Biological networks
Many biological systems are naturally represented as networks because their behaviour depends on patterns of interaction rather than only on the number of components present.
From a graph to a biological network
Graph theory gives the abstract representation
\[G=(V,E).\]To make this biological, we assign meanings to \(V\) and \(E\).
For example, nodes may represent organisms and edges may represent physical contact. Alternatively, nodes may represent species and directed edges may represent feeding relationships.
Different biological meanings of nodes
A node can represent many different biological units:
| Network | Possible node |
|---|---|
| epidemic contact network | individual or group |
| gene-regulatory network | gene or regulatory element |
| protein-interaction network | protein |
| food web | species or functional group |
| neural network | neuron or brain region |
| metapopulation network | habitat patch or population |
The appropriate level depends on the biological question.
Different meanings of edges
Edges may represent contact, movement, predation, competition, mutualism, regulation, biochemical interaction, dispersal or another relationship.
Therefore two networks with the same mathematical adjacency matrix can represent completely different biological mechanisms.
Directed versus undirected interactions
Some biological relationships are naturally symmetric. Physical contact between two individuals may be represented by an undirected edge.
Other interactions have direction. If species \(i\) consumes species \(j\), or gene \(i\) regulates gene \(j\), the direction carries biological information.
With the convention
\[A_{ij}=1\quad\text{when }i\to j,\]a directed network generally satisfies
\[A_{ij}\ne A_{ji}.\]Weighted biological networks
Binary edges record whether an interaction exists. Biological interactions often differ in intensity, so it may be more useful to assign a weight
\[w_{ij}.\]For example, a weight could represent contact duration, movement frequency, interaction strength or biomass flow.
The weighted adjacency matrix is then
\[W=(w_{ij}).\]Signed interactions
Some networks distinguish positive and negative effects. In a regulatory network, an interaction may activate or inhibit. In ecology, one species may benefit while another is harmed.
A signed weight can encode this:
\[w_{ij}>0\quad\text{positive effect},\qquad w_{ij}<0\quad\text{negative effect}.\]Node states
The network gives interaction structure, while each node can also carry a biological state.
For an epidemic network, one might write
\[X_i(t)\in\{S,I,R\}.\]For a gene network, \(X_i(t)\) might represent expression level. For an ecological network, it might represent abundance.
The full model therefore combines topology with changing node states.
Structure and dynamics
It is useful to separate two mathematical layers:
\[\boxed{\text{network structure}+\text{dynamical rules}=\text{network model}.}\]The adjacency matrix describes possible interactions. Differential equations, stochastic transition rates or update rules describe what happens through those interactions.
A generic network-coupled equation
If \(x_i(t)\) is a state variable at node \(i\), a broad class of models can be written schematically as
\[\boxed{\frac{dx_i}{dt}=F_i(x_i)+\sum_j A_{ij}H_{ij}(x_i,x_j)}.\]The term \(F_i\) describes local behaviour at node \(i\), while the sum represents effects transmitted through network connections.
This is a general modelling form rather than one specific biological law.
Static networks
A static network assumes that the edge set does not change during the period being modelled:
\[E(t)=E.\]This may be reasonable when biological dynamics occur much faster than changes in network structure.
Temporal networks
Many biological contacts change through time. Then
\[G(t)=(V,E(t))\]or
\[A=A(t).\]For infection spread, the timing and order of contacts can matter as much as the total number of contacts.
Adaptive networks
In some systems, network structure changes in response to node states. Individuals may avoid infectious contacts, species may alter interactions, or regulatory relationships may change with cellular conditions.
Then the feedback is
\[\boxed{\text{network}\to\text{dynamics}\to\text{network change}}.\]Such systems are often called adaptive or coevolving networks.
Bipartite networks
A bipartite network divides nodes into two sets and permits edges only between the sets.
Examples include plant–pollinator networks, host–parasite networks and affiliation networks connecting individuals to locations.
If the two node sets are \(U\) and \(V\), then
\[E\subseteq U\times V.\]Multilayer networks
A biological system may contain several types of interaction simultaneously. A human contact network, for example, may have household, workplace and social-contact layers.
A multilayer representation keeps these interaction types distinct instead of collapsing them into a single edge set.
Spatial networks
Nodes may have physical locations, and edge probability or weight may depend on spatial distance.
This is common in dispersal, movement and habitat-connectivity models. Network structure then provides a discrete representation of spatial relationships.
Heterogeneity
Real biological networks are rarely perfectly homogeneous. Nodes may differ in degree, edge weight, biological state or intrinsic properties.
This heterogeneity can create outcomes that are missed by models based only on population averages.
Communities and modularity
Biological networks may contain groups of nodes with many internal connections and relatively fewer connections between groups.
Such community structure can slow spreading between groups, localise perturbations or create semi-independent functional modules.
Hubs
A hub is a highly connected node relative to others in the same network.
Hubs can be important for spreading or connectivity, but high degree does not automatically mean greatest importance. The relevant notion of importance depends on the biological process and motivates later centrality measures.
Network data are observations
An observed biological network is usually an estimate of an underlying interaction system. Edges may be missing, falsely detected or measured at different levels of confidence.
Sampling design can strongly affect the apparent degree distribution and other structural properties.
Choosing the correct network representation
A network should be designed around the modelling question. Important choices include whether edges are directed, weighted, signed, static or time-dependent, and whether nodes represent individuals or aggregated groups.
Adding more detail is not always better. Complexity is useful only when it represents biological information relevant to the question.
Networks versus well-mixed models
In a well-mixed population, interactions are often represented through average rates. A network model instead restricts interactions according to explicit edges.
Network models are especially useful when contact heterogeneity, local clustering, communities or particular interaction pathways are expected to affect the outcome.
Examples across mathematical biology
Network methods connect several areas of mathematical biology. Epidemic models use contact networks, ecology uses food webs and mutualistic networks, molecular biology uses regulatory and protein networks, and spatial population biology uses movement and habitat networks.
The mathematical graph concepts remain similar, but the interpretation and dynamical rules change with the biological system.
Transition to contact networks
The next lesson specialises these ideas to populations in which edges represent potentially infectious contacts. This provides the structural basis for epidemic models on networks.