Table S1. Brief explanation of network analysis terms used in the manuscript1
Term
Assortativity coefficient
Explanation
Tendency of the nodes to attach to nodes with similar degrees. A negative value indicates a
tendency of nodes to attach to nodes with different degrees.
Cellular automata
Mathematical model consisting of a regular grid of cells. Each cell has a defined state at a particular
time, which is updated periodically according to the states of the neighbouring cells. Cellular
automata are very useful to model the competition for space of different strains in time.
Connectivity mechanisms
The mechanisms by which nodes connect to form a network.
Connection
Also called an edge. An interaction between two nodes. It can have a direction, if a node acts
towards another, or it can be undirected if both nodes act equally in the interaction. In the
antagonistic interaction network presented in this article, all connections are directed.
Degree
Total number of connections a node has to other nodes in the network. It is the sum of:
the in-degree (receiver degree), or the number of connections coming to a node and
the out-degree (sender degree), or the number of connections stemming from a node.
Degree distribution
Distribution of the degrees of all nodes in the network.
Directed connection
A connection going from one node to another one. Also called directed edge.
Example: A strain antagonizes another one.
Disassortative network
A network where nodes attach preferentially to nodes with different degrees. Disassortative
networks have negative assortativity coefficients
Dyad
A sub-network made of two nodes.
Edge
A connection between two entities (or nodes) in a network. For example, an antagonistic
interaction.
Erdős and Rényi networks
(ER networks)
Random networks commonly used as null models in network analysis.
Hyperimmunity
A state where many strains have antagonistic mechanisms, but most strains are resistant to them.
Interaction density
Fraction of strains with observed interactions relative to the number of possible strain-pairs
Nestedness
Tendency of the nodes to attach with subsets of the connections of more connected nodes. In a
highly nested network, nodes with few connections tend to connect with nodes with many
connections, rather than with nodes with intermediate numbers of connections.
Network diameter
Largest number of antagonistic interactions needed to connect any two strains by the shortest path
between them. To find the diameter one can count the connections needed to connect all pairs, and
take the maximum.
For example, in the three member network:
A->B->C, to connect A with C one has to go from A to B and from B to C. Thus, two connections are
needed to go from A to C, and the network diameter is 2.
Network size
Number of nodes of a network.
Node
An entity in a network, (in our case a strain). They are also called vertexes. Nodes are connected
through edges.
Non-transitive antagonistic
interaction loops
Given a directed interaction from one member towards another one, and the latter with a third one,
the interaction is non-transitive if it is not present from the first towards the third one, or if it is
present from the third towards the first one.
Transitive: A -> B and B -> C implies A -> C
Non-transitive: A -> B and B -> C but A C
Non-transitive: A -> B and B -> C but A <- C
Table S1. Continuation
Preferential attachment
A mechanism of network growth, where new nodes connect to already highly connected nodes with
higher probability than to less connected nodes. The result is that already more connected nodes
become even more connected.
Poisson degree distribution
A distribution that is often used to model the number of events that occur in a fixed time interval
when the events occur at random but with fixed rate. In the case of network analysis, instead of
events, connections are modelled that are formed between pairs of a fixed number of nodes, with a
fixed probability of formation of each node (or a fixed number of total nodes).
Power-law distribution
A distribution of a random variable whose probability density function is characterized by a power of
the variable. These distributions have slowly-decaying tails, which means that large values of the
random variable are observed with non-negligible probabilities.
Receiver degree
See Degree
Scale-free (SF) network
SF networks are characterized by having power-law degree distributions in their right tails. This
implies that they have a few nodes with very large degrees, while most nodes have only small
degrees.
Sender degree
See Degree
Sender determined network
See Sender-receiver asymmetry
Sender-receiver asymmetry
Comparison of the variance of the antagonism and sensitivity degrees to determine weather an
interaction network is on average determined more by the potential of inhibition or by the
sensitivity of strains. Negative values indicate a sender determined interaction network.
Small world network
A type of network where the diameter is small in comparison with the number of nodes, so that all
the nodes can be connected by a small number of edges. In small world networks, the diameter
varies logarithmically with network size.
Sub-network
A subset of the nodes of a network and the connections between them.
Tail of a distribution
Behaviour of a distribution at the extremes. In network analysis, the right tail, on the higher values,
is usually referred to when not indicated otherwise.
Triad
A sub-network made of three nodes.
1. These explanations are intended to help the reader understand the technical terms related to network analysis used in the text, and
are not formal or mathematical definitions.
Table S2. Abbreviated vs. full strain names
Abbreviated name
1A6
1B6
1B7
1B8
1C11
1C9
1D12
1D4
1D6
1D8
1D9
1E10
1E2
1E3
1E6
1E7
1E8
2A1
2A4
2B1
2B4
2B7
2C1
2C2
2C5
2C8
2C9
2D8
2D9
2E4
3E2
3E3
4A4
4B2
4B3
4B8
4D1
Full strain name
J5.1A6
J4.1B6
J4.1B7
J4.1B8
J3.1C11
J3.1C9
J2.1D12
J2.1D4
J2.1D6
J2.1D8
J2.1D9
J1.1E10
J1.1E2
J1.1E3
J1.1E6
J1.1E7
J1.1E8
J5.2A1
J5.2A4
J4.2B1
J4.2B4
J4.2B7
J3.2C1
J3.2C2
J3.2C5
J3.2C8
J3.2C9
J2.2D8
J2.2D9
J1.2E4
J1.3E2
J1.3E3
J5.4A4
J4.4B2
J4.4B3
J4.4B8
J2.4D1
Figure S1. Nestedness analysis matrix. Rows represent antagonist strains and columns represent
sensitive strains. Rows and columns were ordered with the algorithm of Rodríguez-Gironés &
Santamaría (2006) to maximize nestedness visually. Black squares represent antagonistic
interactions from the corresponding rows to the corresponding columns.
Figure S2. Network diameter as a function of network size in simulated networks using our model
(Equations 1 and 2). Values (black dots) are means of 100 simulations. The grey line indicates a
logarithmic fit (y = 0.551log(x) + 1.3182). Logarithmic increase of the network diameter with
network size is a hallmark of small-world type networks (Amaral et al., 2000).
Figure S3. Parameter space analysis of observed and simulated networks. Shades represent triad
counts with different parameter combinations, for the sixteen possible triads (A to P), according to
the corresponding scale above each panel. For parameter space resolution and simulation
numbers see Methods. Red circles are located in the axes according to the values observed in the
actual network. Shades inside these circles indicate the observed triad counts.
1
2
4
3
5
Figure S4. Example of the scale used. Values below the images show the rating assigned to each
image.