## What is a Network?

A network is a collection of components that are connected by relationships between pairs of these components. For instance, a social network is a collection of people with pair-wise acquaintance or friendship-based connections [1].

Similarly, an electric grid network is a collection of generators, substations, and consumers connected by electrical transmission lines [2]. So, in both of these cases, as in all networks, both the individual components as well as the characteristics of their connections will influence how the system behaves.

Many real-world systems can be described as a network. Scientists in many different fields have developed models of some of these systems, including:

- Network models in epidemiology, which help us understand the transmission of infections.
- Ecological networks describing the interactions of the organisms that make up an ecosystem.
- Models of exchange mechanisms used in economics.

Although these systems cover a diverse range of subjects, we can describe them with the same network framework because they contain the two essential elements:

- Nodes – the people, objects, or other components that makeup the network
- Edges – the relationships or connections between the nodes

## What is Network Theory?

These notions of networks, nodes, and edges are formally defined in network theory, an interdisciplinary field of study concerned with networks and network systems. Network theory is part of the wider subject of graph theory, a field in mathematics. By following the conventions defined in network and graph theory, we ensure that we describe our networks in the same way each time. These conventions allow us to leverage insights across the diverse areas in which networks are studied.

Then, once we have defined our network, network theory provides additional tools for analysis. For instance, we can calculate each node’s ‘centrality’ to understand the most influential components [3]. We can look for motifs, similar patterns across the network and form sub-communities or ‘clusters’ within the network [4].

In addition to analyzing our system with these tools, we can also carry information about the network into other areas of study. An example is neural graph networks, which allow us to make predictions using deep learning techniques from machine learning on a network system [5].

## Why are we Interested in Network Modeling?

We can use many frameworks to study a system of multiple interacting components, and using a network model is just one of them. Some other examples are compartmental models and mean-field approximations.

There are advantages and disadvantages to using each method, and some of the benefits of the network’s approach include:

- Using network theory provides a framework to describe a complex system logically. The conventions for encoding the nodes and edges in the model allow us to systematically explain and keep track of many individuals and the links between them.
- Once the system is described in this format, it may be easier to find important network features that we might otherwise have missed. For instance, a common phenomenon observed in network systems is the ‘small-world’ effect; even in systems with many nodes, the largest path between any two nodes in the network can be surprisingly small. Once we have formulated the network model for our system, we can calculate the longest path between any two nodes and see if the small-world effect appears in our system.
- With a network framework, we describe the system in terms of both the individuals or objects that we designate as nodes and the connections between them. We can think about these two features both separately and together, and develop a greater understanding of our system. For instance, we can compare two systems with the same nodes but different edges, or we can compare systems with the same network structure but different characteristics on each node.
- We can study systems with high levels of heterogeneity. The heterogeneity of our systems is the level to which elements are different, and can have different characteristics or behaviors. A system with very low heterogeneity is a mean-field model, which is where we think about identical system components that all communicate with each other equally. However, in a network model, we can include individuals with very different characteristics and relationships.

When we wish to learn more about how our system’s components interact and how these interactions affect their behavior, you can see that taking a network-based approach provides us with some useful tools. We may discover insights that are inaccessible with other modeling approaches.

## How do we Use Network Theory at Olvin?

At Almanac, we process data from over 200 million consumers every day and connect them to around 10 million places. As well as providing accurate visit data and predictions, we also want to help our clients to understand the data in new and powerful ways.

With our new networks product, we create a network model for each metropolitan statistical area. Each point of interest within the metropolitan area is a node in our network, and we form relationships between these nodes by connecting places visited by the same people.

## What does Network Theory Mean for Retailers?

For retailers, networks can have a significant effect on helping to run and manage the day to day operations. It can substantially impact planning new venues and understanding how venues impact each other within an area. But more on that next time.

#### References

[1] Wasserman, S., & Faust, K. (1994). *Social network analysis: Methods and applications* (Vol. 8). Cambridge university press.

[2] Kaplan, S. M. (2009) *Smart Grid: Electrical Power Transmission: Background and Policy Issues*. Congressional Research Service, CRS Report for Congress, R40511.

[3] Stephenson, K., & Zelen, M. (1989). Rethinking centrality: Methods and examples. *Social networks*, *11*(1), 1-37.

[4] Newman, M. E. (2004). Fast algorithm for detecting community structure in networks. *Physical review E*, *69*(6), 066133.

[5] Scarselli, F., Gori, M., Tsoi, A. C., Hagenbuchner, M., & Monfardini, G. (2008). The graph neural network model. *IEEE Transactions on Neural Networks*, *20*(1), 61-80.