Social network

From Wikipedia, the free encyclopedia

A social network is a social structure made up of a set of social actors (such as individuals or organizations), sets of dyadic ties, and other social interactions between actors. The social network perspective provides a set of methods for analyzing the structure of whole social entities as well as a variety of theories explaining the patterns observed in these structures. The study of these structures uses social network analysis to identify local and global patterns, locate influential entities, and examine network dynamics. 

Social networks and the analysis of them is an inherently interdisciplinary academic field which emerged from social psychology, sociology, statistics, and graph theory. Georg Simmel authored early structural theories in sociology emphasizing the dynamics of triads and "web of group affiliations". Jacob Moreno is credited with developing the first sociograms in the 1930s to study interpersonal relationships. These approaches were mathematically formalized in the 1950s and theories and methods of social networks became pervasive in the social and behavioral sciences by the 1980s. Social network analysis is now one of the major paradigms in contemporary sociology, and is also employed in a number of other social and formal sciences. Together with other complex networks, it forms part of the nascent field of network science.

Overview

Evolution graph of a social network: Barabási model.

 

The social network is a theoretical construct useful in the social sciences to study relationships between individuals, groups, organizations, or even entire societies. The term is used to describe a social structure determined by such interactions. The ties through which any given social unit connects represent the convergence of the various social contacts of that unit. This theoretical approach is, necessarily, relational. An axiom of the social network approach to understanding social interaction is that social phenomena should be primarily conceived and investigated through the properties of relations between and within units, instead of the properties of these units themselves. Thus, one common criticism of social network theory is that individual agency is often ignored although this may not be the case in practice (see agent-based modeling). Precisely because many different types of relations, singular or in combination, form these network configurations, network analytics are useful to a broad range of research enterprises. In social science, these fields of study include, but are not limited to anthropology, biology, communication studies, economics, geography, information science, organizational studies, social psychology, sociology, and sociolinguistics.

History

In the late 1890s, both Émile Durkheim and Ferdinand Tönnies foreshadowed the idea of social networks in their theories and research of social groups. Tönnies argued that social groups can exist as personal and direct social ties that either link individuals who share values and belief (Gemeinschaft, German, commonly translated as "community") or impersonal, formal, and instrumental social links (Gesellschaft, German, commonly translated as "society"). Durkheim gave a non-individualistic explanation of social facts, arguing that social phenomena arise when interacting individuals constitute a reality that can no longer be accounted for in terms of the properties of individual actors. Georg Simmel, writing at the turn of the twentieth century, pointed to the nature of networks and the effect of network size on interaction and examined the likelihood of interaction in loosely knit networks rather than groups.

Moreno's sociogram of a 2nd grade class

 

Major developments in the field can be seen in the 1930s by several groups in psychology, anthropology, and mathematics working independently. In psychology, in the 1930s, Jacob L. Moreno began systematic recording and analysis of social interaction in small groups, especially classrooms and work groups (see sociometry). In anthropology, the foundation for social network theory is the theoretical and ethnographic work of Bronislaw Malinowski, Alfred Radcliffe-Brown, and Claude Lévi-Strauss. A group of social anthropologists associated with Max Gluckman and the Manchester School, including John A. Barnes, J. Clyde Mitchell and Elizabeth Bott Spillius, often are credited with performing some of the first fieldwork from which network analyses were performed, investigating community networks in southern Africa, India and the United Kingdom. Concomitantly, British anthropologist S. F. Nadel codified a theory of social structure that was influential in later network analysis. In sociology, the early (1930s) work of Talcott Parsons set the stage for taking a relational approach to understanding social structure. Later, drawing upon Parsons' theory, the work of sociologist Peter Blau provides a strong impetus for analyzing the relational ties of social units with his work on social exchange theory.

By the 1970s, a growing number of scholars worked to combine the different tracks and traditions. One group consisted of sociologist Harrison White and his students at the Harvard University Department of Social Relations. Also independently active in the Harvard Social Relations department at the time were Charles Tilly, who focused on networks in political and community sociology and social movements, and Stanley Milgram, who developed the "six degrees of separation" thesis. Mark Granovetter and Barry Wellman are among the former students of White who elaborated and championed the analysis of social networks.

Beginning in the late 1990s, social network analysis experienced work by sociologists, political scientists, and physicists such as Duncan J. Watts, Albert-László Barabási, Peter Bearman, Nicholas A. Christakis, James H. Fowler, and others, developing and applying new models and methods to emerging data available about online social networks, as well as "digital traces" regarding face-to-face networks.

Levels of analysis

Self-organization of a network, based on Nagler, Levina, & Timme, (2011)

 

Centrality

 

In general, social networks are self-organizing, emergent, and complex, such that a globally coherent pattern appears from the local interaction of the elements that make up the system. These patterns become more apparent as network size increases. However, a global network analysis of, for example, all interpersonal relationships in the world is not feasible and is likely to contain so much information as to be uninformative. Practical limitations of computing power, ethics and participant recruitment and payment also limit the scope of a social network analysis. The nuances of a local system may be lost in a large network analysis, hence the quality of information may be more important than its scale for understanding network properties. Thus, social networks are analyzed at the scale relevant to the researcher's theoretical question. Although levels of analysis are not necessarily mutually exclusive, there are three general levels into which networks may fall: micro-level, meso-level, and macro-level.

Micro level

At the micro-level, social network research typically begins with an individual, snowballing as social relationships are traced, or may begin with a small group of individuals in a particular social context.

Dyadic level: A dyad is a social relationship between two individuals. Network research on dyads may concentrate on structure of the relationship (e.g. multiplexity, strength), social equality, and tendencies toward reciprocity/mutuality. 

Triadic level: Add one individual to a dyad, and you have a triad. Research at this level may concentrate on factors such as balance and transitivity, as well as social equality and tendencies toward reciprocity/mutuality. In the balance theory of Fritz Heider the triad is the key to social dynamics. The discord in a rivalrous love triangle is an example of an unbalanced triad, likely to change to a balanced triad by a change in one of the relations. The dynamics of social friendships in society has been modeled by balancing triads. The study is carried forward with the theory of signed graphs. 

Actor level: The smallest unit of analysis in a social network is an individual in their social setting, i.e., an "actor" or "ego". Egonetwork analysis focuses on network characteristics such as size, relationship strength, density, centrality, prestige and roles such as isolates, liaisons, and bridges. Such analyses, are most commonly used in the fields of psychology or social psychology, ethnographic kinship analysis or other genealogical studies of relationships between individuals. 

Subset level: Subset levels of network research problems begin at the micro-level, but may cross over into the meso-level of analysis. Subset level research may focus on distance and reachability, cliques, cohesive subgroups, or other group actions or behavior.

Meso level

In general, meso-level theories begin with a population size that falls between the micro- and macro-levels. However, meso-level may also refer to analyses that are specifically designed to reveal connections between micro- and macro-levels. Meso-level networks are low density and may exhibit causal processes distinct from interpersonal micro-level networks.

Social network diagram, meso-level

 

Organizations: Formal organizations are social groups that distribute tasks for a collective goal. Network research on organizations may focus on either intra-organizational or inter-organizational ties in terms of formal or informal relationships. Intra-organizational networks themselves often contain multiple levels of analysis, especially in larger organizations with multiple branches, franchises or semi-autonomous departments. In these cases, research is often conducted at a workgroup level and organization level, focusing on the interplay between the two structures. Experiments with networked groups online have documented ways to optimize group-level coordination through diverse interventions, including the addition of autonomous agents to the groups.

Randomly distributed networks: Exponential random graph models of social networks became state-of-the-art methods of social network analysis in the 1980s. This framework has the capacity to represent social-structural effects commonly observed in many human social networks, including general degree-based structural effects commonly observed in many human social networks as well as reciprocity and transitivity, and at the node-level, homophily and attribute-based activity and popularity effects, as derived from explicit hypotheses about dependencies among network ties. Parameters are given in terms of the prevalence of small subgraph configurations in the network and can be interpreted as describing the combinations of local social processes from which a given network emerges. These probability models for networks on a given set of actors allow generalization beyond the restrictive dyadic independence assumption of micro-networks, allowing models to be built from theoretical structural foundations of social behavior.

Examples of a random network and a scale-free network. Each graph has 32 nodes and 32 links. Note the "hubs" (shaded) in the scale-free diagram (on the right).

 

Scale-free networks: A scale-free network is a network whose degree distribution follows a power law, at least asymptotically. In network theory a scale-free ideal network is a random network with a degree distribution that unravels the size distribution of social groups. Specific characteristics of scale-free networks vary with the theories and analytical tools used to create them, however, in general, scale-free networks have some common characteristics. One notable characteristic in a scale-free network is the relative commonness of vertices with a degree that greatly exceeds the average. The highest-degree nodes are often called "hubs", and may serve specific purposes in their networks, although this depends greatly on the social context. Another general characteristic of scale-free networks is the clustering coefficient distribution, which decreases as the node degree increases. This distribution also follows a power law. The Barabási model of network evolution shown above is an example of a scale-free network.

Macro level

Rather than tracing interpersonal interactions, macro-level analyses generally trace the outcomes of interactions, such as economic or other resource transfer interactions over a large population. 

Diagram: section of a large-scale social network

 

Large-scale networks: Large-scale network is a term somewhat synonymous with "macro-level" as used, primarily, in social and behavioral sciences, in economics. Originally, the term was used extensively in the computer sciences.

Complex networks: Most larger social networks display features of social complexity, which involves substantial non-trivial features of network topology, with patterns of complex connections between elements that are neither purely regular nor purely random (see, complexity science, dynamical system and chaos theory), as do biological, and technological networks. Such complex network features include a heavy tail in the degree distribution, a high clustering coefficient, assortativity or disassortativity among vertices, community structure (see stochastic block model), and hierarchical structure. In the case of agency-directed networks these features also include reciprocity, triad significance profile (TSP, see network motif), and other features. In contrast, many of the mathematical models of networks that have been studied in the past, such as lattices and random graphs, do not show these features.

Theoretical links

Imported theories

Various theoretical frameworks have been imported for the use of social network analysis. The most prominent of these are Graph theory, Balance theory, Social comparison theory, and more recently, the Social identity approach.

Indigenous theories

Few complete theories have been produced from social network analysis. Two that have are Structural Role Theory and Heterophily Theory.

The basis of Heterophily Theory was the finding in one study that more numerous weak ties can be important in seeking information and innovation, as cliques have a tendency to have more homogeneous opinions as well as share many common traits. This homophilic tendency was the reason for the members of the cliques to be attracted together in the first place. However, being similar, each member of the clique would also know more or less what the other members knew. To find new information or insights, members of the clique will have to look beyond the clique to its other friends and acquaintances. This is what Granovetter called "the strength of weak ties".

Structural holes

In the context of networks, social capital exists where people have an advantage because of their location in a network. Contacts in a network provide information, opportunities and perspectives that can be beneficial to the central player in the network. Most social structures tend to be characterized by dense clusters of strong connections. Information within these clusters tends to be rather homogeneous and redundant. Non-redundant information is most often obtained through contacts in different clusters. When two separate clusters possess non-redundant information, there is said to be a structural hole between them. Thus, a network that bridges structural holes will provide network benefits that are in some degree additive, rather than overlapping. An ideal network structure has a vine and cluster structure, providing access to many different clusters and structural holes.

Networks rich in structural holes are a form of social capital in that they offer information benefits. The main player in a network that bridges structural holes is able to access information from diverse sources and clusters. For example, in business networks, this is beneficial to an individual's career because he is more likely to hear of job openings and opportunities if his network spans a wide range of contacts in different industries/sectors. This concept is similar to Mark Granovetter's theory of weak ties, which rests on the basis that having a broad range of contacts is most effective for job attainment.

Research clusters

Communication

Communication Studies are often considered a part of both the social sciences and the humanities, drawing heavily on fields such as sociology, psychology, anthropology, information science, biology, political science, and economics as well as rhetoric, literary studies, and semiotics. Many communication concepts describe the transfer of information from one source to another, and can thus be conceived of in terms of a network.

Community

In J.A. Barnes' day, a "community" referred to a specific geographic location and studies of community ties had to do with who talked, associated, traded, and attended church with whom. Today, however, there are extended "online" communities developed through telecommunications devices and social network services. Such devices and services require extensive and ongoing maintenance and analysis, often using network science methods. Community development studies, today, also make extensive use of such methods.

Complex networks

Complex networks require methods specific to modelling and interpreting social complexity and complex adaptive systems, including techniques of dynamic network analysis. Mechanisms such as Dual-phase evolution explain how temporal changes in connectivity contribute to the formation of structure in social networks.

Criminal networks

In criminology and urban sociology, much attention has been paid to the social networks among criminal actors. For example, Andrew Papachristos has studied gang murders as a series of exchanges between gangs. Murders can be seen to diffuse outwards from a single source, because weaker gangs cannot afford to kill members of stronger gangs in retaliation, but must commit other violent acts to maintain their reputation for strength.

Diffusion of innovations

Diffusion of ideas and innovations studies focus on the spread and use of ideas from one actor to another or one culture and another. This line of research seeks to explain why some become "early adopters" of ideas and innovations, and links social network structure with facilitating or impeding the spread of an innovation.

Demography

In demography, the study of social networks has led to new sampling methods for estimating and reaching populations that are hard to enumerate (for example, homeless people or intravenous drug users.) For example, respondent driven sampling is a network-based sampling technique that relies on respondents to a survey recommending further respondents.

Economic sociology

The field of sociology focuses almost entirely on networks of outcomes of social interactions. More narrowly, economic sociology considers behavioral interactions of individuals and groups through social capital and social "markets". Sociologists, such as Mark Granovetter, have developed core principles about the interactions of social structure, information, ability to punish or reward, and trust that frequently recur in their analyses of political, economic and other institutions. Granovetter examines how social structures and social networks can affect economic outcomes like hiring, price, productivity and innovation and describes sociologists' contributions to analyzing the impact of social structure and networks on the economy.

Health care

Analysis of social networks is increasingly incorporated into health care analytics, not only in epidemiological studies but also in models of patient communication and education, disease prevention, mental health diagnosis and treatment, and in the study of health care organizations and systems.

Human ecology

Human ecology is an interdisciplinary and transdisciplinary study of the relationship between humans and their natural, social, and built environments. The scientific philosophy of human ecology has a diffuse history with connections to geography, sociology, psychology, anthropology, zoology, and natural ecology.

Language and linguistics

Studies of language and linguistics, particularly evolutionary linguistics, focus on the development of linguistic forms and transfer of changes, sounds or words, from one language system to another through networks of social interaction. Social networks are also important in language shift, as groups of people add and/or abandon languages to their repertoire.

Literary networks

In the study of literary systems, network analysis has been applied by Anheier, Gerhards and Romo, De Nooy, and Senekal, to study various aspects of how literature functions. The basic premise is that polysystem theory, which has been around since the writings of Even-Zohar, can be integrated with network theory and the relationships between different actors in the literary network, e.g. writers, critics, publishers, literary histories, etc., can be mapped using visualization from SNA.

Organizational studies

Research studies of formal or informal organization relationships, organizational communication, economics, economic sociology, and other resource transfers. Social networks have also been used to examine how organizations interact with each other, characterizing the many informal connections that link executives together, as well as associations and connections between individual employees at different organizations. Intra-organizational networks have been found to affect organizational commitment, organizational identification, interpersonal citizenship behaviour.

Social capital

Social capital is a form of economic and cultural capital in which social networks are central, transactions are marked by reciprocity, trust, and cooperation, and market agents produce goods and services not mainly for themselves, but for a common good. 

Social capital is a sociological concept about the value of social relations and the role of cooperation and confidence to achieve positive outcomes. The term refers to the value one can get from their social ties. For example, newly arrived immigrants can make use of their social ties to established migrants to acquire jobs they may otherwise have trouble getting (e.g., because of unfamiliarity with the local language). A positive relationship exists between social capital and the intensity of social network use. In a dynamic framework, higher activity in a network feeds into higher social capital which itself encourages more activity.

Network position and benefits

In many organizations, members tend to focus their activities inside their own groups, which stifles creativity and restricts opportunities. A player whose network bridges structural holes has an advantage in detecting and developing rewarding opportunities. Such a player can mobilize social capital by acting as a "broker" of information between two clusters that otherwise would not have been in contact, thus providing access to new ideas, opinions and opportunities. British philosopher and political economist John Stuart Mill, writes, "it is hardly possible to overrate the value ... of placing human beings in contact with persons dissimilar to themselves.... Such communication [is] one of the primary sources of progress." Thus, a player with a network rich in structural holes can add value to an organization through new ideas and opportunities. This in turn, helps an individual's career development and advancement. 

A social capital broker also reaps control benefits of being the facilitator of information flow between contacts. In the case of consulting firm Eden McCallum, the founders were able to advance their careers by bridging their connections with former big three consulting firm consultants and mid-size industry firms. By bridging structural holes and mobilizing social capital, players can advance their careers by executing new opportunities between contacts.

There has been research that both substantiates and refutes the benefits of information brokerage. A study of high tech Chinese firms by Zhixing Xiao found that the control benefits of structural holes are "dissonant to the dominant firm-wide spirit of cooperation and the information benefits cannot materialize due to the communal sharing values" of such organizations. However, this study only analyzed Chinese firms, which tend to have strong communal sharing values. Information and control benefits of structural holes are still valuable in firms that are not quite as inclusive and cooperative on the firm-wide level. In 2004, Ronald Burt studied 673 managers who ran the supply chain for one of America's largest electronics companies. He found that managers who often discussed issues with other groups were better paid, received more positive job evaluations and were more likely to be promoted. Thus, bridging structural holes can be beneficial to an organization, and in turn, to an individual's career.

Social media

Computer networks combined with social networking software produces a new medium for social interaction. A relationship over a computerized social networking service can be characterized by context, direction, and strength. The content of a relation refers to the resource that is exchanged. In a computer mediated communication context, social pairs exchange different kinds of information, including sending a data file or a computer program as well as providing emotional support or arranging a meeting. With the rise of electronic commerce, information exchanged may also correspond to exchanges of money, goods or services in the "real" world. Social network analysis methods have become essential to examining these types of computer mediated communication.

In addition, the sheer size and the volatile nature of social media has given rise to new network metrics. A key concern with networks extracted from social media is the lack of robustness of network metrics given missing data.

Quantum complex network

From Wikipedia, the free encyclopedia

Being part of network science the study of quantum complex networks aims to explore the impact of complexity science and network architectures in quantum systems. According to quantum information theory it is possible to improve communication security and data transfer rates by taking advantage of quantum mechanics. In this context the study of quantum complex networks is motivated by the possibility of quantum communications being used on a massive scale in the future. In such case it is likely that quantum communication networks will acquire non trivial features as is common in existing communication networks today.

Motivation

It is theoretically possible to take an advantage of quantum mechanics to create secure and faster communications, namely, quantum key distribution is an application of quantum cryptography that allows for theoretical completely secure communications, and quantum teleportation that can be used to transfer data at higher rate than using only classic channels.

The successful quantum teleportation experiments in 1998 followed by the development of first quantum communication networks in 2004, opened the possibility of quantum communication being used in a large scale in the future. According to findings in network science the topology of the networks is, in most cases, extremely important, and the exiting large scale communication networks today tend to have non-trivial topologies and traits, like small world effect, community structure and scale free properties. The study of networks with quantum properties and complex network topologies, can help us not only to better understand such networks but also how to use the network topology to improve the efficiency of communication networks in the future.

Important concepts

Qubits

In quantum information Qubits are the equivalent to bits in classical systems. A qubit is a property that when measured only can be found to be in one of two states, that is used to transmit information. The polarization of a photon or the nuclear spin are examples of two state systems that can be used as qubits.

Entanglement

Quantum entanglement is a physical phenomenon characterized by a correlation between the quantum states of two or more particles. While entangled particle do not interact in the classical sense, the quantum state of those particle can not be described independently. Particles can be entangle in different degrees, and the maximally entangled state are the ones the maximize the entropy of entanglement. In the context of quantum communication, quantum entanglement qubits are used as a quantum channel capable of transmitting information when combined with a classical channel.

Bell measurement

Bell measurement is joint quantum-mechanical measurement of two qubits, so that after the measurement the two qubits will be maixmailly entangle.

Entanglement swapping

Entanglement swapping is a frequent strategy used in quantum networks that allows the connections in the network to change. Lets us suppose that we have 4 qubits, A B C and D, C and D belong to the same station, while A and C belong to two different stations. Qubit A is entangled with qubit C and qubit B is entangled with qubit D. By performing a bell measurement in qubits A and B, not only the qubits A and B will be entangled but it is also possible to create an entanglement state between qubit C and qubit D, despite the fact that there was never an interactions between them. Following this process the entanglement between qubits A and C, and qubits B and D will be lost. This strategy can be use to shape the connection on the network.

Network structure

While not all models for quantum complex network follow exactly the same structure, usually nodes represent a set of qubits in the same station where operation like Bell measurements and entanglement swapping can be applied. On the other hand, a link between a node ${\displaystyle i}$ and ${\displaystyle j}$ means that a qubit in node ${\displaystyle i}$ is entangled to a qubit in node ${\displaystyle j}$, but those two qubits are in different places, thus physical interactions between them are not possible. Quantum networks where the links are interaction terms instead of entanglement may also be considered but for very different purposes. 

Notation

Each node in the network is in possession of a set of qubits that can be in different states. The most convenient representation for the quantum state of the qubits is the dirac notation and represent the two state of the qubits as ${\displaystyle |0\rangle }$ and ${\displaystyle |1\rangle }$. Two particle are entangled if the joint wave function, ${\displaystyle |\psi _{ij}\rangle }$, can not be decomposed as,

${\displaystyle |\psi _{ij}\rangle =|\phi \rangle _{i}\otimes |\phi \rangle _{j},}$

where ${\displaystyle |\phi \rangle _{i}}$ represents the quantum state of the qubit at node i and ${\displaystyle |\phi \rangle _{j}}$ represents the quantum state of the qubit at node j. Another important concept is maximally entangled states. The four states (the Bell states) that maximize the entropy of entanglement can be written as

${\displaystyle |\Phi _{ij}^{+}\rangle ={\frac {1}{\sqrt {2}}}(|0\rangle _{i}\otimes |0\rangle _{j}+|1\rangle _{i}\otimes |1\rangle _{j}),}$

${\displaystyle |\Phi _{ij}^{-}\rangle ={\frac {1}{\sqrt {2}}}(|0\rangle _{i}\otimes |0\rangle _{j}-|1\rangle _{i}\otimes |1\rangle _{j}),}$

${\displaystyle |\Psi _{ij}^{+}\rangle ={\frac {1}{\sqrt {2}}}(|0\rangle _{i}\otimes |1\rangle _{j}+|1\rangle _{i}\otimes |0\rangle _{j}),}$

${\displaystyle |\Psi _{ij}^{-}\rangle ={\frac {1}{\sqrt {2}}}(|0\rangle _{i}\otimes |1\rangle _{j}-|1\rangle _{i}\otimes |0\rangle _{j}).}$

Models

Quantum random networks

The quantum random network model proposed by Perseguers et al. can be thought of as a quantum version of the Erdős–Rényi model. Instead of the typical links used to represent other complex networks, in the quantum random network model each pair of nodes is connected through a pair of entangled qubits. In this case each node contains ${\displaystyle N-1}$ quibits, one for each other node. In a quantum random network, the degree of entanglement between a pair of nodes, represented by ${\displaystyle p}$, plays a similar role to the parameter ${\displaystyle p}$ in the Erdős–Rényi model. While in the Erdős–Rényi model two nodes form a connection with probability ${\displaystyle p}$, in the context of quantum random networks ${\displaystyle p}$ means the probability of an entangled pair of qubits being successful converted to a maximally entangled state using only local operations and classical communications, called LOCC operations. We can think of maximally entangled qubits as the true links between nodes. 

Using the notation introduced previously, we can represent a pair of entangled qubits connecting the nodes ${\displaystyle i}$ and ${\displaystyle j}$, as

${\displaystyle |\psi _{ij}\rangle ={\sqrt {1-p/2}}|0\rangle _{i}\otimes |0\rangle _{j}+{\sqrt {p/2}}|1\rangle _{i}\otimes |1\rangle _{j},}$

For ${\displaystyle p=0}$ the two qubits are not entangled,

${\displaystyle |\psi _{ij}\rangle =|0\rangle _{i}\otimes |0\rangle _{j},}$

and for ${\displaystyle p=1}$ we obtain the maximally entangled state, given by

${\displaystyle |\psi _{ij}\rangle ={\sqrt {1/2}}(|0\rangle _{i}\otimes |0\rangle _{j}+|1\rangle _{i}\otimes |1\rangle _{j})}$.

For intermediate values of ${\displaystyle p}$, ${\displaystyle 0$

, any entangled state can be, with probability ${\displaystyle p}$, successfully converted to the maximally entangled state using LOCC operations.

One of the main features that distinguish this model from its classic version is the fact the in quantum random networks links are only truly established after measurements in the networks being made, and it is possible to take advantage of this fact to shape the final state of the network. Considering an initial quantum complex network with an infinite number of nodes, Perseguers et al. showed that, by doing the right measurements and entanglement swapping, it is possible to collapse the initial network to a network containing any finite subgraph, provided that ${\displaystyle p}$ scales with ${\displaystyle N}$ as,

${\displaystyle p\sim N^{Z},}$

were ${\displaystyle Z\geq -2}$. This result is contrary to what we find in classic graph theory where the type of subgraphs contained in a network is bounded by the value of ${\displaystyle z}$.

Entanglement Percolation

The goal of entanglement percolation models is to determine if a quantum network is capable of establishing a connection between two arbitrary nodes through entanglement, and to find best the strategies to create those same connections. In a model proposed by Cirac et al. and applied to complex networks by Cuquet et al., nodes are distributed in a lattice, or in a complex network, and each pair of neighbors share two pairs of entangled qubits that can be converted to a maximally entangle qubit pair with probability ${\displaystyle p}$. We can think of maximally entangled qubits as the true links between nodes. According to classic percolation theory, considering a probability ${\displaystyle p}$ of two neighbors being connected, there is a critical ${\displaystyle p}$ designed by ${\displaystyle p_{c}}$, so that if ${\displaystyle p>p_{c}}$ there is a finite probability of existing a path between two random selected node, and for ${\displaystyle p$ the probability of existing a path between two random selected nodes goes to zero. ${\displaystyle p_{c}}$ depends only on the topology of the network. A similar phenomena was found in the model proposed by Cirac et al., where the probability of forming a maximally entangled state between two random selected nodes is zero if ${\displaystyle p$ and finite if ${\displaystyle p>p_{c}}$. The main difference between classic and entangled percolation is that in quantum networks it is possible to change the links in the network, in a way changing the effective topology of the network, as a consequence ${\displaystyle p_{c}}$ will depend on the strategy used to convert partial entangle qubits to maximally connected qubits. A naive approach yields that ${\displaystyle p_{c}}$ for a quantum network is equal to ${\displaystyle p_{c}}$ for a classic network with the same topology. Nevertheless, it was shown that is possible to take advantage of quantum swapping to lower that value, both in regular lattices and complex networks.