933 at a convention of medical scholars. Moreno claimed that "before the advent of sociometry no one knew what the interpersonal structure of a group 'precisely' looked like" (Moreno, 1953). The sociogram was a representation of the social structure of a group of elementary school students. The boys were friends of boys and the girls were friends of girls with the exception of one boy who said he liked a single girl. The feeling was not reciprocated. This network representation of social structure was found so intriguing that it was printed in The New York Times (April 3, 1933, page 17). The sociogram has found many applications and has grown into the field of social network analysis.

Probabilistic theory in network science developed as an offshoot of graph theory with Paul Erdős and Alfréd Rényi's eight famous papers on random graphs. For social networks the exponential random graph model or p* is a notational framework used to represent the probability space of a tie occurring in a social network. An alternate approach to network probability structures is the network probability matrix, which models the probability of edges occurring in a network, based on the historic presence or absence of the edge in a sample of networks.

In 1998, David Krackhardt and Kathleen Carley introduced the idea of a meta-network with the PCANS Model. They suggest that "all organizations are structured along these three domains, Individuals, Tasks, and Resources". Their paper introduced the concept that networks occur across multiple domains and that they are interrelated. This field has grown into another sub-discipline of network science called dynamic network analysis.

More recently other network science efforts have focused on mathematically describing different network topologies. Duncan Watts and Steven Strogatz reconciled empirical data on networks with mathematical representation, describing the small-world network. Albert-László Barabási and Reka Albert discovered scale-free networks, a property that captures the fact that in real network hubs coexist with many small degree vertices, and offered a dynamical model to explain the origin of this scale-free state.

Department of Defense initiatives

The U.S. military first became interested in network-centric warfare as an operational concept based on network science in 1996. John A. Parmentola, the U.S. Army Director for Research and Laboratory Management, proposed to the Army's Board on Science and Technology (BAST) on December 1, 2003 that Network Science become a new Army research area. The BAST, the Division on Engineering and Physical Sciences for the National Research Council (NRC) of the National Academies, serves as a convening authority for the discussion of science and technology issues of importance to the Army and oversees independent Army-related studies conducted by the National Academies. The BAST conducted a study to find out whether identifying and funding a new field of investigation in basic research, Network Science, could help close the gap between what is needed to realize Network-Centric Operations and the current primitive state of fundamental knowledge of networks.

As a result, the BAST issued the NRC study in 2005 titled Network Science (referenced above) that defined a new field of basic research in Network Science for the Army. Based on the findings and recommendations of that study and the subsequent 2007 NRC report titled Strategy for an Army Center for Network Science, Technology, and Experimentation, Army basic research resources were redirected to initiate a new basic research program in Network Science. To build a new theoretical foundation for complex networks, some of the key Network Science research efforts now ongoing in Army laboratories address:


As initiated in 2004 by Frederick I. Moxley with support he solicited from David S. Alberts, the Department of Defense helped to establish the first Network Science Center in conjunction with the U.S. Army at the United States Military Academy (USMA). Under the tutelage of Dr. Moxley and the faculty of the USMA, the first interdisciplinary undergraduate courses in Network Science were taught to cadets at West Point. [3] [4] [5] In order to better instill the tenets of network science among its cadre of future leaders, the USMA has also instituted a five-course undergraduate minor in Network Science. [6]

In 2006, the U.S. Army and the United Kingdom (UK) formed the Network and Information Science International Technology Alliance, a collaborative partnership among the Army Research Laboratory, UK Ministry of Defense and a consortium of industries and universities in the U.S. and UK. The goal of the alliance is to perform basic research in support of Network- Centric Operations across the needs of both nations.

In 2009, the U.S. Army formed the Network Science CTA, a collaborative research alliance among the Army Research Laboratory, CERDEC, and a consortium of about 30 industrial R&D labs and universities in the U.S. The goal of the alliance is to develop a deep understanding of the underlying commonalities among intertwined social/cognitive, information, and communications networks, and as a result improve our ability to analyze, predict, design, and influence complex systems interweaving many kinds of networks.

Subsequently, as a result of these efforts, the U.S. Department of Defense has sponsored numerous research projects that support Network Science.

Network Classification

Deterministic Network

The definition of deterministic network is defined compared with the definition of probabilistic network. In un-weighted deterministic networks, edges either exist or not, usually we use 0 to represent non-existence of an edge while 1 to represent existence of an edge. In weighted deterministic networks, the edge value represents the weight of each edge, for example, the strength level.

Probabilistic Network

In probabilistic networks, values behind each edge represent the likelihood of the existence of each edge. For example, if one edge has a value equals to 0.9, we say the existence probability of this edge is 0.9. [7]

Network properties

Often, networks have certain attributes that can be calculated to analyze the properties & characteristics of the network. The behavior of these network properties often define network models and can be used to analyze how certain models contrast to each other. Many of the definitions for other terms used in network science can be found in Glossary of graph theory.

Size

The size of a network can refer to the number of nodes or, less commonly, the number of edges which (for connected graphs with no multi-edges) can range from (a tree) to (a complete graph). In the case of a simple graph (a network in which at most one (undirected) edge exists between each pair of vertices, and in which no vertices connect to themselves), we have ; for directed graphs (with no self-connected nodes), ; for directed graphs with self-connections allowed, . In the circumstance of a graph within which multiple edges may exist between a pair of vertices, .

Density

The density of a network is defined as a normalized ratio between 0 and 1 of the number of edges to the number of possible edges in a network with nodes. Network density is a measure of the percentage of "optional" edges that exist in the network and can be computed as where and are the minimum and maximum number of edges in a connected network with nodes, respectively. In the case of simple graphs, is given by the binomial coefficient and , giving density . Another possible equation is whereas the ties are unidirectional (Wasserman & Faust 1994). [8] This gives a better overview over the network density, because unidirectional relationships can be measured.

Planar network density

The density of a network, where there is no intersection between edges, is defined as a ratio of the number of edges to the number of possible edges in a network with nodes, given by a graph with no intersecting edges , giving

Average degree

The degree of a node is the number of edges connected to it. Closely related to the density of a network is the average degree, (or, in the case of directed graphs, , the former factor of 2 arising from each edge in an undirected graph contributing to the degree of two distinct vertices). In the ER random graph model () we can compute the expected value of (equal to the expected value of of an arbitrary vertex): a random vertex has other vertices in the network available, and with probability , connects to each. Thus, .

Average shortest path length (or characteristic path length)

The average shortest path length is calculated by finding the shortest path between all pairs of nodes, and taking the average over all paths of the length thereof (the length being the number of intermediate edges contained in the path, i.e., the distance between the two vertices within the graph). This shows us, on average, the number of steps it takes to get from one member of the network to another. The behavior of the expected average shortest path length (that is, the ensemble average of the average shortest path length) as a function of the number of vertices of a random network model defines whether that model exhibits the small-world effect; if it scales as , the model generates small-world nets. For faster-than-logarithmic growth, the model does not produce small worlds. The special case of is known as ultra-small world effect.

Diameter of a network

As another means of measuring network graphs, we can define the diameter of a network as the longest of all the calculated shortest paths in a network. It is the shortest distance between the two most distant nodes in the network. In other words, once the shortest path length from every node to all other nodes is calculated, the diameter is the longest of all the calculated path lengths. The diameter is representative of the linear size of a network. If node A-B-C-D are connected, going from A->D this would be the diameter of 3 (3-hops, 3-links).[ citation needed ]

Clustering coefficient

The clustering coefficient is a measure of an "all-my-friends-know-each-other" property. This is sometimes described as the friends of my friends are my friends. More precisely, the clustering coefficient of a node is the ratio of existing links connecting a node's neighbors to each other to the maximum possible number of such links. The clustering coefficient for the entire network is the average of the clustering coefficients of all the nodes. A high clustering coefficient for a network is another indication of a small world.

The clustering coefficient of the 'th node is

where is the number of neighbours of the 'th node, and is the number of connections between these neighbours. The maximum possible number of connections between neighbors is, then,

From a probabilistic standpoint, the expected local clustering coefficient is the likelihood of a link existing between two arbitrary neighbors of the same node.

Connectedness

The way in which a network is connected plays a large part into how networks are analyzed and interpreted. Networks are classified in four different categories:

Node centrality

Centrality indices produce rankings which seek to identify the most important nodes in a network model. Different centrality indices encode different contexts for the word "importance." The betweenness centrality, for example, considers a node highly important if it form bridges between many other nodes. The eigenvalue centrality, in contrast, considers a node highly important if many other highly important nodes link to it. Hundreds of such measures have been proposed in the literature.

Centrality indices are only accurate for identifying the most important nodes. The measures are seldom, if ever, meaningful for the remainder of network nodes. [9] [10] Also, their indications are only accurate within their assumed context for importance, and tend to "get it wrong" for other contexts. [11] For example, imagine two separate communities whose only link is an edge between the most junior member of each community. Since any transfer from one community to the other must go over this link, the two junior members will have high betweenness centrality. But, since they are junior, (presumably) they have few connections to the "important" nodes in their community, meaning their eigenvalue centrality would be quite low.

Node influence

Limitations to centrality measures have led to the development of more general measures. Two examples are the accessibility, which uses the diversity of random walks to measure how accessible the rest of the network is from a given start node, [12] and the expected force, derived from the expected value of the force of infection generated by a node. [9] Both of these measures can be meaningfully computed from the structure of the network alone.

Community structure

Fig. 1: A sketch of a small network displaying community structure, with three groups of nodes with dense internal connections and sparser connections between groups. Network Community Structure.svg
Fig. 1: A sketch of a small network displaying community structure, with three groups of nodes with dense internal connections and sparser connections between groups.

Nodes in a network may be partitioned into groups representing communities. Depending on the context, communities may be distinct or overlapping. Typically, nodes in such communities will be strongly connected to other nodes in the same community, but weakly connected to nodes outside the community. In the absence of a ground truth describing the community structure of a specific network, several algorithms have been developed to infer possible community structures using either supervised of unsupervised clustering methods.

Network models

Network models serve as a foundation to understanding interactions within empirical complex networks. Various random graph generation models produce network structures that may be used in comparison to real-world complex networks.

Erdős–Rényi random graph model

This Erdos-Renyi model is generated with N = 4 nodes. For each edge in the complete graph formed by all N nodes, a random number is generated and compared to a given probability. If the random number is less than p, an edge is formed on the model. ER model.svg
This Erdős–Rényi model is generated with N = 4 nodes. For each edge in the complete graph formed by all N nodes, a random number is generated and compared to a given probability. If the random number is less than p, an edge is formed on the model.

The Erdős–Rényi model , named for Paul Erdős and Alfréd Rényi, is used for generating random graphs in which edges are set between nodes with equal probabilities. It can be used in the probabilistic method to prove the existence of graphs satisfying various properties, or to provide a rigorous definition of what it means for a property to hold for almost all graphs.

To generate an Erdős–Rényi model two parameters must be specified: the total number of nodes n and the probability p that a random pair of nodes has an edge.

Because the model is generated without bias to particular nodes, the degree distribution is binomial: for a randomly chosen vertex ,

In this model the clustering coefficient is 0 a.s. The behavior of can be broken into three regions.

Subcritical: All components are simple and very small, the largest component has size ;

Critical: ;

Supercritical: where is the positive solution to the equation .

The largest connected component has high complexity. All other components are simple and small .

Configuration model

The configuration model takes a degree sequence [13] [14] or degree distribution [15] (which subsequently is used to generate a degree sequence) as the input, and produces randomly connected graphs in all respects other than the degree sequence. This means that for a given choice of the degree sequence, the graph is chosen uniformly at random from the set of all graphs that comply with this degree sequence. The degree of a randomly chosen vertex is an independent and identically distributed random variable with integer values. When , the configuration graph contains the giant connected component, which has infinite size. [14] The rest of the components have finite sizes, which can be quantified with the notion of the size distribution. The probability that a randomly sampled node is connected to a component of size is given by convolution powers of the degree distribution: [16]

where denotes the degree distribution and . The giant component can be destroyed by randomly removing the critical fraction of all edges. This process is called percolation on random networks. When the second moment of the degree distribution is finite, , this critical edge fraction is given by [17] , and the average vertex-vertex distance in the giant component scales logarithmically with the total size of the network, . [15] In the directed configuration model, the degree of a node is given by two numbers, in-degree and out-degree , and consequently, the degree distribution is two-variate. The expected number of in-edges and out-edges coincides, so that . The directed configuration model contains the giant component iff [18]

Note that and are equal and therefore interchangeable in the latter inequality. The probability that a randomly chosen vertex belongs to a component of size is given by: [19]

for in-components, and

for out-components.

Watts–Strogatz small world model

The Watts and Strogatz model uses the concept of rewiring to achieve its structure. The model generator will iterate through each edge in the original lattice structure. An edge may change its connected vertices according to a given rewiring probability.
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in this example. Watts-Strogatz-rewire.png
The Watts and Strogatz model uses the concept of rewiring to achieve its structure. The model generator will iterate through each edge in the original lattice structure. An edge may change its connected vertices according to a given rewiring probability. in this example.

The Watts and Strogatz model is a random graph generation model that produces graphs with small-world properties.

An initial lattice structure is used to generate a Watts–Strogatz model. Each node in the network is initially linked to its closest neighbors. Another parameter is specified as the rewiring probability. Each edge has a probability that it will be rewired to the graph as a random edge. The expected number of rewired links in the model is .

As the Watts–Strogatz model begins as non-random lattice structure, it has a very high clustering coefficient along with high average path length. Each rewire is likely to create a shortcut between highly connected clusters. As the rewiring probability increases, the clustering coefficient decreases slower than the average path length. In effect, this allows the average path length of the network to decrease significantly with only slightly decreases in clustering coefficient. Higher values of p force more rewired edges, which in effect makes the Watts–Strogatz model a random network.

Barabási–Albert (BA) preferential attachment model

The Barabási–Albert model is a random network model used to demonstrate a preferential attachment or a "rich-get-richer" effect. In this model, an edge is most likely to attach to nodes with higher degrees. The network begins with an initial network of m0 nodes. m0  2 and the degree of each node in the initial network should be at least 1, otherwise it will always remain disconnected from the rest of the network.

In the BA model, new nodes are added to the network one at a time. Each new node is connected to existing nodes with a probability that is proportional to the number of links that the existing nodes already have. Formally, the probability pi that the new node is connected to node i is [20]

where ki is the degree of node i. Heavily linked nodes ("hubs") tend to quickly accumulate even more links, while nodes with only a few links are unlikely to be chosen as the destination for a new link. The new nodes have a "preference" to attach themselves to the already heavily linked nodes.

The degree distribution of the BA Model, which follows a power law. In loglog scale the power law function is a straight line. Barabasi-albert model degree distribution.svg
The degree distribution of the BA Model, which follows a power law. In loglog scale the power law function is a straight line.

The degree distribution resulting from the BA model is scale free, in particular, for large degree it is a power law of the form:

Hubs exhibit high betweenness centrality which allows short paths to exist between nodes. As a result, the BA model tends to have very short average path lengths. The clustering coefficient of this model also tends to 0.

The Barabási–Albert model [21] was developed for undirected networks, aiming to explain the universality of the scale-free property, and applied to a wide range of different networks and applications. The directed version of this model is the Price model [22] [23] which was developed to just citation networks.

Non-linear preferential attachment

In non-linear preferential attachment (NLPA), existing nodes in the network gain new edges proportionally to the node degree raised to a constant positive power, . [24] Formally, this means that the probability that node gains a new edge is given by

If , NLPA reduces to the BA model and is referred to as "linear". If , NLPA is referred to as "sub-linear" and the degree distribution of the network tends to a stretched exponential distribution. If , NLPA is referred to as "super-linear" and a small number of nodes connect to almost all other nodes in the network. For both and , the scale-free property of the network is broken in the limit of infinite system size. However, if is only slightly larger than , NLPA may result in degree distributions which appear to be transiently scale free. [25]

Mediation-driven attachment (MDA) model

In the mediation-driven attachment (MDA) model in which a new node coming with edges picks an existing connected node at random and then connects itself not with that one but with of its neighbors chosen also at random. The probability that the node of the existing node picked is

The factor is the inverse of the harmonic mean (IHM) of degrees of the neighbors of a node . Extensive numerical investigation suggest that for an approximately the mean IHM value in the large limit becomes a constant which means . It implies that the higher the links (degree) a node has, the higher its chance of gaining more links since they can be reached in a larger number of ways through mediators which essentially embodies the intuitive idea of rich get richer mechanism (or the preferential attachment rule of the Barabasi–Albert model). Therefore, the MDA network can be seen to follow the PA rule but in disguise. [26]

However, for it describes the winner takes it all mechanism as we find that almost of the total nodes have degree one and one is super-rich in degree. As value increases the disparity between the super rich and poor decreases and as we find a transition from rich get super richer to rich get richer mechanism.

Fitness model

Another model where the key ingredient is the nature of the vertex has been introduced by Caldarelli et al. [27] Here a link is created between two vertices with a probability given by a linking function of the fitnesses of the vertices involved. The degree of a vertex i is given by [28]

If is an invertible and increasing function of , then the probability distribution is given by

As a result, if the fitnesses are distributed as a power law, then also the node degree does.

Less intuitively with a fast decaying probability distribution as together with a linking function of the kind

with a constant and the Heavyside function, we also obtain scale-free networks.

Such model has been successfully applied to describe trade between nations by using GDP as fitness for the various nodes and a linking function of the kind [29] [30]

Exponential random graph models

Exponential Random Graph Models (ERGMs) are a family of statistical models for analyzing data from social and other networks. [31] The Exponential family is a broad family of models for covering many types of data, not just networks. An ERGM is a model from this family which describes networks.

We adopt the notation to represent a random graph via a set of nodes and a collection of tie variables , indexed by pairs of nodes , where if the nodes are connected by an edge and otherwise.

The basic assumption of ERGMs is that the structure in an observed graph can be explained by a given vector of sufficient statistics which are a function of the observed network and, in some cases, nodal attributes. The probability of a graph in an ERGM is defined by:

where is a vector of model parameters associated with and is a normalising constant.

Network analysis

Social network analysis

Social network analysis examines the structure of relationships between social entities. [32] These entities are often persons, but may also be groups, organizations, nation states, web sites, 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. [33] Amongst many other applications, social network analysis has been used to understand the diffusion of innovations, [34] 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. Similarly, it has been used to study recruitment into political movements and social organizations. It has also been used to conceptualize scientific disagreements as well as academic prestige. In the second language acquisition literature, it has an established history in study abroad research, revealing how peer learner interaction networks influence their language progress. [35] More recently, network analysis (and its close cousin traffic analysis) has gained a significant use in military intelligence, for uncovering insurgent networks of both hierarchical and leaderless nature. [36] [37] In criminology, it is being used to identify influential actors in criminal gangs, offender movements, co-offending, predict criminal activities and make policies. [38]

Dynamic network analysis

Dynamic network analysis examines the shifting structure of relationships among different classes of entities in complex socio-technical systems effects, and reflects social stability and changes such as the emergence of new groups, topics, and leaders. [39] [40] [41] Dynamic Network Analysis focuses on meta-networks composed of multiple types of nodes (entities) and multiple types of links. These entities can be highly varied. Examples include people, organizations, topics, resources, tasks, events, locations, and beliefs.

Dynamic network techniques are particularly useful for assessing trends and changes in networks over time, identification of emergent leaders, and examining the co-evolution of people and ideas.

Biological network analysis

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 content are closely related to social network analysis, but often focusing on local patterns in the network. For example, network motifs are small subgraphs that are over-represented in the network. Activity motifs are similar over-represented patterns in the attributes of nodes and edges in the network that are over represented given the network structure. The analysis of biological networks has led to the development of network medicine, which looks at the effect of diseases in the interactome. [42]

Link analysis is a subset of network analysis, exploring associations between objects. An example may be examining 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 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.

Pandemic analysis

The SIR model is one of the most well known algorithms on predicting the spread of global pandemics within an infectious population.

Susceptible to infected

The formula above describes the "force" of infection for each susceptible unit in an infectious population, where β is equivalent to the transmission rate of said disease.

To track the change of those susceptible in an infectious population:

Infected to recovered

Over time, the number of those infected fluctuates by: the specified rate of recovery, represented by but deducted to one over the average infectious period , the numbered of infectious individuals, , and the change in time, .

Infectious period

Whether a population will be overcome by a pandemic, with regards to the SIR model, is dependent on the value of or the "average people infected by an infected individual."

Several Web search ranking algorithms use link-based centrality metrics, including (in order of appearance) Marchiori's Hyper Search, 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 the structure of collections of web pages. For example, the analysis might be of the interlinking between politicians' web sites or blogs.

PageRank

PageRank works by randomly picking "nodes" or websites and then with a certain probability, "randomly jumping" to other nodes. By randomly jumping to these other nodes, it helps PageRank completely traverse the network as some webpages exist on the periphery and would not as readily be assessed.

Each node, , has a PageRank as defined by the sum of pages that link to times one over the outlinks or "out-degree" of times the "importance" or PageRank of .

Random jumping

As explained above, PageRank enlists random jumps in attempts to assign PageRank to every website on the internet. These random jumps find websites that might not be found during the normal search methodologies such as breadth-first search and depth-first search.

In an improvement over the aforementioned formula for determining PageRank includes adding these random jump components. Without the random jumps, some pages would receive a PageRank of 0 which would not be good.

The first is , or the probability that a random jump will occur. Contrasting is the "damping factor", or .

Another way of looking at it:

Centrality measures

Information about the relative importance of nodes and edges in a graph can be obtained through centrality measures, widely used in disciplines like sociology. Centrality measures are essential when a network analysis has to answer questions such as: "Which nodes in the network should be targeted to ensure that a message or information spreads to all or most nodes in the network?" or conversely, "Which nodes should be targeted to curtail the spread of a disease?". Formally established measures of centrality are degree centrality, closeness centrality, betweenness centrality, eigenvector centrality, and katz centrality. The objective of network analysis generally determines the type of centrality measure(s) to be used. [32]

Spread of content in networks

Content in a complex network can spread via two major methods: conserved spread and non-conserved spread. [43] In conserved spread, 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 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 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.

The SIR model

In 1927, W. O. Kermack and A. G. McKendrick created a model in which they considered a fixed population with only three compartments, susceptible: , infected, , and recovered, . The compartments used for this model consist of three classes:

The flow of this model may be considered as follows:

Using a fixed population, , Kermack and McKendrick derived the following equations:

Several assumptions were made in the formulation of these equations: First, an individual in the population must be considered as having an equal probability as every other individual of contracting the disease with a rate of , which is considered the contact or infection rate of the disease. Therefore, an infected individual makes contact and is able to transmit the disease with others per unit time and the fraction of contacts by an infected with a susceptible is . The number of new infections in unit time per infective then is , giving the rate of new infections (or those leaving the susceptible category) as (Brauer & Castillo-Chavez, 2001). For the second and third equations, consider the population leaving the susceptible class as equal to the number entering the infected class. However, infectives are leaving this class per unit time to enter the recovered/removed class at a rate per unit time (where represents the mean recovery rate, or the mean infective period). These processes which occur simultaneously are referred to as the Law of Mass Action, a widely accepted idea that the rate of contact between two groups in a population is proportional to the size of each of the groups concerned (Daley & Gani, 2005). Finally, it is assumed that the rate of infection and recovery is much faster than the time scale of births and deaths and therefore, these factors are ignored in this model.

More can be read on this model on the Epidemic model page.

The master equation approach

A master equation can express the behaviour of an undirected growing network where, at each time step, a new node is added to the network, linked to an old node (randomly chosen and without preference). The initial network is formed by two nodes and two links between them at time , this configuration is necessary only to simplify further calculations, so at time the network have nodes and links.

The master equation for this network is:

where is the probability to have the node with degree at time , and is the time step when this node was added to the network. Note that there are only two ways for an old node to have links at time :

After simplifying this model, the degree distribution is [44]

Based on this growing network, an epidemic model is developed following a simple rule: Each time the new node is added and after choosing the old node to link, a decision is made: whether or not this new node will be infected. The master equation for this epidemic model is:

where represents the decision to infect () or not (). Solving this master equation, the following solution is obtained: [45]

Multilayer networks

Multilayer networks are networks with multiple kinds of relations. [46] Attempts to model real-world systems as multidimensional networks have been used in various fields such as social network analysis, [47] economics, history, urban and international transport, ecology, psychology, medicine, biology, commerce, climatology, physics, computational neuroscience, operations management, and finance.

Network optimization

Network problems that involve finding an optimal way of doing something are studied under the name of 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 PERT (Program Evaluation & Review Technique).

Interdependent networks

Interdependent networks are networks where the functioning of nodes in one network depends on the functioning of nodes in another network. In nature, networks rarely appear in isolation, rather, usually networks are typically elements in larger systems, and interact with elements in that complex system. Such complex dependencies can have non-trivial effects on one another. A well studied example is the interdependency of infrastructure networks, [48] the power stations which form the nodes of the power grid require fuel delivered via a network of roads or pipes and are also controlled via the nodes of communications network. Though the transportation network does not depend on the power network to function, the communications network does. In such infrastructure networks, the disfunction of a critical number of nodes in either the power network or the communication network can lead to cascading failures across the system with potentially catastrophic result to the whole system functioning. [49] If the two networks were treated in isolation, this important feedback effect would not be seen and predictions of network robustness would be greatly overestimated.

See also

Related Research Articles

In graph theory, an expander graph is a sparse graph that has strong connectivity properties, quantified using vertex, edge or spectral expansion. Expander constructions have spawned research in pure and applied mathematics, with several applications to complexity theory, design of robust computer networks, and the theory of error-correcting codes.

<span class="mw-page-title-main">Scale-free network</span> Network whose degree distribution follows a power law

A scale-free network is a network whose degree distribution follows a power law, at least asymptotically. That is, the fraction P(k) of nodes in the network having k connections to other nodes goes for large values of k as

<span class="mw-page-title-main">Random graph</span> Graph generated by a random process

In mathematics, random graph is the general term to refer to probability distributions over graphs. Random graphs may be described simply by a probability distribution, or by a random process which generates them. The theory of random graphs lies at the intersection between graph theory and probability theory. From a mathematical perspective, random graphs are used to answer questions about the properties of typical graphs. Its practical applications are found in all areas in which complex networks need to be modeled – many random graph models are thus known, mirroring the diverse types of complex networks encountered in different areas. In a mathematical context, random graph refers almost exclusively to the Erdős–Rényi random graph model. In other contexts, any graph model may be referred to as a random graph.

In graph theory, a clustering coefficient is a measure of the degree to which nodes in a graph tend to cluster together. Evidence suggests that in most real-world networks, and in particular social networks, nodes tend to create tightly knit groups characterised by a relatively high density of ties; this likelihood tends to be greater than the average probability of a tie randomly established between two nodes.

<span class="mw-page-title-main">Degree distribution</span>

In the study of graphs and networks, the degree of a node in a network is the number of connections it has to other nodes and the degree distribution is the probability distribution of these degrees over the whole network.

<span class="mw-page-title-main">Giant component</span> Large connected component of a random graph

In network theory, a giant component is a connected component of a given random graph that contains a significant fraction of the entire graph's vertices.

<span class="mw-page-title-main">Barabási–Albert model</span> Scale-free network generation algorithm

The Barabási–Albert (BA) model is an algorithm for generating random scale-free networks using a preferential attachment mechanism. Several natural and human-made systems, including the Internet, the World Wide Web, citation networks, and some social networks are thought to be approximately scale-free and certainly contain few nodes with unusually high degree as compared to the other nodes of the network. The BA model tries to explain the existence of such nodes in real networks. The algorithm is named for its inventors Albert-László Barabási and Réka Albert.

<span class="mw-page-title-main">Watts–Strogatz model</span> Method of generating random small-world graphs

The Watts–Strogatz model is a random graph generation model that produces graphs with small-world properties, including short average path lengths and high clustering. It was proposed by Duncan J. Watts and Steven Strogatz in their article published in 1998 in the Nature scientific journal. The model also became known as the (Watts) beta model after Watts used to formulate it in his popular science book Six Degrees.

<span class="mw-page-title-main">Assortativity</span> Tendency for similar nodes to be connected

Assortativity, or assortative mixing, is a preference for a network's nodes to attach to others that are similar in some way. Though the specific measure of similarity may vary, network theorists often examine assortativity in terms of a node's degree. The addition of this characteristic to network models more closely approximates the behaviors of many real world networks.

<span class="mw-page-title-main">Erdős–Rényi model</span> Two closely related models for generating random graphs

In the mathematical field of graph theory, the Erdős–Rényi model refers to one of two closely related models for generating random graphs or the evolution of a random network. These models are named after Hungarian mathematicians Paul Erdős and Alfréd Rényi, who introduced one of the models in 1959. Edgar Gilbert introduced the other model contemporaneously with and independently of Erdős and Rényi. In the model of Erdős and Rényi, all graphs on a fixed vertex set with a fixed number of edges are equally likely. In the model introduced by Gilbert, also called the Erdős–Rényi–Gilbert model, each edge has a fixed probability of being present or absent, independently of the other edges. These models can be used in the probabilistic method to prove the existence of graphs satisfying various properties, or to provide a rigorous definition of what it means for a property to hold for almost all graphs.

<span class="mw-page-title-main">Random geometric graph</span> In graph theory, the mathematically simplest spatial network

In graph theory, a random geometric graph (RGG) is the mathematically simplest spatial network, namely an undirected graph constructed by randomly placing N nodes in some metric space and connecting two nodes by a link if and only if their distance is in a given range, e.g. smaller than a certain neighborhood radius, r.

<span class="mw-page-title-main">Modularity (networks)</span> Measure of network community structure

Modularity is a measure of the structure of networks or graphs which measures the strength of division of a network into modules. Networks with high modularity have dense connections between the nodes within modules but sparse connections between nodes in different modules. Modularity is often used in optimization methods for detecting community structure in networks. Biological networks, including animal brains, exhibit a high degree of modularity. However, modularity maximization is not statistically consistent, and finds communities in its own null model, i.e. fully random graphs, and therefore it cannot be used to find statistically significant community structures in empirical networks. Furthermore, it has been shown that modularity suffers a resolution limit and, therefore, it is unable to detect small communities.

<span class="mw-page-title-main">Evolving network</span>

Evolving networks are networks that change as a function of time. They are a natural extension of network science since almost all real world networks evolve over time, either by adding or removing nodes or links over time. Often all of these processes occur simultaneously, such as in social networks where people make and lose friends over time, thereby creating and destroying edges, and some people become part of new social networks or leave their networks, changing the nodes in the network. Evolving network concepts build on established network theory and are now being introduced into studying networks in many diverse fields.

<span class="mw-page-title-main">Exponential family random graph models</span>

Exponential Random Graph Models (ERGMs) are a family of statistical models for analyzing data from social and other networks. Examples of networks examined using ERGM include knowledge networks, organizational networks, colleague networks, social media networks, networks of scientific development, and others.

Random walk closeness centrality is a measure of centrality in a network, which describes the average speed with which randomly walking processes reach a node from other nodes of the network. It is similar to the closeness centrality except that the farness is measured by the expected length of a random walk rather than by the shortest path.

<span class="mw-page-title-main">Hyperbolic geometric graph</span>

A hyperbolic geometric graph (HGG) or hyperbolic geometric network (HGN) is a special type of spatial network where (1) latent coordinates of nodes are sprinkled according to a probability density function into a hyperbolic space of constant negative curvature and (2) an edge between two nodes is present if they are close according to a function of the metric (typically either a Heaviside step function resulting in deterministic connections between vertices closer than a certain threshold distance, or a decaying function of hyperbolic distance yielding the connection probability). A HGG generalizes a random geometric graph (RGG) whose embedding space is Euclidean.

<span class="mw-page-title-main">Bianconi–Barabási model</span>

The Bianconi–Barabási model is a model in network science that explains the growth of complex evolving networks. This model can explain that nodes with different characteristics acquire links at different rates. It predicts that a node's growth depends on its fitness and can calculate the degree distribution. The Bianconi–Barabási model is named after its inventors Ginestra Bianconi and Albert-László Barabási. This model is a variant of the Barabási–Albert model. The model can be mapped to a Bose gas and this mapping can predict a topological phase transition between a "rich-get-richer" phase and a "winner-takes-all" phase.

Maximal entropy random walk (MERW) is a popular type of biased random walk on a graph, in which transition probabilities are chosen accordingly to the principle of maximum entropy, which says that the probability distribution which best represents the current state of knowledge is the one with largest entropy. While standard random walk chooses for every vertex uniform probability distribution among its outgoing edges, locally maximizing entropy rate, MERW maximizes it globally by assuming uniform probability distribution among all paths in a given graph.

<span class="mw-page-title-main">Configuration model</span>

In network science, the configuration model is a method for generating random networks from a given degree sequence. It is widely used as a reference model for real-life social networks, because it allows the modeler to incorporate arbitrary degree distributions.

<span class="mw-page-title-main">Soft configuration model</span> Random graph model in applied mathematics

In applied mathematics, the soft configuration model (SCM) is a random graph model subject to the principle of maximum entropy under constraints on the expectation of the degree sequence of sampled graphs. Whereas the configuration model (CM) uniformly samples random graphs of a specific degree sequence, the SCM only retains the specified degree sequence on average over all network realizations; in this sense the SCM has very relaxed constraints relative to those of the CM ("soft" rather than "sharp" constraints). The SCM for graphs of size has a nonzero probability of sampling any graph of size , whereas the CM is restricted to only graphs having precisely the prescribed connectivity structure.

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Further reading